Refinement is complete for implementations

Modal transition systems specify sets of implementations, their refining labelled transition systems, through Larsen & Thomsen’s co-inductive notion of refinement. We demonstrate that refinement precisely captures the identification of a modal transition system with its set of implementations: refinement is reverse containment of sets of implementations. This result extends to models that combine state and event observables and is drawn from a SFP-domain whose elements are equivalence classes of modal transition systems under refinement [HJS04], and abstraction-based finite-model properties proved in this paper. As a corollary, validity checking is model checking for Hennessy-Milner formulas that characterize modal transition systems with bounded computation paths. We finally sketch how techniques developed in this paper can be used to detect inconsistencies between multiple modal transition systems and, if consistent, to verify properties of all common implementations.

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

[2]  K. Namjoshi,et al.  The existence of finite abstractions for branching time model checking , 2004, LICS 2004.

[3]  Flemming Nielson,et al.  Principles of Program Analysis , 1999, Springer Berlin Heidelberg.

[4]  Kim G. Larsen,et al.  A modal process logic , 1988, [1988] Proceedings. Third Annual Information Symposium on Logic in Computer Science.

[5]  Reinhold Heckmann,et al.  Power Domain Constructions , 1991, Sci. Comput. Program..

[6]  Marsha Chechik,et al.  Merging partial behavioural models , 2004, SIGSOFT '04/FSE-12.

[7]  Bruno Courcelle,et al.  Algebraic families of interpretations , 1976, 17th Annual Symposium on Foundations of Computer Science (sfcs 1976).

[8]  Michael Huth.  Beyond image-finiteness: labelled transition systems as a Stone space , 2004, LICS 2004.

[9]  Radha Jagadeesan,et al.  A domain equation for refinement of partial systems , 2004, Mathematical Structures in Computer Science.

[10]  Radha Jagadeesan,et al.  On the Expressiveness of 3-Valued Models , 2002, VMCAI.

[11]  Carl A. Gunter,et al.  The Mixed Powerdomain , 2008 .

[12]  Joseph A. Goguen,et al.  Initial Algebra Semantics and Continuous Algebras , 1977, J. ACM.

[13]  Joseph Sifakis,et al.  Specification and verification of concurrent systems in CESAR , 1982, Symposium on Programming.

[14]  Patrice Godefroid,et al.  Generalized Model Checking: Reasoning about Partial State Spaces , 2000, CONCUR.

[15]  Radha Jagadeesan,et al.  Model checking partial state spaces with 3-valued temporal logics , 2001 .

[16]  M. de Rijke,et al.  Model checking hybrid logics , 2003 .

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

[18]  Radha Jagadeesan,et al.  Abstraction-Based Model Checking Using Modal Transition Systems , 2001, CONCUR.

[19]  Armin Biere,et al.  Symbolic Model Checking without BDDs , 1999, TACAS.

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

[21]  B. V. Fraassen.  Singular Terms, Truth-Value Gaps, and Free Logic , 1966 .

[22]  Michael W. Mislove,et al.  Local compactness and continuous lattices , 1981 .

[23]  Joseph Sifakis,et al.  Automatic Verification Methods for Finite State Systems , 1989, Lecture Notes in Computer Science.

[24]  Edmund M. Clarke,et al.  Design and Synthesis of Synchronization Skeletons Using Branching-Time Temporal Logic , 1981, Logic of Programs.

[25]  Orna Grumberg,et al.  Abstract interpretation of reactive systems , 1997, TOPL.

[26]  Robin Milner,et al.  Algebraic laws for nondeterminism and concurrency , 1985, JACM.

[27]  Radha Jagadeesan,et al.  Automatic Abstraction Using Generalized Model Checking , 2002, CAV.

[28]  Colin Stirling.  Games and Modal Mu-Calculus , 1996, TACAS.

[29]  Pierre Wolper,et al.  Simple on-the-fly automatic verification of linear temporal logic , 1995, PSTV.

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

[31]  Thomas W. Reps,et al.  Semantic minimization of 3-valued propositional formulae , 2002, Proceedings 17th Annual IEEE Symposium on Logic in Computer Science.

[32]  S. C. Kleene,et al.  Introduction to Metamathematics , 1952 .

[33]  Paolo Baldan,et al.  Partializing Stone Spaces using SFP Domains (Extended Abstract) , 1997, TAPSOFT.

[34]  Michael Huth,et al.  Labelled transition systems as a Stone space , 2004, Log. Methods Comput. Sci..

[35]  Susan Eisenbach,et al.  Managing the Evolution of .NET Programs , 2003, FMOODS.

[36]  S. Blamey.  Partial-valued logic , 1980 .