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In mathematics and computer science, an algorithm () is a finite sequence of well-defined instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing calculations and data processing. By making use of artificial intelligence, algorithms can perform automated deductions (referred to as automated reasoning) and use mathematical and logical tests to divert the code through various routes (referred to as automated decision-making). Using human characteristics as descriptors of machines in metaphorical ways was already practiced by Alan Turing with terms such as "memory", "search" and "stimulus".
In contrast, a heuristic is an approach to problem solving that may not be fully specified or may not guarantee correct or optimal results, especially in problem domains where there is no well-defined correct or optimal result.
As an effective method, an algorithm can be expressed within a finite amount of space and time, and in a well-defined formal language for calculating a function. Starting from an initial state and initial input (perhaps empty), the instructions describe a computation that, when executed, proceeds through a finite number of well-defined successive states, eventually producing "output" and terminating at a final ending state. The transition from one state to the next is not necessarily deterministic; some algorithms, known as randomized algorithms, incorporate random input.
History
The concept of algorithm has existed since antiquity. Arithmetic algorithms, such as a division algorithm, were used by ancient Babylonian mathematicians c. 2500 BC and Egyptian mathematicians c. 1550 BC. Greek mathematicians later used algorithms in 240 BC in the sieve of Eratosthenes for finding prime numbers, and the Euclidean algorithm for finding the greatest common divisor of two numbers. Arabic mathematicians such as al-Kindi in the 9th century used cryptographic algorithms for code-breaking, based on frequency analysis.
The word algorithm is derived from the name of the 9th-century Persian mathematician Muḥammad ibn Mūsā al-Khwārizmī, whose nisba (identifying him as from Khwarazm) was Latinized as Algoritmi (Arabized Persian الخوارزمی c. 780–850).
Muḥammad ibn Mūsā al-Khwārizmī was a mathematician, astronomer, geographer, and scholar in the House of Wisdom in Baghdad, whose name means 'the native of Khwarazm', a region that was part of Greater Iran and is now in Uzbekistan. About 825, al-Khwarizmi wrote an Arabic language treatise on the Hindu–Arabic numeral system, which was translated into Latin during the 12th century. The manuscript starts with the phrase Dixit Algorizmi ('Thus spake Al-Khwarizmi'), where "Algorizmi" was the translator's Latinization of Al-Khwarizmi's name. Al-Khwarizmi was the most widely read mathematician in Europe in the late Middle Ages, primarily through another of his books, the Algebra. In late medieval Latin, algorismus, English 'algorism', the corruption of his name, simply meant the "decimal number system". In the 15th century, under the influence of the Greek word ἀριθμός (arithmos), 'number' (cf. 'arithmetic'), the Latin word was altered to algorithmus, and the corresponding English term 'algorithm' is first attested in the 17th century; the modern sense was introduced in the 19th century.
Indian mathematics was predominantly algorithmic.
Algorithms that are representative of the Indian mathematical tradition range from the ancient Śulbasūtrās to the medieval texts of the Kerala School.
In English, the word algorithm was first used in about 1230 and then by Chaucer in 1391. English adopted the French term, but it was not until the late 19th century that "algorithm" took on the meaning that it has in modern English.
Another early use of the word is from 1240, in a manual titled Carmen de Algorismo composed by Alexandre de Villedieu. It begins with:
which translates to:
The poem is a few hundred lines long and summarizes the art of calculating with the new styled Indian dice (Tali Indorum), or Hindu numerals.
A partial formalization of the modern concept of algorithm began with attempts to solve the Entscheidungsproblem (decision problem) posed by David Hilbert in 1928. Later formalizations were framed as attempts to define "effective calculability" or "effective method". Those formalizations included the Gödel–Herbrand–Kleene recursive functions of 1930, 1934 and 1935, Alonzo Church's lambda calculus of 1936, Emil Post's Formulation 1 of 1936, and Alan Turing's Turing machines of 1936–37 and 1939.
Informal definition
An informal definition could be "a set of rules that precisely defines a sequence of operations", which would include all computer programs (including programs that do not perform numeric calculations), and (for example) any prescribed bureaucratic procedure
or cook-book recipe.
In general, a program is only an algorithm if it stops eventually—even though infinite loops may sometimes prove desirable.
A prototypical example of an algorithm is the Euclidean algorithm, which is used to determine the maximum common divisor of two integers; an example (there are others) is described by the flowchart above and as an example in a later section.
offer an informal meaning of the word "algorithm" in the following quotation:
No human being can write fast enough, or long enough, or small enough† ( †"smaller and smaller without limit ... you'd be trying to write on molecules, on atoms, on electrons") to list all members of an enumerably infinite set by writing out their names, one after another, in some notation. But humans can do something equally useful, in the case of certain enumerably infinite sets: They can give explicit instructions for determining the nth member of the set, for arbitrary finite n. Such instructions are to be given quite explicitly, in a form in which they could be followed by a computing machine, or by a human who is capable of carrying out only very elementary operations on symbols.
An "enumerably infinite set" is one whose elements can be put into one-to-one correspondence with the integers. Thus Boolos and Jeffrey are saying that an algorithm implies instructions for a process that "creates" output integers from an arbitrary "input" integer or integers that, in theory, can be arbitrarily large. For example, an algorithm can be an algebraic equation such as y = m + n (i.e., two arbitrary "input variables" m and n that produce an output y), but various authors' attempts to define the notion indicate that the word implies much more than this, something on the order of (for the addition example):
Precise instructions (in a language understood by "the computer") for a fast, efficient, "good" process that specifies the "moves" of "the computer" (machine or human, equipped with the necessary internally contained information and capabilities) to find, decode, and then process arbitrary input integers/symbols m and n, symbols + and = ... and "effectively" produce, in a "reasonable" time, output-integer y at a specified place and in a specified format.
The concept of algorithm is also used to define the notion of decidability—a notion that is central for explaining how formal systems come into being starting from a small set of axioms and rules. In logic, the time that an algorithm requires to complete cannot be measured, as it is not apparently related to the customary physical dimension. From such uncertainties, that characterize ongoing work, stems the unavailability of a definition of algorithm that suits both concrete (in some sense) and abstract usage of the term.
Most algorithms are intended to be implemented as computer programs. However, algorithms are also implemented by other means, such as in a biological neural network (for example, the human brain implementing arithmetic or an insect looking for food), in an electrical circuit, or in a mechanical device.
Formalization
Algorithms are essential to the way computers process data. Many computer programs contain algorithms that detail the specific instructions a computer should perform—in a specific order—to carry out a specified task, such as calculating employees' paychecks or printing students' report cards. Thus, an algorithm can be considered to be any sequence of operations that can be simulated by a Turing-complete system. Authors who assert this thesis include Minsky (1967), Savage (1987) and Gurevich (2000):
Minsky: "But we will also maintain, with Turing ... that any procedure which could "naturally" be called effective, can, in fact, be realized by a (simple) machine. Although this may seem extreme, the arguments ... in its favor are hard to refute".
Gurevich: "… Turing's informal argument in favor of his thesis justifies a stronger thesis: every algorithm can be simulated by a Turing machine … according to Savage [1987], an algorithm is a computational process defined by a Turing machine".Turing machines can define computational processes that do not terminate. The informal definitions of algorithms generally require that the algorithm always terminates. This requirement renders the task of deciding whether a formal procedure is an algorithm impossible in the general case—due to a major theorem of computability theory known as the halting problem.
Typically, when an algorithm is associated with processing information, data can be read from an input source, written to an output device and stored for further processing. Stored data are regarded as part of the internal state of the entity performing the algorithm. In practice, the state is stored in one or more data structures.
For some of these computational processes, the algorithm must be rigorously defined: specified in the way it applies in all possible circumstances that could arise. This means that any conditional steps must be systematically dealt with, case-by-case; the criteria for each case must be clear (and computable).
Because an algorithm is a precise list of precise steps, the order of computation is always crucial to the functioning of the algorithm. Instructions are usually assumed to be listed explicitly, and are described as starting "from the top" and going "down to the bottom"—an idea that is described more formally by flow of control.
So far, the discussion on the formalization of an algorithm has assumed the premises of imperative programming. This is the most common conception—one which attempts to describe a task in discrete, "mechanical" means. Unique to this conception of formalized algorithms is the assignment operation, which sets the value of a variable. It derives from the intuition of "memory" as a scratchpad. An example of such an assignment can be found below.
For some alternate conceptions of what constitutes an algorithm, see functional programming and logic programming.
Expressing algorithms
Algorithms can be expressed in many kinds of notation, including natural languages, pseudocode, flowcharts, drakon-charts, programming languages or control tables (processed by interpreters). Natural language expressions of algorithms tend to be verbose and ambiguous, and are rarely used for complex or technical algorithms. Pseudocode, flowcharts, drakon-charts and control tables are structured ways to express algorithms that avoid many of the ambiguities common in the statements based on natural language. Programming languages are primarily intended for expressing algorithms in a form that can be executed by a computer, but are also often used as a way to define or document algorithms.
There is a wide variety of representations possible and one can express a given Turing machine program as a sequence of machine tables (see finite-state machine, state transition table and control table for more), as flowcharts and drakon-charts (see state diagram for more), or as a form of rudimentary machine code or assembly code called "sets of quadruples" (see Turing machine for more).
Representations of algorithms can be classed into three accepted levels of Turing machine description, as follows:
1 High-level description
"...prose to describe an algorithm, ignoring the implementation details. At this level, we do not need to mention how the machine manages its tape or head."
2 Implementation description
"...prose used to define the way the Turing machine uses its head and the way that it stores data on its tape. At this level, we do not give details of states or transition function."
3 Formal description
Most detailed, "lowest level", gives the Turing machine's "state table".
For an example of the simple algorithm "Add m+n" described in all three levels, see Examples.
Design
Algorithm design refers to a method or a mathematical process for problem-solving and engineering algorithms. The design of algorithms is part of many solution theories of operation research, such as dynamic programming and divide-and-conquer. Techniques for designing and implementing algorithm designs are also called algorithm design patterns, with examples including the template method pattern and the decorator pattern.
One of the most important aspects of algorithm design is resource (run-time, memory usage) efficiency; the big O notation is used to describe e.g. an algorithm's run-time growth as the size of its input increases.
Typical steps in the development of algorithms:
Problem definition
Development of a model
Specification of the algorithm
Designing an algorithm
Checking the correctness of the algorithm
Analysis of algorithm
Implementation of algorithm
Program testing
Documentation preparation
Computer algorithms
"Elegant" (compact) programs, "good" (fast) programs : The notion of "simplicity and elegance" appears informally in Knuth and precisely in Chaitin:
Knuth: " ... we want good algorithms in some loosely defined aesthetic sense. One criterion ... is the length of time taken to perform the algorithm .... Other criteria are adaptability of the algorithm to computers, its simplicity and elegance, etc."
Chaitin: " ... a program is 'elegant,' by which I mean that it's the smallest possible program for producing the output that it does"
Chaitin prefaces his definition with: "I'll show you can't prove that a program is 'elegant—such a proof would solve the Halting problem (ibid).
Algorithm versus function computable by an algorithm: For a given function multiple algorithms may exist. This is true, even without expanding the available instruction set available to the programmer. Rogers observes that "It is ... important to distinguish between the notion of algorithm, i.e. procedure and the notion of function computable by algorithm, i.e. mapping yielded by procedure. The same function may have several different algorithms".
Unfortunately, there may be a tradeoff between goodness (speed) and elegance (compactness)—an elegant program may take more steps to complete a computation than one less elegant. An example that uses Euclid's algorithm appears below.
Computers (and computors), models of computation: A computer (or human "computor") is a restricted type of machine, a "discrete deterministic mechanical device" that blindly follows its instructions. Melzak's and Lambek's primitive models reduced this notion to four elements: (i) discrete, distinguishable locations, (ii) discrete, indistinguishable counters (iii) an agent, and (iv) a list of instructions that are effective relative to the capability of the agent.
Minsky describes a more congenial variation of Lambek's "abacus" model in his "Very Simple Bases for Computability". Minsky's machine proceeds sequentially through its five (or six, depending on how one counts) instructions unless either a conditional IF-THEN GOTO or an unconditional GOTO changes program flow out of sequence. Besides HALT, Minsky's machine includes three assignment (replacement, substitution) operations: ZERO (e.g. the contents of location replaced by 0: L ← 0), SUCCESSOR (e.g. L ← L+1), and DECREMENT (e.g. L ← L − 1). Rarely must a programmer write "code" with such a limited instruction set. But Minsky shows (as do Melzak and Lambek) that his machine is Turing complete with only four general types of instructions: conditional GOTO, unconditional GOTO, assignment/replacement/substitution, and HALT. However, a few different assignment instructions (e.g. DECREMENT, INCREMENT, and ZERO/CLEAR/EMPTY for a Minsky machine) are also required for Turing-completeness; their exact specification is somewhat up to the designer. The unconditional GOTO is a convenience; it can be constructed by initializing a dedicated location to zero e.g. the instruction " Z ← 0 "; thereafter the instruction IF Z=0 THEN GOTO xxx is unconditional.
Simulation of an algorithm: computer (computor) language: Knuth advises the reader that "the best way to learn an algorithm is to try it . . . immediately take pen and paper and work through an example". But what about a simulation or execution of the real thing? The programmer must translate the algorithm into a language that the simulator/computer/computor can effectively execute. Stone gives an example of this: when computing the roots of a quadratic equation the computor must know how to take a square root. If they don't, then the algorithm, to be effective, must provide a set of rules for extracting a square root.
This means that the programmer must know a "language" that is effective relative to the target computing agent (computer/computor).
But what model should be used for the simulation? Van Emde Boas observes "even if we base complexity theory on abstract instead of concrete machines, arbitrariness of the choice of a model remains. It is at this point that the notion of simulation enters". When speed is being measured, the instruction set matters. For example, the subprogram in Euclid's algorithm to compute the remainder would execute much faster if the programmer had a "modulus" instruction available rather than just subtraction (or worse: just Minsky's "decrement").
Structured programming, canonical structures: Per the Church–Turing thesis, any algorithm can be computed by a model known to be Turing complete, and per Minsky's demonstrations, Turing completeness requires only four instruction types—conditional GOTO, unconditional GOTO, assignment, HALT. Kemeny and Kurtz observe that, while "undisciplined" use of unconditional GOTOs and conditional IF-THEN GOTOs can result in "spaghetti code", a programmer can write structured programs using only these instructions; on the other hand "it is also possible, and not too hard, to write badly structured programs in a structured language". Tausworthe augments the three Böhm-Jacopini canonical structures: SEQUENCE, IF-THEN-ELSE, and WHILE-DO, with two more: DO-WHILE and CASE. An additional benefit of a structured program is that it lends itself to proofs of correctness using mathematical induction.
Canonical flowchart symbols: The graphical aide called a flowchart, offers a way to describe and document an algorithm (and a computer program of one). Like the program flow of a Minsky machine, a flowchart always starts at the top of a page and proceeds down. Its primary symbols are only four: the directed arrow showing program flow, the rectangle (SEQUENCE, GOTO), the diamond (IF-THEN-ELSE), and the dot (OR-tie). The Böhm–Jacopini canonical structures are made of these primitive shapes. Sub-structures can "nest" in rectangles, but only if a single exit occurs from the superstructure. The symbols, and their use to build the canonical structures are shown in the diagram.
Examples
Algorithm example
One of the simplest algorithms is to find the largest number in a list of numbers of random order. Finding the solution requires looking at every number in the list. From this follows a simple algorithm, which can be stated in a high-level description in English prose, as:
High-level description:
If there are no numbers in the set then there is no highest number.
Assume the first number in the set is the largest number in the set.
For each remaining number in the set: if this number is larger than the current largest number, consider this number to be the largest number in the set.
When there are no numbers left in the set to iterate over, consider the current largest number to be the largest number of the set.
(Quasi-)formal description:
Written in prose but much closer to the high-level language of a computer program, the following is the more formal coding of the algorithm in pseudocode or pidgin code:
Input: A list of numbers L.
Output: The largest number in the list L.
if L.size = 0 return null
largest ← L[0]
for each item in L, do
if item > largest, then
largest ← item
return largest
Euclid's algorithm
In mathematics, the Euclidean algorithm, or Euclid's algorithm, is an efficient method for computing the greatest common divisor (GCD) of two integers (numbers), the largest number that divides them both without a remainder. It is named after the ancient Greek mathematician Euclid, who first described it in his Elements (c. 300 BC). It is one of the oldest algorithms in common use. It can be used to reduce fractions to their simplest form, and is a part of many other number-theoretic and cryptographic calculations.
Euclid poses the problem thus: "Given two numbers not prime to one another, to find their greatest common measure". He defines "A number [to be] a multitude composed of units": a counting number, a positive integer not including zero. To "measure" is to place a shorter measuring length s successively (q times) along longer length l until the remaining portion r is less than the shorter length s. In modern words, remainder r = l − q×s, q being the quotient, or remainder r is the "modulus", the integer-fractional part left over after the division.
For Euclid's method to succeed, the starting lengths must satisfy two requirements: (i) the lengths must not be zero, AND (ii) the subtraction must be "proper"; i.e., a test must guarantee that the smaller of the two numbers is subtracted from the larger (or the two can be equal so their subtraction yields zero).
Euclid's original proof adds a third requirement: the two lengths must not be prime to one another. Euclid stipulated this so that he could construct a reductio ad absurdum proof that the two numbers' common measure is in fact the greatest. While Nicomachus' algorithm is the same as Euclid's, when the numbers are prime to one another, it yields the number "1" for their common measure. So, to be precise, the following is really Nicomachus' algorithm.
Computer language for Euclid's algorithm
Only a few instruction types are required to execute Euclid's algorithm—some logical tests (conditional GOTO), unconditional GOTO, assignment (replacement), and subtraction.
A location is symbolized by upper case letter(s), e.g. S, A, etc.
The varying quantity (number) in a location is written in lower case letter(s) and (usually) associated with the location's name. For example, location L at the start might contain the number l = 3009.
An inelegant program for Euclid's algorithm
The following algorithm is framed as Knuth's four-step version of Euclid's and Nicomachus', but, rather than using division to find the remainder, it uses successive subtractions of the shorter length s from the remaining length r until r is less than s. The high-level description, shown in boldface, is adapted from Knuth 1973:2–4:
INPUT:
[Into two locations L and S put the numbers l and s that represent the two lengths]:
INPUT L, S
[Initialize R: make the remaining length r equal to the starting/initial/input length l]:
R ← L
E0: [Ensure r ≥ s.]
[Ensure the smaller of the two numbers is in S and the larger in R]:
IF R > S THEN
the contents of L is the larger number so skip over the exchange-steps 4, 5 and 6:
GOTO step 7
ELSE
swap the contents of R and S.
L ← R (this first step is redundant, but is useful for later discussion).
