## A NATURAL AXIOMATIZATION OF COMPUTABILITY AND PROOF OF CHURCH’S THESIS

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@MISC{Dershowitz_anatural,

author = {Nachum Dershowitz and Yuri Gurevich},

title = {A NATURAL AXIOMATIZATION OF COMPUTABILITY AND PROOF OF CHURCH’S THESIS},

year = {}

}

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### Abstract

Abstract. Church’s Thesis asserts that the only numeric functions that can be calculated by effective means are the recursive ones, which are the same, extensionally, as the Turingcomputable numeric functions. The Abstract State Machine Theorem states that every classical algorithm is behaviorally equivalent to an abstract state machine. This theorem presupposes three natural postulates about algorithmic computation. Here, we show that augmenting those postulates with an additional requirement regarding basic operations gives a natural axiomatization of computability and a proof of Church’s Thesis, as Gödel and others suggested may be possible. In a similar way, but with a different set of basic operations, one can prove Turing’s Thesis, characterizing the effective string functions, and—in particular—the effectively-computable functions on string representations of numbers.

### Citations

1155 |
On computable numbers, with an application to the Entscheidungsproblem
- Turing
- 1936
(Show Context)
Citation Context ...r Post [76, p. 418], it “is not a matter of mathematical proof but of psychological analysis of the mental processes involved in combinatory mathematical processes”. Turing, in his seminal 1936 paper =-=[107]-=-, analyzed human computation “from the bottom up”, building complex procedures from the most primitive of operations on single symbols. He asserted that computation proceeds by sequential symbol manip... |

837 |
Theory of recursive functions and effective computability
- Rogers
- 1967
(Show Context)
Citation Context ...scheidungsproblems ist für die Theorie aller Gebiete, deren Sätze überhaupt einer logischen Entwickelbarkeit aus endlich vielen Axiomen fähig sind, von grundsätzlicher Wichtigkeit.” 3 Nowadays (e.g., =-=[82]-=-) it is more common to work with the nonnegative integers, including zero, though in the past (e.g., [23]) it was common to deal only with the positive integers. The difference is immaterial for discu... |

216 | Mathematical Logic - Shoenfield - 1967 |

100 | A note on the entscheidungsproblem - Church - 1936 |

98 | Java and the Java Virtual Machine: Definition, Verification, Validation
- Stärk, Schmid, et al.
- 2001
(Show Context)
Citation Context ...strings, or graphs, or sets would require a daunting amount of encoding. In our setting, however, this poses no problem; witness the detailed structuralrepresentations of C and Java given in [43] and =-=[105]-=-, respectively. Even algorithms computing purely numeric functions or pure string functions typically involve additional types of objects. So, to define effectiveness of numerical calculations or of s... |

97 |
Trial and error predicates and the solution to a problem of mostowski
- Putnam
- 1965
(Show Context)
Citation Context ... Postulate I (Sequential time). An algorithm is a state-transition system. Its transitions are partial functions. Continuous (analogue) processes, transfinite computation sequences (involving limits) =-=[78, 40]-=-, nondeterministic transitions, and nonprocedural input-output specifications are thereby excluded from consideration. Classical algorithms, of the sort Church was considering, never leave room for ch... |

89 |
Limiting recursion
- Gold
- 1965
(Show Context)
Citation Context ... Postulate I (Sequential time). An algorithm is a state-transition system. Its transitions are partial functions. Continuous (analogue) processes, transfinite computation sequences (involving limits) =-=[78, 40]-=-, nondeterministic transitions, and nonprocedural input-output specifications are thereby excluded from consideration. Classical algorithms, of the sort Church was considering, never leave room for ch... |

84 | The Semantics of the C Programming Language - Gurevich, Huggins - 1992 |

76 | Grundzüge der theoretischen Logik - Hilbert, Ackermann - 1928 |

76 |
Der Wahrheitsbegriff in den formalisierten Sprachen", Studia Philosophica I, 261-405. German translation of Tarski
- Tarski
- 1935
(Show Context)
Citation Context ...erested here in characterizing arbitrary algorithms, we ought not limit the form of states a priori. Accordingly, we let the states of state-transition systems be first-order structures with equality =-=[106]-=-, firstorder structures being the most general thing mathematicians have in their arsenal for representing discrete states. It will simplify matters if the relations (predicates) of a first-order stru... |

