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Correspondences between Classical, Intuitionistic and Uniform Provability [article]

Gopalan Nadathur
1998 arXiv   pre-print
We then study the reduction of classical and, derivatively, intuitionistic provability to uniform provability via the addition to the assumption set of the negation of the formula to be proved.  ...  Using this result, we identify the richest versions of the so-called abstract logic programming languages in classical and intuitionistic logic.  ...  This work has been supported in part by NSF Grants CCR-92-08465 and CCR-9596119.  ... 
arXiv:cs/9809015v1 fatcat:xq7fafxp7ffdhidselr2u6k6om

Correspondences between classical, intuitionistic and uniform provability

Gopalan Nadathur
2000 Theoretical Computer Science  
We then study the reduction of classical and, derivatively, intuitionistic provability to uniform provability via the addition to the assumption set of the negation of the formula to be proved.  ...  Using this result, we identify the richest versions of the socalled abstract logic programming languages in classical and intuitionistic logic.  ...  This work has been supported in part by NSF Grants CCR-92-08465 and CCR-9596119.  ... 
doi:10.1016/s0304-3975(99)00177-2 fatcat:4g6rfu7vxvblpccuyzbd74ywbu

Page 8394 of Mathematical Reviews Vol. , Issue 2000m [page]

2000 Mathematical Reviews  
Norman Danner (1-UCLA; Los Angeles, CA) 2000m:03142 03F07 03B20 68N17 Nadathur, Gopalan (1-CHI-C; Chicago, IL) Correspondences between classical, intuitionistic and uniform provability.  ...  The second question which is considered is the relationship 03 MATHEMATICAL LOGIC AND FOUNDATIONS 8394 between classical and intuitionistic provability on the one hand and uniform provability on the other  ... 

Editorial

2000 Theoretical Computer Science  
There is a long history of work in proof-search in a variety of systems of logic, including classical, intuitionistic, relevant, linear and modal systems, at the propositional, ÿrst-and higher-order levels  ...  Such work has ranged from the most abstract to the most practical and has employed the full spectrum of logical techniques, from proof theory, model theory and recursion theory.  ...  The paper Correspondences between Classical; Intuitionistic and Uniform Provability starts from an analysis of the inference rules used and provides a characterization of the cases in which classical provability  ... 
doi:10.1016/s0304-3975(99)00168-1 fatcat:pw6ussamo5dobmb6ebviodz7gu

Uniform provability realization of intuitionistic logic, modality and λ-terms

Sergei N. Artemov
1999 Electronical Notes in Theoretical Computer Science  
In addition, LP subsumes the -calculus, modal -calculus, combinatory logic and provides a uniform provability realization of modality a n d -terms.  ...  The intended meaning of intuitionistic logic is explained by the Brouwer-Heyting-Kolmogorov ( BHK) p r o vability semantics which informally de nes intuitionistic truth as provability and speci es the  ...  First of all, following Kleene ( 10] ) we distinguish between intuitionistic and classical understanding of realizability s e m a n tics for intuitionistic theories.  ... 
doi:10.1016/s1571-0661(04)00100-8 fatcat:ms423mkdgjh2lgj5oume2ouv4a

Proof-terms for classical and intuitionistic resolution

E Ritter
2000 Journal of Logic and Computation  
A translation between the classical sequent calculus and classical resolution due to Mints is used to lift the conditions to classical resolution, thereby giving a characterization of the intuitionistic  ...  of search spaces in both classical and intuitionistic logic.  ...  is intuitionistically provable if there is a weakly uniform classical sequent derivation such that the corresponding -term is intuitionistic.  ... 
doi:10.1093/logcom/10.2.173 fatcat:qe5fgbmzi5at7myw77dlxrkzhq

On the intuitionistic force of classical search

Eike Ritter, David Pym, Lincoln Wallen
2000 Theoretical Computer Science  
One successful approach is to view the non-classical logic as a perturbation on search in classical logic and characterize when a least-commitment (classical) search yields su cient evidence for provability  ...  As an application, we develop a proof procedure based on the natural extension of the notion of uniform proof to the multiple-conclusioned classical sequent calculus Harrop fragment of intuitionistic logic  ...  of proof procedures" and via an Advanced Fellowship held by Pym.  ... 
doi:10.1016/s0304-3975(99)00178-4 fatcat:bc26d3pcs5fojbmexgaed6ddbu

A uniform semantic proof for cut-elimination and completeness of various first and higher order logics

Mitsuhiro Okada
2002 Theoretical Computer Science  
, substructural, and standard logics uniformly, and for both their intuitionistic and classical versions uniformly.  ...  Sci. 50 (1987)) to intuitionistic and higher-order phase semantics.  ...  This paper is the full and revised version of the ÿrst part of Okada [20] (the extended abstract "Phase Semantics for Higher Order Completeness, Cut-Elimination and Normalization Proofs" appeared in  ... 
doi:10.1016/s0304-3975(02)00024-5 fatcat:dwlmhsceknfytalgc5pud5cuum