R ← S
S ← L
E1: [Find remainder]: Until the remaining length r in R is less than the shorter length s in S, repeatedly subtract the measuring number s in S from the remaining length r in R.
IF S > R THEN
done measuring so
GOTO 10
ELSE
measure again,
R ← R − S
[Remainder-loop]:
GOTO 7.
E2: [Is the remainder zero?]: EITHER (i) the last measure was exact, the remainder in R is zero, and the program can halt, OR (ii) the algorithm must continue: the last measure left a remainder in R less than measuring number in S.
IF R = 0 THEN
done so
GOTO step 15
ELSE
CONTINUE TO step 11,
E3: [Interchange s and r]: The nut of Euclid's algorithm. Use remainder r to measure what was previously smaller number s; L serves as a temporary location.
L ← R
R ← S
S ← L
[Repeat the measuring process]:
GOTO 7
OUTPUT:
[Done. S contains the greatest common divisor]:
PRINT S
DONE:
HALT, END, STOP.
An elegant program for Euclid's algorithm
The flowchart of "Elegant" can be found at the top of this article. In the (unstructured) Basic language, the steps are numbered, and the instruction LET [] = [] is the assignment instruction symbolized by ←.
5 REM Euclid's algorithm for greatest common divisor
6 PRINT "Type two integers greater than 0"
10 INPUT A,B
20 IF B=0 THEN GOTO 80
30 IF A > B THEN GOTO 60
40 LET B=B-A
50 GOTO 20
60 LET A=A-B
70 GOTO 20
80 PRINT A
90 END
How "Elegant" works: In place of an outer "Euclid loop", "Elegant" shifts back and forth between two "co-loops", an A > B loop that computes A ← A − B, and a B ≤ A loop that computes B ← B − A. This works because, when at last the minuend M is less than or equal to the subtrahend S (Difference = Minuend − Subtrahend), the minuend can become s (the new measuring length) and the subtrahend can become the new r (the length to be measured); in other words the "sense" of the subtraction reverses.
The following version can be used with programming languages from the C-family:
// Euclid's algorithm for greatest common divisor
int euclidAlgorithm (int A, int B){
A=abs(A);
B=abs(B);
while (B!=0){
while (A>B) A=A-B;
B=B-A;
}
return A;
}
Testing the Euclid algorithms
Does an algorithm do what its author wants it to do? A few test cases usually give some confidence in the core functionality. But tests are not enough. For test cases, one source uses 3009 and 884. Knuth suggested 40902, 24140. Another interesting case is the two relatively prime numbers 14157 and 5950.
But "exceptional cases" must be identified and tested. Will "Inelegant" perform properly when R > S, S > R, R = S? Ditto for "Elegant": B > A, A > B, A = B? (Yes to all). What happens when one number is zero, both numbers are zero? ("Inelegant" computes forever in all cases; "Elegant" computes forever when A = 0.) What happens if negative numbers are entered? Fractional numbers? If the input numbers, i.e. the domain of the function computed by the algorithm/program, is to include only positive integers including zero, then the failures at zero indicate that the algorithm (and the program that instantiates it) is a partial function rather than a total function. A notable failure due to exceptions is the Ariane 5 Flight 501 rocket failure (June 4, 1996).
Proof of program correctness by use of mathematical induction: Knuth demonstrates the application of mathematical induction to an "extended" version of Euclid's algorithm, and he proposes "a general method applicable to proving the validity of any algorithm". Tausworthe proposes that a measure of the complexity of a program be the length of its correctness proof.
Measuring and improving the Euclid algorithms
Elegance (compactness) versus goodness (speed): With only six core instructions, "Elegant" is the clear winner, compared to "Inelegant" at thirteen instructions. However, "Inelegant" is faster (it arrives at HALT in fewer steps). Algorithm analysis indicates why this is the case: "Elegant" does two conditional tests in every subtraction loop, whereas "Inelegant" only does one. As the algorithm (usually) requires many loop-throughs, on average much time is wasted doing a "B = 0?" test that is needed only after the remainder is computed.
Can the algorithms be improved?: Once the programmer judges a program "fit" and "effective"—that is, it computes the function intended by its author—then the question becomes, can it be improved?
The compactness of "Inelegant" can be improved by the elimination of five steps. But Chaitin proved that compacting an algorithm cannot be automated by a generalized algorithm; rather, it can only be done heuristically; i.e., by exhaustive search (examples to be found at Busy beaver), trial and error, cleverness, insight, application of inductive reasoning, etc. Observe that steps 4, 5 and 6 are repeated in steps 11, 12 and 13. Comparison with "Elegant" provides a hint that these steps, together with steps 2 and 3, can be eliminated. This reduces the number of core instructions from thirteen to eight, which makes it "more elegant" than "Elegant", at nine steps.
The speed of "Elegant" can be improved by moving the "B=0?" test outside of the two subtraction loops. This change calls for the addition of three instructions (B = 0?, A = 0?, GOTO). Now "Elegant" computes the example-numbers faster; whether this is always the case for any given A, B, and R, S would require a detailed analysis.
Algorithmic analysis
It is frequently important to know how much of a particular resource (such as time or storage) is theoretically required for a given algorithm. Methods have been developed for the analysis of algorithms to obtain such quantitative answers (estimates); for example, an algorithm which adds up the elements of a list of n numbers would have a time requirement of O(n), using big O notation. At all times the algorithm only needs to remember two values: the sum of all the elements so far, and its current position in the input list. Therefore, it is said to have a space requirement of O(1), if the space required to store the input numbers is not counted, or O(n) if it is counted.
Different algorithms may complete the same task with a different set of instructions in less or more time, space, or 'effort' than others. For example, a binary search algorithm (with cost O(log n)) outperforms a sequential search (cost O(n) ) when used for table lookups on sorted lists or arrays.
Formal versus empirical
The analysis, and study of algorithms is a discipline of computer science, and is often practiced abstractly without the use of a specific programming language or implementation. In this sense, algorithm analysis resembles other mathematical disciplines in that it focuses on the underlying properties of the algorithm and not on the specifics of any particular implementation. Usually pseudocode is used for analysis as it is the simplest and most general representation. However, ultimately, most algorithms are usually implemented on particular hardware/software platforms and their algorithmic efficiency is eventually put to the test using real code. For the solution of a "one off" problem, the efficiency of a particular algorithm may not have significant consequences (unless n is extremely large) but for algorithms designed for fast interactive, commercial or long life scientific usage it may be critical. Scaling from small n to large n frequently exposes inefficient algorithms that are otherwise benign.
Empirical testing is useful because it may uncover unexpected interactions that affect performance. Benchmarks may be used to compare before/after potential improvements to an algorithm after program optimization.
Empirical tests cannot replace formal analysis, though, and are not trivial to perform in a fair manner.
Execution efficiency
To illustrate the potential improvements possible even in well-established algorithms, a recent significant innovation, relating to FFT algorithms (used heavily in the field of image processing), can decrease processing time up to 1,000 times for applications like medical imaging. In general, speed improvements depend on special properties of the problem, which are very common in practical applications. Speedups of this magnitude enable computing devices that make extensive use of image processing (like digital cameras and medical equipment) to consume less power.
Classification
There are various ways to classify algorithms, each with its own merits.
By implementation
One way to classify algorithms is by implementation means.
Recursion
A recursive algorithm is one that invokes (makes reference to) itself repeatedly until a certain condition (also known as termination condition) matches, which is a method common to functional programming. Iterative algorithms use repetitive constructs like loops and sometimes additional data structures like stacks to solve the given problems. Some problems are naturally suited for one implementation or the other. For example, towers of Hanoi is well understood using recursive implementation. Every recursive version has an equivalent (but possibly more or less complex) iterative version, and vice versa.
Logical
An algorithm may be viewed as controlled logical deduction. This notion may be expressed as: Algorithm = logic + control. The logic component expresses the axioms that may be used in the computation and the control component determines the way in which deduction is applied to the axioms. This is the basis for the logic programming paradigm. In pure logic programming languages, the control component is fixed and algorithms are specified by supplying only the logic component. The appeal of this approach is the elegant semantics: a change in the axioms produces a well-defined change in the algorithm.
Serial, parallel or distributed
Algorithms are usually discussed with the assumption that computers execute one instruction of an algorithm at a time. Those computers are sometimes called serial computers. An algorithm designed for such an environment is called a serial algorithm, as opposed to parallel algorithms or distributed algorithms. Parallel algorithms take advantage of computer architectures where several processors can work on a problem at the same time, whereas distributed algorithms utilize multiple machines connected with a computer network. Parallel or distributed algorithms divide the problem into more symmetrical or asymmetrical subproblems and collect the results back together. The resource consumption in such algorithms is not only processor cycles on each processor but also the communication overhead between the processors. Some sorting algorithms can be parallelized efficiently, but their communication overhead is expensive. Iterative algorithms are generally parallelizable. Some problems have no parallel algorithms and are called inherently serial problems.
Deterministic or non-deterministic
Deterministic algorithms solve the problem with exact decision at every step of the algorithm whereas non-deterministic algorithms solve problems via guessing although typical guesses are made more accurate through the use of heuristics.
Exact or approximate
While many algorithms reach an exact solution, approximation algorithms seek an approximation that is closer to the true solution. The approximation can be reached by either using a deterministic or a random strategy. Such algorithms have practical value for many hard problems. One of the examples of an approximate algorithm is the Knapsack problem, where there is a set of given items. Its goal is to pack the knapsack to get the maximum total value. Each item has some weight and some value. Total weight that can be carried is no more than some fixed number X. So, the solution must consider weights of items as well as their value.
Quantum algorithm
They run on a realistic model of quantum computation. The term is usually used for those algorithms which seem inherently quantum, or use some essential feature of Quantum computing such as quantum superposition or quantum entanglement.
By design paradigm
Another way of classifying algorithms is by their design methodology or paradigm. There is a certain number of paradigms, each different from the other. Furthermore, each of these categories includes many different types of algorithms. Some common paradigms are:
Brute-force or exhaustive search
This is the naive method of trying every possible solution to see which is best.
Divide and conquer
A divide and conquer algorithm repeatedly reduces an instance of a problem to one or more smaller instances of the same problem (usually recursively) until the instances are small enough to solve easily. One such example of divide and conquer is merge sorting. Sorting can be done on each segment of data after dividing data into segments and sorting of entire data can be obtained in the conquer phase by merging the segments. A simpler variant of divide and conquer is called a decrease and conquer algorithm, which solves an identical subproblem and uses the solution of this subproblem to solve the bigger problem. Divide and conquer divides the problem into multiple subproblems and so the conquer stage is more complex than decrease and conquer algorithms. An example of a decrease and conquer algorithm is the binary search algorithm.
Search and enumeration
Many problems (such as playing chess) can be modeled as problems on graphs. A graph exploration algorithm specifies rules for moving around a graph and is useful for such problems. This category also includes search algorithms, branch and bound enumeration and backtracking.
Randomized algorithm
Such algorithms make some choices randomly (or pseudo-randomly). They can be very useful in finding approximate solutions for problems where finding exact solutions can be impractical (see heuristic method below). For some of these problems, it is known that the fastest approximations must involve some randomness. Whether randomized algorithms with polynomial time complexity can be the fastest algorithms for some problems is an open question known as the P versus NP problem. There are two large classes of such algorithms:
Monte Carlo algorithms return a correct answer with high-probability. E.g. RP is the subclass of these that run in polynomial time.
Las Vegas algorithms always return the correct answer, but their running time is only probabilistically bound, e.g. ZPP.
Reduction of complexity
This technique involves solving a difficult problem by transforming it into a better-known problem for which we have (hopefully) asymptotically optimal algorithms. The goal is to find a reducing algorithm whose complexity is not dominated by the resulting reduced algorithm's. For example, one selection algorithm for finding the median in an unsorted list involves first sorting the list (the expensive portion) and then pulling out the middle element in the sorted list (the cheap portion). This technique is also known as transform and conquer.
Back tracking
In this approach, multiple solutions are built incrementally and abandoned when it is determined that they cannot lead to a valid full solution.
Optimization problems
For optimization problems there is a more specific classification of algorithms; an algorithm for such problems may fall into one or more of the general categories described above as well as into one of the following:
Linear programming
When searching for optimal solutions to a linear function bound to linear equality and inequality constraints, the constraints of the problem can be used directly in producing the optimal solutions. There are algorithms that can solve any problem in this category, such as the popular simplex algorithm. Problems that can be solved with linear programming include the maximum flow problem for directed graphs. If a problem additionally requires that one or more of the unknowns must be an integer then it is classified in integer programming. A linear programming algorithm can solve such a problem if it can be proved that all restrictions for integer values are superficial, i.e., the solutions satisfy these restrictions anyway. In the general case, a specialized algorithm or an algorithm that finds approximate solutions is used, depending on the difficulty of the problem.
Dynamic programming
When a problem shows optimal substructures—meaning the optimal solution to a problem can be constructed from optimal solutions to subproblems—and overlapping subproblems, meaning the same subproblems are used to solve many different problem instances, a quicker approach called dynamic programming avoids recomputing solutions that have already been computed. For example, Floyd–Warshall algorithm, the shortest path to a goal from a vertex in a weighted graph can be found by using the shortest path to the goal from all adjacent vertices. Dynamic programming and memoization go together. The main difference between dynamic programming and divide and conquer is that subproblems are more or less independent in divide and conquer, whereas subproblems overlap in dynamic programming. The difference between dynamic programming and straightforward recursion is in caching or memoization of recursive calls. When subproblems are independent and there is no repetition, memoization does not help; hence dynamic programming is not a solution for all complex problems. By using memoization or maintaining a table of subproblems already solved, dynamic programming reduces the exponential nature of many problems to polynomial complexity.
The greedy method
A greedy algorithm is similar to a dynamic programming algorithm in that it works by examining substructures, in this case not of the problem but of a given solution. Such algorithms start with some solution, which may be given or have been constructed in some way, and improve it by making small modifications. For some problems they can find the optimal solution while for others they stop at local optima, that is, at solutions that cannot be improved by the algorithm but are not optimum. The most popular use of greedy algorithms is for finding the minimal spanning tree where finding the optimal solution is possible with this method. Huffman Tree, Kruskal, Prim, Sollin are greedy algorithms that can solve this optimization problem.
The heuristic method
In optimization problems, heuristic algorithms can be used to find a solution close to the optimal solution in cases where finding the optimal solution is impractical. These algorithms work by getting closer and closer to the optimal solution as they progress. In principle, if run for an infinite amount of time, they will find the optimal solution. Their merit is that they can find a solution very close to the optimal solution in a relatively short time. Such algorithms include local search, tabu search, simulated annealing, and genetic algorithms. Some of them, like simulated annealing, are non-deterministic algorithms while others, like tabu search, are deterministic. When a bound on the error of the non-optimal solution is known, the algorithm is further categorized as an approximation algorithm.
By field of study
Every field of science has its own problems and needs efficient algorithms. Related problems in one field are often studied together. Some example classes are search algorithms, sorting algorithms, merge algorithms, numerical algorithms, graph algorithms, string algorithms, computational geometric algorithms, combinatorial algorithms, medical algorithms, machine learning, cryptography, data compression algorithms and parsing techniques.
Fields tend to overlap with each other, and algorithm advances in one field may improve those of other, sometimes completely unrelated, fields. For example, dynamic programming was invented for optimization of resource consumption in industry but is now used in solving a broad range of problems in many fields.
By complexity
Algorithms can be classified by the amount of time they need to complete compared to their input size:
Constant time: if the time needed by the algorithm is the same, regardless of the input size. E.g. an access to an array element.
Logarithmic time: if the time is a logarithmic function of the input size. E.g. binary search algorithm.
Linear time: if the time is proportional to the input size. E.g. the traverse of a list.
Polynomial time: if the time is a power of the input size. E.g. the bubble sort algorithm has quadratic time complexity.
Exponential time: if the time is an exponential function of the input size. E.g. Brute-force search.
Some problems may have multiple algorithms of differing complexity, while other problems might have no algorithms or no known efficient algorithms. There are also mappings from some problems to other problems. Owing to this, it was found to be more suitable to classify the problems themselves instead of the algorithms into equivalence classes based on the complexity of the best possible algorithms for them.
Continuous algorithms
The adjective "continuous" when applied to the word "algorithm" can mean:
An algorithm operating on data that represents continuous quantities, even though this data is represented by discrete approximations—such algorithms are studied in numerical analysis; or
An algorithm in the form of a differential equation that operates continuously on the data, running on an analog computer.
Legal issues
Algorithms, by themselves, are not usually patentable. In the United States, a claim consisting solely of simple manipulations of abstract concepts, numbers, or signals does not constitute "processes" (USPTO 2006), and hence algorithms are not patentable (as in Gottschalk v. Benson). However practical applications of algorithms are sometimes patentable. For example, in Diamond v. Diehr, the application of a simple feedback algorithm to aid in the curing of synthetic rubber was deemed patentable. The patenting of software is highly controversial, and there are highly criticized patents involving algorithms, especially data compression algorithms, such as Unisys' LZW patent.
Additionally, some cryptographic algorithms have export restrictions (see export of cryptography).
History: Development of the notion of "algorithm"
Ancient Near East
The earliest evidence of algorithms is found in the Babylonian mathematics of ancient Mesopotamia (modern Iraq). A Sumerian clay tablet found in Shuruppak near Baghdad and dated to circa 2500 BC described the earliest division algorithm. During the Hammurabi dynasty circa 1800-1600 BC, Babylonian clay tablets described algorithms for computing formulas. Algorithms were also used in Babylonian astronomy. Babylonian clay tablets describe and employ algorithmic procedures to compute the time and place of significant astronomical events.
Algorithms for arithmetic are also found in ancient Egyptian mathematics, dating back to the Rhind Mathematical Papyrus circa 1550 BC. Algorithms were later used in ancient Hellenistic mathematics. Two examples are the Sieve of Eratosthenes, which was described in the Introduction to Arithmetic by Nicomachus, and the Euclidean algorithm, which was first described in Euclid's Elements (c. 300 BC).
Discrete and distinguishable symbols
Tally-marks: To keep track of their flocks, their sacks of grain and their money the ancients used tallying: accumulating stones or marks scratched on sticks or making discrete symbols in clay. Through the Babylonian and Egyptian use of marks and symbols, eventually Roman numerals and the abacus evolved (Dilson, p. 16–41). Tally marks appear prominently in unary numeral system arithmetic used in Turing machine and Post–Turing machine computations.
