In mathematics and computer science, an algorithm ( ) is an unambiguous specification of how to solve a class of problems. Algorithms can perform calculation, data processing and automated reasoning tasks.
An algorithm is an effective method that can be expressed within a finite amount of space and time"Any classical mathematical algorithm, for example, can be described in a finite number of English words" (Rogers 1987:2). and in a well-defined formal languageWell defined with respect to the agent that executes the algorithm: "There is a computing agent, usually human, which can react to the instructions and carry out the computations" (Rogers 1987:2). for calculating a function."an algorithm is a procedure for computing a function (with respect to some chosen notation for integers) ... this limitation (to numerical functions) results in no loss of generality", (Rogers 1987:1). Starting from an initial state and initial input (perhaps Empty string),"An algorithm has zero or more inputs, i.e., quantity which are given to it initially before the algorithm begins" (Knuth 1973:5). the instructions describe a computation that, when executed, proceeds through a finite"A procedure which has all the characteristics of an algorithm except that it possibly lacks finiteness may be called a 'computational method'" (Knuth 1973:5). number of well-defined successive states, eventually producing "output""An algorithm has one or more outputs, i.e. quantities which have a specified relation to the inputs" (Knuth 1973:5). 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.Whether or not a process with random interior processes (not including the input) is an algorithm is debatable. Rogers opines that: "a computation is carried out in a discrete stepwise fashion, without use of continuous methods or analogue devices . . . carried forward deterministically, without resort to random methods or devices, e.g., dice" Rogers 1987:2.
The concept of algorithm has existed for centuries; however, a partial formalization of what would become the modern algorithm began with attempts to solve the Entscheidungsproblem (the "decision problem") posed by David Hilbert in 1928. Subsequent formalizations were framed as attempts to define "effective calculability"Kleene 1943 in Davis 1965:274 or "effective method";Rosser 1939 in Davis 1965:225 those formalizations included the Gödel–Jacques 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 1936–7 and 1939. Giving a formal definition of algorithms, corresponding to the intuitive notion, remains a challenging problem.
Al-Khwārizmī (خوارزمی, c. 780–850) was a Persian people mathematician, astronomer, geographer, and scholar in the House of Wisdom in Baghdad, whose name means 'the native of Khwarezm', a region that was part of Greater Iran and is now in Uzbekistan. About 825, he wrote a treatise in the Arabic language, which was translated into Latin in the 12th century under the title Algoritmi de numero Indorum. This title means "Algoritmi on the numbers of the Indians", where "Algoritmi" was the translator's Latinization of Al-Khwarizmi's name.
In English, it was first used in about 1230 and then by Geoffrey Chaucer in 1391. English adopted the French term, but it wasn't 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 thus:
which translates as:
The poem is a few hundred lines long and summarizes the art of calculating with the new style of Indian dice, or Talibus Indorum, or Hindu numerals.
A prototypical example of an algorithm is the Euclidean algorithm to determine the maximum common divisor of two integers; an example (there are others) is described by the flow chart above and as an example in a later section.
offer an informal meaning of the word 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.Boolos and Jeffrey 1974,1999:19
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. Thus an algorithm can be an algebraic equation such as y = m + n – 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):
The concept of algorithm is also used to define the notion of decidability. That notion is central for explaining how come into being starting from a small set of and rules. In logic, the time that an algorithm requires to complete cannot be measured, as it is not apparently related with our 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.
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 , an algorithm is a computational process defined by a Turing machine".Gurevich 2000:1, 3
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 .
For some such computational process, the algorithm must be rigorously defined: specified in the way it applies in all possible circumstances that could arise. That is, 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 control flow.
So far, this discussion of the formalization of an algorithm has assumed the premises of imperative programming. This is the most common conception, and it attempts to describe a task in discrete, "mechanical" means. Unique to this conception of formalized algorithms is the assignment operation, setting the value of a variable. It derives from the intuition of "memory" as a scratchpad. There is an example below of such an assignment.
For some alternate conceptions of what constitutes an algorithm see functional programming and logic programming.
There is a wide variety of representations possible and one can express a given Turing machine program as a sequence of machine tables (see more at finite-state machine, state transition table and control table), as flowcharts and DRAKON (see more at state diagram), or as a form of rudimentary machine code or assembly code called "sets of quadruples" (see more at Turing machine).
Representations of algorithms can be classed into three accepted levels of Turing machine description:Sipser 2006:157
For an example of the simple algorithm "Add m+n" described in all three levels, see Algorithm#Examples.
In computer systems, an algorithm is basically an instance of logic written in software by software developers to be effective for the intended "target" computer(s) to produce output from given (perhaps null) input. An optimal algorithm, even running in old hardware, would produce faster results than a non-optimal (higher time complexity) algorithm for the same purpose, running in more efficient hardware; that is why algorithms, like computer hardware, are considered technology.
