Results 1  10
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15
Computational foundations of basic recursive function theory
 Journal of Theoretical Computer Science
, 1993
"... ..."
A mathematical modeling of pure, recursive algorithms
 Logic at Botik ’89
, 1989
"... This paper follows previous work on the Formal Language of Recursion FLR and develops intensional (algorithmic) semantics for it: the intension of a term t on a structure A is a recursor, a set–theoretic object which represents the (abstract, recursive) algorithm defined by t on A. Main results are ..."
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Cited by 9 (6 self)
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This paper follows previous work on the Formal Language of Recursion FLR and develops intensional (algorithmic) semantics for it: the intension of a term t on a structure A is a recursor, a set–theoretic object which represents the (abstract, recursive) algorithm defined by t on A. Main results are the soundness of the reduction calculus of FLR (which models faithful, algorithm–preserving compilation) for this semantics, and the robustness of the class of algorithms assigned to a structure under algorithm adjunction. This is the second in a sequence of papers begun with [16] in which we develop a foundation for the theory of computation based on a mathematical modeling of recursive algorithms. The general features, aims and methodological assumptions of this program were discussed and illustrated by examples in the preliminary, expository report [15]. In [16] we studied the formal language of recursion FLR which is the main technical tool for this work, we developed several alternative denotational semantics for it and we established a key unique termination theorem for a reduction calculus which models faithful (algorithm–preserving) compilation. Here we will define the intensional semantics of FLR for structures with given (pure) recursors, the set–theoretic objects we use to model pure (side–effect–free) algorithms: the intension of a term t on each structure A is a recursor which models the algorithm expressed by t on A. Basic results of the paper During the preparation of this paper the author was partially supported by an NSF
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 comp ..."
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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.
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.
Elementary algorithms and their implementations
"... In the sequence of articles [3, 5, 4, 6, 7], Moschovakis has proposed a mathematical modeling of the notion of algorithm—a settheoretic “definition ” of algorithms, much like the “definition ” of real numbers as Dedekind cuts on the rationals or that of random variables as measurable functions on a ..."
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In the sequence of articles [3, 5, 4, 6, 7], Moschovakis has proposed a mathematical modeling of the notion of algorithm—a settheoretic “definition ” of algorithms, much like the “definition ” of real numbers as Dedekind cuts on the rationals or that of random variables as measurable functions on a probability space. The aim is to provide a traditional foundation for the theory of algorithms, a development of it within axiomatic set theory in the same way as analysis and probability theory are rigorously developed within set theory on the basis of the set theoretic modeling of their basic notions. A characteristic feature of this approach is the adoption of a very abstract notion of algorithm which takes recursion as a primitive operation, and is so wide as to admit “nonimplementable ” algorithms: implementations are special, restricted algorithms (which include the familiar models of computation, e.g., Turing and random access machines), and an algorithm is implementable if it is reducible to an implementation. Our main aim here is to investigate the important relation between an
unknown title
"... On the expressive power of existential quantification in polynomialtime computability (Extended abstract) ..."
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On the expressive power of existential quantification in polynomialtime computability (Extended abstract)
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 comput ..."
Abstract
 Add to MetaCart
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...