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Coinductive Models of Finite Computing Agents
 Electronic Notes in Theoretical Computer Science
, 1999
"... This paper explores the role of coinductive methods in modeling nite interactive computing agents. The computational extension of computing agents from algorithms to interaction parallels the mathematical extension of set theory and algebra from inductive to coinductive models. Maximal xed points ..."
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Cited by 13 (6 self)
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This paper explores the role of coinductive methods in modeling nite interactive computing agents. The computational extension of computing agents from algorithms to interaction parallels the mathematical extension of set theory and algebra from inductive to coinductive models. Maximal xed points are shown to play a role in models of observation that parallels minimal xed points in inductive mathematics. The impact of interactive (coinductive) models on Church's thesis and the connection between incompleteness and greater expressiveness are examined. A nal section shows that actual software systems are interactive rather than algorithmic. Coinductive models could become as important as inductive models for software technology as computer applications become increasingly interactive.
Gödel's Dialectica interpretation and its twoway stretch
 in Computational Logic and Proof Theory (G. Gottlob et al eds.), Lecture Notes in Computer Science 713
, 1997
"... this article has appeared in Computational Logic and Proof Theory (Proc. 3 ..."
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Cited by 3 (1 self)
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this article has appeared in Computational Logic and Proof Theory (Proc. 3
Incompleteness, Complexity, Randomness and Beyond
, 2001
"... The Library is composed of an... infinite number of hexagonal galleries... [it] includes all verbal structures, all variations permitted by the twentyfive orthographical symbols, but not a single example of absolute nonsense.... These phrases, at first glance incoherent, can no doubt be justified i ..."
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The Library is composed of an... infinite number of hexagonal galleries... [it] includes all verbal structures, all variations permitted by the twentyfive orthographical symbols, but not a single example of absolute nonsense.... These phrases, at first glance incoherent, can no doubt be justified in a cryptographical or allegorical manner; such a justification is verbal and, ex hypothesi, already figures in the Library.... The certitude that some shelf in some hexagon held precious books and that these precious books were inaccessible seemed almost intolerable. A blasphemous sect suggested that... all men should juggle letters and symbols until they constructed, by an improbable gift of chance, these canonical books... but the Library is... useless, incorruptible, secret. Jorge Luis Borges, “The Library of Babel” Gödel’s Incompleteness Theorems have the same scientific status as Einstein’s principle of relativity, Heisenberg’s uncertainty principle, and Watson and Crick’s double helix model of DNA. Our aim is to discuss some new faces of the incompleteness phenomenon unveiled by an informationtheoretic approach to randomness and recent developments in quantum computing.
Physical unknowables
, 2008
"... Different types of physical unknowables are discussed. Provable unknowables are derived from reduction to problems which are known to be recursively unsolvable. Recent series solutions to the nbody problem and related to it, chaotic systems, may have no computable radius of convergence. Quantum unk ..."
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Different types of physical unknowables are discussed. Provable unknowables are derived from reduction to problems which are known to be recursively unsolvable. Recent series solutions to the nbody problem and related to it, chaotic systems, may have no computable radius of convergence. Quantum unknowables include the random occurrence of single events, complementarity and value indefiniteness.
The Gödel Paradox and Wittgenstein’s Reasons 1. The Implausibile Wittgenstein Wittgenstein’s notorious comments on Gödel’s First Incompleteness Theorem in the
"... Remarks on the Foundations of Mathematics were dismissed by early commentators, such as Kreisel, Anderson, Dummett, and Bernays, as an unfortunate episode in the career of a great philosopher. It appears that Wittgenstein had in his sights only the informal account of the Theorem, presented by Gödel ..."
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Remarks on the Foundations of Mathematics were dismissed by early commentators, such as Kreisel, Anderson, Dummett, and Bernays, as an unfortunate episode in the career of a great philosopher. It appears that Wittgenstein had in his sights only the informal account of the Theorem, presented by Gödel in the introduction of his celebrated 1931 paper, and was misguided by it (not that he was the only one: because of the misunderstandings it originated, Helmer said that that exposition “without any claim to complete precision”1 is the only mistake in Gödel’s paper). It is claimed that Wittgenstein erroneously considered essential the natural language interpretation of the Gödel sentence, whose undecidability within (the modified system considered by Gödel, taken from) Russell and Whitehead’s Principia mathematica is at the core of the First Theorem, as claiming “I am not provable”. On the contrary, Gödel’s proof can be framed in syntactic terms in which no extramathematical interpretation of the formulas is required. Commentators were particularly struck by the fact that Wittgenstein seems to take the Gödel formula as a paradoxical sentence, not too different from the usual Liar – and Gödel’s proof itself, therefore, as the deduction of an inconsistency: 11. Let us suppose I prove the unprovability (in Russell’s system) of P; then by this proof I have proved P. Now if this proof were one in Russell’s system – I should in this case have proved at once that it belonged and did not belong to Russell’s system. – That is what comes of making up such