Sufficient conditions for cut elimination with complexity analysis

Abstract Sufficient conditions for first-order-based sequent calculi to admit cut elimination by a Schutte–Tait style cut elimination proof are established. The worst case complexity of the cut elimination is analysed. The obtained upper bound is parameterized by a quantity related to the calculus. The conditions are general enough to be satisfied by a wide class of sequent calculi encompassing, among others, some sequent calculi presentations for the first order and the propositional versions of classical and intuitionistic logic, classical and intuitionistic modal logic S4, and classical and intuitionistic linear logic and some of its fragments. Moreover the conditions are such that there is an algorithm for checking if they are satisfied by a sequent calculus.

[1]  Helmut Schwichtenberg,et al.  Basic proof theory, Second Edition , 2000 .

[2]  Patrick Lincoln,et al.  Linear logic , 1992, SIGA.

[3]  Jon Barwise,et al.  The Syntax and Semantics of Infinitary Languages , 1968 .

[4]  G. Gentzen.  Untersuchungen über das logische Schließen. I , 1935 .

[5]  Alexander Leitsch,et al.  Cut Normal Forms and Proof Complexity , 1999, Ann. Pure Appl. Log..

[6]  Albert Visser.  The unprovability of small inconsistency , 1993, Arch. Math. Log..

[7]  Wenhui Zhang.  Cut Elimination and Automatic Proof Procedures , 1991, Theor. Comput. Sci..

[8]  William W. Tait,et al.  Normal derivability in classical logic , 1968 .

[9]  P. Pudlák.  Chapter VIII - The Lengths of Proofs , 1998 .

[10]  Grigori Mints,et al.  Indexed systems of sequents and cut-elimination , 1997, J. Philos. Log..

[11]  Arnon Avron,et al.  Non-deterministic Multiple-valued Structures , 2005, J. Log. Comput..

[12]  Greg Restall,et al.  An Introduction to Substructural Logics , 2000 .

[13]  Agata Ciabattoni,et al.  Automated Generation of Analytic Calculi for Logics with Linearity , 2004, CSL.

[14]  Kazushige Terui,et al.  Towards a Semantic Characterization of Cut-Elimination , 2006, Stud Logica.

[15]  Kazushige Terui,et al.  Which structural rules admit cut elimination? An algebraic criterion , 2005, Journal of Symbolic Logic.

[16]  Philipp Gerhardy.  The Role of Quantifier Alternations in Cut Elimination , 2005, Notre Dame J. Formal Log..

[17]  G. A. Miller,et al.  MATHEMATISCHE ZEITSCHRIFT. , 1920, Science.

[18]  Elaine Pimentel,et al.  Using Linear Logic to Reason about Sequent Systems , 2002, TABLEAUX.

[19]  Richard Statman,et al.  Intuitionistic Propositional Logic is Polynomial-Space Complete , 1979, Theor. Comput. Sci..

[20]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[21]  J. Lambek.  The Mathematics of Sentence Structure , 1958 .

[22]  Wenhui Zhang.  Depth of Proofs, Depth of Cut-Formulas and Complexity of Cut Formulas , 1994, Theor. Comput. Sci..

[23]  Samuel R. Buss,et al.  Chapter I - An Introduction to Proof Theory , 1998 .

[24]  Helmut Schwichtenberg,et al.  Basic proof theory , 1996, Cambridge tracts in theoretical computer science.

[25]  Jiirg Hudelmaier,et al.  Bounds for cut elimination in intuitionistic propositional logic , 1992, Arch. Math. Log..

[26]  Sara Negri,et al.  Structural proof theory , 2001 .

[27]  Stephen A. Cook,et al.  The Relative Efficiency of Propositional Proof Systems , 1979, Journal of Symbolic Logic.

[28]  Cristina Sernadas,et al.  Modal Sequent Calculi Labelled with Truth Values: Cut Elimination , 2005, Log. J. IGPL.

[29]  Roy Dyckhoff,et al.  Contraction-free sequent calculi for intuitionistic logic , 1992, Journal of Symbolic Logic.

[30]  Alessandra Carbone,et al.  Duplication of Directed Graphs and Exponential Blow Up of Proofs , 1999, Ann. Pure Appl. Log..

[31]  Alexander Leitsch,et al.  Cut-elimination and Redundancy-elimination by Resolution , 2000, J. Symb. Comput..

[32]  S. Buss.  Handbook of proof theory , 1998 .

[33]  M. Nivat.  Fiftieth volume of theoretical computer science , 1988 .

[34]  Nuel Belnap,et al.  Display logic , 1982, J. Philos. Log..

[35]  Alexander Leitsch,et al.  Comparing the Complexity of Cut-Elimination Methods , 2001, Proof Theory in Computer Science.

[36]  Lutz Straßburger,et al.  Non-commutativity and MELL in the Calculus of Structures , 2001, CSL.

[37]  Arnon Avron,et al.  Canonical Propositional Gentzen-Type Systems , 2001, IJCAR.