Uniform Provability in Classical Logic [article]

Gopalan Nadathur
1998 arXiv   pre-print
Our analysis sheds light on the relationship between these mechanisms and the notion of uniform proofs.  ...  The resulting proof system and the uniform provability property that holds of it are used to outline a proof procedure for classical logic.  ...  The distinction between the two notions of provability notwithstanding, there is a correspondence between the classical provability of a sequent of the kind being considered and the intuitionistic provability  ... 
arXiv:cs/9809014v1 fatcat:vqmbhrpjqrhhjjodqm3i2hru2m

Proof-search in type-theoretic languages: an introduction

Didier Galmiche, David J. Pym
2000 Theoretical Computer Science  
points of departure such as a deÿnition of a type-theoretic language or the relationship between languages and proof-objects.  ...  The strong connections between di erent proof-search methods in logics, type theories and logical frameworks, together with their impact on programming and implementation issues, are central in this context  ...  Acknowledgements We are grateful to Maurice Nivat for encouraging us to produce this article and for acting as a referee. Any remaining errors are our responsibility.  ... 
doi:10.1016/s0304-3975(99)00169-3 fatcat:gy4nmmsupfeype23pw3jzbrvwu

A Tutorial on Proof Theoretic Foundations of Logic Programming [chapter]

Paola Bruscoli, Alessio Guglielmi
2003 Lecture Notes in Computer Science  
logic programming is about designing logic programming languages via the proof theoretic notion of uniform provability.  ...  This tutorial intends to expose the main ideas of this discipline in the most direct and simple way.  ...  On the other hand, we believe that understanding this relatively new methodology in the old, familiar playground of classical logic is the best way of getting a feeling for it without having to invest  ... 
doi:10.1007/978-3-540-24599-5_9 fatcat:pgtdgyhgdjfnpiueiv5abwlchm

Index

2000 Theoretical Computer Science  
On the intuitionistic force of classical search (1}2) 299}333 Underwood, J., see J.L. Caldwell (1}2) 55} 90 Wallen, L., see E. Ritter (1}2) 299}333 Elsevier Science B.V.  ...  ., Correspondences between classical, intuitionistic and uniform provability (1}2) 273}298 Pfenning, F., see I. Cervesato (1}2) 133}163 Pym, D., see E. Ritter (1}2) 299}333 Pym, D.J., see D.  ...  Pym and L. Wallen, On the intuitionistic force of classical search (1}2) 299}333 Underwood, J., see J.L. Caldwell (1}2) 55} 90 Wallen, L., see E. Ritter (1}2) 299}333 Elsevier Science B.V.  ... 
doi:10.1016/s0304-3975(99)00309-6 fatcat:6gx6x6avzfbkfnjybbriiogmvm

Can Turing machines capture everything we can compute? [article]

Bhupinder Singh Anand
2003 arXiv   pre-print
If we define classical foundational concepts constructively, and introduce non-algorithmic effective methods into classical mathematics, then we can bridge the chasm between truth and provability, and  ...  We use square brackets to distinguish between the uninterpreted string [F] of a formal system, and the symbolic expression "F" that corresponds to it under a given interpretation that unambiguously assigns  ...  that, since mathematics is, indeed, indisputably accepted as the language that most effectively expresses and communicates intuitive truth, the chasm between formal truth and provability must, of necessity  ... 
arXiv:math/0304379v2 fatcat:obyu3cct25f5zjidaivx7rckqq

A uniform proof procedure for classical and non-classical logics [chapter]

Jens Otten, Christoph Kreitz
1996 Lecture Notes in Computer Science  
Our uniform path-checking algorithm operates on arbitrary (non-normal form) formulae and generalizes Bibel's connection method for classical logic and formulae in clause-form.  ...  We present a proof procedure for classical and non-classical logics.  ...  In this paper we present a proof procedure which allows a uniform treatment of classical, intuitionistic, and modal logics.  ... 
doi:10.1007/3-540-61708-6_70 fatcat:5tvjssygyrgwtmthlk73gjbuyy

A focused approach to combining logics

Chuck Liang, Dale Miller
2011 Annals of Pure and Applied Logic  
We examine certain circumstances, for example, in which a classical lemma can be used in an intuitionistic proof while preserving intuitionistic provability.  ...  Focused sequent calculi for classical, intuitionistic, and multiplicative-additive linear logics are derived as fragments of the host system by varying the sensitivity of specialized structural rules to  ...  But meaningful distinctions between classical and linear provability are sustained. Consider A (A ⊥ ⊗ A ⊥ ), which is provable if A is a classical formula but not if A is linear.  ... 
doi:10.1016/j.apal.2011.01.012 fatcat:oqd4umjx3bad3kcj2qhboxua5u
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