Manipulation of symbols as "place holders" for numbers: algebra
Muhammad ibn Mūsā al-Khwārizmī, a Persian mathematician, wrote the Al-jabr in the 9th century. The terms "algorism" and "algorithm" are derived from the name al-Khwārizmī, while the term "algebra" is derived from the book Al-jabr. In Europe, the word "algorithm" was originally used to refer to the sets of rules and techniques used by Al-Khwarizmi to solve algebraic equations, before later being generalized to refer to any set of rules or techniques. This eventually culminated in Leibniz's notion of the calculus ratiocinator (ca 1680):
Cryptographic algorithms
The first cryptographic algorithm for deciphering encrypted code was developed by Al-Kindi, a 9th-century Arab mathematician, in A Manuscript On Deciphering Cryptographic Messages. He gave the first description of cryptanalysis by frequency analysis, the earliest codebreaking algorithm.
Mechanical contrivances with discrete states
The clock: Bolter credits the invention of the weight-driven clock as "The key invention [of Europe in the Middle Ages]", in particular, the verge escapement that provides us with the tick and tock of a mechanical clock. "The accurate automatic machine" led immediately to "mechanical automata" beginning in the 13th century and finally to "computational machines"—the difference engine and analytical engines of Charles Babbage and Countess Ada Lovelace, mid-19th century. Lovelace is credited with the first creation of an algorithm intended for processing on a computer—Babbage's analytical engine, the first device considered a real Turing-complete computer instead of just a calculator—and is sometimes called "history's first programmer" as a result, though a full implementation of Babbage's second device would not be realized until decades after her lifetime.
Logical machines 1870 – Stanley Jevons' "logical abacus" and "logical machine": The technical problem was to reduce Boolean equations when presented in a form similar to what is now known as Karnaugh maps. Jevons (1880) describes first a simple "abacus" of "slips of wood furnished with pins, contrived so that any part or class of the [logical] combinations can be picked out mechanically ... More recently, however, I have reduced the system to a completely mechanical form, and have thus embodied the whole of the indirect process of inference in what may be called a Logical Machine" His machine came equipped with "certain moveable wooden rods" and "at the foot are 21 keys like those of a piano [etc.] ...". With this machine he could analyze a "syllogism or any other simple logical argument".
This machine he displayed in 1870 before the Fellows of the Royal Society. Another logician John Venn, however, in his 1881 Symbolic Logic, turned a jaundiced eye to this effort: "I have no high estimate myself of the interest or importance of what are sometimes called logical machines ... it does not seem to me that any contrivances at present known or likely to be discovered really deserve the name of logical machines"; see more at Algorithm characterizations. But not to be outdone he too presented "a plan somewhat analogous, I apprehend, to Prof. Jevon's abacus ... [And] [a]gain, corresponding to Prof. Jevons's logical machine, the following contrivance may be described. I prefer to call it merely a logical-diagram machine ... but I suppose that it could do very completely all that can be rationally expected of any logical machine".
Jacquard loom, Hollerith punch cards, telegraphy and telephony – the electromechanical relay: Bell and Newell (1971) indicate that the Jacquard loom (1801), precursor to Hollerith cards (punch cards, 1887), and "telephone switching technologies" were the roots of a tree leading to the development of the first computers. By the mid-19th century the telegraph, the precursor of the telephone, was in use throughout the world, its discrete and distinguishable encoding of letters as "dots and dashes" a common sound. By the late 19th century the ticker tape (ca 1870s) was in use, as was the use of Hollerith cards in the 1890 U.S. census. Then came the teleprinter (ca. 1910) with its punched-paper use of Baudot code on tape.
Telephone-switching networks of electromechanical relays (invented 1835) was behind the work of George Stibitz (1937), the inventor of the digital adding device. As he worked in Bell Laboratories, he observed the "burdensome' use of mechanical calculators with gears. "He went home one evening in 1937 intending to test his idea... When the tinkering was over, Stibitz had constructed a binary adding device".
Davis (2000) observes the particular importance of the electromechanical relay (with its two "binary states" open and closed):
It was only with the development, beginning in the 1930s, of electromechanical calculators using electrical relays, that machines were built having the scope Babbage had envisioned."
Mathematics during the 19th century up to the mid-20th century
Symbols and rules: In rapid succession, the mathematics of George Boole (1847, 1854), Gottlob Frege (1879), and Giuseppe Peano (1888–1889) reduced arithmetic to a sequence of symbols manipulated by rules. Peano's The principles of arithmetic, presented by a new method (1888) was "the first attempt at an axiomatization of mathematics in a symbolic language".
But Heijenoort gives Frege (1879) this kudos: Frege's is "perhaps the most important single work ever written in logic. ... in which we see a " 'formula language', that is a lingua characterica, a language written with special symbols, "for pure thought", that is, free from rhetorical embellishments ... constructed from specific symbols that are manipulated according to definite rules". The work of Frege was further simplified and amplified by Alfred North Whitehead and Bertrand Russell in their Principia Mathematica (1910–1913).
The paradoxes: At the same time a number of disturbing paradoxes appeared in the literature, in particular, the Burali-Forti paradox (1897), the Russell paradox (1902–03), and the Richard Paradox. The resultant considerations led to Kurt Gödel's paper (1931)—he specifically cites the paradox of the liar—that completely reduces rules of recursion to numbers.
Effective calculability: In an effort to solve the Entscheidungsproblem defined precisely by Hilbert in 1928, mathematicians first set about to define what was meant by an "effective method" or "effective calculation" or "effective calculability" (i.e., a calculation that would succeed). In rapid succession the following appeared: Alonzo Church, Stephen Kleene and J.B. Rosser's λ-calculus a finely honed definition of "general recursion" from the work of Gödel acting on suggestions of Jacques Herbrand (cf. Gödel's Princeton lectures of 1934) and subsequent simplifications by Kleene. Church's proof that the Entscheidungsproblem was unsolvable, Emil Post's definition of effective calculability as a worker mindlessly following a list of instructions to move left or right through a sequence of rooms and while there either mark or erase a paper or observe the paper and make a yes-no decision about the next instruction. Alan Turing's proof of that the Entscheidungsproblem was unsolvable by use of his "a- [automatic-] machine"—in effect almost identical to Post's "formulation", J. Barkley Rosser's definition of "effective method" in terms of "a machine". Kleene's proposal of a precursor to "Church thesis" that he called "Thesis I", and a few years later Kleene's renaming his Thesis "Church's Thesis" and proposing "Turing's Thesis".
Emil Post (1936) and Alan Turing (1936–37, 1939)
Emil Post (1936) described the actions of a "computer" (human being) as follows:
"...two concepts are involved: that of a symbol space in which the work leading from problem to answer is to be carried out, and a fixed unalterable set of directions.
His symbol space would be
"a two-way infinite sequence of spaces or boxes... The problem solver or worker is to move and work in this symbol space, being capable of being in, and operating in but one box at a time.... a box is to admit of but two possible conditions, i.e., being empty or unmarked, and having a single mark in it, say a vertical stroke.
"One box is to be singled out and called the starting point. ...a specific problem is to be given in symbolic form by a finite number of boxes [i.e., INPUT] being marked with a stroke. Likewise, the answer [i.e., OUTPUT] is to be given in symbolic form by such a configuration of marked boxes...
"A set of directions applicable to a general problem sets up a deterministic process when applied to each specific problem. This process terminates only when it comes to the direction of type (C ) [i.e., STOP]". See more at Post–Turing machine
Alan Turing's work preceded that of Stibitz (1937); it is unknown whether Stibitz knew of the work of Turing. Turing's biographer believed that Turing's use of a typewriter-like model derived from a youthful interest: "Alan had dreamt of inventing typewriters as a boy; Mrs. Turing had a typewriter, and he could well have begun by asking himself what was meant by calling a typewriter 'mechanical'". Given the prevalence of Morse code and telegraphy, ticker tape machines, and teletypewriters we might conjecture that all were influences.
Turing—his model of computation is now called a Turing machine—begins, as did Post, with an analysis of a human computer that he whittles down to a simple set of basic motions and "states of mind". But he continues a step further and creates a machine as a model of computation of numbers.
"Computing is normally done by writing certain symbols on paper. We may suppose this paper is divided into squares like a child's arithmetic book...I assume then that the computation is carried out on one-dimensional paper, i.e., on a tape divided into squares. I shall also suppose that the number of symbols which may be printed is finite...
"The behavior of the computer at any moment is determined by the symbols which he is observing, and his "state of mind" at that moment. We may suppose that there is a bound B to the number of symbols or squares which the computer can observe at one moment. If he wishes to observe more, he must use successive observations. We will also suppose that the number of states of mind which need be taken into account is finite...
"Let us imagine that the operations performed by the computer to be split up into 'simple operations' which are so elementary that it is not easy to imagine them further divided."
Turing's reduction yields the following:
"The simple operations must therefore include:
"(a) Changes of the symbol on one of the observed squares
"(b) Changes of one of the squares observed to another square within L squares of one of the previously observed squares.
"It may be that some of these change necessarily invoke a change of state of mind. The most general single operation must, therefore, be taken to be one of the following:
"(A) A possible change (a) of symbol together with a possible change of state of mind.
"(B) A possible change (b) of observed squares, together with a possible change of state of mind"
"We may now construct a machine to do the work of this computer."
A few years later, Turing expanded his analysis (thesis, definition) with this forceful expression of it:
"A function is said to be "effectively calculable" if its values can be found by some purely mechanical process. Though it is fairly easy to get an intuitive grasp of this idea, it is nevertheless desirable to have some more definite, mathematical expressible definition ... [he discusses the history of the definition pretty much as presented above with respect to Gödel, Herbrand, Kleene, Church, Turing, and Post] ... We may take this statement literally, understanding by a purely mechanical process one which could be carried out by a machine. It is possible to give a mathematical description, in a certain normal form, of the structures of these machines. The development of these ideas leads to the author's definition of a computable function, and to an identification of computability † with effective calculability ... .
"† We shall use the expression "computable function" to mean a function calculable by a machine, and we let "effectively calculable" refer to the intuitive idea without particular identification with any one of these definitions".
J.B. Rosser (1939) and S.C. Kleene (1943)
J. Barkley Rosser defined an 'effective [mathematical] method' in the following manner (italicization added):
"'Effective method' is used here in the rather special sense of a method each step of which is precisely determined and which is certain to produce the answer in a finite number of steps. With this special meaning, three different precise definitions have been given to date. [his footnote #5; see discussion immediately below]. The simplest of these to state (due to Post and Turing) says essentially that an effective method of solving certain sets of problems exists if one can build a machine which will then solve any problem of the set with no human intervention beyond inserting the question and (later) reading the answer. All three definitions are equivalent, so it doesn't matter which one is used. Moreover, the fact that all three are equivalent is a very strong argument for the correctness of any one." (Rosser 1939:225–226)
Rosser's footnote No. 5 references the work of (1) Church and Kleene and their definition of λ-definability, in particular, Church's use of it in his An Unsolvable Problem of Elementary Number Theory (1936); (2) Herbrand and Gödel and their use of recursion, in particular, Gödel's use in his famous paper On Formally Undecidable Propositions of Principia Mathematica and Related Systems I (1931); and (3) Post (1936) and Turing (1936–37) in their mechanism-models of computation.
Stephen C. Kleene defined as his now-famous "Thesis I" known as the Church–Turing thesis. But he did this in the following context (boldface in original):
"12. Algorithmic theories... In setting up a complete algorithmic theory, what we do is to describe a procedure, performable for each set of values of the independent variables, which procedure necessarily terminates and in such manner that from the outcome we can read a definite answer, "yes" or "no," to the question, "is the predicate value true?"" (Kleene 1943:273)
History after 1950
A number of efforts have been directed toward further refinement of the definition of "algorithm", and activity is on-going because of issues surrounding, in particular, foundations of mathematics (especially the Church–Turing thesis) and philosophy of mind (especially arguments about artificial intelligence). For more, see Algorithm characterizations.
See also
Abstract machine
Algorithm engineering
Algorithm characterizations
Algorithmic bias
Algorithmic composition
Algorithmic entities
Algorithmic synthesis
Algorithmic technique
Algorithmic topology
Garbage in, garbage out
Introduction to Algorithms (textbook)
List of algorithms
List of algorithm general topics
List of important publications in theoretical computer science – Algorithms
Regulation of algorithms
Theory of computation
Computability theory
Computational complexity theory
Notes
Bibliography
Bell, C. Gordon and Newell, Allen (1971), Computer Structures: Readings and Examples, McGraw–Hill Book Company, New York. .
Includes an excellent bibliography of 56 references.
,
: cf. Chapter 3 Turing machines where they discuss "certain enumerable sets not effectively (mechanically) enumerable".
Campagnolo, M.L., Moore, C., and Costa, J.F. (2000) An analog characterization of the subrecursive functions. In Proc. of the 4th Conference on Real Numbers and Computers, Odense University, pp. 91–109
Reprinted in The Undecidable, p. 89ff. The first expression of "Church's Thesis". See in particular page 100 (The Undecidable) where he defines the notion of "effective calculability" in terms of "an algorithm", and he uses the word "terminates", etc.
Reprinted in The Undecidable, p. 110ff. Church shows that the Entscheidungsproblem is unsolvable in about 3 pages of text and 3 pages of footnotes.
Davis gives commentary before each article. Papers of Gödel, Alonzo Church, Turing, Rosser, Kleene, and Emil Post are included; those cited in the article are listed here by author's name.
Davis offers concise biographies of Leibniz, Boole, Frege, Cantor, Hilbert, Gödel and Turing with von Neumann as the show-stealing villain. Very brief bios of Joseph-Marie Jacquard, Babbage, Ada Lovelace, Claude Shannon, Howard Aiken, etc.
,
Yuri Gurevich, Sequential Abstract State Machines Capture Sequential Algorithms, ACM Transactions on Computational Logic, Vol 1, no 1 (July 2000), pp. 77–111. Includes bibliography of 33 sources.
, 3rd edition 1976[?], (pbk.)
, . Cf. Chapter "The Spirit of Truth" for a history leading to, and a discussion of, his proof.
Presented to the American Mathematical Society, September 1935. Reprinted in The Undecidable, p. 237ff. Kleene's definition of "general recursion" (known now as mu-recursion) was used by Church in his 1935 paper An Unsolvable Problem of Elementary Number Theory that proved the "decision problem" to be "undecidable" (i.e., a negative result).
Reprinted in The Undecidable, p. 255ff. Kleene refined his definition of "general recursion" and proceeded in his chapter "12. Algorithmic theories" to posit "Thesis I" (p. 274); he would later repeat this thesis (in Kleene 1952:300) and name it "Church's Thesis"(Kleene 1952:317) (i.e., the Church thesis).
Kosovsky, N.K. Elements of Mathematical Logic and its Application to the theory of Subrecursive Algorithms, LSU Publ., Leningrad, 1981
A.A. Markov (1954) Theory of algorithms. [Translated by Jacques J. Schorr-Kon and PST staff] Imprint Moscow, Academy of Sciences of the USSR, 1954 [i.e., Jerusalem, Israel Program for Scientific Translations, 1961; available from the Office of Technical Services, U.S. Dept. of Commerce, Washington] Description 444 p. 28 cm. Added t.p. in Russian Translation of Works of the Mathematical Institute, Academy of Sciences of the USSR, v. 42. Original title: Teoriya algerifmov. [QA248.M2943 Dartmouth College library. U.S. Dept. of Commerce, Office of Technical Services, number OTS .]
Minsky expands his "...idea of an algorithm – an effective procedure..." in chapter 5.1 Computability, Effective Procedures and Algorithms. Infinite machines.
Reprinted in The Undecidable, pp. 289ff. Post defines a simple algorithmic-like process of a man writing marks or erasing marks and going from box to box and eventually halting, as he follows a list of simple instructions. This is cited by Kleene as one source of his "Thesis I", the so-called Church–Turing thesis.
Reprinted in The Undecidable, p. 223ff. Herein is Rosser's famous definition of "effective method": "...a method each step of which is precisely predetermined and which is certain to produce the answer in a finite number of steps... a machine which will then solve any problem of the set with no human intervention beyond inserting the question and (later) reading the answer" (p. 225–226, The Undecidable)
Cf. in particular the first chapter titled: Algorithms, Turing Machines, and Programs. His succinct informal definition: "...any sequence of instructions that can be obeyed by a robot, is called an algorithm" (p. 4).
. Corrections, ibid, vol. 43(1937) pp. 544–546. Reprinted in The Undecidable, p. 116ff. Turing's famous paper completed as a Master's dissertation while at King's College Cambridge UK.
Reprinted in The Undecidable, pp. 155ff. Turing's paper that defined "the oracle" was his PhD thesis while at Princeton.
United States Patent and Trademark Office (2006), 2106.02 **>Mathematical Algorithms: 2100 Patentability, Manual of Patent Examining Procedure (MPEP). Latest revision August 2006
Further reading
Knuth, Donald E. (2000). Selected Papers on Analysis of Algorithms. Stanford, California: Center for the Study of Language and Information.
Knuth, Donald E. (2010). Selected Papers on Design of Algorithms. Stanford, California: Center for the Study of Language and Information.
External links
Dictionary of Algorithms and Data Structures – National Institute of Standards and Technology
Algorithm repositories
The Stony Brook Algorithm Repository – State University of New York at Stony Brook
Collected Algorithms of the ACM – Association for Computing Machinery
The Stanford GraphBase – Stanford University
Articles with example pseudocode
Mathematical logic
Theoretical computer science | Algorithm |
In chemistry, an alcohol is a type of organic compound that carries at least one hydroxyl functional group (−OH) bound to a saturated carbon atom. The term alcohol originally referred to the primary alcohol ethanol (ethyl alcohol), which is used as a drug and is the main alcohol present in alcoholic drinks. An important class of alcohols, of which methanol and ethanol are the simplest members, includes all compounds for which the general formula is . Simple monoalcohols that are the subject of this article include primary (), secondary () and tertiary () alcohols.
The suffix -ol appears in the IUPAC chemical name of all substances where the hydroxyl group is the functional group with the highest priority. When a higher priority group is present in the compound, the prefix hydroxy- is used in its IUPAC name. The suffix -ol in non-IUPAC names (such as paracetamol or cholesterol) also typically indicates that the substance is an alcohol. However, many substances that contain hydroxyl functional groups (particularly sugars, such as glucose and sucrose) have names which include neither the suffix -ol, nor the prefix hydroxy-.
History
The inflammable nature of the exhalations of wine was already known to ancient natural philosophers such as Aristotle (384–322 BCE), Theophrastus (c. 371–287 BCE), and Pliny the Elder (23/24–79 CE). However, this did not immediately lead to the isolation of alcohol, even despite the development of more advanced distillation techniques in second- and third-century Roman Egypt. An important recognition, first found in one of the writings attributed to Jābir ibn Ḥayyān (ninth century CE), was that by adding salt to boiling wine, which increases the wine's relative volatility, the flammability of the resulting vapors may be enhanced. The distillation of wine is attested in Arabic works attributed to al-Kindī (c. 801–873 CE) and to al-Fārābī (c. 872–950), and in the 28th book of al-Zahrāwī's (Latin: Abulcasis, 936–1013) Kitāb al-Taṣrīf (later translated into Latin as Liber servatoris). In the twelfth century, recipes for the production of aqua ardens ("burning water", i.e., alcohol) by distilling wine with salt started to appear in a number of Latin works, and by the end of the thirteenth century it had become a widely known substance among Western European chemists.
