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A NATURAL AXIOMATIZATION OF COMPUTABILITY AND PROOF OF CHURCH’S THESIS
"... 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 e ..."
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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 effectivelycomputable functions on string representations of numbers.
Prospects for mathematical logic in the twentyfirst century
 BULLETIN OF SYMBOLIC LOGIC
, 2002
"... The four authors present their speculations about the future developments of mathematical logic in the twentyfirst century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently. ..."
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Cited by 8 (0 self)
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The four authors present their speculations about the future developments of mathematical logic in the twentyfirst century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently.
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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Cited by 2 (0 self)
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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.
Looking Ahead
"... Abstract. We discuss some of the opportunities and problems which may confront the field of automated reasoning in the years ahead. We focus on various issues related to the development of a Universal Automated Information System for Science and Technology, and the problem of developing institutiona ..."
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Abstract. We discuss some of the opportunities and problems which may confront the field of automated reasoning in the years ahead. We focus on various issues related to the development of a Universal Automated Information System for Science and Technology, and the problem of developing institutional support for longterm projects. 1
The Arrow System Philosophy
, 2000
"... This proposal introduces a unified system for computations based on a cybernetic theory introduced here as modellevel reflection. It provides a specification of what the Arrow system is and what it does for software systems. It outlines the theoretical basis for the system in terms of analysis of m ..."
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This proposal introduces a unified system for computations based on a cybernetic theory introduced here as modellevel reflection. It provides a specification of what the Arrow system is and what it does for software systems. It outlines the theoretical basis for the system in terms of analysis of modern programming systems and their reflective capabilities. The system transcends the limitations of stateoftheart reflection systems due to the current restricted notion of metasystems. The system can represent any intuitive concept, and can manage the interactions among arbitrary domains of knowledge. Because of these properties, the user may delegate arbitrary aspects of its own operation and management to itself, with reliable and reusable results. The nature of the system is ideal for computations in a unified field of knowledge, and as such provides an ideal basis for the migration of knowledge to new contexts and uses.
The ChurchTuring Thesis∗
, 1999
"... The ChurchTuring thesis (CTT) establishes a strong relation between effectively calculable (computable) functions and Turing computable (recursive) functions, actually, this is not just a relation, this is an equivalence between them. In section 2 we present a short history about the CTT. We beg ..."
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The ChurchTuring thesis (CTT) establishes a strong relation between effectively calculable (computable) functions and Turing computable (recursive) functions, actually, this is not just a relation, this is an equivalence between them. In section 2 we present a short history about the CTT. We begin pre
LIMITATIONS ON OUR UNDERSTANDING OF THE BEHAVIOR OF SIMPLIFIED PHYSICAL SYSTEMS
, 2008
"... Results going back to Turing and Gödel provide us with limitations on our ability to algorithmically decide the truth or falsity of mathematical assertions in a number of important mathematical contexts. Here we adapt some of this earlier work to very simplified mathematical models of discrete dete ..."
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Results going back to Turing and Gödel provide us with limitations on our ability to algorithmically decide the truth or falsity of mathematical assertions in a number of important mathematical contexts. Here we adapt some of this earlier work to very simplified mathematical models of discrete deterministic physical systems involving a few moving bodies (twelve point masses) in potentially infinite one dimensional space. There are two kinds of such limiting results that must be carefully distinguished. Results of the first kind state the nonexistence of any algorithm for determining whether any statement among a given set of statements is true or false. Results of the second kind are much deeper and present much greater challenges. They point to specific statements A, where we can neither prove nor refute A using accepted principles of mathematical reasoning. We give a brief survey of these limiting results. These include limiting results of the first kind: from number theory, group theory, and topology, in mathematics, and from idealized computing devices in theoretical computer science. We present a new limiting result of the first kind for simplified physical systems. We conjecture some related limiting results of the second kind, for simplified physical systems.
Arithmetic and the Incompleteness Theorems
, 2000
"... this paper please consult me first, via my home page. ..."
NatureLike Computation and a Measure of Programmability
, 2012
"... I will propose an alternative behavioural definition of computation (naturelike) based on whether a system is capable of reacting to the environment—the input—as reflected in a measure of programmability. This will be done by using a phase transition coefficient previously defined in an attempt to ..."
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I will propose an alternative behavioural definition of computation (naturelike) based on whether a system is capable of reacting to the environment—the input—as reflected in a measure of programmability. This will be done by using a phase transition coefficient previously defined in an attempt to characterise the dynamical behaviour of cellular automata and other rulebased systems. The transition coefficient measures the sensitivity of a system to external stimuli and will be used to define the susceptibility of a system to being (efficiently) programmed.