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A formalization and proof of the extended church turing thesis
Church–Turing thesis. In computability theory, the Church–Turing thesis (also known as computability thesis , [1] the Turing–Church thesis , [2] the Church–Turing conjecture , Church's thesis , Church's conjecture , and Turing's thesis ) is a hypothesis about the nature of computable functions. It states that a function on the natural numbers can be calculated by an effective method if and only if it is computable by a Turing machine. The thesis is named after American mathematician Alonzo Church and the British mathematician Alan Turing. Before the precise definition of computable function, mathematicians often used the informal term effectively calculable to describe functions that are computable by paperandpencil methods. In the 1930s, several independent attempts were made to formalize the notion of computability: In 1933, Kurt Gödel, with Jacques Herbrand, created a formal definition of a >[ citation needed ] , Alan Turing created a theoretical model for machines, now called Turing machines, that could carry out calculations from inputs by manipulating symbols on a tape. Given a suitable encoding of the natural numbers as sequences of symbols, a function on the natural numbers is called Turing computable if some Turing machine computes the corresponding function on encoded natural numbers. Church [3] and Turing [4] [6] proved that these three formally defined classes of computable functions coincide: a function is λcomputable if and only if it is Turing computable if and only if it is general recursive . This has led mathematicians and computer scientists to believe that the concept of computability is accurately characterized by these three equivalent processes. Other formal attempts to characterize computability have subsequently strengthened this belief (see below). On the other hand, the Church–Turing thesis states that the above three formallydefined classes of computable functions coincide with the informal notion of an effectively calculable function. Since, as an informal notion, the concept of effective calculability does not have a formal definition, the thesis, although it has nearuniversal acceptance, cannot be formally proven. Since its inception, variations on the original thesis have arisen, including statements about what can physically be realized by a computer in our universe (physical ChurchTuring thesis) and what can be efficiently computed (Church–Turing thesis (complexity theory)). These variations are not due to Church or Turing, but arise from later work in complexity theory and digital physics. The thesis also has implications for the philosophy of mind (see below). Contents. Statement in Church's and Turing's words [ edit ] J. B. Rosser (1939) addresses the notion of "effective computability" as follows: "Clearly the existence of CC and RC (Church's and Rosser's proofs) presupposes a precise definition of 'effective'. 'Effective method' is here used in the rather special sense of a method each step of which is precisely predetermined and which is certain to produce the answer in a finite number of steps". [7] Thus the adverbadjective "effective" is used in a sense of "1a: producing a dec >[8] [9] In the following, the words "effectively calculable" will mean "produced by any intuitively 'effective' means whatsoever" and "effectively computable" will mean "produced by a Turingmachine or equivalent mechanical device". Turing's "definitions" given in a footnote in his 1938 Ph.D. thesis Systems of Logic Based on Ordinals , supervised by Church, are virtually the same: † We shall use the expression "computable function" to mean a function calculable by a machine, and let "effectively calculable" refer to the intuitive >[10] The thesis can be stated as: Every effectively calculable function is a computable function . [11] Turing stated it this way: It was stated . that "a function is effectively calculable if its values can be found by some purely mechanical process". We may take this literally, understanding that by a purely mechanical process one which could be carried out by a machine. The development . leads to . an identification of computability † with effective calculability. [ † is the footnote quoted above.] [10] History [ edit ] One of the important problems for logicians in the 1930s was the Entsche >[12] , which asked whether there was a mechanical procedure for separating mathematical truths from mathematical falsehoods. This quest required that the notion of "algorithm" or "effective calculability" be pinned down, at least well enough for the quest to begin. [13] But from the very outset Alonzo Church's attempts began with a debate that continues to this day. [14] Was [ clarify ] the notion of "effective calculability" to be (i) an "axiom or axioms" in an axiomatic system, (ii) merely a definition that "identified" two or more propositions, (iii) an empirical hypothesis to be verified by observation of natural events, or (iv) just a proposal for the sake of argument (i.e. a "thesis"). Circa 1930–1952 [ edit ] In the course of studying the problem, Church and his student Stephen Kleene introduced the notion of λdefinable functions, and