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The ChurchTuring Thesis over Arbitrary Domains
, 2008
"... The ChurchTuring 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 pu ..."
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The ChurchTuring 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 nontrivial 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.
Another Example of Higher Order Randomness
 FUNDAMENTA INFORMATICAE
, 2002
"... We consider the notion of algorithmic randomness relative to an oracle. We prove that the probability # that a program for infinite computations (a program that never halts) outputs a cofinite set is random in the second jump of the halting problem. Indeed, we prove that # is exactly as random as ..."
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We consider the notion of algorithmic randomness relative to an oracle. We prove that the probability # that a program for infinite computations (a program that never halts) outputs a cofinite set is random in the second jump of the halting problem. Indeed, we prove that # is exactly as random as the halting probability of a universal machine equipped with an oracle for the second jump of the halting problem, in spite of the fact that # is defined without considering oracles.
Gold’s theorem and cognitive science
 Philosophy of Science
, 2004
"... A variety of inaccurate claims about Gold’s Theorem have appeared in the cognitive science literature. I begin by characterizing the logic of this theorem and its proof. I then examine several claims about Gold’s Theorem, and I show why they are false. Finally, I assess the significance of Gold’s Th ..."
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A variety of inaccurate claims about Gold’s Theorem have appeared in the cognitive science literature. I begin by characterizing the logic of this theorem and its proof. I then examine several claims about Gold’s Theorem, and I show why they are false. Finally, I assess the significance of Gold’s Theorem for cognitive science.
A Formalization of the ChurchTuring Thesis for StateTransition Models
"... Abstract. Our goal is to formalize the ChurchTuring Thesis for a very large class of computational models. Specifically, the notion of an “effective model of computation ” over an arbitrary countable domain is axiomatized. This is accomplished by modifying Gurevich’s “Abstract State Machine ” postu ..."
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Abstract. Our goal is to formalize the ChurchTuring Thesis for a very large class of computational models. Specifically, the notion of an “effective model of computation ” over an arbitrary countable domain is axiomatized. This is accomplished by modifying Gurevich’s “Abstract State Machine ” postulates for statetransition systems. A proof is provided that all models satisfying our axioms, regardless of underlying data structure—and including all standard statetransition models—are equivalent to (up to isomorphism), or weaker than, Turing machines. To allow the comparison of arbitrary models operating over arbitrary domains, we employ a quasiordering on computational models, based on their extensionality. LCMs can do anything that could be described as “rule of thumb ” or “purely mechanical”.... This is sufficiently well established that it is now agreed amongst logicians that “calculable by means of an LCM” is the correct accurate rendering of such phrases. 1
A natural axiomatization of Church’s thesis
, 2007
"... The Abstract State Machine Thesis asserts that every classical algorithm is behaviorally equivalent to an abstract state machine. This thesis has been shown to follow from three natural postulates about algorithmic computation. Here, we prove that augmenting those postulates with an additional requ ..."
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The Abstract State Machine Thesis asserts that every classical algorithm is behaviorally equivalent to an abstract state machine. This thesis has been shown to follow from three natural postulates about algorithmic computation. Here, we prove that augmenting those postulates with an additional requirement regarding basic operations implies Church’s Thesis, namely, 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). In particular, this gives a natural axiomatization of Church’s Thesis, as Gödel and others suggested may be possible.
Foundations for a SelfReflective, ContextAware Semantic Representation of Mathematical Specifications
"... 5.2 The representation of informal mathematical text....... 83 5.3 The representation of optimization problems.......... 86 ..."
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5.2 The representation of informal mathematical text....... 83 5.3 The representation of optimization problems.......... 86
Linear Recursive Functions
"... With the recent trend of analysing the process of computation through the linear logic looking glass, it is well understood that the ability to copy and erase data is essential in order to obtain a Turingcomplete computation model. However, erasing and copying do not need to be explicitly included ..."
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With the recent trend of analysing the process of computation through the linear logic looking glass, it is well understood that the ability to copy and erase data is essential in order to obtain a Turingcomplete computation model. However, erasing and copying do not need to be explicitly included in Turingcomplete computation models: in this paper we show that the class of partial recursive functions that are syntactically linear (that is, partial recursive functions where no argument is erased or copied) is Turingcomplete.
Proving Church’s Thesis (Abstract)
"... The talk reflects recent joint work with Nachum Dershowitz [4]. In 1936, Church suggested that the recursive functions, which had been defined by Gödel earlier that decade, adequately capture the intuitive notion of a computable (“effectively calculable”) numerical function1 [2]. Independently Turin ..."
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The talk reflects recent joint work with Nachum Dershowitz [4]. In 1936, Church suggested that the recursive functions, which had been defined by Gödel earlier that decade, adequately capture the intuitive notion of a computable (“effectively calculable”) numerical function1 [2]. Independently Turing argued that, for stringstostrings functions, the same goal is achieved by his machines [11]. The modern form of Church’s thesis is due to Church’s student Kleene. It asserts that every computable numerical partial function is partial recursive. (Originally Church spoke of total functions.) Kleene thought that the thesis as unprovable: “Since our original notion of effective calculability... is a somewhat vague intuitive one, the thesis cannot be proved ” [7]. But he presented evidence in favor of the thesis. By far the strongest argument was Turing’s analysis [11] of “the sorts of operations which a human computer could perform, working according to preassigned instructions ” [7]. The argument convinced Gödel who thought the idea “that this really is the correct
Theories of arithmetics in finite models
"... We investigate theories of initial segments of the standard models for arithmetics. It is easy to see that if the ordering relation is definable in the standard model then the decidability results can be transferred from the infinite model into the finite models. On the contrary we show that the Σ2– ..."
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We investigate theories of initial segments of the standard models for arithmetics. It is easy to see that if the ordering relation is definable in the standard model then the decidability results can be transferred from the infinite model into the finite models. On the contrary we show that the Σ2–theory of multiplication is undecidable in finite models. We show that this result is optimal by proving that the Σ1–theory of multiplication and order is decidable in finite models as well as in the standard model. We show also that the exponentiation function is definable in finite models by a formula of arithmetic with multiplication and that one can define in finite models the arithmetic of addition and multiplication with the concatenation operation. We consider also the spectrum problem. We show that the spectrum of arithmetic with multiplication and arithmetic with exponentiation is strictly contained in the spectrum of arithmetic with addition and multiplication. 1