The works of Taddeo Alderotti (1223–1296) describe a method for concentrating alcohol involving repeated fractional distillation through a water-cooled still, by which an alcohol purity of 90% could be obtained. The medicinal properties of ethanol were studied by Arnald of Villanova (1240–1311 CE) and John of Rupescissa (c. 1310–1366), the latter of whom regarded it as a life-preserving substance able to prevent all diseases (the aqua vitae or "water of life", also called by John the quintessence of wine).
Nomenclature
Etymology
The word "alcohol" is from the Arabic kohl (), a powder used as an eyeliner. Al- is the Arabic definite article, equivalent to the in English. Alcohol was originally used for the very fine powder produced by the sublimation of the natural mineral stibnite to form antimony trisulfide . It was considered to be the essence or "spirit" of this mineral. It was used as an antiseptic, eyeliner, and cosmetic. The meaning of alcohol was extended to distilled substances in general, and then narrowed to ethanol, when "spirits" was a synonym for hard liquor.
Bartholomew Traheron, in his 1543 translation of John of Vigo, introduces the word as a term used by "barbarous" authors for "fine powder." Vigo wrote: "the barbarous auctours use alcohol, or (as I fynde it sometymes wryten) alcofoll, for moost fine poudre."
The 1657 Lexicon Chymicum, by William Johnson glosses the word as "antimonium sive stibium." By extension, the word came to refer to any fluid obtained by distillation, including "alcohol of wine," the distilled essence of wine. Libavius in Alchymia (1594) refers to "vini alcohol vel vinum alcalisatum". Johnson (1657) glosses alcohol vini as "quando omnis superfluitas vini a vino separatur, ita ut accensum ardeat donec totum consumatur, nihilque fæcum aut phlegmatis in fundo remaneat." The word's meaning became restricted to "spirit of wine" (the chemical known today as ethanol) in the 18th century and was extended to the class of substances so-called as "alcohols" in modern chemistry after 1850.
The term ethanol was invented in 1892, blending "ethane" with the "-ol" ending of "alcohol", which was generalized as a libfix.
Systematic names
IUPAC nomenclature is used in scientific publications and where precise identification of the substance is important, especially in cases where the relative complexity of the molecule does not make such a systematic name unwieldy. In naming simple alcohols, the name of the alkane chain loses the terminal e and adds the suffix -ol, e.g., as in "ethanol" from the alkane chain name "ethane". When necessary, the position of the hydroxyl group is indicated by a number between the alkane name and the -ol: propan-1-ol for , propan-2-ol for . If a higher priority group is present (such as an aldehyde, ketone, or carboxylic acid), then the prefix hydroxy-is used, e.g., as in 1-hydroxy-2-propanone ().
In cases where the hydroxy group is bonded to an sp2 carbon on an aromatic ring, the molecule is classified separately as a phenol and is named using the IUPAC rules for naming phenols. Phenols have distinct properties and are not classified as alcohols.
Common names
In other less formal contexts, an alcohol is often called with the name of the corresponding alkyl group followed by the word "alcohol", e.g., methyl alcohol, ethyl alcohol. Propyl alcohol may be n-propyl alcohol or isopropyl alcohol, depending on whether the hydroxyl group is bonded to the end or middle carbon on the straight propane chain. As described under systematic naming, if another group on the molecule takes priority, the alcohol moiety is often indicated using the "hydroxy-" prefix.
Alcohols are then classified into primary, secondary (sec-, s-), and tertiary (tert-, t-), based upon the number of carbon atoms connected to the carbon atom that bears the hydroxyl functional group. (The respective numeric shorthands 1°, 2°, and 3° are also sometimes used in informal settings.) The primary alcohols have general formulas . The simplest primary alcohol is methanol (), for which R=H, and the next is ethanol, for which , the methyl group. Secondary alcohols are those of the form RR'CHOH, the simplest of which is 2-propanol (). For the tertiary alcohols the general form is RR'R"COH. The simplest example is tert-butanol (2-methylpropan-2-ol), for which each of R, R', and R" is . In these shorthands, R, R', and R" represent substituents, alkyl or other attached, generally organic groups.
In archaic nomenclature, alcohols can be named as derivatives of methanol using "-carbinol" as the ending. For instance, can be named trimethylcarbinol.
Applications
Alcohols have a long history of myriad uses. For simple mono-alcohols, which is the focus on this article, the following are most important industrial alcohols:
methanol, mainly for the production of formaldehyde and as a fuel additive
ethanol, mainly for alcoholic beverages, fuel additive, solvent
1-propanol, 1-butanol, and isobutyl alcohol for use as a solvent and precursor to solvents
C6–C11 alcohols used for plasticizers, e.g. in polyvinylchloride
fatty alcohol (C12–C18), precursors to detergents
Methanol is the most common industrial alcohol, with about 12 million tons/y produced in 1980. The combined capacity of the other alcohols is about the same, distributed roughly equally.
Toxicity
With respect to acute toxicity, simple alcohols have low acute toxicities. Doses of several milliliters are tolerated. For pentanols, hexanols, octanols and longer alcohols, LD50 range from 2–5 g/kg (rats, oral). Methanol and ethanol are less acutely toxic. All alcohols are mild skin irritants.
The metabolism of methanol (and ethylene glycol) is affected by the presence of ethanol, which has a higher affinity for liver alcohol dehydrogenase. In this way methanol will be excreted intact in urine.
Physical properties
In general, the hydroxyl group makes alcohols polar. Those groups can form hydrogen bonds to one another and to most other compounds. Owing to the presence of the polar OH alcohols are more water-soluble than simple hydrocarbons. Methanol, ethanol, and propanol are miscible in water. Butanol, with a four-carbon chain, is moderately soluble.
Because of hydrogen bonding, alcohols tend to have higher boiling points than comparable hydrocarbons and ethers. The boiling point of the alcohol ethanol is 78.29 °C, compared to 69 °C for the hydrocarbon hexane, and 34.6 °C for diethyl ether.
Occurrence in nature
Simple alcohols are found widely in nature. Ethanol is the most prominent because it is the product of fermentation, a major energy-producing pathway. Other simple alcohols, chiefly fusel alcohols, are formed in only trace amounts. More complex alcohols however are pervasive, as manifested in sugars, some amino acids, and fatty acids.
Production
Ziegler and oxo processes
In the Ziegler process, linear alcohols are produced from ethylene and triethylaluminium followed by oxidation and hydrolysis. An idealized synthesis of 1-octanol is shown:
The process generates a range of alcohols that are separated by distillation.
Many higher alcohols are produced by hydroformylation of alkenes followed by hydrogenation. When applied to a terminal alkene, as is common, one typically obtains a linear alcohol:
Such processes give fatty alcohols, which are useful for detergents.
Hydration reactions
Some low molecular weight alcohols of industrial importance are produced by the addition of water to alkenes. Ethanol, isopropanol, 2-butanol, and tert-butanol are produced by this general method. Two implementations are employed, the direct and indirect methods. The direct method avoids the formation of stable intermediates, typically using acid catalysts. In the indirect method, the alkene is converted to the sulfate ester, which is subsequently hydrolyzed. The direct hydration using ethylene (ethylene hydration) or other alkenes from cracking of fractions of distilled crude oil.
Hydration is also used industrially to produce the diol ethylene glycol from ethylene oxide.
Biological routes
Ethanol is obtained by fermentation using glucose produced from sugar from the hydrolysis of starch, in the presence of yeast and temperature of less than 37 °C to produce ethanol. For instance, such a process might proceed by the conversion of sucrose by the enzyme invertase into glucose and fructose, then the conversion of glucose by the enzyme complex zymase into ethanol and carbon dioxide.
Several species of the benign bacteria in the intestine use fermentation as a form of anaerobic metabolism. This metabolic reaction produces ethanol as a waste product. Thus, human bodies contain some quantity of alcohol endogenously produced by these bacteria. In rare cases, this can be sufficient to cause "auto-brewery syndrome" in which intoxicating quantities of alcohol are produced.
Like ethanol, butanol can be produced by fermentation processes. Saccharomyces yeast are known to produce these higher alcohols at temperatures above . The bacterium Clostridium acetobutylicum can feed on cellulose to produce butanol on an industrial scale.
Substitution
Primary alkyl halides react with aqueous NaOH or KOH mainly to primary alcohols in nucleophilic aliphatic substitution. (Secondary and especially tertiary alkyl halides will give the elimination (alkene) product instead). Grignard reagents react with carbonyl groups to secondary and tertiary alcohols. Related reactions are the Barbier reaction and the Nozaki-Hiyama reaction.
Reduction
Aldehydes or ketones are reduced with sodium borohydride or lithium aluminium hydride (after an acidic workup). Another reduction by aluminiumisopropylates is the Meerwein-Ponndorf-Verley reduction. Noyori asymmetric hydrogenation is the asymmetric reduction of β-keto-esters.
Hydrolysis
Alkenes engage in an acid catalysed hydration reaction using concentrated sulfuric acid as a catalyst that gives usually secondary or tertiary alcohols. The hydroboration-oxidation and oxymercuration-reduction of alkenes are more reliable in organic synthesis. Alkenes react with NBS and water in halohydrin formation reaction. Amines can be converted to diazonium salts, which are then hydrolyzed.
The formation of a secondary alcohol via reduction and hydration is shown:
Reactions
Deprotonation
With aqueous pKa values of around 16–19, they are, in general, slightly weaker acids than water. With strong bases such as sodium hydride or sodium they form salts called alkoxides, with the general formula RO− M+.
The acidity of alcohols is strongly affected by solvation. In the gas phase, alcohols are more acidic than in water. In DMSO, alcohols (and water) have a pKa of around 29–32. As a consequence, alkoxides (and hydroxide) are powerful bases and nucleophiles (e.g., for the Williamson ether synthesis) in this solvent. In particular, RO– or HO– in DMSO can be used to generate significant equilibrium concentrations of acetylide ions through the deprotonation of alkynes (see Favorskii reaction).
Nucleophilic substitution
The OH group is not a good leaving group in nucleophilic substitution reactions, so neutral alcohols do not react in such reactions. However, if the oxygen is first protonated to give , the leaving group (water) is much more stable, and the nucleophilic substitution can take place. For instance, tertiary alcohols react with hydrochloric acid to produce tertiary alkyl halides, where the hydroxyl group is replaced by a chlorine atom by unimolecular nucleophilic substitution. If primary or secondary alcohols are to be reacted with hydrochloric acid, an activator such as zinc chloride is needed. In alternative fashion, the conversion may be performed directly using thionyl chloride.[1]
Alcohols may, likewise, be converted to alkyl bromides using hydrobromic acid or phosphorus tribromide, for example:
In the Barton-McCombie deoxygenation an alcohol is deoxygenated to an alkane with tributyltin hydride or a trimethylborane-water complex in a radical substitution reaction.
Dehydration
Meanwhile, the oxygen atom has lone pairs of nonbonded electrons that render it weakly basic in the presence of strong acids such as sulfuric acid. For example, with methanol:
Upon treatment with strong acids, alcohols undergo the E1 elimination reaction to produce alkenes. The reaction, in general, obeys Zaitsev's Rule, which states that the most stable (usually the most substituted) alkene is formed. Tertiary alcohols eliminate easily at just above room temperature, but primary alcohols require a higher temperature.
This is a diagram of acid catalysed dehydration of ethanol to produce ethylene:
A more controlled elimination reaction requires the formation of the xanthate ester.
Protonolysis
Tertiary alcohols react with strong acids to generate carbocations. The reaction is related to their dehydration, e.g. isobutylene from tert-butyl alcohol. A special kind of dehydration reaction involves triphenylmethanol and especially its amine-substituted derivatives. When treated with acid, these alcohols lose water to give stable carbocations, which are commercial dyes.
Esterification
Alcohol and carboxylic acids react in the so-called Fischer esterification. The reaction usually requires a catalyst, such as concentrated sulfuric acid:
Other types of ester are prepared in a similar manner for example, tosyl (tosylate) esters are made by reaction of the alcohol with p-toluenesulfonyl chloride in pyridine.
Oxidation
Primary alcohols () can be oxidized either to aldehydes (R-CHO) or to carboxylic acids (). The oxidation of secondary alcohols (R1R2CH-OH) normally terminates at the ketone (R1R2C=O) stage. Tertiary alcohols (R1R2R3C-OH) are resistant to oxidation.
The direct oxidation of primary alcohols to carboxylic acids normally proceeds via the corresponding aldehyde, which is transformed via an aldehyde hydrate () by reaction with water before it can be further oxidized to the carboxylic acid.
Reagents useful for the transformation of primary alcohols to aldehydes are normally also suitable for the oxidation of secondary alcohols to ketones. These include Collins reagent and Dess-Martin periodinane. The direct oxidation of primary alcohols to carboxylic acids can be carried out using potassium permanganate or the Jones reagent.
See also
Enol
Ethanol fuel
Fatty alcohol
Index of alcohol-related articles
List of alcohols
Lucas test
Polyol
Rubbing alcohol
Sugar alcohol
Transesterification
Citations
General references
Antiseptics
Functional groups | Alcohol (chemistry) |
Algorithms for calculating variance play a major role in computational statistics. A key difficulty in the design of good algorithms for this problem is that formulas for the variance may involve sums of squares, which can lead to numerical instability as well as to arithmetic overflow when dealing with large values.
Naïve algorithm
A formula for calculating the variance of an entire population of size N is:
Using Bessel's correction to calculate an unbiased estimate of the population variance from a finite sample of n observations, the formula is:
Therefore, a naïve algorithm to calculate the estimated variance is given by the following:
Let
For each datum :
This algorithm can easily be adapted to compute the variance of a finite population: simply divide by N instead of n − 1 on the last line.
Because and can be very similar numbers, cancellation can lead to the precision of the result to be much less than the inherent precision of the floating-point arithmetic used to perform the computation. Thus this algorithm should not be used in practice, and several alternate, numerically stable, algorithms have been proposed. This is particularly bad if the standard deviation is small relative to the mean.
Computing shifted data
The variance is invariant with respect to changes in a location parameter, a property which can be used to avoid the catastrophic cancellation in this formula.
with any constant, which leads to the new formula
the closer is to the mean value the more accurate the result will be, but just choosing a value inside the
samples range will guarantee the desired stability. If the values are small then there are no problems with the sum of its squares, on the contrary, if they are large it necessarily means that the variance is large as well. In any case the second term in the formula is always smaller than the first one therefore no cancellation may occur.
If just the first sample is taken as the algorithm can be written in Python programming language as
def shifted_data_variance(data):
if len(data) < 2:
return 0.0
K = data[0]
n = Ex = Ex2 = 0.0
for x in data:
n = n + 1
Ex += x - K
Ex2 += (x - K) * (x - K)
variance = (Ex2 - (Ex * Ex) / n) / (n - 1)
# use n instead of (n-1) if want to compute the exact variance of the given data
# use (n-1) if data are samples of a larger population
return variance
This formula also facilitates the incremental computation that can be expressed as
K = n = Ex = Ex2 = 0.0
def add_variable(x):
global K, n, Ex, Ex2
if n == 0:
K = x
n += 1
Ex += x - K
Ex2 += (x - K) * (x - K)
def remove_variable(x):
global K, n, Ex, Ex2
n -= 1
Ex -= x - K
Ex2 -= (x - K) * (x - K)
def get_mean():
global K, n, Ex
return K + Ex / n
def get_variance():
global n, Ex, Ex2
return (Ex2 - (Ex * Ex) / n) / (n - 1)
Two-pass algorithm
An alternative approach, using a different formula for the variance, first computes the sample mean,
and then computes the sum of the squares of the differences from the mean,
where s is the standard deviation. This is given by the following code:
def two_pass_variance(data):
n = sum1 = sum2 = 0
for x in data:
n += 1
sum1 += x
mean = sum1 / n
for x in data:
sum2 += (x - mean) * (x - mean)
variance = sum2 / (n - 1)
return variance
This algorithm is numerically stable if n is small. However, the results of both of these simple algorithms ("naïve" and "two-pass") can depend inordinately on the ordering of the data and can give poor results for very large data sets due to repeated roundoff error in the accumulation of the sums. Techniques such as compensated summation can be used to combat this error to a degree.
Welford's online algorithm
It is often useful to be able to compute the variance in a single pass, inspecting each value only once; for example, when the data is being collected without enough storage to keep all the values, or when costs of memory access dominate those of computation. For such an online algorithm, a recurrence relation is required between quantities from which the required statistics can be calculated in a numerically stable fashion.
The following formulas can be used to update the mean and (estimated) variance of the sequence, for an additional element xn. Here, denotes the sample mean of the first n samples , their biased sample variance, and their unbiased sample variance.
These formulas suffer from numerical instability, as they repeatedly subtract a small number from a big number which scales with n. A better quantity for updating is the sum of squares of differences from the current mean, , here denoted :
This algorithm was found by Welford, and it has been thoroughly analyzed. It is also common to denote and .
An example Python implementation for Welford's algorithm is given below.
# For a new value newValue, compute the new count, new mean, the new M2.
# mean accumulates the mean of the entire dataset
# M2 aggregates the squared distance from the mean
# count aggregates the number of samples seen so far
def update(existingAggregate, newValue):
(count, mean, M2) = existingAggregate
count += 1
delta = newValue - mean
mean += delta / count
delta2 = newValue - mean
M2 += delta * delta2
return (count, mean, M2)
# Retrieve the mean, variance and sample variance from an aggregate
def finalize(existingAggregate):
(count, mean, M2) = existingAggregate
if count < 2:
return float("nan")
else:
(mean, variance, sampleVariance) = (mean, M2 / count, M2 / (count - 1))
return (mean, variance, sampleVariance)
This algorithm is much less prone to loss of precision due to catastrophic cancellation, but might not be as efficient because of the division operation inside the loop. For a particularly robust two-pass algorithm for computing the variance, one can first compute and subtract an estimate of the mean, and then use this algorithm on the residuals.
The parallel algorithm below illustrates how to merge multiple sets of statistics calculated online.
Weighted incremental algorithm
The algorithm can be extended to handle unequal sample weights, replacing the simple counter n with the sum of weights seen so far. West (1979) suggests this incremental algorithm:
def weighted_incremental_variance(data_weight_pairs):
w_sum = w_sum2 = mean = S = 0
for x, w in data_weight_pairs:
w_sum = w_sum + w
w_sum2 = w_sum2 + w * w
mean_old = mean
mean = mean_old + (w / w_sum) * (x - mean_old)
S = S + w * (x - mean_old) * (x - mean)
population_variance = S / w_sum
# Bessel's correction for weighted samples
# Frequency weights
sample_frequency_variance = S / (w_sum - 1)
# Reliability weights
sample_reliability_variance = S / (w_sum - w_sum2 / w_sum)
Parallel algorithm
Chan et al. note that Welford's online algorithm detailed above is a special case of an algorithm that works for combining arbitrary sets and :
.