67 |
Evolving algebras 1993:Lipari guide, in Specification and Validation Methods
- Gurevich
- 1995
(Show Context)
Citation Context ...rmulate the three Sequential Postulates rigorously, motivate each of them, and adduce support for them from the classical literature. In Section 3, we recall the definition of abstract state machines =-=[41]-=-, and the fact that they emulate any algorithm obeying those postulates [42].NATURAL AXIOMATIZATION OF COMPUTABILITY 307 (See also [80].) These machines will play a central part in our proof. Then, i... |

64 |
Storage modification machines
- Schönhage
- 1980
(Show Context)
Citation Context ...107]: “I think that it will be agreed that the two-dimensional character of paper is no essential of computation.” Kolmogorov–Uspensky’s model [57], and later variants of the “pointer machine” (e.g., =-=[85]-=-), use graphs as a more free-form representation of state. Gandy [35] suggests hereditarily finite sets as a generic data structure for the same purpose. Since we are interested here in characterizing... |

63 |
Computable algebra, general theory and theory of computable fields
- Rabin
- 1960
(Show Context)
Citation Context ...r (cf. [81, 18]). A similar approach to effectiveness in non-numerical domains D is taken in the field of computable algebra, namely that operations on D are tracked by homomorphic images in N (e.g., =-=[34, 64, 79]-=-), but our conditions on the encoding are noticeably weaker. Remark 6.2. When, in addition to ρN being recursive, the image ρN(N)it gives of the natural numbers is recursive (as in [64, 79]), then the... |

44 | Recursive predicates and quantifiers - Kleene - 1943 |

43 | On undecidable propositions of formal mathematical systems - Gödel - 1965 |

40 | Finitary Combinatory Processes – Formulation I - Post - 1936 |

36 |
Algebraic specifications of computable and semicomputable data types, Theoretical Computer Science 50
- Bergstra, Tucker
- 1987
(Show Context)
Citation Context ...stratification function σ0 for encoded natural numbers is recursive. If σ −1 0 is also recursive, then all purely numeric functions over D are in fact ρ-recursive, as explained in Remark 6.2. Compare =-=[5]-=-. Example 6.10. One convenient stratification mapping is obtained by partitioning the natural numbers into m residue classes Nj modulo m, corresponding to the m sorts of D. The mappings and their inve... |

35 | The Origins and the Development of the ASM Method for High Level System Design and Analysis". 2002
- Börger
(Show Context)
Citation Context ...directly as lambda terms while preserving their structure. Indeed, lambda calculus is strong enough to do this. . . .” 13 Indeed, the experience of a wide range of abstract state machine applications =-=[20]-=- also supports this claim: In all cases, it has been possible to specify software faithfully on the precise proper level of abstraction, without introducing unnecessary details, or divulging internal ... |

30 |
Fredkin and Tommaso Toffoli, "Conservative Logic
- Edward
- 1982
(Show Context)
Citation Context ...y finite sets, but applies to arbitrary state-transition systems with arbitrary structures for states. For a recent critique of Gandy, see [87]. For another set of (informal) physical postulates, see =-=[32]-=-. We should point out that, nowadays, one deals daily with more flexible notions of algorithm, such as interactive and distributed computations. To capture such non-sequential processes and non-classi... |

27 | Classical recursion theory: the theory of functions and sets of natural numbers - Odifreddi - 1989 |

25 |
Church’s thesis and principles for mechanisms. The Kleene Symposium
- Gandy
- 1980
(Show Context)
Citation Context ...n doing this (although some interesting partial results have been obtained). Kalmár [46] and Rogers [82, p. 20] (and, more recently, Folina [31]) argued against provability of the thesis, while Gandy =-=[35]-=- and Mendelson [69, 70] notion with your λ-definability gives a stronger conviction of the adequacy of these concepts for expressing the popular meaning of ‘effective calculability’.” 11 Levin was Kol... |

23 | The impact of the lambda calculus in logic and computer science
- Barendregt
- 1997
(Show Context)
Citation Context ...nger than the seventy years that have elapsed since Church proposed identifying effectiveness with recursiveness, but still those physical theories were eventually found lacking. As Barendregt writes =-=[2]-=-: One may wonder why doubting Church’s Thesis is not a completely academic question. This becomes clear by realizing that [Skolem in 1923] had introduced the class of primitive recursive functions tha... |