they were able to prove that several large >[15] The debate began when Church proposed to Gödel that one should define the "effectively computable" functions as the λdefinable functions. Gödel, however, was not convinced and called the proposal "thoroughly unsatisfactory". [16] Rather, in correspondence with Church (c. 1934–35), Gödel proposed axiomatizing the notion of "effective calculability"; indeed, in a 1935 letter to Kleene, Church reported that: But Gödel offered no further gu >[18] Next, it was necessary to >has a "normal form" [ clarify ] . [19] Many years later in a letter to Davis (c. 1965), Gödel sa >[20] By 1963–64 Gödel would disavow Herbrand–Gödel recursion and the λcalculus in favor of the Turing machine as the definition of "algorithm" or "mechanical procedure" or "formal system". [21] A hypothesis leading to a natural law? : In late 1936 Alan Turing's paper (also proving that the Entsche >[22] On the other hand, Emil Post's 1936 paper had appeared and was certified independent of Turing's work. [23] Post strongly disagreed with Church's "identification" of effective computability with the λcalculus and recursion, stating: Actually the work already done by Church and others carries this >[24] Rather, he regarded the notion of "effective calculability" as merely a "working hypothesis" that might lead by inductive reasoning to a "natural law" rather than by "a definition or an axiom". [25] This >[26] Thus Post in his 1936 paper was also discounting Kurt Gödel's suggestion to Church in 1934–35 that the thesis might be expressed as an axiom or set of axioms. [17] Turing adds another definition, Rosser equates all three : Within just a short time, Turing's 1936–37 paper "On Computable Numbers, with an Application to the Entsche >[22] appeared. In it he stated another notion of "effective computability" with the introduction of his amachines (now known as the Turing machine abstract computational model). And in a proofsketch added as an "Appendix" to his 1936–37 paper, Turing showed that the >[27] Church was quick to recognise how compelling Turing's analysis was. In his review of Turing's paper he made clear that Turing's notion made "the >[28] In a few years (1939) Turing would propose, like Church and Kleene before him, that his formal definition of mechanical computing agent was the correct one. [29] Thus, by 1939, both Church (1934) and Turing (1939) had indiv >[30] neither framed their statements as theses . Rosser (1939) formally identified the three notionsasdefinitions: All three definitions are equivalent, so it does not matter which one is used. [31] Kleene proposes Church's Thesis : This left the overt expression of a "thesis" to Kleene. In his 1943 paper Recursive Predicates and Quantifiers Kleene proposed his "THESIS I": This heuristic fact [general recursive functions are effectively calculable] . led Church to state the following thesis( 22 ). The same thesis is implicit in Turing's description of computing machines( 23 ). THESIS I. Every effectively calculable function (effectively dec >[32] recursive [Kleene's italics] Since a precise mathematical definition of the term effectively calculable (effectively dec >[33] ( 22 ) references Church 1936; [ not specific enough to verify ] ( 23 ) references Turing 1936–7 Kleene goes on to note that: the thesis has the character of an hypothesis—a point emphasized by Post and by Church( 24 ). If we cons >[33] (24) references Post 1936 of Post and Church's Formal definitions in the theory of ordinal numbers , Fund. Math . vol 28 (1936) pp.11–21 (see ref. #2, Davis 1965:286). Kleene's Church–Turing Thesis : A few years later (1952) Kleene, who switched from presenting his work in the mathematical terminology of the lambda calculus of his phd advisor Alonzo Church to the theory of general recursive functions of his other teacher Kurt Gödel, would overtly name the Church–Turing thesis in his correction of Turing's paper "The Word Problem in SemiGroups with Cancellation", [34] defend, and express the two "theses" and then "identify" them (show equivalence) by use of his Theorem XXX: Heuristic evidence and other considerations led Church 1936 to propose the following thesis. Thesis I. Every effectively calculable function (effectively dec >[35] Theorem XXX: The following >[36] Turing's thesis: Turing's thesis that every function which would naturally be regarded as computable is computable under his definition, i.e. by one of his machines, is equivalent to Church's thesis by Theorem XXX. [36] Later developments [ edit ] An attempt to understand the notion of "effective computability" better led Robin Gandy (Turing's student and friend) in 1980 to analyze machine computation (as opposed to humancomputation acted out by a Turing machine). Gandy's curiosity about, and analysis of, cellular automata (including Conway's game of life), parallelism, and crystalline automata, led him to propose four "principles (or constraints) . which it is argued, any machine must satisfy". [37] His mostimportant fourth, "the principle of causality" is based on the "finite velocity of propagation of effects and signals; contemporary physics rejects the possibility of instantaneous action at a distance". [38] From these principles and some