Incompleteness of states w.r.t. traces in model checking

Cousot and Cousot introduced and studied a general past/future-time specification language, called -calculus, featuring a natural time-symmetric trace-based semantics. The standard state-based semantics of the -calculus is an abstract interpretation of its trace-based semantics, which turns out to be incomplete, that is trace-incomplete, even for finite systems. As a consequence, standard state-based model checking of the -calculus is incomplete w.r.t. trace-based model checking. This paper shows that any refinement or abstraction of the domain of sets of states induces a corresponding semantics which is still trace-incomplete for any propositional fragment of the -calculus. This derives from a number of results, one for each incomplete logical/temporal connective of the -calculus, that characterize the structure of models, i.e., transition systems, whose corresponding state-based semantics of the -calculus is trace-complete. lete.

[1]  Francesco Ranzato,et al.  An Abstract Interpretation Perspective on Linear vs. Branching Time , 2005, APLAS.

[2]  Patrick Cousot,et al.  Abstract interpretation: a unified lattice model for static analysis of programs by construction or approximation of fixpoints , 1977, POPL.

[3]  Moshe Y. Vardi.  Branching vs. Linear Time: Final Showdown , 2001, TACAS.

[4]  Patrick Cousot,et al.  Systematic design of program analysis frameworks , 1979, POPL.

[5]  M. Maidi.  The common fragment of CTL and LTL , 2000, Proceedings 41st Annual Symposium on Foundations of Computer Science.

[6]  Bernhard Steffen,et al.  Model-Checking: A Tutorial Introduction , 1999, SAS.

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

[8]  Roberto Giacobazzi,et al.  Making abstract interpretations complete , 2000, JACM.

[9]  Francesco Ranzato,et al.  On the Completeness of Model Checking , 2001, ESOP.

[10]  Charles H. Bennett,et al.  Logical reversibility of computation , 1973 .

[11]  C. Freund.  Incompleteness , 1888, The Hospital.

[12]  Patrick Cousot,et al.  Temporal abstract interpretation , 2000, POPL '00.

[13]  Francesco Ranzato,et al.  Making Abstract Model Checking Strongly Preserving , 2002, SAS.

[14]  Orna Kupferman,et al.  Relating linear and branching model checking , 1998, PROCOMET.

[15]  Monika Maidl,et al.  The Common Fragment of CTL and LTL , 2000, FOCS.

[16]  Helmut Veith,et al.  Tree-like counterexamples in model checking , 2002, Proceedings 17th Annual IEEE Symposium on Logic in Computer Science.

[17]  Roberto Giacobazzi,et al.  Incompleteness, Counterexamples, and Refinements in Abstract Model-Checking , 2001, SAS.

[18]  Edmund M. Clarke,et al.  Counterexample-guided abstraction refinement , 2003, 10th International Symposium on Temporal Representation and Reasoning, 2003 and Fourth International Conference on Temporal Logic. Proceedings..

[19]  Edmund M. Clarke,et al.  Counterexample-Guided Abstraction Refinement , 2000, CAV.

[20]  Joseph Y. Halpern,et al.  “Sometimes” and “not never” revisited: on branching versus linear time temporal logic , 1986, JACM.

[21]  Francesco Ranzato,et al.  Strong Preservation as Completeness in Abstract Interpretation , 2004, ESOP.

[22]  Edmund M. Clarke,et al.  Model Checking , 1999, Handbook of Automated Reasoning.

[23]  Edmund M. Clarke,et al.  Expressibility results for linear-time and branching-time logics , 1988, REX Workshop.

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

[25]  David A. Schmidt.  From Trace Sets to Modal-Transition Systems by Stepwise Abstract Interpretation , 2003 .

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

[27]  Roberto Giacobazzi,et al.  States vs. Traces in Model Checking by Abstract Interpretation , 2002, SAS.

[28]  Leslie Lamport,et al.  "Sometime" is sometimes "not never": on the temporal logic of programs , 1980, POPL '80.

[29]  Radha Jagadeesan,et al.  Modal Transition Systems: A Foundation for Three-Valued Program Analysis , 2001, ESOP.

[30]  Helmut Veith,et al.  Counterexample-guided abstraction refinement for symbolic model checking , 2003, JACM.

[31]  Francesco Ranzato,et al.  An Abstract Interpretation-Based Refinement Algorithm for Strong Preservation , 2005, TACAS.

[32]  Edmund M. Clarke,et al.  Model checking and abstraction , 1994, TOPL.