Rule Refinement for Semantic Tableau Calculi

This paper investigates refinement techniques for semantic tableau calculi. The focus is on techniques to reduce branching in inference rules and thus allow more effective ways of carrying out deductions. We introduce an easy to apply, general principle of atomic rule refinement, which depends on a purely syntactic condition that can be easily verified. The refinement has a wide scope, for example, it is immediately applicable to inference rules associated with frame conditions of modal logics, or declarations of role properties in description logics, and it allows for routine development of hypertableau-like calculi for logics with disjunction and negation. The techniques are illustrated on Humberstone’s modal logic \({{\mathrm{K}_m}(\lnot )}\) with modal operators defined with respect to both accessibility and inaccessibility, for which two refined calculi are given.

[1]  Renate A. Schmidt,et al.  Axiomatic and Tableau-Based Reasoning for Kt(H, R) , 2014, Advances in Modal Logic.

[2]  Christoph Weidenbach,et al.  Computing Small Clause Normal Forms , 2001, Handbook of Automated Reasoning.

[3]  R. Goodstein.  FIRST-ORDER LOGIC , 1969 .

[4]  Tinko Tinchev,et al.  Modal Environment for Boolean Speculations , 1987 .

[5]  Renate A. Schmidt,et al.  A bi-intuitionistic modal logic: Foundations and automation , 2016, J. Log. Algebraic Methods Program..

[6]  Renate A. Schmidt,et al.  Using tableau to decide description logics with full role negation and identity , 2012, ACM Trans. Comput. Log..

[7]  Renate A. Schmidt,et al.  A Tableau Method for Checking Rule Admissibility in S4 , 2010, M4M.

[8]  Ullrich Hustadt,et al.  Resolution-Based Methods for Modal Logics , 2000, Log. J. IGPL.

[9]  Renate A. Schmidt,et al.  Modal Tableau Systems with Blocking and Congruence Closure , 2015, TABLEAUX.

[10]  Renate A. Schmidt,et al.  A Refined Tableau Calculus with Controlled Blocking for the Description Logic SHOI , 2013, Description Logics.

[11]  Rajeev Goré,et al.  The Tableau Workbench , 2009, Electron. Notes Theor. Comput. Sci..

[12]  Renate A. Schmidt,et al.  Automated Synthesis of Tableau Calculi , 2009, TABLEAUX.

[13]  Ullrich Hustadt,et al.  Simplification and Backjumping in Modal Tableau , 1998, TABLEAUX.

[14]  M. Fitting.  Proof Methods for Modal and Intuitionistic Logics , 1983 .

[15]  Reiner Hähnle,et al.  A-Ordered Tableaux , 1996, J. Log. Comput..

[16]  Lloyd Humberstone.  The modal logic of 'all and only' , 1987, Notre Dame J. Formal Log..

[17]  Patrick Blackburn,et al.  Hybrid languages , 1995, J. Log. Lang. Inf..

[18]  Peter Baumgartner,et al.  Hyper Tableaux , 1996, JELIA.

[19]  Boris Motik,et al.  Hypertableau Reasoning for Description Logics , 2009, J. Artif. Intell. Res..

[20]  Barbara Petit,et al.  Semantics of Typed Lambda-Calculus with Constructors , 2010, Log. Methods Comput. Sci..

[21]  Ulrike Sattler,et al.  The Complexity of Reasoning with Boolean Modal Logics , 2000, Advances in Modal Logic.

[22]  Rajeev Goré,et al.  Tableau Methods for Modal and Temporal Logics , 1999 .

[23]  Marcello D'Agostino,et al.  The Taming of the Cut. Classical Refutations with Analytic Cut , 1994, J. Log. Comput..