Mechanised Reasoning and Model Generation for Extended Modal Logics

The approach presented in this overview paper exploits that modal logics can be seen to be fragments of first-order logic and deductive methods can be developed and studied within the framework of first-order resolution. We focus on a class of extended modal logics very similar in spirit to propositional dynamic logic and closely related to description logics. We review and discuss the development of decision procedures for decidable extended modal logics and look at methods for automatically generating models.

[1]  Patrice Enjalbert,et al.  Modal Theorem Proving: An Equational Viewpoint , 1989, IJCAI.

[2]  Andreas Herzig.  Raisonnement automatique en logique modale et algorithmes d'unification , 1989 .

[3]  Joseph Douglas Horton,et al.  Merge Path Improvements for Minimal Model Hyper Tableaux , 1999, TABLEAUX.

[4]  Christoph Weidenbach,et al.  Optimised Functional Translation and Resolution , 1998 .

[5]  Willard Van Orman Quine,et al.  Algebraic Logic and Predicate Functors , 1971 .

[6]  Alexander Leitsch,et al.  The Resolution Calculus , 1997, Texts in Theoretical Computer Science An EATCS Series.

[7]  Ullrich Hustadt,et al.  Issues of Decidability for Description Logics in the Framework of Resolution , 1998, FTP.

[8]  Patrick Brézillon,et al.  Lecture Notes in Artificial Intelligence , 1999 .

[9]  George Gargov,et al.  A Note on Boolean Modal Logic , 1990 .

[10]  Dov M. Gabbay,et al.  Labelled deduction , 2000 .

[11]  Harald Ganzinger,et al.  A Resolution-Based Decision Procedure for Extensions of K4 , 1998, Advances in Modal Logic.

[12]  Hans Jürgen Ohlbach.  Combining Hilbert Style and Semantic Reasoning in a Resolution Framework , 1998, CADE.

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

[14]  Roy Dyckhoff.  Automated Reasoning with Analytic Tableaux and Related Methods , 2000, Lecture Notes in Computer Science.

[15]  Henrik Sahlqvist.  Completeness and Correspondence in the First and Second Order Semantics for Modal Logic , 1975 .

[16]  András Simon,et al.  Undecidable Varieties of Semilattice - ordered Semigroups, of Boolean Algebras with Operators, and logics extending Lambek Calculus , 1993, Log. J. IGPL.

[17]  Harald Ganzinger.  Proceedings of the 16th International Conference on Automated Deduction (CADE-16) , 1999 .

[18]  Alan Robinson,et al.  Handbook of automated reasoning , 2001 .

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

[20]  Hans de Nivelle.  Splitting Through New Proposition Symbols , 2001, LPAR.

[21]  Stéphane Demri.  A Hierarchy of Backward Translations: Applications to Modal Logics , 1995, WOCFAI.

[22]  Ullrich Hustadt,et al.  Computational Space Efficiency and Minimal Model Generation for Guarded Formulae , 2001, LPAR.

[23]  Dov M. Gabbay,et al.  Logic, Language, and Reasoning Essays in Honour of Dov Gabbay , 1999 .

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

[25]  Ullrich Hustadt,et al.  Hyperresolution for guarded formulae , 2003, J. Symb. Comput..

[26]  Ullrich Hustadt,et al.  Using Resolution for Testing Modal Satisfiability and Building Models , 2002, Journal of Automated Reasoning.

[27]  David A. McAllester,et al.  Automated Deduction - CADE-17 , 2000, Lecture Notes in Computer Science.

[28]  Miyuki Koshimura,et al.  Efficient Minimal Model Generation Using Branching Lemmas , 2000, CADE.

[29]  Stephan Schulz.  System Abstract: E 0.61 , 2001, IJCAR.

[30]  Andreas Nonnengart,et al.  First-Order Modal Logic Theorem Proving and Functional Simulation , 1993, IJCAI.

[31]  William McCune,et al.  Automated Deduction—CADE-14 , 1997, Lecture Notes in Computer Science.

