A Natural Axiomatization of Computability and Proof of Church's Thesis

Dershowitz, Nachum; Gurevich, Yuri

doi:10.2178/bsl/1231081370

Cited by 97 publications

(77 citation statements)

References 56 publications

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“…The easing of the requirements on fully-defined equality and other functions also lends strong support to the contention -put forth in [8,10] -that the Church-Turing Thesis is provably true from first principles. In addition to the sequential postulates, the arguments require that initial states contain only free constructors and functions that can be programmed from them (plus input).…”

Section: Discussionmentioning

confidence: 62%

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Exact Exploration and Hanging Algorithms

Blass

Dershowitz

Gurevich

2010

Computer Science Logic

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Abstract. Recent analysis of sequential algorithms resulted in their axiomatization and in a representation theorem stating that, for any sequential algorithm, there is an abstract state machine (ASM) with the same states, initial states and state transitions. That analysis, however, abstracted from details of intra-step computation, and the ASM, produced in the proof of the representation theorem, may and often does explore parts of the state unexplored by the algorithm. We refine the analysis, the axiomatization and the representation theorem. Emulating a step of the given algorithm, the ASM, produced in the proof of the new representation theorem, explores exactly the part of the state explored by the algorithm. That frugality pays off when state exploration is costly. The algorithm may be a high-level specification, and a simple function call on the abstraction level of the algorithm may hide expensive interaction with the environment. Furthermore, the original analysis presumed that state functions are total. Now we allow state functions, including equality, to be partial so that a function call may cause the algorithm as well as the ASM to hang. Since the emulating ASM does not make any superfluous function calls, it hangs only if the algorithm does.[T]he monotony of equality can only lead us to boredom.-Francis Picabia

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Section: Discussionmentioning

confidence: 62%

“…Careful analysis of the notion of algorithm [13] and an examination of the intent of the founders of the field of computability [10] have demonstrated that the sequential postulates are true of all sequential, deterministic algorithms, the only kind envisioned by the pioneers of the field. In Sects.…”

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confidence: 99%

Exact Exploration and Hanging Algorithms

Blass

Dershowitz

Gurevich

2010

Computer Science Logic

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“…A careful analysis of the notion of algorithm in [25] and an examination of the intent of the founders of the field of computability in [18] demonstrate that the Sequential Postulates are in fact true of all ordinary, sequential algorithms, the (only) kind envisioned by the pioneers of the field. In other words, all classical algorithms satisfy Postulates I, II, and III.…”

Section: Postulate III (Effective Transitions)mentioning

confidence: 99%

“…The only means that an algorithm has with which to reference locations is via terms, since the values themselves are abstract entities. If every referenced location has the same value in two states, then the behavior of the algorithm must be the same for both of those states.This postulate-with its fixed, finite set of critical terms-precludes programs of infinite size (like an infinite table lookup) or which are input-dependent.A careful analysis of the notion of algorithm in [25] and an examination of the intent of the founders of the field of computability in [18] demonstrate that the Sequential Postulates are in fact true of all ordinary, sequential algorithms, the (only) kind envisioned by the pioneers of the field. In other words, all classical algorithms satisfy Postulates I, II, and III.…”

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confidence: 99%

The Generic Model of Computation

Dershowitz

2012

Electron. Proc. Theor. Comput. Sci.

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Over the past two decades, Yuri Gurevich and his colleagues have formulated axiomatic foundations for the notion of algorithm, be it classical, interactive, or parallel, and formalized them in the new generic framework of abstract state machines. This approach has recently been extended to suggest a formalization of the notion of effective computation over arbitrary countable domains. The central notions are summarized herein. BackgroundAbstract state machines (ASMs), invented by Yuri Gurevich [24], constitute a most general model of computation, one that can operate on any desired level of abstraction of data structures and native operations. All (ordinary) models of computation are instances of this one generic paradigm. Here, we give an overview of the foundational considerations underlying the model (cobbled together primarily from [18,3,12]). 1 Programs (of the sequential, non-interactive variety) in this formalism are built from three components:• There are generalized assignments f (s 1 , . . . , s n ) := t, where f is any function symbol (in the vocabulary of the program) and the s i and t are arbitrary terms (in that vocabulary).• Statements may be prefaced by a conditional test, if C then P or if C then P else Q, where C is a propositional combination of equalities between terms.• Program statements may be composed in parallel, following the keyword do, short for do in parallel.An ASM program describes a single transition step; its statements are executed repeatedly, as a unit, until no assignments have their conditions enabled. (Additional constructs beyond these are needed for interaction and large-scale parallelism, which are not dealt with here.) As a simple example, consider the program shown as Algorithm 1, describing a version of selection sort, where F(0), . . . , F(n − 1) contain values to be sorted, F being a unary function symbol. Initially, n ≥ 1 is the quantity of values to be sorted, i is set to 0, and j to 1. The brackets indicate statements that are executed in parallel. The program proceeds by repeatedly modifying the values of i and j, as well as of locations in F, referring to terms F(i) and F( j). When all conditions fail, that is, when j = n and i + 1 = n, the values in F have been sorted vis-à-vis the black-box relation ">". The program halts, as there is nothing left to do. (Declarations and initializations for program constants and variables are not shown.)This sorting program is not partial to any particular representation of the natural numbers 1, 2, etc., which are being used to index F. Whether an implementation uses natural language, or decimal numbers, 1 For a video lecture of Gurevich's on this subject, see http://www.youtube.com/v/7XfA5EhH7Bc. Generic Model of ComputationAlgorithm 1 An abstract-state-machine program for sorting.Algorithm 2 An abstract-state-machine program for bisection search.or binary strings is immaterial, as long as addition behaves as expected (and equality and disequality, too). Furthermore, the program will work regardless of the domain from which...

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“…It is also worth mentioning that the axiomatic approach allowed researchers to prove the ChurchTuring Thesis for an algorithmic class that satisfies very simple initial axioms [44,45]. These axioms form a system C considered in the previous section and this system provides a definite perspective on different classes of algorithms, ensuring that in these classes the Church-Turing Thesis is true, i.e., it is a theorem.…”

Section: How To Navigate In the Algorithmic Multiversementioning

confidence: 99%

Unconventional Algorithms: Complementarity of Axiomatics and Construction

Dodig-Crnković

Burgin

2012

Entropy

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Abstract:In this paper, we analyze axiomatic and constructive issues of unconventional computations from a methodological and philosophical point of view. We explain how the new models of algorithms and unconventional computations change the algorithmic universe, making it open and allowing increased flexibility and expressive power that augment creativity. At the same time, the greater power of new types of algorithms also results in the greater complexity of the algorithmic universe, transforming it into the algorithmic multiverse and demanding new tools for its study. That is why we analyze new powerful tools brought forth by local mathematics, local logics, logical varieties and the axiomatic theory of algorithms, automata and computation. We demonstrate how these new tools allow efficient navigation in the algorithmic multiverse. Further work includes study of natural computation by unconventional algorithms and constructive approaches.

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A Natural Axiomatization of Computability and Proof of Church's Thesis

Cited by 97 publications

References 56 publications

Exact Exploration and Hanging Algorithms

Exact Exploration and Hanging Algorithms

The Generic Model of Computation

Unconventional Algorithms: Complementarity of Axiomatics and Construction

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