23 |
2002a, ‘The Church–Turing Thesis’, The Stanford Encyclopedia of Philosophy (Fall 2002
- Copeland
(Show Context)
Citation Context ...ion, whether physical or biological, must violate (at least) one of the Sequential Postulates, and/or must include at least one non-recursive function in its initial states. See [74, pp. 101–123] and =-=[26]-=- for discussions of these and other variants of Church’s Thesis. Finally, the question of what effectiveness means for computations over arbitrary, non-numerical domains is taken up in [64, 62, 17, 19... |

21 |
Mathematische Probleme: Vortrag, gehalten auf dem internationalen Mathematiker-Kongreß zu Paris 1900 (in German). Available at http://wikilivres.info/wiki/Mathematische_ Probleme (viewed Dec
- Hilbert
- 2011
(Show Context)
Citation Context ...nd with rational integral numerical coefficients: To devise a process according to which it can be determined in a finite number of operations whether the equation is solvable in rational integers. 1 =-=[44]-=- The Entscheidungsproblem [decision problem for first-order logic] is solved when we know a procedure that allows for any given logical expression to decide by finitely many operations its validity or... |

16 |
2003): On Gurevich’s Theorem on Sequential Algorithms
- Reisig
- 2010
(Show Context)
Citation Context ... Section 3, we recall the definition of abstract state machines [41], and the fact that they emulate any algorithm obeying those postulates [42].NATURAL AXIOMATIZATION OF COMPUTABILITY 307 (See also =-=[80]-=-.) These machines will play a central part in our proof. Then, in Section 4, we turn Church’s Thesis into a precise mathematical statement and explain why the fact that only the recursive functions ca... |

14 | Beiträge zur Algebra der Logik, insbesondere zum Entscheidungsproblem - Behmann - 1922 |

13 |
The optimal approach to recursive programs
- Manna, Shamir
- 1977
(Show Context)
Citation Context ...ute a function that is, in general, less defined than the least fixed point. Recursive definition have, besides their least fixed-point solution, a unique “optimal” (maximally consistent) fixed point =-=[65]-=-, which (though not necessarily computable) is, in general, more defined than the least fixed point and could also be taken as the intended semantics of the definition. Even after specifying that the ... |

9 | Algorithms: Main Ideas and Applications - Semenov - 1993 |

9 |
Completeness before Post: Bernays, Hilbert, and the development of propositional logic
- Zach
- 1999
(Show Context)
Citation Context ...ly on information about the limited “active part” of S and affects this part only. Much earlier, in 1922, Behmann [4, p. 166] expressed the stepwise nature of algorithmic activity by saying (cited in =-=[113]-=-): 16 A completely determined general [set of] instructions shall be exhibited, according to which the correctness or falsity of an arbitrary given claim, which can be formulated with purely logical m... |

8 | Y.: When are two algorithms the same
- Blass, Dershowitz, et al.
- 2009
(Show Context)
Citation Context ...lementations” of such recursive definitions that we have set out to characterize here. In [8], it is argued that such a machine-independent implementation does not reduce the level of abstraction; in =-=[7]-=-, it is argued that there is no “one size fits all” notion of equivalence between algorithms—or between faithful implementations of recursive definitions. Remark 2.3. Though Turing’s (human) computers... |

8 | The prospects for mathematical logic in the twenty-first century, the independent presentations included in panel discussion at The Annual Meeting of the Association for Symbolic Logic held in Urbana–Champaign
- Buss, Kechris, et al.
- 2000
(Show Context)
Citation Context ...te axiomatization has previously been presented in the literature. In fact, the challenge of proving Church’s Thesis is first in Shore’s list of “pie-in-the-sky problems” for the twenty-first century =-=[21]-=-. Whereas Kripke [60, p. 14] feels that it is “a very difficult task”, Friedman [33] predicted that sometime in this century, “There will be an unexpected striking discovery that any model of computat... |

8 | Choiceless polynomial time, counting and the CaiFürer-Immerman graphs
- Dawar, Richerby, et al.
(Show Context)
Citation Context ...ounded Exploration Postulate. In this sense, we are truer to the claim of Kolmogorov 32 The explicit bound on rank is removed in Sieg’s more recent work [96, 97, 99, 100]. 33 Indeed, the algorithm in =-=[30]-=- (determining the “parity” of certain graphs) cannot be naturally encoded as hereditarily finite sets of bounded rank, as shown there.NATURAL AXIOMATIZATION OF COMPUTABILITY 341 and Uspensky [57, 16]... |