additional constraints—(1a) a lower bound on the linear dimensions of any of the parts, (1b) an upper bound on speed of propagation (the velocity of light), (2) discrete progress of the machine, and (3) deterministic behavior—he produces a theorem that "What can be calculated by a device satisfying principles I–IV is computable." [39] In the late 1990s Wilfried Sieg analyzed Turing's and Gandy's notions of "effective calculability" with the intent of "sharpening the informal notion, formulating its general features axiomatically, and investigating the axiomatic framework". [40] In his 1997 and 2002 work Sieg presents a series of constraints on the behavior of a computor —"a human computing agent who proceeds mechanically". These constraints reduce to: "(B.1) (Boundedness) There is a fixed bound on the number of symbolic configurations a computor can immediately recognize. "(B.2) (Boundedness) There is a fixed bound on the number of internal states a computor can be in. "(L.1) (Locality) A computor can change only elements of an observed symbolic configuration. "(L.2) (Locality) A computor can shift attention from one symbolic configuration to another one, but the new observed configurations must be within a bounded distance of the immediately previously observed configuration. "(D) (Determinacy) The immediately recognizable (sub)configuration determines uniquely the next computation step (and >[41] The matter remains in active discussion within the academic community. [42] [43] The thesis as a definition [ edit ] The thesis can be viewed as nothing but an ordinary mathematical definition. Comments by Gödel on the subject suggest this view, e.g. "the correct definition of mechanical computability was established beyond any doubt by Turing". [44] The case for viewing the thesis as nothing more than a definition is made explicitly by Robert I. Soare, [45] where it is also argued that Turing's definition of computability is no less likely to be correct than the epsilondelta definition of a continuous function. Success of the thesis [ edit ] Other formalisms (bes >[46] In the 1950s Hao Wang and Martin Davis greatly simplified the onetape Turingmachine model (see Post–Turing machine). Marvin Minsky expanded the model to two or more tapes and greatly simplified the tapes into "updown counters", which Melzak and Lambek further evolved into what is now known as the counter machine model. In the late 1960s and early 1970s researchers expanded the counter machine model into the register machine, a close cousin to the modern notion of the computer. Other models include combinatory logic and Markov algorithms. Gurevich adds the pointer machine model of Kolmogorov and Uspensky (1953, 1958): ". they just wanted to . convince themselves that there is no way to extend the notion of computable function." [47] All these contributions involve proofs that the models are computationally equivalent to the Turing machine; such models are said to be Turing complete. Because all these different attempts at formalizing the concept of "effective calculability/computability" have yielded equivalent results, it is now generally assumed that the Church–Turing thesis is correct. In fact, Gödel (1936) proposed something stronger than this; he observed that there was something "absolute" about the concept of "reckonable in S 1 ": It may also be shown that a function which is computable ['reckonable'] in one of the systems S i , or even in a system of transfinite type, is already computable [reckonable] in S 1 . Thus the concept 'computable' ['reckonable'] is in a certain definite sense 'absolute', while practically all other familiar metamathematical concepts (e.g. provable, definable, etc.) depend quite essentially on the system to which they are defined . [48] Informal usage in proofs [ edit ] Proofs in computability theory often invoke the Church–Turing thesis in an informal way to establish the computability of functions while avo >[49] To establish that a function is computable by Turing machine, it is usually considered sufficient to give an informal English description of how the function can be effectively computed, and then conclude "by the Church–Turing thesis" that the function is Turing computable (equivalently, partial recursive). Dirk van Dalen gives the following example for the sake of illustrating this informal use of the Church–Turing thesis: [50] EXAMPLE: Each infinite RE set contains an infinite recursive set. Proof: Let A be infinite RE. We list the elements of A effectively, n 0 , n 1 , n 2 , n 3 , . From this list we extract an increasing sublist: put m 0 =n 0 , after finitely many steps we find an n k such that n k > m 0 , put m 1 =n k . We repeat this procedure to find m 2 > m 1 , etc. this yields an effective listing of the subset B= 0 ,m 1 ,m 2 . > of A, with the property m i Variations [ edit ] The success of the Church–Turing thesis prompted variations of the thesis to be proposed. For example, the physical Church–Turing thesis states: "All physically computable functions are Turingcomputable." [51] : 101. The Church–Turing thesis says nothing about the efficiency with which one model of computation can simulate another. It has been proved for instance that a (multitape) universal