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The logic of recursive equations
 the Journal of Symbolic Logic
, 1998
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Cited by 13 (5 self)
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JSTOR is a notforprofit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.
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 10 (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...
The Journal of Symbolic Logic
 the Journal of Symbolic Logic
, 1998
"... We study logical systems for reasoning about equations involving recursive de#nitions. In particular, we are interested in #propositional# fragments of the functional language of recursion FLR [18, 17], i.e., without the value passing or abstraction allowed in FLR. The #pure,# propositional fragme ..."
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We study logical systems for reasoning about equations involving recursive de#nitions. In particular, we are interested in #propositional# fragments of the functional language of recursion FLR [18, 17], i.e., without the value passing or abstraction allowed in FLR. The #pure,# propositional fragment FLR 0 turns out to coincide with the iteration theories of [1]. Our main focus here concerns the sharp contrast between the simple class of valid identities and the very complexconsequence relation over several natural classes of models.
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
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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...