Correctness of an Incremental and Worst-Case Optimal Decision Procedure for Modal Logic with Eventualities

We present a simple theory explaining the construction and the correctness of an incremental and worst-case optimal decision procedure for modal logic with eventualities. The procedure gives an abstract account of important aspects of Gor\'e and Widmann's PDL prover. Starting from an input formula, the procedure grows a Pratt-style graph tableau until the tableau proves or disproves the satisfiability of the formula. The procedure provides a basis for practical provers since satisfiability and unsatisfiability of formulas can often be determined with small tableaux.

[1]  Jerzy Tiuryn,et al.  Logics of Programs , 1991, Handbook of Theoretical Computer Science, Volume B: Formal Models and Sematics.

[2]  Bernhard Beckert,et al.  Dynamic Logic , 2007, The KeY Approach.

[3]  Vaughan R. Pratt,et al.  SEMANTICAL CONSIDERATIONS ON FLOYD-HOARE LOGIC , 1976, FOCS 1976.

[4]  Rajeev Goré,et al.  Optimal Tableaux for Propositional Dynamic Logic with Converse , 2010, ArXiv.

[5]  Gert Smolka,et al.  Clausal Graph Tableaux for Hybrid Logic with Eventualities and Difference , 2010, LPAR.

[6]  Ullrich Hustadt,et al.  A Comparison of Solvers for Propositional Dynamic Logic , 2010, PAAR@IJCAR.

[7]  Giuseppe De Giacomo,et al.  Combining Deduction and Model Checking into Tableaux and Algorithms for Converse-PDL , 2000, Inf. Comput..

[8]  Richard E. Ladner,et al.  Propositional Dynamic Logic of Regular Programs , 1979, J. Comput. Syst. Sci..

[9]  Vaughan R. Pratt,et al.  A Near-Optimal Method for Reasoning about Action , 1980, J. Comput. Syst. Sci..

[10]  M. de Rijke,et al.  Modal Logic , 2001, Cambridge Tracts in Theoretical Computer Science.

[11]  Gert Smolka,et al.  Terminating Tableaux for Hybrid Logic with Eventualities , 2010, IJCAR.

[12]  Vaughan R. Pratt,et al.  Models of program logics , 1979, 20th Annual Symposium on Foundations of Computer Science (sfcs 1979).

[13]  Rajeev Goré,et al.  An Optimal On-the-Fly Tableau-Based Decision Procedure for PDL-Satisfiability , 2009, CADE.

[14]  Chang Liu,et al.  Term rewriting and all that , 2000, SOEN.

[15]  Franz Baader.  Augmenting Concept Languages by Transitive Closure of Roles: An Alternative to Terminological Cycles , 1991, IJCAI.

[16]  Zohar Manna,et al.  Logics of Programs , 1981, Lecture Notes in Computer Science.

[17]  Yde Venema,et al.  Dynamic Logic by David Harel, Dexter Kozen and Jerzy Tiuryn. The MIT Press, Cambridge, Massachusetts. Hardback: ISBN 0–262–08289–6, $50, xv + 459 pages , 2002, Theory and Practice of Logic Programming.