Reasoning about The Past with Two-Way Automata

The Μ-calculus can be viewed as essentially the “ultimate” program logic, as it expressively subsumes all propositional program logics, including dynamic logics, process logics, and temporal logics. It is known that the satisfiability problem for the Μ-calculus is EXPTIME-complete. This upper bound, however, is known for a version of the logic that has only forward modalities, which express weakest preconditions, but not backward modalities, which express strongest postconditions. Our main result in this paper is an exponential time upper bound for the satisfiability problem of the Μ-calculus with both forward and backward modalities. To get this result we develop a theory of two-way alternating automata on infinite trees.

[1]  David E. Muller,et al.  Alternating Automata on Infinite Trees , 1987, Theor. Comput. Sci..

[2]  S. Safra On The Complexity of w-Automata , 1988 .

[3]  E. Allen Emerson,et al.  The complexity of tree automata and logics of programs , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

[4]  Amir Pnueli,et al.  On the synthesis of a reactive module , 1989, POPL '89.

[5]  Phokion G. Kolaitis,et al.  On the expressive power of variable-confined logics , 1996, Proceedings 11th Annual IEEE Symposium on Logic in Computer Science.

[6]  Amir Pnueli,et al.  The Glory of the Past , 1985, Logic of Programs.

[7]  Moshe Y. Vardi On the Complexity of Bounded-Variable Queries. , 1995, PODS 1995.

[8]  Edmund M. Clarke,et al.  Characterizing Correctness Properties of Parallel Programs Using Fixpoints , 1980, ICALP.

[9]  Moshe Y. Vardi Sometimes and Not Never Re-revisited: On Branching Versus Linear Time , 1998, CONCUR.

[10]  Amir Pnueli,et al.  In Transition From Global to Modular Temporal Reasoning about Programs , 1989, Logics and Models of Concurrent Systems.

[11]  Moshe Y. Vardi Alternating Automata: Unifying Truth and Validity Checking for Temporal Logics , 1997, CADE.

[12]  David E. Muller,et al.  Weak alternating automata give a simple explanation of why most temporal and dynamic logics are decidable in exponential time , 1988, [1988] Proceedings. Third Annual Information Symposium on Logic in Computer Science.

[13]  Dexter Kozen,et al.  RESULTS ON THE PROPOSITIONAL’p-CALCULUS , 2001 .

[14]  Martin Otto,et al.  Undecidability Results on Two-Variable Logics , 1997, STACS.

[15]  Orna Kupferman,et al.  Weak alternating automata and tree automata emptiness , 1998, STOC '98.

[16]  E. Allen Emerson,et al.  Tree automata, mu-calculus and determinacy , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.

[17]  Larry J. Stockmeyer,et al.  Improved upper and lower bounds for modal logics of programs , 1985, STOC '85.

[18]  Maurizio Lenzerini,et al.  Concept Language with Number Restrictions and Fixpoints, and its Relationship with Mu-calculus , 1994, ECAI.

[19]  R. Cleaveland Eecient Local Model-checking for Fragments of the Modal -calculus , 1996 .

[20]  Robert S. Streett Propositional Dynamic Logic of looping and converse , 1981, STOC '81.

[21]  David Michael Ritchie Park Finiteness is Mu-Ineffable , 1976, Theor. Comput. Sci..

[22]  Pierre Wolper,et al.  An automata-theoretic approach to branching-time model checking , 2000, JACM.

[23]  Chin-Laung Lei,et al.  Modalities for Model Checking: Branching Time Logic Strikes Back , 1987, Sci. Comput. Program..

[24]  Charanjit S. Jutla,et al.  Determinization and Memoryless Winning Strategies , 1997, Inf. Comput..

[25]  Rohit Parikh,et al.  A Decision Procedure for the Propositional µ-Calculus , 1983, Logic of Programs.

[26]  Vaughan R. Pratt,et al.  A decidable mu-calculus: Preliminary report , 1981, 22nd Annual Symposium on Foundations of Computer Science (sfcs 1981).

[27]  Giuseppe De Giacomo,et al.  Tableaux and Algorithms for Propositional Dynamic Logic with Converse , 1996, CADE.

[28]  E. Allen Emerson,et al.  An Automata Theoretic Decision Procedure for the Propositional Mu-Calculus , 1989, Inf. Comput..

[29]  Pierre Wolper,et al.  A temporal logic for reasoning about partially ordered computations (Extended Abstract) , 1984, PODC '84.

[30]  Robert S. Streett,et al.  Propositional Dynamic Logic of Looping and Converse Is Elementarily Decidable , 1982, Inf. Control..

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

[32]  A. Prasad Sistla,et al.  On Model-Checking for Fragments of µ-Calculus , 1993, CAV.

[33]  Dexter Kozen,et al.  A finite model theorem for the propositional μ-calculus , 1988, Stud Logica.

[34]  Moshe Y. Vardi A temporal fixpoint calculus , 1988, POPL '88.

[35]  Edmund M. Clarke,et al.  Symbolic Model Checking: 10^20 States and Beyond , 1990, Inf. Comput..

[36]  Giora Slutzki,et al.  Alternating Tree Automata , 1983, Theoretical Computer Science.

[37]  Maurizio Lenzerini,et al.  Description Logics with Inverse Roles, Functional Restrictions, and N-ary Relations , 1994, JELIA.

[38]  Anuj Dawary,et al.  Innnitary Logic and Inductive Deenability over Finite Structures , 1995 .

[39]  Wolfgang Thomas,et al.  Languages, Automata, and Logic , 1997, Handbook of Formal Languages.

[40]  S. Safra,et al.  On the complexity of omega -automata , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

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

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

[43]  Pierre Wolper,et al.  Automata theoretic techniques for modal logics of programs: (Extended abstract) , 1984, STOC '84.

[44]  Amir Pnueli,et al.  Applications of Temporal Logic to the Specification and Verification of Reactive Systems: A Survey of Current Trends , 1986, Current Trends in Concurrency.

[45]  Rance Cleaveland,et al.  A linear-time model-checking algorithm for the alternation-free modal mu-calculus , 1993, Formal Methods Syst. Des..

[46]  Dexter Kozen,et al.  Results on the Propositional µ-Calculus , 1982, ICALP.

[47]  E. Muller David,et al.  Alternating automata on infinite trees , 1987 .

[48]  Shmuel Safra,et al.  Exponential determinization for ω-automata with strong-fairness acceptance condition (extended abstract) , 1992, STOC '92.

[49]  Amir Pnueli,et al.  Once and for all [temporal logic] , 1995, Proceedings of Tenth Annual IEEE Symposium on Logic in Computer Science.

[50]  Michael Yoeli,et al.  Methodology and System for Practical Formal Verification of Reactive Hardware , 1994, CAV.