Methods of Cut-Elimination
" This is the first book on cut-elimination in first-order predicate logic from an algorithmic point of view. Instead of just proving the existence of cut-free proofs, it focuses on the algorithmic methods transforming proofs with arbitrary cuts to proofs with only atomic cuts (atomic cut norma...
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description | " This is the first book on cut-elimination in first-order predicate logic from an algorithmic point of view. Instead of just proving the existence of cut-free proofs, it focuses on the algorithmic methods transforming proofs with arbitrary cuts to proofs with only atomic cuts (atomic cut normal forms, so-called ACNFs). The first part investigates traditional reductive methods from the point of view of proof rewriting. Within this general framework, generalizations of Gentzen's and Sch""utte-Tait's cut-elimination methods are defined and shown terminating with ACNFs of the original proof. Moreover, a complexity theoretic comparison of Gentzen's and Tait's methods is given. The core of the book centers around the cut-elimination method CERES (cut elimination by resolution) developed by the authors. CERES is based on the resolution calculus and radically differs from the reductive cut-elimination methods. The book shows that CERES asymptotically outperforms all reductive methods based on Gentzen's cut-reduction rules. It obtains this result by heavy use of subsumption theorems in clause logic. Moreover, several applications of CERES are given (to interpolation, complexity analysis of cut-elimination, generalization of proofs, and to the analysis of real mathematical proofs). Lastly, the book demonstrates that CERES can be extended to nonclassical logics, in particular to finitely-valued logics and to G""odel logic. " |
doi_str_mv | 10.1007/978-94-007-0320-9 |
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Instead of just proving the existence of cut-free proofs, it focuses on the algorithmic methods transforming proofs with arbitrary cuts to proofs with only atomic cuts (atomic cut normal forms, so-called ACNFs). The first part investigates traditional reductive methods from the point of view of proof rewriting. Within this general framework, generalizations of Gentzen's and Sch""utte-Tait's cut-elimination methods are defined and shown terminating with ACNFs of the original proof. Moreover, a complexity theoretic comparison of Gentzen's and Tait's methods is given. The core of the book centers around the cut-elimination method CERES (cut elimination by resolution) developed by the authors. CERES is based on the resolution calculus and radically differs from the reductive cut-elimination methods. The book shows that CERES asymptotically outperforms all reductive methods based on Gentzen's cut-reduction rules. It obtains this result by heavy use of subsumption theorems in clause logic. Moreover, several applications of CERES are given (to interpolation, complexity analysis of cut-elimination, generalization of proofs, and to the analysis of real mathematical proofs). Lastly, the book demonstrates that CERES can be extended to nonclassical logics, in particular to finitely-valued logics and to G""odel logic. "</description><edition>1. Aufl.</edition><identifier>ISBN: 9789400703209</identifier><identifier>ISBN: 9400703201</identifier><identifier>ISBN: 9400703198</identifier><identifier>ISBN: 9789400703193</identifier><identifier>EISBN: 9400703201</identifier><identifier>EISBN: 9789400703209</identifier><identifier>DOI: 10.1007/978-94-007-0320-9</identifier><identifier>OCLC: 704395501</identifier><language>eng</language><publisher>Dordrecht: Springer Nature</publisher><subject>Automatic theorem proving ; Mathematical Logic and Formal Languages ; Mathematical Logic and Foundations ; Mathematics ; Mathematics and Statistics ; Mathematics. 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Moreover, several applications of CERES are given (to interpolation, complexity analysis of cut-elimination, generalization of proofs, and to the analysis of real mathematical proofs). Lastly, the book demonstrates that CERES can be extended to nonclassical logics, in particular to finitely-valued logics and to G""odel logic. "</description><subject>Automatic theorem proving</subject><subject>Mathematical Logic and Formal Languages</subject><subject>Mathematical Logic and Foundations</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Mathematics. General principles</subject><subject>Proof theory</subject><isbn>9789400703209</isbn><isbn>9400703201</isbn><isbn>9400703198</isbn><isbn>9789400703193</isbn><isbn>9400703201</isbn><isbn>9789400703209</isbn><fulltext>true</fulltext><rsrctype>book</rsrctype><creationdate>2011</creationdate><recordtype>book</recordtype><sourceid>I4C</sourceid><recordid>eNpVkMtOwzAQRY0QiFL6AbCqWCCxCB2_6yVE5SEVsUFsLce129A0LnEK4u9xCEIwm5krnTsvhE4xXGEAOVFymimWpTIDSiBTe-hYsSQ7hffRKAG_Wh2igQRGFeeAj9AoxldIwaRgmA_Q2aNrV2ERx8GP812bzapyU9amLUN9gg68qaIb_eQhermdPef32fzp7iG_nmeGccFFhilw4oAokAvlvZOceOcFsYW3gI1nHHyhLCFGCOamzKvCCCsNnlKw1GI6RJd9YxPX7iOuQtVG_V65IoR11P-OSeykZ-O2Keula3RPYdDdbzpaK6a7sjPoznHRO7ZNeNu52OrvxtbVbWMqPbvJhRCSduB5D9rS1Iug04CNaT51LwmlisAfyERTlXWpN6EOy8ZsV1FzljZNi34BxZR1Yw</recordid><startdate>2011</startdate><enddate>2011</enddate><creator>Baaz, Matthias</creator><general>Springer Nature</general><general>Springer Science + Business Media</general><general>Springer Netherlands</general><general>Springer</general><scope>I4C</scope><scope>08O</scope></search><sort><creationdate>2011</creationdate><title>Methods of Cut-Elimination</title><author>Baaz, Matthias</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-a45656-13052e02907d9ffe752fef62cbfc01af450fb9c22a664e84f9ba6c7a1830c3c13</frbrgroupid><rsrctype>books</rsrctype><prefilter>books</prefilter><language>eng</language><creationdate>2011</creationdate><topic>Automatic theorem proving</topic><topic>Mathematical Logic and Formal Languages</topic><topic>Mathematical Logic and Foundations</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Mathematics. General principles</topic><topic>Proof theory</topic><toplevel>online_resources</toplevel><creatorcontrib>Baaz, Matthias</creatorcontrib><collection>Casalini Torrossa eBook Single Purchase</collection><collection>ciando eBooks</collection></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Baaz, Matthias</au><au>Leitsch, Alexander</au><format>book</format><genre>book</genre><ristype>BOOK</ristype><btitle>Methods of Cut-Elimination</btitle><seriestitle>Trends in Logic</seriestitle><date>2011</date><risdate>2011</risdate><volume>34</volume><isbn>9789400703209</isbn><isbn>9400703201</isbn><isbn>9400703198</isbn><isbn>9789400703193</isbn><eisbn>9400703201</eisbn><eisbn>9789400703209</eisbn><abstract>" This is the first book on cut-elimination in first-order predicate logic from an algorithmic point of view. 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Moreover, several applications of CERES are given (to interpolation, complexity analysis of cut-elimination, generalization of proofs, and to the analysis of real mathematical proofs). Lastly, the book demonstrates that CERES can be extended to nonclassical logics, in particular to finitely-valued logics and to G""odel logic. "</abstract><cop>Dordrecht</cop><pub>Springer Nature</pub><doi>10.1007/978-94-007-0320-9</doi><oclcid>704395501</oclcid><tpages>285</tpages><edition>1. Aufl.</edition></addata></record> |
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subjects | Automatic theorem proving Mathematical Logic and Formal Languages Mathematical Logic and Foundations Mathematics Mathematics and Statistics Mathematics. General principles Proof theory |
title | Methods of Cut-Elimination |
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