"Elegant" (compact) programs, "good" (fast) programs : The notion of "simplicity and elegance" appears informally in Knuth and precisely in Chaitin:
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".Rogers 1987:1–2
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"In his essay "Calculations by Man and Machine: Conceptual Analysis" Seig 2002:390 credits this distinction to Robin Gandy, cf Wilfred Seig, et al., 2002 Reflections on the foundations of mathematics: Essays in honor of Solomon Feferman, Association for Symbolic Logic, A. K Peters Ltd, Natick, MA.) is a restricted type of machine, a "discrete deterministic mechanical device"cf Gandy 1980:126, Robin Gandy Church's Thesis and Principles for Mechanisms appearing on pp. 123–148 in Jon Barwise et al. 1980 The Kleene Symposium, North-Holland Publishing Company. that blindly follows its instructions.A "robot": "A computer is a robot that performs any task that can be described as a sequence of instructions." cf Stone 1972:3 Melzak's and Lambek's primitive modelsLambek's "abacus" is a "countably infinite number of locations (holes, wires etc.) together with an unlimited supply of counters (pebbles, beads, etc). The locations are distinguishable, the counters are not". The holes have unlimited capacity, and standing by is an agent who understands and is able to carry out the list of instructions" (Lambek 1961:295). Lambek references Melzak who defines his Q-machine as "an indefinitely large number of locations . . . an indefinitely large supply of counters distributed among these locations, a program, and an operator whose sole purpose is to carry out the program" (Melzak 1961:283). B-B-J (loc. cit.) add the stipulation that the holes are "capable of holding any number of stones" (p. 46). Both Melzak and Lambek appear in The Canadian Mathematical Bulletin, vol. 4, no. 3, September 1961. reduced this notion to four elements: (i) discrete, distinguishable locations, (ii) discrete, indistinguishable countersIf no confusion results, the word "counters" can be dropped, and a location can be said to contain a single "number". (iii) an agent, and (iv) a list of instructions that are effective relative to the capability of the agent."We say that an instruction is effective if there is a procedure that the robot can follow in order to determine precisely how to obey the instruction." (Stone 1972:6)
Minsky describes a more congenial variation of Lambek's "abacus" model in his "Very Simple Bases for Computability".cf Minsky 1967: Chapter 11 "Computer models" and Chapter 14 "Very Simple Bases for Computability" pp. 255–281 in particular Minsky 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)cf Knuth 1973:3. 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).But always preceded by IF–THEN to avoid improper subtraction. 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".Knuth 1973:4 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.Stone 1972:5. Methods for extracting roots are not trivial: see Methods of computing square roots.
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".
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".John G. Kemeny and Thomas E. Kurtz 1985 Back to Basic: The History, Corruption, and Future of the Language, Addison-Wesley Publishing Company, Inc. Reading, MA, . Tausworthe augments the three Böhm-Jacopini canonical structures:Tausworthe 1977:101 SEQUENCE, IF-THEN-ELSE, and WHILE-DO, with two more: DO-WHILE and CASE.Tausworthe 1977:142 An additional benefit of a structured program is that it lends itself to proofs of correctness using mathematical induction.Knuth 1973 section 1.2.1, expanded by Tausworthe 1977 at pages 100ff and Chapter 9.1
Canonical flowchart symbolscf Tausworthe 1977: The graphical aide called a flowchart offers a way to describe and document an algorithm (and a computer program of one). Like 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.
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 English prose, as:
Input: A list of numbers ''L''. Output: The largest number in the list ''L''.
'''if''' ''L.size'' = 0 '''return''' null ''largest'' ← ''L'' '''for each''' ''item'' '''in''' ''L'', '''do''' '''if''' ''item'' > ''largest'', '''then''' ''largest'' ← ''item'' '''return''' ''largest''
Euclid's algorithm to compute the greatest common divisor (GCD) to two numbers appears as Proposition II in Book VII ("Elementary Number Theory") of his Elements.Heath 1908:300; Hawking's Dover 2005 edition derives from Heath. Euclid poses the problem thus: "Given two numbers not prime to one another, to find their greatest common measure". He defines "A number to 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." 'Let CD, measuring BF, leave FA less than itself.' This is a neat abbreviation for saying, measure along BA successive lengths equal to CD until a point F is reached such that the length FA remaining is less than CD; in other words, let BF be the largest exact multiple of CD contained in BA" (Heath 1908:297) 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 modern treatments using division in the algorithm, see Hardy and Wright 1979:180, Knuth 1973:2 (Volume 1), plus more discussion of Euclid's algorithm in Knuth 1969:293–297 (Volume 2).
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 (alternately, 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.Euclid covers this question in his Proposition 1. 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.