Turing machine only suffers a logarithmic slowdown factor in simulating any Turing machine. [52] A variation of the Church–Turing thesis addresses whether an arbitrary but "reasonable" model of computation can be efficiently simulated. This is called the feasibility thesis , [53] also known as the ( >[54] "A probabilistic Turing machine can efficiently simulate any realistic model of computation." The word 'efficiently' here means up to polynomialtime reductions. This thesis was originally called computational complexitytheoretic Church–Turing thesis by Ethan Bernstein and Umesh Vazirani (1997). The complexitytheoretic Church–Turing thesis, then, posits that all 'reasonable' models of computation yield the same >" 'Reasonable' machines can simulate each other within a polynomially bounded overhead in time and a constantfactor overhead in space." [55] The thesis originally appeared in a paper at STOC'84, which was the first paper to show that polynomialtime overhead and constantspace overhead could be simultaneously achieved for a simulation of a Random Access Machine on a Turing machine. [56] If BQP is shown to be a strict superset of BPP, it would inval >[54] "A quantum Turing machine can efficiently simulate any realistic model of computation." Eugene Eberbach and Peter Wegner claim that the Church–Turing thesis is sometimes interpreted too broadly, stating "the broader assertion that algorithms precisely capture what can be computed is inval >[57] [ page needed ] They claim that forms of computation not captured by the thesis are relevant today, terms which they call superTuring computation. Philosophical implications [ edit ] Philosophers have interpreted the Church–Turing thesis as having implications for the philosophy of mind. [58] [59] B. Jack Copeland states that it is an open empirical question whether there are actual deterministic physical processes that, in the long run, elude simulation by a Turing machine; furthermore, he states that it is an open empirical question whether any such processes are involved in the working of the human brain. [60] There are also some important open questions which cover the relationship between the Church–Turing thesis and physics, and the possibility of hypercomputation. When applied to physics, the thesis has several possible meanings: The universe is equivalent to a Turing machine; thus, computing nonrecursive functions is physically impossible. This has been termed the strong Church–Turing thesis, or Church–Turing–Deutsch principle, and is a foundation of digital physics. The universe is not equivalent to a Turing machine (i.e., the laws of physics are not Turingcomputable), but incomputable physical events are not "harnessable" for the construction of a hypercomputer. For example, a universe in which physics involves random real numbers, as opposed to computable reals, would fall into this category. The universe is a hypercomputer, and it is possible to build physical devices to harness this property and calculate nonrecursive functions. For example, it is an open question whether all quantum mechanical events are Turingcomputable, although it is known that rigorous models such as quantum Turing machines are equivalent to deterministic Turing machines. (They are not necessarily efficiently equivalent; see above.) John Lucas and Roger Penrose have suggested that the human mind might be the result of some kind of quantummechanically enhanced, "nonalgorithmic" computation. [61][62] There are many other technical possibilities which fall outside or between these three categories, but these serve to illustrate the range of the concept. Philosophical aspects of the thesis, regarding both physical and biological computers, are also discussed in Odifreddi's 1989 textbook on recursion theory. [63] : 101123. Noncomputable functions [ edit ] One can formally define functions that are not computable. A wellknown example of such a function is the Busy Beaver function. This function takes an input n and returns the largest number of symbols that a Turing machine with n states can print before halting, when run with no input. Finding an upper bound on the busy beaver function is equivalent to solving the halting problem, a problem known to be unsolvable by Turing machines. Since the busy beaver function cannot be computed by Turing machines, the Church–Turing thesis states that this function cannot be effectively computed by any method. Several computational models allow for the computation of (ChurchTuring) noncomputable functions. These are known as hypercomputers. Mark Burgin argues that superrecursive algorithms such as inductive Turing machines disprove the Church–Turing thesis. [64] [ page needed ] His argument relies on a definition of algorithm broader than the ordinary one, so that noncomputable functions obtained from some inductive Turing machines are called computable. This interpretation of the Church–Turing thesis differs from the interpretation commonly accepted in computability theory, discussed above. The argument that superrecursive algorithms are indeed algorithms in the sense of the Church–Turing thesis has not found broad acceptance within the computability research community.
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