[32]  Renate A. Schmidt.  Decidability by Resolution for Propositional Modal Logics , 2004, Journal of Automated Reasoning.

[33]  Andrei Voronkov,et al.  Splitting Without Backtracking , 2001, IJCAI.

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

[35]  Jaime G. Carbonell,et al.  Automated Deduction — CADE-16 , 2002, Lecture Notes in Computer Science.

[36]  Stephan Schulz.  System Abstract: E 0.3 , 1999, CADE.

[37]  Christian G. Fermüller,et al.  Resolution Decision Procedures , 2001, Handbook of Automated Reasoning.

[38]  Stephan Schulz,et al.  E - a brainiac theorem prover , 2002, AI Commun..

[39]  Christian G. Fermüller,et al.  Resolution Methods for the Decision Problem , 1993, Lecture Notes in Computer Science.

[40]  Christoph Weidenbach,et al.  Soft Typing for Ordered Resolution , 1997, CADE.

[41]  F. Massacci.  Single Step Tableaux for Modal Logics Computational Properties, Complexity and Methodology , 2000 .

[42]  Johan van Benthem,et al.  Halldén-completeness by gluing of Kripke frames , 1983, Notre Dame J. Formal Log..

[43]  Andrei Voronkov,et al.  Automated Deduction—CADE-18 , 2002, Lecture Notes in Computer Science.

[44]  Ullrich Hustadt,et al.  MSPASS: Modal Reasoning by Translation and First-Order Resolution , 2000, TABLEAUX.

[45]  Thierry Boy de la Tour.  An Optimality Result for Clause Form Translation , 1992, J. Symb. Comput..

[46]  Matteo Baldoni,et al.  Normal multimodal logics with interaction axioms , 2000 .

[47]  Hans Jürgen.  Semantics-Based Translation Methods for Modal Logics , 1991 .

[48]  Thierry Bajou,et al.  De La Tour , 1985 .

[49]  Fabio Massacci,et al.  Single Step Tableaux for Modal Logics , 2000, Journal of Automated Reasoning.

[50]  Laurent Catach.  Normal Multimodal Logics , 1988, AAAI.

[51]  Vincent Danos,et al.  The Structure of Exponentials: Uncovering the Dynamics of Linear Logic Proofs , 1993, Kurt Gödel Colloquium.

[52]  M. de Rijke,et al.  Encoding Two-Valued Nonclassical Logics in Classical Logic , 2001, Handbook of Automated Reasoning.

[53]  David A. Plaisted,et al.  A Structure-Preserving Clause Form Translation , 1986, J. Symb. Comput..

[54]  Stéphane Demri,et al.  Deciding Regular Grammar Logics with Converse Through First-Order Logic , 2003, J. Log. Lang. Inf..

[55]  Stéphane Demri,et al.  Cooperation between Direct Method and Translation Method in Non Classical Logics: Some Results in Propositional S5 , 1993, IJCAI.

[56]  Fausto Giunchiglia,et al.  Sat vs. Translation Based Decision Procedures for Modal Logics: a Comparative Evaluation Istituto per La Ricerca Scientifica E Tecnologica Sat vs. Translation Based Decision Procedures for Modal Logics: a Comparative Evaluation , 1998 .

[57]  Hans Jürgen Ohlbach,et al.  Translation Methods for Non-Classical Logics: An Overview , 1993, Log. J. IGPL.

[58]  Christoph Weidenbach,et al.  Combining Superposition, Sorts and Splitting , 2001, Handbook of Automated Reasoning.

[59]  Ilkka Niemelä.  A Tableau Calculus for Minimal Model Reasoning , 1996, TABLEAUX.

[60]  Ullrich Hustadt,et al.  An empirical analysis of modal theorem provers , 1999, J. Appl. Non Class. Logics.

[61]  Ullrich Hustadt.  Resolution based decision procedures for subclasses of first-order logic , 1999 .