[[File:Euclids-algorithm-example-1599-650.gif|350px|thumb|right|A graphical expression of Euclid's algorithm to find the greatest common divisor for 1599 and 650. ]]
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:
[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
[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 6 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: 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: 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: 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
[Done. S contains the greatest common divisor]: PRINT S
HALT, END, STOP.
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".Knuth 1973:13–18. He credits "the formulation of algorithm-proving in terms of assertions and induction" to R. W. Floyd, Peter Naur, C. A. R. Hoare, H. H. Goldstine and J. von Neumann. Tausworth 1977 borrows Knuth's Euclid example and extends Knuth's method in section 9.1 Formal Proofs (pages 288–298). Tausworthe proposes that a measure of the complexity of a program be the length of its correctness proof.Tausworthe 1997:294
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;Breakdown occurs when an algorithm tries to compact itself. Success would solve the Halting problem. rather, it can only be done ; 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.
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 lookup table on sorted lists or arrays.
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.
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.
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.
Additionally, some cryptographic algorithms have export restrictions (see export of cryptography).
Researcher, Andrew Tutt, argues that algorithms should be overseen by a specialist regulatory agency, similar to FDA. His academic work emphasizes that the rise of increasingly complex algorithms calls for the need to think about the effects of algorithms today. Due to the nature and complexity of algorithms, it will prove to be difficult to hold algorithms accountable under criminal law. Tutt recognizes that while some algorithms will be beneficial to help meet technological demand, others should not be used or sold if they fail to meet safety requirements. Thus, for Tutt, algorithms will require "closer forms of federal uniformity, expert judgment, political independence, and pre-market review to prevent the introduction of unacceptably dangerous algorithms into the market". The issue of algorithmic accountability (the responsibility of algorithm designers to provide evidence of potential or realised harms) is of particular relevance in the field of dynamic and non-linearly programmed systems, e.g. artificial neural networks, deep learning, and genetic algorithms (see Explainable AI).
Logical machines 1870—Stanley Jevons' "logical abacus" and "logical machine": The technical problem was to reduce when presented in a form similar to what are now known as . 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".All quotes from W. Stanley Jevons 1880 Elementary Lessons in Logic: Deductive and Inductive, Macmillan and Co., London and New York. Republished as a googlebook; cf Jevons 1880:199–201. Louis Couturat 1914 the Algebra of Logic, The Open Court Publishing Company, Chicago and London. Republished as a googlebook; cf Couturat 1914:75–76 gives a few more details; interestingly he compares this to a typewriter as well as a piano. Jevons states that the account is to be found at Jan . 20, 1870 The Proceedings of the Royal Society.
This machine he displayed in 1870 before the Fellows of the Royal Society.Jevons 1880:199–200 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 again, 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".All quotes from John Venn 1881 Symbolic Logic, Macmillan and Co., London. Republished as a googlebook. cf Venn 1881:120–125. The interested reader can find a deeper explanation in those pages.
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.Bell and Newell diagram 1971:39, cf. Davis 2000 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 (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".* Melina Hill, Valley News Correspondent, A Tinkerer Gets a Place in History, Valley News West Lebanon NH, Thursday March 31, 1983, page 13.
Davis (2000) observes the particular importance of the electromechanical relay (with its two "binary states" open and closed):
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".van Heijenoort's commentary on Frege's Begriffsschrift, a formula language, modeled upon that of arithmetic, for pure thought in van Heijenoort 1967:1 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.Dixon 1906, cf. Kleene 1952:36–40 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-calculuscf. footnote in Alonzo Church 1936a in Davis 1965:90 and 1936b in Davis 1965:110 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.Kleene 1935–6 in Davis 1965:237ff, Kleene 1943 in Davis 1965:255ff Church's proofChurch 1936 in Davis 1965:88ff 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.cf. "Formulation I", Post 1936 in Davis 1965:289–290 Alan Turing's proof of that the Entscheidungsproblem was unsolvable by use of his "a- automatic- machine"Turing 1936–7 in Davis 1965:116ff—in effect almost identical to Post's "formulation", J. Barkley Rosser's definition of "effective method" in terms of "a machine".Rosser 1939 in Davis 1965:226 S. C. Kleene's proposal of a precursor to "Church thesis" that he called "Thesis I",Kleene 1943 in Davis 1965:273–274 and a few years later Kleene's renaming his Thesis "Church's Thesis"Kleene 1952:300, 317 and proposing "Turing's Thesis".Kleene 1952:376
Emil Post (1936) described the actions of a "computer" (human being) as follows:
His symbol space would be
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.Turing 1936–7:116
Turing's reduction yields the following:
A few years later, Turing expanded his analysis (thesis, definition) with this forceful expression of it:
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):