This may be useful when, for example, multiple processing units may be assigned to discrete parts of the input.
Chan's method for estimating the mean is numerically unstable when and both are large, because the numerical error in is not scaled down in the way that it is in the case. In such cases, prefer .
def parallel_variance(n_a, avg_a, M2_a, n_b, avg_b, M2_b):
n = n_a + n_b
delta = avg_b - avg_a
M2 = M2_a + M2_b + delta ** 2 * n_a * n_b / n
var_ab = M2 / (n - 1)
return var_ab
This can be generalized to allow parallelization with AVX, with GPUs, and computer clusters, and to covariance.
Example
Assume that all floating point operations use standard IEEE 754 double-precision arithmetic. Consider the sample (4, 7, 13, 16) from an infinite population. Based on this sample, the estimated population mean is 10, and the unbiased estimate of population variance is 30. Both the naïve algorithm and two-pass algorithm compute these values correctly.
Next consider the sample (, , , ), which gives rise to the same estimated variance as the first sample. The two-pass algorithm computes this variance estimate correctly, but the naïve algorithm returns 29.333333333333332 instead of 30.
While this loss of precision may be tolerable and viewed as a minor flaw of the naïve algorithm, further increasing the offset makes the error catastrophic. Consider the sample (, , , ). Again the estimated population variance of 30 is computed correctly by the two-pass algorithm, but the naïve algorithm now computes it as −170.66666666666666. This is a serious problem with naïve algorithm and is due to catastrophic cancellation in the subtraction of two similar numbers at the final stage of the algorithm.
Higher-order statistics
Terriberry extends Chan's formulae to calculating the third and fourth central moments, needed for example when estimating skewness and kurtosis:
Here the are again the sums of powers of differences from the mean , giving
For the incremental case (i.e., ), this simplifies to:
By preserving the value , only one division operation is needed and the higher-order statistics can thus be calculated for little incremental cost.
An example of the online algorithm for kurtosis implemented as described is:
def online_kurtosis(data):
n = mean = M2 = M3 = M4 = 0
for x in data:
n1 = n
n = n + 1
delta = x - mean
delta_n = delta / n
delta_n2 = delta_n * delta_n
term1 = delta * delta_n * n1
mean = mean + delta_n
M4 = M4 + term1 * delta_n2 * (n*n - 3*n + 3) + 6 * delta_n2 * M2 - 4 * delta_n * M3
M3 = M3 + term1 * delta_n * (n - 2) - 3 * delta_n * M2
M2 = M2 + term1
# Note, you may also calculate variance using M2, and skewness using M3
# Caution: If all the inputs are the same, M2 will be 0, resulting in a division by 0.
kurtosis = (n * M4) / (M2 * M2) - 3
return kurtosis
Pébaÿ
further extends these results to arbitrary-order central moments, for the incremental and the pairwise cases, and subsequently Pébaÿ et al.
for weighted and compound moments. One can also find there similar formulas for covariance.
Choi and Sweetman
offer two alternative methods to compute the skewness and kurtosis, each of which can save substantial computer memory requirements and CPU time in certain applications. The first approach is to compute the statistical moments by separating the data into bins and then computing the moments from the geometry of the resulting histogram, which effectively becomes a one-pass algorithm for higher moments. One benefit is that the statistical moment calculations can be carried out to arbitrary accuracy such that the computations can be tuned to the precision of, e.g., the data storage format or the original measurement hardware. A relative histogram of a random variable can be constructed in the conventional way: the range of potential values is
divided into bins and the number of occurrences within each bin are counted and plotted such that the area of each rectangle equals the portion of the sample values within that bin:
where and represent the frequency and the relative frequency at bin and is the total area of the histogram. After this normalization, the raw moments and central moments of can be calculated from the relative histogram:
where the superscript indicates the moments are calculated from the histogram. For constant bin width these two expressions can be simplified using :
The second approach from Choi and Sweetman is an analytical methodology to combine statistical moments from individual segments of a time-history such that the resulting overall moments are those of the complete time-history. This methodology could be used for parallel computation of statistical moments with subsequent combination of those moments, or for combination of statistical moments computed at sequential times.
If sets of statistical moments are known:
for , then each can
be expressed in terms of the equivalent raw moments:
where is generally taken to be the duration of the time-history, or the number of points if is constant.
The benefit of expressing the statistical moments in terms of is that the sets can be combined by addition, and there is no upper limit on the value of .
where the subscript represents the concatenated time-history or combined . These combined values of can then be inversely transformed into raw moments representing the complete concatenated time-history
Known relationships between the raw moments () and the central moments ()
are then used to compute the central moments of the concatenated time-history. Finally, the statistical moments of the concatenated history are computed from the central moments:
Covariance
Very similar algorithms can be used to compute the covariance.
Naïve algorithm
The naïve algorithm is
For the algorithm above, one could use the following Python code:
def naive_covariance(data1, data2):
n = len(data1)
sum12 = 0
sum1 = sum(data1)
sum2 = sum(data2)
for i1, i2 in zip(data1, data2):
sum12 += i1 * i2
covariance = (sum12 - sum1 * sum2 / n) / n
return covariance
With estimate of the mean
As for the variance, the covariance of two random variables is also shift-invariant, so given any two constant values and it can be written:
and again choosing a value inside the range of values will stabilize the formula against catastrophic cancellation as well as make it more robust against big sums. Taking the first value of each data set, the algorithm can be written as:
def shifted_data_covariance(data_x, data_y):
n = len(data_x)
if n < 2:
return 0
kx = data_x[0]
ky = data_y[0]
Ex = Ey = Exy = 0
for ix, iy in zip(data_x, data_y):
Ex += ix - kx
Ey += iy - ky
Exy += (ix - kx) * (iy - ky)
return (Exy - Ex * Ey / n) / n
Two-pass
The two-pass algorithm first computes the sample means, and then the covariance:
The two-pass algorithm may be written as:
def two_pass_covariance(data1, data2):
n = len(data1)
mean1 = sum(data1) / n
mean2 = sum(data2) / n
covariance = 0
for i1, i2 in zip(data1, data2):
a = i1 - mean1
b = i2 - mean2
covariance += a * b / n
return covariance
A slightly more accurate compensated version performs the full naive algorithm on the residuals. The final sums and should be zero, but the second pass compensates for any small error.
Online
A stable one-pass algorithm exists, similar to the online algorithm for computing the variance, that computes co-moment :
The apparent asymmetry in that last equation is due to the fact that , so both update terms are equal to . Even greater accuracy can be achieved by first computing the means, then using the stable one-pass algorithm on the residuals.
Thus the covariance can be computed as
def online_covariance(data1, data2):
meanx = meany = C = n = 0
for x, y in zip(data1, data2):
n += 1
dx = x - meanx
meanx += dx / n
meany += (y - meany) / n
C += dx * (y - meany)
population_covar = C / n
# Bessel's correction for sample variance
sample_covar = C / (n - 1)
A small modification can also be made to compute the weighted covariance:
def online_weighted_covariance(data1, data2, data3):
meanx = meany = 0
wsum = wsum2 = 0
C = 0
for x, y, w in zip(data1, data2, data3):
wsum += w
wsum2 += w * w
dx = x - meanx
meanx += (w / wsum) * dx
meany += (w / wsum) * (y - meany)
C += w * dx * (y - meany)
population_covar = C / wsum
# Bessel's correction for sample variance
# Frequency weights
sample_frequency_covar = C / (wsum - 1)
# Reliability weights
sample_reliability_covar = C / (wsum - wsum2 / wsum)
Likewise, there is a formula for combining the covariances of two sets that can be used to parallelize the computation:
Weighted batched version
A version of the weighted online algorithm that does batched updated also exists: let
denote the weights, and write
The covariance can then be computed as
See also
Kahan summation algorithm
Squared deviations from the mean
Yamartino method
References
External links
Statistical algorithms
Statistical deviation and dispersion
Articles with example pseudocode
Articles with example Python (programming language) code | Algorithms for calculating variance |
Ascorbic acid is an organic compound with formula , originally called hexuronic acid. It is a white solid, but impure samples can appear yellowish. It dissolves well in water to give mildly acidic solutions. It is a mild reducing agent.
Ascorbic acid exists as two enantiomers (mirror-image isomers), commonly denoted "" (for "levo") and "" (for "dextro"). The isomer is the one most often encountered: it occurs naturally in many foods, and is one form ("vitamer") of vitamin C, an essential nutrient for humans and many animals. Deficiency of vitamin C causes scurvy, formerly a major disease of sailors in long sea voyages. It is used as a food additive and a dietary supplement for its antioxidant properties. The "" form can be made via chemical synthesis but has no significant biological role.
History
The antiscorbutic properties of certain foods were demonstrated in the 18th century by James Lind. In 1907, Axel Holst and Theodor Frølich discovered that the antiscorbutic factor was a water-soluble chemical substance, distinct from the one that prevented beriberi. Between 1928 and 1932, Albert Szent-Györgyi isolated a candidate for this substance, which he called it "hexuronic acid", first from plants and later from animal adrenal glands. In 1932 Charles Glen King confirmed that it was indeed the antiscorbutic factor.
In 1933, sugar chemist Walter Norman Haworth, working with samples of "hexuronic acid" that Szent-Györgyi had isolated from paprika and sent him in the previous year, deduced the correct structure and optical-isomeric nature of the compound, and in 1934 reported its first synthesis. In reference to the compound's antiscorbutic properties, Haworth and Szent-Györgyi proposed to rename it "a-scorbic acid" for the compound, and later specifically -ascorbic acid. Because of their work, in 1937 the Nobel Prizes for chemistry and medicine were awarded to Haworth and Szent-Györgyi, respectively.
Chemical properties
Acidity
Ascorbic acid is a vinylogous acid and forms the ascorbate anion when deprotonated on one of the hydroxyls. This property is characteristic of reductones: enediols with a carbonyl group adjacent to the enediol group, namely with the group –C(OH)=C(OH)–C(=O)–. The ascorbate anion is stabilized by electron delocalization that results from resonance between two forms:
For this reason, ascorbic acid is much more acidic than would be expected if the compound contained only isolated hydroxyl groups.
Salts
The ascorbate anion forms salts, such as sodium ascorbate, calcium ascorbate, and potassium ascorbate.
Esters
Ascorbic acid can also react with organic acids as an alcohol forming esters such as ascorbyl palmitate and ascorbyl stearate.
Nucleophilic attack
Nucleophilic attack of ascorbic acid on a proton results in a 1,3-diketone:
Oxidation
The ascorbate ion is the predominant species at typical biological pH values. It is a mild reducing agent and antioxidant. It is oxidized with loss of one electron to form a radical cation and then with loss of a second electron to form dehydroascorbic acid. It typically reacts with oxidants of the reactive oxygen species, such as the hydroxyl radical.
Ascorbic acid is special because it can transfer a single electron, owing to the resonance-stabilized nature of its own radical ion, called semidehydroascorbate. The net reaction is:
RO• + → RO− + C6H7O → ROH + C6H6O6
On exposure to oxygen, ascorbic acid will undergo further oxidative decomposition to various products including diketogulonic acid, xylonic acid, threonic acid and oxalic acid.
Reactive oxygen species are damaging to animals and plants at the molecular level due to their possible interaction with nucleic acids, proteins, and lipids. Sometimes these radicals initiate chain reactions. Ascorbate can terminate these chain radical reactions by electron transfer. The oxidized forms of ascorbate are relatively unreactive and do not cause cellular damage.
However, being a good electron donor, excess ascorbate in the presence of free metal ions can not only promote but also initiate free radical reactions, thus making it a potentially dangerous pro-oxidative compound in certain metabolic contexts.
Ascorbic acid and its sodium, potassium, and calcium salts are commonly used as antioxidant food additives. These compounds are water-soluble and, thus, cannot protect fats from oxidation: For this purpose, the fat-soluble esters of ascorbic acid with long-chain fatty acids (ascorbyl palmitate or ascorbyl stearate) can be used as food antioxidants.
Other reactions
It creates volatile compounds when mixed with glucose and amino acids in 90 °C.
It is a cofactor in tyrosine oxidation.
Uses
Food additive
The main use of -ascorbic acid and its salts is as food additives, mostly to combat oxidation. It is approved for this purpose in the EU with E number E300, USA, Australia, and New Zealand)
Dietary supplement
Another major use of -ascorbic acid is as dietary supplement.
Niche, non-food uses
Ascorbic acid is easily oxidized and so is used as a reductant in photographic developer solutions (among others) and as a preservative.
In fluorescence microscopy and related fluorescence-based techniques, ascorbic acid can be used as an antioxidant to increase fluorescent signal and chemically retard dye photobleaching.
It is also commonly used to remove dissolved metal stains, such as iron, from fiberglass swimming pool surfaces.
In plastic manufacturing, ascorbic acid can be used to assemble molecular chains more quickly and with less waste than traditional synthesis methods.
Heroin users are known to use ascorbic acid as a means to convert heroin base to a water-soluble salt so that it can be injected.
As justified by its reaction with iodine, it is used to negate the effects of iodine tablets in water purification. It reacts with the sterilized water, removing the taste, color, and smell of the iodine. This is why it is often sold as a second set of tablets in most sporting goods stores as Potable Aqua-Neutralizing Tablets, along with the potassium iodide tablets.
Intravenous high-dose ascorbate is being used as a chemotherapeutic and biological response modifying agent. Currently it is still under clinical trials.
It is sometimes used as a urinary acidifier to enhance the antiseptic effect of methenamine.
Synthesis
Natural biosynthesis of vitamin C occurs in many plants, and animals, by a variety of processes.
Industrial preparation
Eighty percent of the world's supply of ascorbic acid is produced in China.
Ascorbic acid is prepared in industry from glucose in a method based on the historical Reichstein process. In the first of a five-step process, glucose is catalytically hydrogenated to sorbitol, which is then oxidized by the microorganism Acetobacter suboxydans to sorbose. Only one of the six hydroxy groups is oxidized by this enzymatic reaction. From this point, two routes are available. Treatment of the product with acetone in the presence of an acid catalyst converts four of the remaining hydroxyl groups to acetals. The unprotected hydroxyl group is oxidized to the carboxylic acid by reaction with the catalytic oxidant TEMPO (regenerated by sodium hypochlorite — bleaching solution). Historically, industrial preparation via the Reichstein process used potassium permanganate as the bleaching solution. Acid-catalyzed hydrolysis of this product performs the dual function of removing the two acetal groups and ring-closing lactonization. This step yields ascorbic acid. Each of the five steps has a yield larger than 90%.
A more biotechnological process, first developed in China in the 1960s, but further developed in the 1990s, bypasses the use of acetone-protecting groups. A second genetically modified microbe species, such as mutant Erwinia, among others, oxidises sorbose into 2-ketogluconic acid (2-KGA), which can then undergo ring-closing lactonization via dehydration. This method is used in the predominant process used by the ascorbic acid industry in China, which supplies 80% of world's ascorbic acid. American and Chinese researchers are competing to engineer a mutant that can carry out a one-pot fermentation directly from glucose to 2-KGA, bypassing both the need for a second fermentation and the need to reduce glucose to sorbitol.
There exists a -ascorbic acid, which does not occur in nature but can be synthesized artificially. To be specific, -ascorbate is known to participate in many specific enzyme reactions that require the correct enantiomer (-ascorbate and not -ascorbate). -Ascorbic acid has a specific rotation of [α] = +23°.
Determination
The traditional way to analyze the ascorbic acid content is the process of titration with an oxidizing agent, and several procedures have been developed.
The popular iodometry approach uses iodine in the presence of a starch indicator. Iodine is reduced by ascorbic acid, and, when all the ascorbic acid has reacted, the iodine is then in excess, forming a blue-black complex with the starch indicator. This indicates the end-point of the titration.
As an alternative, ascorbic acid can be treated with iodine in excess, followed by back titration with sodium thiosulfate using starch as an indicator.
This iodometric method has been revised to exploit reaction of ascorbic acid with iodate and iodide in acid solution. Electrolyzing the solution of potassium iodide produces iodine, which reacts with ascorbic acid. The end of process is determined by potentiometric titration in a manner similar to Karl Fischer titration. The amount of ascorbic acid can be calculated by Faraday's law.
Another alternative uses N-bromosuccinimide (NBS) as the oxidizing agent, in the presence of potassium iodide and starch. The NBS first oxidizes the ascorbic acid; when the latter is exhausted, the NBS liberates the iodine from the potassium iodide, which then forms the blue-black complex with starch.
See also
Colour retention agent
Erythorbic acid: a diastereomer of ascorbic acid.
Mineral ascorbates: salts of ascorbic acid
Acids in wine
Notes and references
Further reading
External links
IPCS Poisons Information Monograph (PIM) 046
Interactive 3D-structure of vitamin C with details on the x-ray structure
Organic acids
Antioxidants
Dietary antioxidants
Coenzymes
Corrosion inhibitors
Furanones
Vitamers
Vitamin C
Biomolecules
3-Hydroxypropenals | Chemistry of ascorbic acid |
In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolically as . The process of solving for antiderivatives is called antidifferentiation (or indefinite integration), and its opposite operation is called differentiation, which is the process of finding a derivative. Antiderivatives are often denoted by capital Roman letters such as and .
Antiderivatives are related to definite integrals through the fundamental theorem of calculus: the definite integral of a function over a closed interval where the function is Riemann integrable is equal to the difference between the values of an antiderivative evaluated at the endpoints of the interval.
In physics, antiderivatives arise in the context of rectilinear motion (e.g., in explaining the relationship between position, velocity and acceleration). The discrete equivalent of the notion of antiderivative is antidifference.
Examples
The function is an antiderivative of , since the derivative of is , and since the derivative of a constant is zero, will have an infinite number of antiderivatives, such as , etc. Thus, all the antiderivatives of can be obtained by changing the value of in , where is an arbitrary constant known as the constant of integration. Essentially, the graphs of antiderivatives of a given function are vertical translations of each other, with each graph's vertical location depending upon the value .
More generally, the power function has antiderivative if , and if .
In physics, the integration of acceleration yields velocity plus a constant. The constant is the initial velocity term that would be lost upon taking the derivative of velocity, because the derivative of a constant term is zero. This same pattern applies to further integrations and derivatives of motion (position, velocity, acceleration, and so on).
