## The Church-Turing Thesis over Arbitrary Domains (2008)

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Citations: | 12 - 9 self |

### BibTeX

@MISC{Boker08thechurch-turing,

author = {Udi Boker and Nachum Dershowitz},

title = {The Church-Turing Thesis over Arbitrary Domains },

year = {2008}

}

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### Abstract

The Church-Turing Thesis has been the subject of many variations and interpretations over the years. Specifically, there are versions that refer only to functions over the natural numbers (as Church and Kleene did), while others refer to functions over arbitrary domains (as Turing intended). Our purpose is to formalize and analyze the thesis when referring to functions over arbitrary domains. First, we must handle the issue of domain representation. We show that, prima facie, the thesis is not well defined for arbitrary domains, since the choice of representation of the domain might have a non-trivial influence. We overcome this problem in two steps: (1) phrasing the thesis for entire computational models, rather than for a single function; and (2) proving a “completeness” property of the recursive functions and Turing machines with respect to domain representations. In the second part, we propose an axiomatization of an “effective model of computation” over an arbitrary countable domain. This axiomatization is based on Gurevich’s postulates for sequential algorithms. A proof is provided showing that all models satisfying these axioms, regardless of underlying data structure, are of equivalent computational power to, or weaker than, Turing machines.

### Citations

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(Show Context)
Citation Context ... numeric functions be identified with Gödel and Herbrand’s general recursive functions, or – equivalently – with Church and Kleene’s lambda-definable functions of positive integers. Similarly, Turing =-=[22]-=- suggested that his computational model, namely, Turing machines, could compute anything that might be mechanically computable, but his interests extended beyond numeric functions. Church’s original t... |

837 | Theory of recursive functions and effective computability - Rogers - 1967 |

454 |
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Citation Context ...ces Yao, that two-counter machines cannot compute the function λn.2 n . Since two-counter machines can simulate all the recursive functions via some proper injective representation (see, for example, =-=[12]-=-), it follows that two-counter machines can “enlarge” their computational power via some representations. A reasonable direction might have been to restrict the representation to bijections between do... |

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114 | Sequential abstract state machines capture sequential algorithms
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(Show Context)
Citation Context ...sis over arbitrary domains, we investigate the general class of “effective computational models”. We proffer an axiomatization of this class, based on Gurevich’s postulates for a sequential algorithm =-=[8]-=-. The thesis is then proved, in the sense that a proof is provided that all models satisfying these axioms are equivalent to, or weaker than, Turing machines. The specifics of our effectiveness axioms... |

94 | Computability and Complexity from a Programming Perspective - Jones - 1997 |

56 |
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(Show Context)
Citation Context ... effective sequential deterministic symbol manipulation: finite internal states; finite symbol space; external memory that can be represented linearly; finite observability; and local action. 4sGandy =-=[7]-=-, and later Sieg and Byrnes [20], define a model whose states are described by hereditarily finite sets. Effectiveness of Gandy machines is achieved by bounding the rank (depth) of states, insisting t... |

45 | Recursive predicates and quantifiers - Kleene - 1943 |

21 |
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(Show Context)
Citation Context ...ch most people would agree are evidently true. It might be possible to prove Church’s Thesis from such axioms. In fact, Gödel has also been reported (by Church in a letter to Kleene cited by Davis in =-=[6]-=-) to have thought “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”. Axiom... |

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15 |
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(Show Context)
Citation Context ...se Gurevich’s more general “Abstract State Machines” (ASMs) [8] as our starting point. (Some of the problems of incorporating the Gandy model under the abstract state machine rubric are dealt with in =-=[1]-=-.) Overview. The first part of this paper, Sect. 2, deals with the issue of domain representation. In Sect. 2.1, we show that checking for the computability of a single function over an arbitrary doma... |

15 |
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(Show Context)
Citation Context ...stic symbol manipulation: finite internal states; finite symbol space; external memory that can be represented linearly; finite observability; and local action. 4sGandy [7], and later Sieg and Byrnes =-=[20]-=-, define a model whose states are described by hereditarily finite sets. Effectiveness of Gandy machines is achieved by bounding the rank (depth) of states, insisting that they be unambiguously assemb... |

12 |
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(Show Context)
Citation Context ...r Solution. Our approach to overcoming the representation problem is to ask about effectiveness of a set of functions over the domain of interest, rather than of a single function. As Myhill observed =-=[14]-=-: undecidability is a property of classes of problems, not of individual problems. The Church-Turing Thesis, interpreted accordingly, asserts that there is no effective computational model that is mor... |

11 | The Theory of Computability - Sommerhalder, Westrhenen - 1988 |

7 |
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(Show Context)
Citation Context ...s, like Gödel numbering, Church numerals, unary representation of numbers, etc. This is also true of the usual handling of representations in the context of the Church-Turing Thesis. Richard Montague =-=[13]-=- raises the problem of representation when applying Turing’s notion of computability to other domains, as well as the circularity in choosing a “computable representation”. Stewart Shapiro [18] raises... |

7 |
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(Show Context)
Citation Context ...ntague [13] raises the problem of representation when applying Turing’s notion of computability to other domains, as well as the circularity in choosing a “computable representation”. Stewart Shapiro =-=[18]-=- raises the very same problem of representation when applying computability to number-theoretic functions. He suggests a definition of an “acceptable notation” (an acceptable string representation of ... |

6 | N.: Comparing computational power
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(Show Context)
Citation Context ...rictly contains them – all depending on the choice of domain representation. 2sFortunately, this cannot be the case with Turing machines (nor with the recursive functions), as we have demonstrated in =-=[3]-=-, where we proved that Turing machines are “complete” in the sense that if some model is equivalent to, or weaker than, Turing machines under one representation, then no other representation (no matte... |

6 | Church’s thesis and the conceptual analysis of computability
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(Show Context)
Citation Context ...esis over arbitrary domains, we suggest, in Section 2.5, a definition of an “effective representation”, resembling Shapiro’s notion of “acceptable notation”. Michael Rescorla claims in a recent paper =-=[15]-=- that the Church-Turing Thesis has inherent circularity because of the above problem of representing numbers by strings. He is not satisfied with Shapiro’s definition of an acceptable notation, findin... |

6 |
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(Show Context)
Citation Context ...ivalent to one of its strict supermodels. That is, a representation might allow to “enlarge” a model, adding some “new” functions to it. Consider two-counter machines. It was shown by Rich Schroeppel =-=[17]-=-, and independently by Frances Yao, that two-counter machines cannot compute the function λn.2 n . Since two-counter machines can simulate all the recursive functions via some proper injective represe... |

3 | How to compare the power of computational models
- Boker, Dershowitz
- 2005
(Show Context)
Citation Context ... This is done, for example, by Rogers [16, p. 27], Sommerhalder [21, p. 30], and Cutland [5, p. 24]. Our notion of comparing computational power is very similar to this. To the best of our knowledge, =-=[2, 3]-=- are the first papers to point out and handle the possible influence of the representation on the extensionality of computational models. As for the axiomatization of effectiveness, several different ... |

1 | to Compare the Power of Computational Models - Blass, Gurevich, et al. |

1 | W.M.: A notion of effectiveness in arbitrary structures. The Journal of Symbolic Logic 33(4), 577–602 - Addison-Wesley, MA - 1968 |