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Physics, Topology, Logic and Computation: A Rosetta Stone
, 2009
"... Category theory is a very general formalism, but there is a certain special way that physicists use categories which turns out to have close analogues in topology, logic and computation. A category has objects and morphisms, which represent things and ways to go between things. In physics, the objec ..."
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Category theory is a very general formalism, but there is a certain special way that physicists use categories which turns out to have close analogues in topology, logic and computation. A category has objects and morphisms, which represent things and ways to go between things. In physics, the objects are often physical systems, and the morphisms are processes turning a state of one physical system into a state of another system — perhaps
Hilbert’s Program Then and Now
, 2005
"... Hilbert’s program is, in the first instance, a proposal and a research program in the philosophy and foundations of mathematics. It was formulated in the early 1920s by German mathematician David Hilbert (1862–1943), and was pursued by him and his collaborators at the University of Göttingen and els ..."
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Hilbert’s program is, in the first instance, a proposal and a research program in the philosophy and foundations of mathematics. It was formulated in the early 1920s by German mathematician David Hilbert (1862–1943), and was pursued by him and his collaborators at the University of Göttingen and elsewhere in the 1920s
A Note On The GödelGentzen Translation
, 1998
"... . We give a variant of the GodelGentzennegative translation, and a syntactic characterization which entails conservativity result for formulas. The GodelGentzennegative translation, and its variants, of classical logic into intuitionistic logic are well known (see [5], [4], [3] and [6]). Danie ..."
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. We give a variant of the GodelGentzennegative translation, and a syntactic characterization which entails conservativity result for formulas. The GodelGentzennegative translation, and its variants, of classical logic into intuitionistic logic are well known (see [5], [4], [3] and [6]). Daniel Leivant gave a systematization of the conservative extension results, not only for predicate logic but also for mathematical theories (see [7]; see [9, 2.3] for a comprehensive exposition of the negative translation as well as conservativity results). He gave a simple syntactic characterization of certain theories and formula A, for which ` c A implies `m A (or ` i A), where ` c , ` i and `m denote derivabilities in classical, intuitionistic and minimal logics, respectively. We give here a variant of the GodelGentzennegative translation, and another syntactic characterization which entails conservativity result for formulas. We refer to Gentzen's natural deduction systems for classical...
NonConstructive Computational Mathematics
 Journal of Automated Reasoning
, 1995
"... We describe a nonconstructive extension to Primitive Recursive Arithmetic, both abstractly, and as implemented on the BoyerMoore prover. Abstractly, this extension is obtained by adding the unbounded ¯ operator applied to primitive recursive functions; doing so, one can define the Ackermann functi ..."
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We describe a nonconstructive extension to Primitive Recursive Arithmetic, both abstractly, and as implemented on the BoyerMoore prover. Abstractly, this extension is obtained by adding the unbounded ¯ operator applied to primitive recursive functions; doing so, one can define the Ackermann function and prove the consistency of Primitive Recursive Arithmetic. The implementation does not mention the ¯ operator explicitly, but has the strength to define the ¯ operator through the builtin functions EVAL$ and V&C$. x1. INTRODUCTION This paper is a mixture of theory and practice. The theory begins with the notions of constructivism and finitism in the philosophy of mathematics. As with all philosophical notions, these cannot appear directly in a mathematical theorem or a computer program, but they have been useful guides over the past hundred years to discovering mathematical results, and more recently, to designing computer implementations. Informally, a constructivist only believes in...
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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this article has appeared in Computational Logic and Proof Theory (Proc. 3
PROOF INTERPRETATIONS AND MAJORIZABILITY
"... Abstract. In the last fifteen years, the traditional proof interpretations of modified realizability and functional (dialectica) interpretation in finitetype arithmetic have been adapted by taking into account majorizability considerations. One of such adaptations, the monotone functional interpret ..."
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Abstract. In the last fifteen years, the traditional proof interpretations of modified realizability and functional (dialectica) interpretation in finitetype arithmetic have been adapted by taking into account majorizability considerations. One of such adaptations, the monotone functional interpretation of Ulrich Kohlenbach, has been at the center of a vigorous program in applied proof theory dubbed proof mining. We discuss some of the traditional and majorizability interpretations, including the recent bounded interpretations, and focus on the main theoretical techniques behind proof mining. Contents