Uses and properties
Antiderivatives can be used to compute definite integrals, using the fundamental theorem of calculus: if is an antiderivative of the integrable function over the interval , then:
Because of this, each of the infinitely many antiderivatives of a given function may be called the "indefinite integral" of f and written using the integral symbol with no bounds:
If is an antiderivative of , and the function is defined on some interval, then every other antiderivative of differs from by a constant: there exists a number such that for all . is called the constant of integration. If the domain of is a disjoint union of two or more (open) intervals, then a different constant of integration may be chosen for each of the intervals. For instance
is the most general antiderivative of on its natural domain
Every continuous function has an antiderivative, and one antiderivative is given by the definite integral of with variable upper boundary:
Varying the lower boundary produces other antiderivatives (but not necessarily all possible antiderivatives). This is another formulation of the fundamental theorem of calculus.
There are many functions whose antiderivatives, even though they exist, cannot be expressed in terms of elementary functions (like polynomials, exponential functions, logarithms, trigonometric functions, inverse trigonometric functions and their combinations). Examples of these are
From left to right, the functions are the error function, the Fresnel function, the sine integral, the logarithmic integral function and Sophomore's dream. For a more detailed discussion, see also Differential Galois theory.
Techniques of integration
Finding antiderivatives of elementary functions is often considerably harder than finding their derivatives (indeed, there is no pre-defined method for computing indefinite integrals). For some elementary functions, it is impossible to find an antiderivative in terms of other elementary functions. To learn more, see elementary functions and nonelementary integral.
There exist many properties and techniques for finding antiderivatives. These include, among others:
The linearity of integration (which breaks complicated integrals into simpler ones)
Integration by substitution, often combined with trigonometric identities or the natural logarithm
The inverse chain rule method (a special case of integration by substitution)
Integration by parts (to integrate products of functions)
Inverse function integration (a formula that expresses the antiderivative of the inverse of an invertible and continuous function , in terms of the antiderivative of and of ).
The method of partial fractions in integration (which allows us to integrate all rational functions—fractions of two polynomials)
The Risch algorithm
Additional techniques for multiple integrations (see for instance double integrals, polar coordinates, the Jacobian and the Stokes' theorem)
Numerical integration (a technique for approximating a definite integral when no elementary antiderivative exists, as in the case of )
Algebraic manipulation of integrand (so that other integration techniques, such as integration by substitution, may be used)
Cauchy formula for repeated integration (to calculate the -times antiderivative of a function)
Computer algebra systems can be used to automate some or all of the work involved in the symbolic techniques above, which is particularly useful when the algebraic manipulations involved are very complex or lengthy. Integrals which have already been derived can be looked up in a table of integrals.
Of non-continuous functions
Non-continuous functions can have antiderivatives. While there are still open questions in this area, it is known that:
Some highly pathological functions with large sets of discontinuities may nevertheless have antiderivatives.
In some cases, the antiderivatives of such pathological functions may be found by Riemann integration, while in other cases these functions are not Riemann integrable.
Assuming that the domains of the functions are open intervals:
A necessary, but not sufficient, condition for a function to have an antiderivative is that have the intermediate value property. That is, if is a subinterval of the domain of and is any real number between and , then there exists a between and such that . This is a consequence of Darboux's theorem.
The set of discontinuities of must be a meagre set. This set must also be an F-sigma set (since the set of discontinuities of any function must be of this type). Moreover, for any meagre F-sigma set, one can construct some function having an antiderivative, which has the given set as its set of discontinuities.
If has an antiderivative, is bounded on closed finite subintervals of the domain and has a set of discontinuities of Lebesgue measure 0, then an antiderivative may be found by integration in the sense of Lebesgue. In fact, using more powerful integrals like the Henstock–Kurzweil integral, every function for which an antiderivative exists is integrable, and its general integral coincides with its antiderivative.
If has an antiderivative on a closed interval , then for any choice of partition if one chooses sample points as specified by the mean value theorem, then the corresponding Riemann sum telescopes to the value .
However if is unbounded, or if is bounded but the set of discontinuities of has positive Lebesgue measure, a different choice of sample points may give a significantly different value for the Riemann sum, no matter how fine the partition. See Example 4 below.
Some examples
See also
Antiderivative (complex analysis)
Formal antiderivative
Jackson integral
Lists of integrals
Symbolic integration
Area
Notes
References
Further reading
Introduction to Classical Real Analysis, by Karl R. Stromberg; Wadsworth, 1981 (see also)
Historical Essay On Continuity Of Derivatives by Dave L. Renfro
External links
Wolfram Integrator — Free online symbolic integration with Mathematica
Mathematical Assistant on Web — symbolic computations online. Allows users to integrate in small steps (with hints for next step (integration by parts, substitution, partial fractions, application of formulas and others), powered by Maxima
Function Calculator from WIMS
Integral at HyperPhysics
Antiderivatives and indefinite integrals at the Khan Academy
Integral calculator at Symbolab
The Antiderivative at MIT
Introduction to Integrals at SparkNotes
Antiderivatives at Harvy Mudd College
Integral calculus
Linear operators in calculus | Antiderivative |
An acid–base reaction is a chemical reaction that occurs between an acid and a base. It can be used to determine pH. Several theoretical frameworks provide alternative conceptions of the reaction mechanisms and their application in solving related problems; these are called the acid–base theories, for example, Brønsted–Lowry acid–base theory.
Their importance becomes apparent in analyzing acid–base reactions for gaseous or liquid species, or when acid or base character may be somewhat less apparent. The first of these concepts was provided by the French chemist Antoine Lavoisier, around 1776.
It is important to think of the acid-base reaction models as theories that complement each other. For example, the current Lewis model has the broadest definition of what an acid and base are, with the Brønsted-Lowry theory being a subset of what acids and bases are, and the Arrhenius theory being the most restrictive.
Acid–base definitions
Historic development
The concept of an acid-base reaction was first proposed in 1754 by Guillaume-François Rouelle, who introduced the word "base" into chemistry to mean a substance which reacts with an acid to give it solid form (as a salt). Bases are mostly bitter in nature.
Lavoisier's oxygen theory of acids
The first scientific concept of acids and bases was provided by Lavoisier in around 1776. Since Lavoisier's knowledge of strong acids was mainly restricted to oxoacids, such as (nitric acid) and (sulfuric acid), which tend to contain central atoms in high oxidation states surrounded by oxygen, and since he was not aware of the true composition of the hydrohalic acids (HF, HCl, HBr, and HI), he defined acids in terms of their containing oxygen, which in fact he named from Greek words meaning "acid-former" (from the Greek ὀξύς (oxys) meaning "acid" or "sharp" and γεινομαι (geinomai) meaning "engender"). The Lavoisier definition held for over 30 years, until the 1810 article and subsequent lectures by Sir Humphry Davy in which he proved the lack of oxygen in , H2Te, and the hydrohalic acids. However, Davy failed to develop a new theory, concluding that "acidity does not depend upon any particular elementary substance, but upon peculiar arrangement of various substances". One notable modification of oxygen theory was provided by Jöns Jacob Berzelius, who stated that acids are oxides of nonmetals while bases are oxides of metals.
Liebig's hydrogen theory of acids
In 1838, Justus von Liebig proposed that an acid is a hydrogen-containing compound whose hydrogen can be replaced by a metal. This redefinition was based on his extensive work on the chemical composition of organic acids, finishing the doctrinal shift from oxygen-based acids to hydrogen-based acids started by Davy. Liebig's definition, while completely empirical, remained in use for almost 50 years until the adoption of the Arrhenius definition.
Arrhenius definition
The first modern definition of acids and bases in molecular terms was devised by Svante Arrhenius. A hydrogen theory of acids, it followed from his 1884 work with Friedrich Wilhelm Ostwald in establishing the presence of ions in aqueous solution and led to Arrhenius receiving the Nobel Prize in Chemistry in 1903.
As defined by Arrhenius:
an Arrhenius acid is a substance that dissociates in water to form hydrogen ions (H+); that is, an acid increases the concentration of H+ ions in an aqueous solution.
This causes the protonation of water, or the creation of the hydronium (H3O+) ion. Thus, in modern times, the symbol H+ is interpreted as a shorthand for H3O+, because it is now known that a bare proton does not exist as a free species in aqueous solution. This is the species which is measured by pH indicators to measure the acidity or basicity of a solution.
an Arrhenius base is a substance that dissociates in water to form hydroxide (OH−) ions; that is, a base increases the concentration of OH− ions in an aqueous solution."
The Arrhenius definitions of acidity and alkalinity are restricted to aqueous solutions and are not valid for most non-aqueous solutions, and refer to the concentration of the solvent ions. Under this definition, pure H2SO4 and HCl dissolved in toluene are not acidic, and molten NaOH and solutions of calcium amide in liquid ammonia are not alkaline. This led to the development of the Brønsted-Lowry theory and subsequent Lewis theory to account for these non-aqueous exceptions.
Overall, to qualify as an Arrhenius acid, upon the introduction to water, the chemical must either cause, directly or otherwise:
an increase in the aqueous hydronium concentration, or
a decrease in the aqueous hydroxide concentration.
Conversely, to qualify as an Arrhenius base, upon the introduction to water, the chemical must either cause, directly or otherwise:
a decrease in the aqueous hydronium concentration, or
an increase in the aqueous hydroxide concentration.
The reaction of an acid with a base is called a neutralization reaction. The products of this reaction are a salt and water.
acid + base → salt + water
In this traditional representation an acid–base neutralization reaction is formulated as a double-replacement reaction. For example, the reaction of hydrochloric acid, HCl, with sodium hydroxide, NaOH, solutions produces a solution of sodium chloride, NaCl, and some additional water molecules.
HCl(aq) + NaOH(aq) → NaCl(aq) + H2O
The modifier (aq) in this equation was implied by Arrhenius, rather than included explicitly. It indicates that the substances are dissolved in water. Though all three substances, HCl, NaOH and NaCl are capable of existing as pure compounds, in aqueous solutions they are fully dissociated into the aquated ions H+, Cl−, Na+ and OH−.
Brønsted–Lowry definition
The Brønsted–Lowry definition, formulated in 1923, independently by Johannes Nicolaus Brønsted in Denmark and Martin Lowry in England, is based upon the idea of protonation of bases through the deprotonation of acids – that is, the ability of acids to "donate" hydrogen ions (H+)—otherwise known as protons—to bases, which "accept" them."Removal and addition of a proton from the nucleus of an atom does not occur – it would require very much more energy than is involved in the dissociation of acids."
An acid–base reaction is, thus, the removal of a hydrogen ion from the acid and its addition to the base. The removal of a hydrogen ion from an acid produces its conjugate base, which is the acid with a hydrogen ion removed. The reception of a proton by a base produces its conjugate acid, which is the base with a hydrogen ion added.
Unlike the previous definitions, the Brønsted–Lowry definition does not refer to the formation of salt and solvent, but instead to the formation of conjugate acids and conjugate bases, produced by the transfer of a proton from the acid to the base. In this approach, acids and bases are fundamentally different in behavior from salts, which are seen as electrolytes, subject to the theories of Debye, Onsager, and others. An acid and a base react not to produce a salt and a solvent, but to form a new acid and a new base. The concept of neutralization is thus absent. Brønsted–Lowry acid–base behavior is formally independent of any solvent, making it more all-encompassing than the Arrhenius model. The calculation of pH under the Arrhenius model depended on alkalis (bases) dissolving in water (aqueous solution). The Brønsted–Lowry model expanded what could be pH tested using insoluble and soluble solutions (gas, liquid, solid).
The general formula for acid–base reactions according to the Brønsted–Lowry definition is:
HA + B → BH+ + A−
where HA represents the acid, B represents the base, BH+ represents the conjugate acid of B, and A− represents the conjugate base of HA.
For example, a Brønsted-Lowry model for the dissociation of hydrochloric acid (HCl) in aqueous solution would be the following:
HCl + H2O H3O+ + Cl−
The removal of H+ from the HCl produces the chloride ion, Cl−, the conjugate base of the acid. The addition of H+ to the H2O (acting as a base) forms the hydronium ion, H3O+, the conjugate acid of the base.
Water is amphoteric—that is, it can act as both an acid and a base. The Brønsted-Lowry model explains this, showing the dissociation of water into low concentrations of hydronium and hydroxide ions:
H2O + H2O H3O+ + OH−
This equation is demonstrated in the image below:
Here, one molecule of water acts as an acid, donating an H+ and forming the conjugate base, OH−, and a second molecule of water acts as a base, accepting the H+ ion and forming the conjugate acid, H3O+.
As an example of water acting as an acid, consider an aqueous solution of pyridine, C5H5N.
C5H5N + H2O [C5H5NH]+ + OH−
In this example, a water molecule is split into a hydrogen ion, which is donated to a pyridine molecule, and a hydroxide ion.
In the Brønsted-Lowry model, the solvent does not necessarily have to be water, as is required by the Arrhenius Acid-Base model. For example, consider what happens when acetic acid, CH3COOH, dissolves in liquid ammonia.
+ +
An H+ ion is removed from acetic acid, forming its conjugate base, the acetate ion, CH3COO−. The addition of an H+ ion to an ammonia molecule of the solvent creates its conjugate acid, the ammonium ion, .
The Brønsted–Lowry model calls hydrogen-containing substances (like HCl) acids. Thus, some substances, which many chemists considered to be acids, such as SO3 or BCl3, are excluded from this classification due to lack of hydrogen. Gilbert N. Lewis wrote in 1938, "To restrict the group of acids to those substances that contain hydrogen interferes as seriously with the systematic understanding of chemistry as would the restriction of the term oxidizing agent to substances containing oxygen." Furthermore, KOH and KNH2 are not considered Brønsted bases, but rather salts containing the bases OH− and .
Lewis definition
The hydrogen requirement of Arrhenius and Brønsted–Lowry was removed by the Lewis definition of acid–base reactions, devised by Gilbert N. Lewis in 1923, in the same year as Brønsted–Lowry, but it was not elaborated by him until 1938. Instead of defining acid–base reactions in terms of protons or other bonded substances, the Lewis definition defines a base (referred to as a Lewis base) to be a compound that can donate an electron pair, and an acid (a Lewis acid) to be a compound that can receive this electron pair.
For example, boron trifluoride, BF3 is a typical Lewis acid. It can accept a pair of electrons as it has a vacancy in its octet. The fluoride ion has a full octet and can donate a pair of electrons. Thus
BF3 + F− →
is a typical Lewis acid, Lewis base reaction. All compounds of group 13 elements with a formula AX3 can behave as Lewis acids. Similarly, compounds of group 15 elements with a formula DY3, such as amines, NR3, and phosphines, PR3, can behave as Lewis bases. Adducts between them have the formula X3A←DY3 with a dative covalent bond, shown symbolically as ←, between the atoms A (acceptor) and D (donor). Compounds of group 16 with a formula DX2 may also act as Lewis bases; in this way, a compound like an ether, R2O, or a thioether, R2S, can act as a Lewis base. The Lewis definition is not limited to these examples. For instance, carbon monoxide acts as a Lewis base when it forms an adduct with boron trifluoride, of formula F3B←CO.
Adducts involving metal ions are referred to as co-ordination compounds; each ligand donates a pair of electrons to the metal ion. The reaction
[Ag(H2O)4]+ + 2NH3 → [Ag(NH3)2]+ + 4H2O
can be seen as an acid–base reaction in which a stronger base (ammonia) replaces a weaker one (water)
The Lewis and Brønsted–Lowry definitions are consistent with each other since the reaction
H+ + OH− H2O
is an acid–base reaction in both theories.
Solvent system definition
One of the limitations of the Arrhenius definition is its reliance on water solutions. Edward Curtis Franklin studied the acid–base reactions in liquid ammonia in 1905 and pointed out the similarities to the water-based Arrhenius theory. Albert F.O. Germann, working with liquid phosgene, , formulated the solvent-based theory in 1925, thereby generalizing the Arrhenius definition to cover aprotic solvents.
Germann pointed out that in many solutions, there are ions in equilibrium with the neutral solvent molecules:
solvonium ions: a generic name for positive ions. (The term solvonium has replaced the older term lyonium ions: positive ions formed by protonation of solvent molecules.)
solvate ions: a generic name for negative ions. (The term solvate has replaced the older term lyate ions: negative ions formed by deprotonation of solvent molecules.)
For example, water and ammonia undergo such dissociation into hydronium and hydroxide, and ammonium and amide, respectively:
2 +
2 +
Some aprotic systems also undergo such dissociation, such as dinitrogen tetroxide into nitrosonium and nitrate, antimony trichloride into dichloroantimonium and tetrachloroantimonate, and phosgene into chlorocarboxonium and chloride:
+
2 +
+
A solute that causes an increase in the concentration of the solvonium ions and a decrease in the concentration of solvate ions is defined as an acid. A solute that causes an increase in the concentration of the solvate ions and a decrease in the concentration of the solvonium ions is defined as a base.
Thus, in liquid ammonia, (supplying ) is a strong base, and (supplying ) is a strong acid. In liquid sulfur dioxide (), thionyl compounds (supplying ) behave as acids, and sulfites (supplying ) behave as bases.
The non-aqueous acid–base reactions in liquid ammonia are similar to the reactions in water:
+ →
+ →
Nitric acid can be a base in liquid sulfuric acid:
+ 2 → + + 2
The unique strength of this definition shows in describing the reactions in aprotic solvents; for example, in liquid :
+ → +
Because the solvent system definition depends on the solute as well as on the solvent itself, a particular solute can be either an acid or a base depending on the choice of the solvent: is a strong acid in water, a weak acid in acetic acid, and a weak base in fluorosulfonic acid; this characteristic of the theory has been seen as both a strength and a weakness, because some substances (such as and ) have been seen to be acidic or basic on their own right. On the other hand, solvent system theory has been criticized as being too general to be useful. Also, it has been thought that there is something intrinsically acidic about hydrogen compounds, a property not shared by non-hydrogenic solvonium salts.
Lux–Flood definition
This acid–base theory was a revival of oxygen theory of acids and bases, proposed by German chemist Hermann Lux in 1939, further improved by Håkon Flood circa 1947 and is still used in modern geochemistry and electrochemistry of molten salts. This definition describes an acid as an oxide ion () acceptor and a base as an oxide ion donor. For example:
+ →
+ →
+ → + 2
This theory is also useful in the systematisation of the reactions of noble gas compounds, especially the xenon oxides, fluorides, and oxofluorides.
Usanovich definition
Mikhail Usanovich developed a general theory that does not restrict acidity to hydrogen-containing compounds, but his approach, published in 1938, was even more general than Lewis theory. Usanovich's theory can be summarized as defining an acid as anything that accepts negative species or donates positive ones, and a base as the reverse. This defined the concept of redox (oxidation-reduction) as a special case of acid–base reactions
Some examples of Usanovich acid–base reactions include:
+ → 2 + (species exchanged: anion)
+ → 6 + 2 (species exchanged: 3 anions)
+ → 2 + 2 (species exchanged: 2 electrons)
Rationalizing the strength of Lewis acid–base interactions
HSAB theory
In 1963, Ralph Pearson proposed a qualitative concept known as the Hard and Soft Acids and Bases principle. later made quantitative with help of Robert Parr in 1984. 'Hard' applies to species that are small, have high charge states, and are weakly polarizable. 'Soft' applies to species that are large, have low charge states and are strongly polarizable. Acids and bases interact, and the most stable interactions are hard–hard and soft–soft. This theory has found use in organic and inorganic chemistry.