8 | From Mathematics to Philosophy, Routledge and Kegan - Wang - 1974 |

7 |
Towards a general theory of computability
- Montague
(Show Context)
Citation Context ...e to the conditions of the imitation game, the interrogator will not be able to take any advantage of this difference.” The first attempt, as far as we know, at characterizing analogue computation is =-=[72]-=-. 15 Cf. Knuth in 1966 [54]: “Algorithms are concepts which have existence apart from any programming language. . . . I believe algorithms were present long before Turing et al. formulated them, just ... |

7 |
by Recursive Step: Church’s analysis of effective calculability
- Sieg, Step
- 1997
(Show Context)
Citation Context ...r conviction of the adequacy of these concepts for expressing the popular meaning of ‘effective calculability’.” 11 Levin was Kolmogorov’s student.306 NACHUM DERSHOWITZ AND YURI GUREVICH (along with =-=[88, 90, 60, 94, 6]-=-) argued in favor of the possibility of axiomatizing effectivity. Kreisel described the discovery of “evident axioms about constructive functions” as “one of the really important open problems” [58] a... |

7 |
K-Graph machines: generalizing Turing’s machines and arguments
- Sieg, Byrnes
- 1996
(Show Context)
Citation Context ...limited complexity” of operations and on the locality of information needed to determine the next state, but he too gave no precise characterizations. Kolmogorov and Uspensky [57] and Sieg and Byrnes =-=[101]-=- proposed sufficient conditions on labeled graphs to ensure boundedness of complexity and locality of action for Kolmogorov-like machines, but their conditions are overly restrictive, and cannot chara... |

6 |
Second Thoughts about Church's Thesis and Mathematical Proofs
- Mendelson
- 1990
(Show Context)
Citation Context ...ugh some interesting partial results have been obtained). Kalmár [46] and Rogers [82, p. 20] (and, more recently, Folina [31]) argued against provability of the thesis, while Gandy [35] and Mendelson =-=[69, 70]-=- notion with your λ-definability gives a stronger conviction of the adequacy of these concepts for expressing the popular meaning of ‘effective calculability’.” 11 Levin was Kolmogorov’s student.306 ... |

6 | Church’s thesis and the conceptual analysis of computability
- Rescorla
- 2007
(Show Context)
Citation Context ... the successor function s on the natural numbers is ρ-recursive, then ρN is in fact recursive, since ρ(0) is some constant and ρ(s(n)) = ̂s(ρ(n)), where ̂s is the recursive witness for successor (cf. =-=[81, 18]-=-). A similar approach to effectiveness in non-numerical domains D is taken in the field of computable algebra, namely that operations on D are tracked by homomorphic images in N (e.g., [34, 64, 79]), ... |

6 |
An Informal Exposition of Proofs of Godel’s theorem and Church’s Theorem
- Rosser
- 1939
(Show Context)
Citation Context ... will tell us what steps to perform next. Furthermore, we view computation as proceeding deterministically,“leaving no place to arbitrariness” [66, p. 1]. As Rosser, also a student of Church, puts it =-=[83]-=-: “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. . . . An ef... |

6 |
A two counter machine cannot calculate 2 n
- Schroeppel
- 1972
(Show Context)
Citation Context ...e will make use of the following variant of Corollary 4.6, which uses this slightly more general notion of recursive function and which allows for arbitrarily many recursive oracles: 30 See [3] (also =-=[86]-=-) for the fundamental weakness of a two-counter machine, as compared to a machine with three counters or more.NATURAL AXIOMATIZATION OF COMPUTABILITY 329 Corollary 4.10. Every partial function comput... |

5 |
Church’s Thesis: Prelude to a Proof
- Folina
- 1998
(Show Context)
Citation Context ...ctively computable, is almost universally held. (Goodstein is an exception [36, n. 29]; Kalmár is not [6, n. 10].) Kleene claims to have proved this (e.g., [50, p. 300]), though some (e.g., Folina in =-=[31]-=-) contend that such an argument, inasmuch as it too involves the informal notion of effectiveness, should not be designated a “proof”. Cf. Gödel’s assertion [38, p. 44] that primitive recursive functi... |

