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Computational Foundations of Basic Recursive Function Theory
 Theoretical Computer Science
, 1988
"... The theory of computability, or basic recursive function theory as it is often called, is usually motivated and developed using Church's Thesis. Here we show that there is an alternative computability theory in which some of the basic results on unsolvability become more absolute, results on complet ..."
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Cited by 20 (7 self)
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The theory of computability, or basic recursive function theory as it is often called, is usually motivated and developed using Church's Thesis. Here we show that there is an alternative computability theory in which some of the basic results on unsolvability become more absolute, results on completeness become simpler, and many of the central concepts become more abstract. In this approach computations are viewed as mathematical objects, and the major theorems in recursion theory may be classified according to which axioms about computation are needed to prove them. The theory is a typed theory of functions over the natural numbers, and there are unsolvable problems in this setting independent of the existence of indexings. The unsolvability results are interpreted to show that the partial function concept, so important in computer science, serves to distinguish between classical and constructive type theories (in a different way than does the decidability concept as expressed in the ...
On Founding the Theory of Algorithms
, 1998
"... machines and implementations The first definition of an abstract machine was given by Turing, in the classic [20]. Without repeating here the wellknown definition (e.g., see [6]), 13 we recall that each Turing machine M is equipped with a "semiinfinite tape" which it uses both to compute and al ..."
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Cited by 9 (4 self)
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machines and implementations The first definition of an abstract machine was given by Turing, in the classic [20]. Without repeating here the wellknown definition (e.g., see [6]), 13 we recall that each Turing machine M is equipped with a "semiinfinite tape" which it uses both to compute and also to communicate with its environment: To determine the value f(n) (if any) of the partial function 14 f : N * N computed by M , we put n on the tape in some standard way, e.g., by placing n + 1 consecutive 1s at its beginning; we start the machine in some specified, initial, internal state q 0 and looking at the leftmost end of the tape; and we wait until the machine stops (if it does), at which time the value f(n) can be read off the tape, by counting the successive 1s at the left end. Turing argued that the numbertheoretic functions which can (in principle) be computed by any deterministic, physical device are exactly those which can be computed by a Turing machine, and the correspon...
What Is an Algorithm
 SOFSEM, Lecture Notes in
"... We present a twopart exposition on the notion of algorithm and foundational analyses of computation. The first part is below, and the second is here: ..."
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Cited by 3 (2 self)
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We present a twopart exposition on the notion of algorithm and foundational analyses of computation. The first part is below, and the second is here:
Guarded Quantification in Least Fixed Point Logic
, 2002
"... We develop a variant of Least Fixed Point logic based on First Order logic with a relaxed version of guarded quantification. We develop a Game Theoretic Semantics of this logic, and find that under reasonable conditions, guarding quantification does not reduce the expressibility of Least Fixed Point ..."
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Cited by 2 (1 self)
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We develop a variant of Least Fixed Point logic based on First Order logic with a relaxed version of guarded quantification. We develop a Game Theoretic Semantics of this logic, and find that under reasonable conditions, guarding quantification does not reduce the expressibility of Least Fixed Point logic. But guarding quantification increases worstcase time complexity.
An Intensional Investigation of Parallelism
, 1994
"... Denotational semantics is usually extensional in that it deals only with input/output properties of programs by making the meaning of a program a function. Intensional semantics maps a program into an algorithm, thus enabling one to reason about complexity, order of evaluation, degree of parallelism ..."
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Denotational semantics is usually extensional in that it deals only with input/output properties of programs by making the meaning of a program a function. Intensional semantics maps a program into an algorithm, thus enabling one to reason about complexity, order of evaluation, degree of parallelism, efficiencyimproving program transformations, etc. I propose to develop intensional models for a number of parallel programming languages. The semantics will be implemented, resulting in a programming language of parallel algorithms, called CDSP. Applications of CDSP will be developed to determine its suitability for actual use. The thesis will hopefully make both theoretical and practical contributions: as a foundational study of parallelism by looking at the expressive power of various constructs, and with the design, implementation, and applications of an intensional parallel programming language. 1 Introduction Denotational semantics has now been around for about 25 years, which makes...
On the Expressive Power of Existential Quantification in PolynomialTime Computability
"... this paper to study the expressive power of bounded existential quantification in polynomialtime computability. Our goal was to characterize nondeterministic polynomialtime computations in a machineindependent way. The following considerations are intended to make our idea clear. Let # be the fin ..."
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this paper to study the expressive power of bounded existential quantification in polynomialtime computability. Our goal was to characterize nondeterministic polynomialtime computations in a machineindependent way. The following considerations are intended to make our idea clear. Let # be the finite alphabet
On Founding the Theory of Algorithms
, 1998
"... machines and implementations The first definition of an abstract machine was given by Turing, in the classic [20]. Without repeating here the wellknown definition (e.g., see [6]), we recall that each Turing machine M is equipped with a "semiinfinite tape" which it uses both to compute and also ..."
Abstract
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machines and implementations The first definition of an abstract machine was given by Turing, in the classic [20]. Without repeating here the wellknown definition (e.g., see [6]), we recall that each Turing machine M is equipped with a "semiinfinite tape" which it uses both to compute and also to communicate with its environment: To determine the value f(n) (if any) of the partial function f : N * N computed by M , we put n on the tape in some standard way, e.g., by placing n + 1 consecutive 1s at its beginning; we start the machine in some specified, initial, internal state q 0 and looking at the leftmost end of the tape; and we wait until the machine stops (if it does), at which time the value f(n) can be read off the tape, by counting the successive 1s at the left end. Turing argued that the numbertheoretic functions which can (in principle) be computed by any deterministic, physical device are exactly those which can be computed by a Turing machine, and the corresponding version of this claim for partial functions has come to be known as the ChurchTuring Thesis, because an equivalent claim was made by Church at about the same time. Turing's brilliant analysis of "mechanical computation" in [20] and a huge body of work in the last sixty years has established the truth of the ChurchTuring Thesis beyond reasonable doubt; it is of immense importance in the derivation of foundationally significant undecidability results from technical theorems about Turing machines, and it has been called "the first natural law of pure mathematics." Turing machines capture the notion of mechanical computability of numbertheoretic functions, by the ChurchTuring Thesis, but they do not model faith It has also been suggested that we do not need algorithms, only the equival...
WHAT IS AN ALGORITHM? (REVISED)
"... Abstract. We put the title problem and Church’s thesis into a proper perspective, and we address some common misconceptions about Turing’s analysis of computation. In addition, we comment on two approaches to the title problem, one well known among philosophers and another well known among logicians ..."
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Abstract. We put the title problem and Church’s thesis into a proper perspective, and we address some common misconceptions about Turing’s analysis of computation. In addition, we comment on two approaches to the title problem, one well known among philosophers and another well known among logicians.