ECW model
The ECW model created by Russell S. Drago is a quantitative model that describes and predicts the strength of Lewis acid base interactions, −ΔH. The model assigned E and C parameters to many Lewis acids and bases. Each acid is characterized by an EA and a CA. Each base is likewise characterized by its own EB and CB. The E and C parameters refer, respectively, to the electrostatic and covalent contributions to the strength of the bonds that the acid and base will form. The equation is
−ΔH = EAEB + CACB + WThe W term represents a constant energy contribution for acid–base reaction such as the cleavage of a dimeric acid or base. The equation predicts reversal of acids and base strengths. The graphical presentations of the equation show that there is no single order of Lewis base strengths or Lewis acid strengths.
Acid–base equilibrium
The reaction of a strong acid with a strong base is essentially a quantitative reaction. For example,
HCl(aq) + Na(OH)(aq) → H2O + NaCl(aq)
In this reaction both the sodium and chloride ions are spectators as the neutralization reaction,
H+ + OH− → H2O
does not involve them. With weak bases addition of acid is not quantitative because a solution of a weak base is a buffer solution. A solution of a weak acid is also a buffer solution. When a weak acid reacts with a weak base an equilibrium mixture is produced. For example, adenine, written as AH, can react with a hydrogen phosphate ion, .
AH + A− +
The equilibrium constant for this reaction can be derived from the acid dissociation constants of adenine and of the dihydrogen phosphate ion.
[A−] [H+] = Ka1[AH]
[] [H+] = Ka2[]
The notation [X] signifies "concentration of X". When these two equations are combined by eliminating the hydrogen ion concentration, an expression for the equilibrium constant, K is obtained.
[A−] [] = K[AH] []; K'' =
Acid–alkali reaction
An acid–alkali reaction is a special case of an acid–base reaction, where the base used is also an alkali. When an acid reacts with an alkali salt (a metal hydroxide), the product is a metal salt and water. Acid–alkali reactions are also neutralization reactions.
In general, acid–alkali reactions can be simplified to
(aq) + (aq) →
by omitting spectator ions.
Acids are in general pure substances that contain hydrogen cations () or cause them to be produced in solutions. Hydrochloric acid () and sulfuric acid () are common examples. In water, these break apart into ions:
→ (aq) + (aq)
→ (aq) + (aq)
The alkali breaks apart in water, yielding dissolved hydroxide ions:
→ (aq) + (aq)
See also
Acid–base titration
Deprotonation
Donor number
Electron configuration
Gutmann–Beckett method
Lewis structure
Nucleophilic substitution
Neutralization (chemistry)
Protonation
Redox reactions
Resonance (chemistry)
Notes
References
Sources
External links
Acid-base Physiology: an on-line text
John W. Kimball's online Biology book section of acid and bases.
Acids
Bases (chemistry)
Acid-base chemistry
Equilibrium chemistry | Acid–base reaction |
Biochemistry or biological chemistry, is the study of chemical processes within and relating to living organisms. A sub-discipline of both chemistry and biology, biochemistry may be divided into three fields: structural biology, enzymology and metabolism. Over the last decades of the 20th century, biochemistry has become successful at explaining living processes through these three disciplines. Almost all areas of the life sciences are being uncovered and developed through biochemical methodology and research. Biochemistry focuses on understanding the chemical basis which allows biological molecules to give rise to the processes that occur within living cells and between cells, in turn relating greatly to the understanding of tissues and organs, as well as organism structure and function. Biochemistry is closely related to molecular biology, which is the study of the molecular mechanisms of biological phenomena.
Much of biochemistry deals with the structures, bonding, functions, and interactions of biological macromolecules, such as proteins, nucleic acids, carbohydrates, and lipids. They provide the structure of cells and perform many of the functions associated with life. The chemistry of the cell also depends upon the reactions of small molecules and ions. These can be inorganic (for example, water and metal ions) or organic (for example, the amino acids, which are used to synthesize proteins). The mechanisms used by cells to harness energy from their environment via chemical reactions are known as metabolism. The findings of biochemistry are applied primarily in medicine, nutrition and agriculture. In medicine, biochemists investigate the causes and cures of diseases. Nutrition studies how to maintain health and wellness and also the effects of nutritional deficiencies. In agriculture, biochemists investigate soil and fertilizers. Improving crop cultivation, crop storage, and pest control are also goals.
History
At its most comprehensive definition, biochemistry can be seen as a study of the components and composition of living things and how they come together to become life. In this sense, the history of biochemistry may therefore go back as far as the ancient Greeks. However, biochemistry as a specific scientific discipline began sometime in the 19th century, or a little earlier, depending on which aspect of biochemistry is being focused on. Some argued that the beginning of biochemistry may have been the discovery of the first enzyme, diastase (now called amylase), in 1833 by Anselme Payen, while others considered Eduard Buchner's first demonstration of a complex biochemical process alcoholic fermentation in cell-free extracts in 1897 to be the birth of biochemistry. Some might also point as its beginning to the influential 1842 work by Justus von Liebig, Animal chemistry, or, Organic chemistry in its applications to physiology and pathology, which presented a chemical theory of metabolism, or even earlier to the 18th century studies on fermentation and respiration by Antoine Lavoisier. Many other pioneers in the field who helped to uncover the layers of complexity of biochemistry have been proclaimed founders of modern biochemistry. Emil Fischer, who studied the chemistry of proteins, and F. Gowland Hopkins, who studied enzymes and the dynamic nature of biochemistry, represent two examples of early biochemists.
The term "biochemistry" itself is derived from a combination of biology and chemistry. In 1877, Felix Hoppe-Seyler used the term (biochemie in German) as a synonym for physiological chemistry in the foreword to the first issue of Zeitschrift für Physiologische Chemie (Journal of Physiological Chemistry) where he argued for the setting up of institutes dedicated to this field of study. The German chemist Carl Neuberg however is often cited to have coined the word in 1903, while some credited it to Franz Hofmeister.
It was once generally believed that life and its materials had some essential property or substance (often referred to as the "vital principle") distinct from any found in non-living matter, and it was thought that only living beings could produce the molecules of life. In 1828, Friedrich Wöhler published a paper on his serendipitous urea synthesis from potassium cyanate and ammonium sulfate; some regarded that as a direct overthrow of vitalism and the establishment of organic chemistry. However, the Wöhler synthesis has sparked controversy as some reject the death of vitalism at his hands. Since then, biochemistry has advanced, especially since the mid-20th century, with the development of new techniques such as chromatography, X-ray diffraction, dual polarisation interferometry, NMR spectroscopy, radioisotopic labeling, electron microscopy and molecular dynamics simulations. These techniques allowed for the discovery and detailed analysis of many molecules and metabolic pathways of the cell, such as glycolysis and the Krebs cycle (citric acid cycle), and led to an understanding of biochemistry on a molecular level.
Another significant historic event in biochemistry is the discovery of the gene, and its role in the transfer of information in the cell. In the 1950s, James D. Watson, Francis Crick, Rosalind Franklin and Maurice Wilkins were instrumental in solving DNA structure and suggesting its relationship with the genetic transfer of information. In 1958, George Beadle and Edward Tatum received the Nobel Prize for work in fungi showing that one gene produces one enzyme. In 1988, Colin Pitchfork was the first person convicted of murder with DNA evidence, which led to the growth of forensic science. More recently, Andrew Z. Fire and Craig C. Mello received the 2006 Nobel Prize for discovering the role of RNA interference (RNAi), in the silencing of gene expression.
Starting materials: the chemical elements of life
Around two dozen chemical elements are essential to various kinds of biological life. Most rare elements on Earth are not needed by life (exceptions being selenium and iodine), while a few common ones (aluminum and titanium) are not used. Most organisms share element needs, but there are a few differences between plants and animals. For example, ocean algae use bromine, but land plants and animals do not seem to need any. All animals require sodium, but some plants do not. Plants need boron and silicon, but animals may not (or may need ultra-small amounts).
Just six elements—carbon, hydrogen, nitrogen, oxygen, calcium and phosphorus—make up almost 99% of the mass of living cells, including those in the human body (see composition of the human body for a complete list). In addition to the six major elements that compose most of the human body, humans require smaller amounts of possibly 18 more.
Biomolecules
The 4 main classes of molecules in bio-chemistry (often called biomolecules) are carbohydrates, lipids, proteins, and nucleic acids. Many biological molecules are polymers: in this terminology, monomers are relatively small macromolecules that are linked together to create large macromolecules known as polymers. When monomers are linked together to synthesize a biological polymer, they undergo a process called dehydration synthesis. Different macromolecules can assemble in larger complexes, often needed for biological activity.
Carbohydrates
Two of the main functions of carbohydrates are energy storage and providing structure. One of the common sugars known as glucose is carbohydrate, but not all carbohydrates are sugars. There are more carbohydrates on Earth than any other known type of biomolecule; they are used to store energy and genetic information, as well as play important roles in cell to cell interactions and communications.
The simplest type of carbohydrate is a monosaccharide, which among other properties contains carbon, hydrogen, and oxygen, mostly in a ratio of 1:2:1 (generalized formula CnH2nOn, where n is at least 3). Glucose (C6H12O6) is one of the most important carbohydrates; others include fructose (C6H12O6), the sugar commonly associated with the sweet taste of fruits, and deoxyribose (C5H10O4), a component of DNA. A monosaccharide can switch between acyclic (open-chain) form and a cyclic form. The open-chain form can be turned into a ring of carbon atoms bridged by an oxygen atom created from the carbonyl group of one end and the hydroxyl group of another. The cyclic molecule has a hemiacetal or hemiketal group, depending on whether the linear form was an aldose or a ketose.
In these cyclic forms, the ring usually has 5 or 6 atoms. These forms are called furanoses and pyranoses, respectively—by analogy with furan and pyran, the simplest compounds with the same carbon-oxygen ring (although they lack the carbon-carbon double bonds of these two molecules). For example, the aldohexose glucose may form a hemiacetal linkage between the hydroxyl on carbon 1 and the oxygen on carbon 4, yielding a molecule with a 5-membered ring, called glucofuranose. The same reaction can take place between carbons 1 and 5 to form a molecule with a 6-membered ring, called glucopyranose. Cyclic forms with a 7-atom ring called heptoses are rare.
Two monosaccharides can be joined together by a glycosidic or ester bond into a disaccharide through a dehydration reaction during which a molecule of water is released. The reverse reaction in which the glycosidic bond of a disaccharide is broken into two monosaccharides is termed hydrolysis. The best-known disaccharide is sucrose or ordinary sugar, which consists of a glucose molecule and a fructose molecule joined together. Another important disaccharide is lactose found in milk, consisting of a glucose molecule and a galactose molecule. Lactose may be hydrolysed by lactase, and deficiency in this enzyme results in lactose intolerance.
When a few (around three to six) monosaccharides are joined, it is called an oligosaccharide (oligo- meaning "few"). These molecules tend to be used as markers and signals, as well as having some other uses. Many monosaccharides joined together form a polysaccharide. They can be joined together in one long linear chain, or they may be branched. Two of the most common polysaccharides are cellulose and glycogen, both consisting of repeating glucose monomers. Cellulose is an important structural component of plant's cell walls and glycogen is used as a form of energy storage in animals.
Sugar can be characterized by having reducing or non-reducing ends. A reducing end of a carbohydrate is a carbon atom that can be in equilibrium with the open-chain aldehyde (aldose) or keto form (ketose). If the joining of monomers takes place at such a carbon atom, the free hydroxy group of the pyranose or furanose form is exchanged with an OH-side-chain of another sugar, yielding a full acetal. This prevents opening of the chain to the aldehyde or keto form and renders the modified residue non-reducing. Lactose contains a reducing end at its glucose moiety, whereas the galactose moiety forms a full acetal with the C4-OH group of glucose. Saccharose does not have a reducing end because of full acetal formation between the aldehyde carbon of glucose (C1) and the keto carbon of fructose (C2).
Lipids
Lipids comprise a diverse range of molecules and to some extent is a catchall for relatively water-insoluble or nonpolar compounds of biological origin, including waxes, fatty acids, fatty-acid derived phospholipids, sphingolipids, glycolipids, and terpenoids (e.g., retinoids and steroids). Some lipids are linear, open-chain aliphatic molecules, while others have ring structures. Some are aromatic (with a cyclic [ring] and planar [flat] structure) while others are not. Some are flexible, while others are rigid.
Lipids are usually made from one molecule of glycerol combined with other molecules. In triglycerides, the main group of bulk lipids, there is one molecule of glycerol and three fatty acids. Fatty acids are considered the monomer in that case, and may be saturated (no double bonds in the carbon chain) or unsaturated (one or more double bonds in the carbon chain).
Most lipids have some polar character in addition to being largely nonpolar. In general, the bulk of their structure is nonpolar or hydrophobic ("water-fearing"), meaning that it does not interact well with polar solvents like water. Another part of their structure is polar or hydrophilic ("water-loving") and will tend to associate with polar solvents like water. This makes them amphiphilic molecules (having both hydrophobic and hydrophilic portions). In the case of cholesterol, the polar group is a mere –OH (hydroxyl or alcohol). In the case of phospholipids, the polar groups are considerably larger and more polar, as described below.
Lipids are an integral part of our daily diet. Most oils and milk products that we use for cooking and eating like butter, cheese, ghee etc., are composed of fats. Vegetable oils are rich in various polyunsaturated fatty acids (PUFA). Lipid-containing foods undergo digestion within the body and are broken into fatty acids and glycerol, which are the final degradation products of fats and lipids. Lipids, especially phospholipids, are also used in various pharmaceutical products, either as co-solubilisers (e.g., in parenteral infusions) or else as drug carrier components (e.g., in a liposome or transfersome).
Proteins
Proteins are very large molecules—macro-biopolymers—made from monomers called amino acids. An amino acid consists of an alpha carbon atom attached to an amino group, –NH2, a carboxylic acid group, –COOH (although these exist as –NH3+ and –COO− under physiologic conditions), a simple hydrogen atom, and a side chain commonly denoted as "–R". The side chain "R" is different for each amino acid of which there are 20 standard ones. It is this "R" group that made each amino acid different, and the properties of the side-chains greatly influence the overall three-dimensional conformation of a protein. Some amino acids have functions by themselves or in a modified form; for instance, glutamate functions as an important neurotransmitter. Amino acids can be joined via a peptide bond. In this dehydration synthesis, a water molecule is removed and the peptide bond connects the nitrogen of one amino acid's amino group to the carbon of the other's carboxylic acid group. The resulting molecule is called a dipeptide, and short stretches of amino acids (usually, fewer than thirty) are called peptides or polypeptides. Longer stretches merit the title proteins. As an example, the important blood serum protein albumin contains 585 amino acid residues.
Proteins can have structural and/or functional roles. For instance, movements of the proteins actin and myosin ultimately are responsible for the contraction of skeletal muscle. One property many proteins have is that they specifically bind to a certain molecule or class of molecules—they may be extremely selective in what they bind. Antibodies are an example of proteins that attach to one specific type of molecule. Antibodies are composed of heavy and light chains. Two heavy chains would be linked to two light chains through disulfide linkages between their amino acids. Antibodies are specific through variation based on differences in the N-terminal domain.
The enzyme-linked immunosorbent assay (ELISA), which uses antibodies, is one of the most sensitive tests modern medicine uses to detect various biomolecules. Probably the most important proteins, however, are the enzymes. Virtually every reaction in a living cell requires an enzyme to lower the activation energy of the reaction. These molecules recognize specific reactant molecules called substrates; they then catalyze the reaction between them. By lowering the activation energy, the enzyme speeds up that reaction by a rate of 1011 or more; a reaction that would normally take over 3,000 years to complete spontaneously might take less than a second with an enzyme. The enzyme itself is not used up in the process and is free to catalyze the same reaction with a new set of substrates. Using various modifiers, the activity of the enzyme can be regulated, enabling control of the biochemistry of the cell as a whole.
The structure of proteins is traditionally described in a hierarchy of four levels. The primary structure of a protein consists of its linear sequence of amino acids; for instance, "alanine-glycine-tryptophan-serine-glutamate-asparagine-glycine-lysine-…". Secondary structure is concerned with local morphology (morphology being the study of structure). Some combinations of amino acids will tend to curl up in a coil called an α-helix or into a sheet called a β-sheet; some α-helixes can be seen in the hemoglobin schematic above. Tertiary structure is the entire three-dimensional shape of the protein. This shape is determined by the sequence of amino acids. In fact, a single change can change the entire structure. The alpha chain of hemoglobin contains 146 amino acid residues; substitution of the glutamate residue at position 6 with a valine residue changes the behavior of hemoglobin so much that it results in sickle-cell disease. Finally, quaternary structure is concerned with the structure of a protein with multiple peptide subunits, like hemoglobin with its four subunits. Not all proteins have more than one subunit.
Ingested proteins are usually broken up into single amino acids or dipeptides in the small intestine and then absorbed. They can then be joined to form new proteins. Intermediate products of glycolysis, the citric acid cycle, and the pentose phosphate pathway can be used to form all twenty amino acids, and most bacteria and plants possess all the necessary enzymes to synthesize them. Humans and other mammals, however, can synthesize only half of them. They cannot synthesize isoleucine, leucine, lysine, methionine, phenylalanine, threonine, tryptophan, and valine. Because they must be ingested, these are the essential amino acids. Mammals do possess the enzymes to synthesize alanine, asparagine, aspartate, cysteine, glutamate, glutamine, glycine, proline, serine, and tyrosine, the nonessential amino acids. While they can synthesize arginine and histidine, they cannot produce it in sufficient amounts for young, growing animals, and so these are often considered essential amino acids.
If the amino group is removed from an amino acid, it leaves behind a carbon skeleton called an α-keto acid. Enzymes called transaminases can easily transfer the amino group from one amino acid (making it an α-keto acid) to another α-keto acid (making it an amino acid). This is important in the biosynthesis of amino acids, as for many of the pathways, intermediates from other biochemical pathways are converted to the α-keto acid skeleton, and then an amino group is added, often via transamination. The amino acids may then be linked together to form a protein.
A similar process is used to break down proteins. It is first hydrolyzed into its component amino acids. Free ammonia (NH3), existing as the ammonium ion (NH4+) in blood, is toxic to life forms. A suitable method for excreting it must therefore exist. Different tactics have evolved in different animals, depending on the animals' needs. Unicellular organisms simply release the ammonia into the environment. Likewise, bony fish can release the ammonia into the water where it is quickly diluted. In general, mammals convert the ammonia into urea, via the urea cycle.