4 | Interactive small-step algorithms I: Axiomatization, and II: Abstract state machines and the characterization theorem
- BLASS, GUREVICH, et al.
(Show Context)
Citation Context ... of algorithm, such as interactive and distributed computations. To capture such non-sequential processes and non-classical algorithms, additional postulates are required. For these developments, see =-=[9, 10, 11, 12, 13, 15, 16, 37]-=-. We also do not address the question of the computational capabilities of the human mind, what Shagrir [87, p. 223] refers to as “The Human version of the Church–Turing Thesis” (more generally called... |

4 |
Gödel didn’t have Church’s thesis
- Why
- 1982
(Show Context)
Citation Context ....) Hence, it remains of importance to provide a small number of convincing postulates in support of Church’s Thesis. Indeed, Gödel has been reported (by Church in a letter to Kleene cited by Davis in =-=[28]-=-) to have believed “that it might be possible . . . to state a set of axioms which would embody the generally accepted properties of [effective calculability], and to do something on that basis”. As e... |

4 |
Algorithm and program; information and data
- Knuth
- 1966
(Show Context)
Citation Context ...imitation game, the interrogator will not be able to take any advantage of this difference.” The first attempt, as far as we know, at characterizing analogue computation is [72]. 15 Cf. Knuth in 1966 =-=[54]-=-: “Algorithms are concepts which have existence apart from any programming language. . . . I believe algorithms were present long before Turing et al. formulated them, just as the concept of the numbe... |

4 |
O ponyatii algoritma [On the concept of algorithm], Uspekhi Matematicheskikh Nauk [Russian Mathematical Surveys
- Kolmogorov
- 1953
(Show Context)
Citation Context ...it is non-trivial to reduce Turing’s analysis to a few general axioms. Subsequent models of computation did not add much force to Turing’s arguments, with the possible exception of Kolmogorov’s model =-=[56, 57]-=-, which, according to Leonid Levin [personal communication], 11 was inspired by an analysis of computation in physical space-time. This model partly addresses the issue of computations that compute wi... |

4 |
A notion of effectiveness in arbitrary structures
- Lambert
- 1968
(Show Context)
Citation Context ... be taken into account is finite. The reasons for 18 The idea of providing the basic operations of recursively-defined functions over arbitrary domains by means of a logical structure also appears in =-=[62]-=-.NATURAL AXIOMATIZATION OF COMPUTABILITY 319 this are of the same character as those which restrict the number of symbols. This finiteness requirement is expressed in more general terms by Kolmogorov... |

4 |
Church’s Thesis
- Shapiro
- 1998
(Show Context)
Citation Context ...r conviction of the adequacy of these concepts for expressing the popular meaning of ‘effective calculability’.” 11 Levin was Kolmogorov’s student.306 NACHUM DERSHOWITZ AND YURI GUREVICH (along with =-=[88, 90, 60, 94, 6]-=-) argued in favor of the possibility of axiomatizing effectivity. Kreisel described the discovery of “evident axioms about constructive functions” as “one of the really important open problems” [58] a... |

4 |
Mechanical procedures and mathematical experience. Mathematics and mind
- Sieg
- 1994
(Show Context)
Citation Context ...107] and its subsequent generalizations by Post [77] and Kolmogorov [56, 57] are on an informal level. Gandy [35] was the first to attempt an axiomatization, and was followed in this endeavor by Sieg =-=[93, 94, 97, 96, 99, 100]-=-; though their axioms are formal, they are expressed on the level of a specific representation of states (namely, hereditarily finite sets). In contrast, our axioms of effective computation are, at th... |

3 |
Proving Church's Thesis
- Black
- 2000
(Show Context)
Citation Context ...r conviction of the adequacy of these concepts for expressing the popular meaning of ‘effective calculability’.” 11 Levin was Kolmogorov’s student.306 NACHUM DERSHOWITZ AND YURI GUREVICH (along with =-=[88, 90, 60, 94, 6]-=-) argued in favor of the possibility of axiomatizing effectivity. Kreisel described the discovery of “evident axioms about constructive functions” as “one of the really important open problems” [58] a... |

3 |
Algorithms vs. machines”, Bulletin of the European Association for Theoretical Computer
- Blass, Gurevich
- 2002
(Show Context)
Citation Context ...here can be significant algorithmic distinctions and performance differences between reasonable methods of computing that least fixed point, such as “call by name” and “call by need”. So, for us (see =-=[8]-=-), recursion equations are a partial specification of desired properties of the algorithm in question, not the algorithm itself. Nevertheless, if one does accept Moschovakis’s point of view, then it i... |