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

[63]  Johan van Benthem,et al.  Modal Languages and Bounded Fragments of Predicate Logic , 1998, J. Philos. Log..

[64]  Ulrike Sattler,et al.  A Suggestion for an n-ary Description Logic , 1999, Description Logics.

[65]  Ullrich Hustadt,et al.  On the relationship between decidable fragments, non-classical logics, and description logics , 2002, Description Logics.

[66]  J. Gustafsson.  An Implementation and Optimization of an Algorithm for Reducing Formulae in Second-Order Logic , 1996 .

[67]  Ullrich Hustadt,et al.  A Principle for Incorporating Axioms into the First-Order Translation of Modal Formulae , 2003, CADE.

[68]  Ullrich Hustadt,et al.  A Resolution Decision Procedure for Fluted Logic , 2000, CADE.

[69]  François Bry,et al.  Positive Unit Hyperresolution Tableaux and Their Application to Minimal Model Generation , 2004, Journal of Automated Reasoning.

[70]  Lloyd Humberstone.  Inaccessible worlds , 1983, Notre Dame J. Formal Log..

[71]  Harald Ganzinger,et al.  Resolution Theorem Proving , 2001, Handbook of Automated Reasoning.

[72]  Frank Wolter,et al.  Monodic fragments of first-order temporal logics: 2000-2001 A.D , 2001, LPAR.

[73]  Tinko Tinchev,et al.  PDL with Data Constants , 1985, Inf. Process. Lett..

[74]  Phokion G. Kolaitis,et al.  On the Decision Problem for Two-Variable First-Order Logic , 1997, Bulletin of Symbolic Logic.

[75]  Ian Green,et al.  System Description: Proof Planning in Higher-Order Logic with Lambda-Clam , 1998, CADE.

[76]  Elmar Eder,et al.  Relative complexities of first order calculi , 1992, Artificial intelligence = Künstliche Intelligenz.

[77]  Harald Ganzinger,et al.  A superposition decision procedure for the guarded fragment with equality , 1999, Proceedings. 14th Symposium on Logic in Computer Science (Cat. No. PR00158).

[78]  Harald Ganzinger,et al.  Superposition with Simplification as a Desision Procedure for the Monadic Class with Equality , 1993, Kurt Gödel Colloquium.

[79]  Harald Ganzinger,et al.  Rewrite-Based Equational Theorem Proving with Selection and Simplification , 1994, J. Log. Comput..

[80]  Dov M. Gabbay,et al.  Quantifier Elimination in Second-Order Predicate Logic , 1992, KR.

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

[82]  Ullrich Hustadt,et al.  A New Clausal Class Decidable by Hyperresolution , 2002, CADE.

[83]  Maarten de Rijke,et al.  Tree-based Heuristics in Modal Theorem Proving , 2000, ECAI.

[84]  Jacques D. Fleuriot,et al.  IsaPlanner: A Prototype Proof Planner in Isabelle , 2003, CADE.

[85]  Christoph Weidenbach,et al.  On Generating Small Clause Normal Forms , 1998, CADE.

[86]  Maarten de Rijke,et al.  Deciding the guarded fragments by resolution , 2003, J. Symb. Comput..

[87]  Renate A. Schmidt,et al.  Functional Translation and Second-Order Frame Properties of Modal Logics , 1997, J. Log. Comput..

[88]  Gernot Salzer,et al.  Automated Deduction in Classical and Non-Classical Logics , 2002, Lecture Notes in Computer Science.

[89]  Peter F. Patel-Schneider,et al.  FaCT and DLP , 1998, TABLEAUX.

[90]  Patrick Doherty,et al.  Computing Circumscription Revisited: A Reduction Algorithm , 1997, Journal of Automated Reasoning.

[91]  Ullrich Hustadt,et al.  Maslov's Class K Revisited , 1999, CADE.

[92]  Ullrich Hustadt,et al.  On the Relation of Resolution and Tableaux Proof Systems for Description Logics , 1999, IJCAI.