In order to determine whether two proteins are related, or in other words to decide whether they are homologous or not, scientists use sequence-comparison methods. Methods like sequence alignments and structural alignments are powerful tools that help scientists identify homologies between related molecules. The relevance of finding homologies among proteins goes beyond forming an evolutionary pattern of protein families. By finding how similar two protein sequences are, we acquire knowledge about their structure and therefore their function.
Nucleic acids
Nucleic acids, so-called because of their prevalence in cellular nuclei, is the generic name of the family of biopolymers. They are complex, high-molecular-weight biochemical macromolecules that can convey genetic information in all living cells and viruses. The monomers are called nucleotides, and each consists of three components: a nitrogenous heterocyclic base (either a purine or a pyrimidine), a pentose sugar, and a phosphate group.
The most common nucleic acids are deoxyribonucleic acid (DNA) and ribonucleic acid (RNA). The phosphate group and the sugar of each nucleotide bond with each other to form the backbone of the nucleic acid, while the sequence of nitrogenous bases stores the information. The most common nitrogenous bases are adenine, cytosine, guanine, thymine, and uracil. The nitrogenous bases of each strand of a nucleic acid will form hydrogen bonds with certain other nitrogenous bases in a complementary strand of nucleic acid (similar to a zipper). Adenine binds with thymine and uracil, thymine binds only with adenine, and cytosine and guanine can bind only with one another. Adenine and Thymine & Adenine and Uracil contains two hydrogen Bonds, while Hydrogen Bonds formed between cytosine and guanine are three in number.
Aside from the genetic material of the cell, nucleic acids often play a role as second messengers, as well as forming the base molecule for adenosine triphosphate (ATP), the primary energy-carrier molecule found in all living organisms. Also, the nitrogenous bases possible in the two nucleic acids are different: adenine, cytosine, and guanine occur in both RNA and DNA, while thymine occurs only in DNA and uracil occurs in RNA.
Metabolism
Carbohydrates as energy source
Glucose is an energy source in most life forms. For instance, polysaccharides are broken down into their monomers by enzymes (glycogen phosphorylase removes glucose residues from glycogen, a polysaccharide). Disaccharides like lactose or sucrose are cleaved into their two component monosaccharides.
Glycolysis (anaerobic)
Glucose is mainly metabolized by a very important ten-step pathway called glycolysis, the net result of which is to break down one molecule of glucose into two molecules of pyruvate. This also produces a net two molecules of ATP, the energy currency of cells, along with two reducing equivalents of converting NAD+ (nicotinamide adenine dinucleotide: oxidized form) to NADH (nicotinamide adenine dinucleotide: reduced form). This does not require oxygen; if no oxygen is available (or the cell cannot use oxygen), the NAD is restored by converting the pyruvate to lactate (lactic acid) (e.g., in humans) or to ethanol plus carbon dioxide (e.g., in yeast). Other monosaccharides like galactose and fructose can be converted into intermediates of the glycolytic pathway.
Aerobic
In aerobic cells with sufficient oxygen, as in most human cells, the pyruvate is further metabolized. It is irreversibly converted to acetyl-CoA, giving off one carbon atom as the waste product carbon dioxide, generating another reducing equivalent as NADH. The two molecules acetyl-CoA (from one molecule of glucose) then enter the citric acid cycle, producing two molecules of ATP, six more NADH molecules and two reduced (ubi)quinones (via FADH2 as enzyme-bound cofactor), and releasing the remaining carbon atoms as carbon dioxide. The produced NADH and quinol molecules then feed into the enzyme complexes of the respiratory chain, an electron transport system transferring the electrons ultimately to oxygen and conserving the released energy in the form of a proton gradient over a membrane (inner mitochondrial membrane in eukaryotes). Thus, oxygen is reduced to water and the original electron acceptors NAD+ and quinone are regenerated. This is why humans breathe in oxygen and breathe out carbon dioxide. The energy released from transferring the electrons from high-energy states in NADH and quinol is conserved first as proton gradient and converted to ATP via ATP synthase. This generates an additional 28 molecules of ATP (24 from the 8 NADH + 4 from the 2 quinols), totaling to 32 molecules of ATP conserved per degraded glucose (two from glycolysis + two from the citrate cycle). It is clear that using oxygen to completely oxidize glucose provides an organism with far more energy than any oxygen-independent metabolic feature, and this is thought to be the reason why complex life appeared only after Earth's atmosphere accumulated large amounts of oxygen.
Gluconeogenesis
In vertebrates, vigorously contracting skeletal muscles (during weightlifting or sprinting, for example) do not receive enough oxygen to meet the energy demand, and so they shift to anaerobic metabolism, converting glucose to lactate.
The combination of glucose from noncarbohydrates origin, such as fat and proteins. This only happens when glycogen supplies in the liver are worn out. The pathway is a crucial reversal of glycolysis from pyruvate to glucose and can utilize many sources like amino acids, glycerol and Krebs Cycle. Large scale protein and fat catabolism usually occur when those suffer from starvation or certain endocrine disorders. The liver regenerates the glucose, using a process called gluconeogenesis. This process is not quite the opposite of glycolysis, and actually requires three times the amount of energy gained from glycolysis (six molecules of ATP are used, compared to the two gained in glycolysis). Analogous to the above reactions, the glucose produced can then undergo glycolysis in tissues that need energy, be stored as glycogen (or starch in plants), or be converted to other monosaccharides or joined into di- or oligosaccharides. The combined pathways of glycolysis during exercise, lactate's crossing via the bloodstream to the liver, subsequent gluconeogenesis and release of glucose into the bloodstream is called the Cori cycle.
Relationship to other "molecular-scale" biological sciences
Researchers in biochemistry use specific techniques native to biochemistry, but increasingly combine these with techniques and ideas developed in the fields of genetics, molecular biology, and biophysics. There is not a defined line between these disciplines. Biochemistry studies the chemistry required for biological activity of molecules, molecular biology studies their biological activity, genetics studies their heredity, which happens to be carried by their genome. This is shown in the following schematic that depicts one possible view of the relationships between the fields:
Biochemistry is the study of the chemical substances and vital processes occurring in live organisms. Biochemists focus heavily on the role, function, and structure of biomolecules. The study of the chemistry behind biological processes and the synthesis of biologically active molecules are applications of biochemistry. Biochemistry studies life at the atomic and molecular level.
Genetics is the study of the effect of genetic differences in organisms. This can often be inferred by the absence of a normal component (e.g. one gene). The study of "mutants" – organisms that lack one or more functional components with respect to the so-called "wild type" or normal phenotype. Genetic interactions (epistasis) can often confound simple interpretations of such "knockout" studies.
Molecular biology is the study of molecular underpinnings of the biological phenomena, focusing on molecular synthesis, modification, mechanisms and interactions. The central dogma of molecular biology, where genetic material is transcribed into RNA and then translated into protein, despite being oversimplified, still provides a good starting point for understanding the field. This concept has been revised in light of emerging novel roles for RNA.
'Chemical biology' seeks to develop new tools based on small molecules that allow minimal perturbation of biological systems while providing detailed information about their function. Further, chemical biology employs biological systems to create non-natural hybrids between biomolecules and synthetic devices (for example emptied viral capsids that can deliver gene therapy or drug molecules).
See also
Lists
Important publications in biochemistry (chemistry)
List of biochemistry topics
List of biochemists
List of biomolecules
See also
Astrobiology
Biochemistry (journal)
Biological Chemistry (journal)
Biophysics
Chemical ecology
Computational biomodeling
Dedicated bio-based chemical
EC number
Hypothetical types of biochemistry
International Union of Biochemistry and Molecular Biology
Metabolome
Metabolomics
Molecular biology
Molecular medicine
Plant biochemistry
Proteolysis
Small molecule
Structural biology
TCA cycle
Notes
a. Fructose is not the only sugar found in fruits. Glucose and sucrose are also found in varying quantities in various fruits, and sometimes exceed the fructose present. For example, 32% of the edible portion of a date is glucose, compared with 24% fructose and 8% sucrose. However, peaches contain more sucrose (6.66%) than they do fructose (0.93%) or glucose (1.47%).
References
Cited literature
Further reading
Fruton, Joseph S. Proteins, Enzymes, Genes: The Interplay of Chemistry and Biology. Yale University Press: New Haven, 1999.
Keith Roberts, Martin Raff, Bruce Alberts, Peter Walter, Julian Lewis and Alexander Johnson, Molecular Biology of the Cell
4th Edition, Routledge, March, 2002, hardcover, 1616 pp.
3rd Edition, Garland, 1994,
2nd Edition, Garland, 1989,
Kohler, Robert. From Medical Chemistry to Biochemistry: The Making of a Biomedical Discipline. Cambridge University Press, 1982.
External links
The Virtual Library of Biochemistry, Molecular Biology and Cell Biology
Biochemistry, 5th ed. Full text of Berg, Tymoczko, and Stryer, courtesy of NCBI.
SystemsX.ch – The Swiss Initiative in Systems Biology
Full text of Biochemistry by Kevin and Indira, an introductory biochemistry textbook.
Biotechnology
Molecular biology | Biochemistry |
Bayesian probability is an interpretation of the concept of probability, in which, instead of frequency or propensity of some phenomenon, probability is interpreted as reasonable expectation representing a state of knowledge or as quantification of a personal belief.
The Bayesian interpretation of probability can be seen as an extension of propositional logic that enables reasoning with hypotheses; that is, with propositions whose truth or falsity is unknown. In the Bayesian view, a probability is assigned to a hypothesis, whereas under frequentist inference, a hypothesis is typically tested without being assigned a probability.
Bayesian probability belongs to the category of evidential probabilities; to evaluate the probability of a hypothesis, the Bayesian probabilist specifies a prior probability. This, in turn, is then updated to a posterior probability in the light of new, relevant data (evidence). The Bayesian interpretation provides a standard set of procedures and formulae to perform this calculation.
The term Bayesian derives from the 18th-century mathematician and theologian Thomas Bayes, who provided the first mathematical treatment of a non-trivial problem of statistical data analysis using what is now known as Bayesian inference. Mathematician Pierre-Simon Laplace pioneered and popularized what is now called Bayesian probability.
Bayesian methodology
Bayesian methods are characterized by concepts and procedures as follows:
The use of random variables, or more generally unknown quantities, to model all sources of uncertainty in statistical models including uncertainty resulting from lack of information (see also aleatoric and epistemic uncertainty).
The need to determine the prior probability distribution taking into account the available (prior) information.
The sequential use of Bayes' formula: when more data become available, calculate the posterior distribution using Bayes' formula; subsequently, the posterior distribution becomes the next prior.
While for the frequentist, a hypothesis is a proposition (which must be either true or false) so that the frequentist probability of a hypothesis is either 0 or 1, in Bayesian statistics, the probability that can be assigned to a hypothesis can also be in a range from 0 to 1 if the truth value is uncertain.
Objective and subjective Bayesian probabilities
Broadly speaking, there are two interpretations of Bayesian probability. For objectivists, who interpret probability as an extension of logic, probability quantifies the reasonable expectation that everyone (even a "robot") who shares the same knowledge should share in accordance with the rules of Bayesian statistics, which can be justified by Cox's theorem. For subjectivists, probability corresponds to a personal belief. Rationality and coherence allow for substantial variation within the constraints they pose; the constraints are justified by the Dutch book argument or by decision theory and de Finetti's theorem. The objective and subjective variants of Bayesian probability differ mainly in their interpretation and construction of the prior probability.
History
The term Bayesian derives from Thomas Bayes (1702–1761), who proved a special case of what is now called Bayes' theorem in a paper titled "An Essay towards solving a Problem in the Doctrine of Chances". In that special case, the prior and posterior distributions were beta distributions and the data came from Bernoulli trials. It was Pierre-Simon Laplace (1749–1827) who introduced a general version of the theorem and used it to approach problems in celestial mechanics, medical statistics, reliability, and jurisprudence. Early Bayesian inference, which used uniform priors following Laplace's principle of insufficient reason, was called "inverse probability" (because it infers backwards from observations to parameters, or from effects to causes). After the 1920s, "inverse probability" was largely supplanted by a collection of methods that came to be called frequentist statistics.
In the 20th century, the ideas of Laplace developed in two directions, giving rise to objective and subjective currents in Bayesian practice.
Harold Jeffreys' Theory of Probability (first published in 1939) played an important role in the revival of the Bayesian view of probability, followed by works by Abraham Wald (1950) and Leonard J. Savage (1954). The adjective Bayesian itself dates to the 1950s; the derived Bayesianism, neo-Bayesianism is of 1960s coinage. In the objectivist stream, the statistical analysis depends on only the model assumed and the data analysed. No subjective decisions need to be involved. In contrast, "subjectivist" statisticians deny the possibility of fully objective analysis for the general case.
In the 1980s, there was a dramatic growth in research and applications of Bayesian methods, mostly attributed to the discovery of Markov chain Monte Carlo methods and the consequent removal of many of the computational problems, and to an increasing interest in nonstandard, complex applications. While frequentist statistics remains strong (as demonstrated by the fact that much of undergraduate teaching is based on it ), Bayesian methods are widely accepted and used, e.g., in the field of machine learning.
Justification of Bayesian probabilities
The use of Bayesian probabilities as the basis of Bayesian inference has been supported by several arguments, such as Cox axioms, the Dutch book argument, arguments based on decision theory and de Finetti's theorem.
Axiomatic approach
Richard T. Cox showed that Bayesian updating follows from several axioms, including two functional equations and a hypothesis of differentiability. The assumption of differentiability or even continuity is controversial; Halpern found a counterexample based on his observation that the Boolean algebra of statements may be finite. Other axiomatizations have been suggested by various authors with the purpose of making the theory more rigorous.
Dutch book approach
The Dutch book argument was proposed by de Finetti; it is based on betting. A Dutch book is made when a clever gambler places a set of bets that guarantee a profit, no matter what the outcome of the bets. If a bookmaker follows the rules of the Bayesian calculus in the construction of his odds, a Dutch book cannot be made.
However, Ian Hacking noted that traditional Dutch book arguments did not specify Bayesian updating: they left open the possibility that non-Bayesian updating rules could avoid Dutch books. For example, Hacking writes "And neither the Dutch book argument, nor any other in the personalist arsenal of proofs of the probability axioms, entails the dynamic assumption. Not one entails Bayesianism. So the personalist requires the dynamic assumption to be Bayesian. It is true that in consistency a personalist could abandon the Bayesian model of learning from experience. Salt could lose its savour."
In fact, there are non-Bayesian updating rules that also avoid Dutch books (as discussed in the literature on "probability kinematics" following the publication of Richard C. Jeffreys' rule, which is itself regarded as Bayesian). The additional hypotheses sufficient to (uniquely) specify Bayesian updating are substantial and not universally seen as satisfactory.
Decision theory approach
A decision-theoretic justification of the use of Bayesian inference (and hence of Bayesian probabilities) was given by Abraham Wald, who proved that every admissible statistical procedure is either a Bayesian procedure or a limit of Bayesian procedures. Conversely, every Bayesian procedure is admissible.
Personal probabilities and objective methods for constructing priors
Following the work on expected utility theory of Ramsey and von Neumann, decision-theorists have accounted for rational behavior using a probability distribution for the agent. Johann Pfanzagl completed the Theory of Games and Economic Behavior by providing an axiomatization of subjective probability and utility, a task left uncompleted by von Neumann and Oskar Morgenstern: their original theory supposed that all the agents had the same probability distribution, as a convenience. Pfanzagl's axiomatization was endorsed by Oskar Morgenstern: "Von Neumann and I have anticipated ... [the question whether probabilities] might, perhaps more typically, be subjective and have stated specifically that in the latter case axioms could be found from which could derive the desired numerical utility together with a number for the probabilities (cf. p. 19 of The Theory of Games and Economic Behavior). We did not carry this out; it was demonstrated by Pfanzagl ... with all the necessary rigor".
Ramsey and Savage noted that the individual agent's probability distribution could be objectively studied in experiments. Procedures for testing hypotheses about probabilities (using finite samples) are due to Ramsey (1931) and de Finetti (1931, 1937, 1964, 1970). Both Bruno de Finetti and Frank P. Ramsey acknowledge their debts to pragmatic philosophy, particularly (for Ramsey) to Charles S. Peirce.
The "Ramsey test" for evaluating probability distributions is implementable in theory, and has kept experimental psychologists occupied for a half century.
This work demonstrates that Bayesian-probability propositions can be falsified, and so meet an empirical criterion of Charles S. Peirce, whose work inspired Ramsey. (This falsifiability-criterion was popularized by Karl Popper.)
Modern work on the experimental evaluation of personal probabilities uses the randomization, blinding, and Boolean-decision procedures of the Peirce-Jastrow experiment. Since individuals act according to different probability judgments, these agents' probabilities are "personal" (but amenable to objective study).
Personal probabilities are problematic for science and for some applications where decision-makers lack the knowledge or time to specify an informed probability-distribution (on which they are prepared to act). To meet the needs of science and of human limitations, Bayesian statisticians have developed "objective" methods for specifying prior probabilities.
Indeed, some Bayesians have argued the prior state of knowledge defines the (unique) prior probability-distribution for "regular" statistical problems; cf. well-posed problems. Finding the right method for constructing such "objective" priors (for appropriate classes of regular problems) has been the quest of statistical theorists from Laplace to John Maynard Keynes, Harold Jeffreys, and Edwin Thompson Jaynes. These theorists and their successors have suggested several methods for constructing "objective" priors (Unfortunately, it is not clear how to assess the relative "objectivity" of the priors proposed under these methods):
Maximum entropy
Transformation group analysis
Reference analysis
Each of these methods contributes useful priors for "regular" one-parameter problems, and each prior can handle some challenging statistical models (with "irregularity" or several parameters). Each of these methods has been useful in Bayesian practice. Indeed, methods for constructing "objective" (alternatively, "default" or "ignorance") priors have been developed by avowed subjective (or "personal") Bayesians like James Berger (Duke University) and José-Miguel Bernardo (Universitat de València), simply because such priors are needed for Bayesian practice, particularly in science. The quest for "the universal method for constructing priors" continues to attract statistical theorists.
Thus, the Bayesian statistician needs either to use informed priors (using relevant expertise or previous data) or to choose among the competing methods for constructing "objective" priors.
See also
Bertrand paradox—a paradox in classical probability
De Finetti's game—a procedure for evaluating someone's subjective probability
QBism—an interpretation of quantum mechanics based on subjective Bayesian probability
Reference class problem
An Essay towards solving a Problem in the Doctrine of Chances
Monty Hall problem
Bayesian epistemology
References
Bibliography
(translation of de Finetti, 1931)
(translation of de Finetti, 1937, above)
, , two volumes.
Goertz, Gary and James Mahoney. 2012. A Tale of Two Cultures: Qualitative and Quantitative Research in the Social Sciences. Princeton University Press.
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