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Characteristic formulae for fixed-point semantics: a general framework

Published online by Cambridge University Press:  28 February 2012

LUCA ACETO ,
ANNA INGOLFSDOTTIR ,
PAUL BLAIN LEVY  and
JOSHUA SACK
LUCA ACETO
Affiliation:
School of Computer Science, Reykjavik University, IS-101 Reykjavík, Iceland Email: luca@ru.is, annai@ru.is, joshua@ru.is
ANNA INGOLFSDOTTIR
Affiliation:
School of Computer Science, Reykjavik University, IS-101 Reykjavík, Iceland Email: luca@ru.is, annai@ru.is, joshua@ru.is
PAUL BLAIN LEVY
Affiliation:
University of Birmingham, Birmingham B15 2TT, UK Email: pbl@cs.bham.ac.uk
JOSHUA SACK
Affiliation:
School of Computer Science, Reykjavik University, IS-101 Reykjavík, Iceland Email: luca@ru.is, annai@ru.is, joshua@ru.is
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Abstract

The concurrency theory literature offers a wealth of examples of characteristic-formula constructions for various behavioural relations over finite labelled transition systems and Kripke structures that are defined in terms of fixed points of suitable functions. Such constructions and their proofs of correctness have been developed independently, but have a common underlying structure. This paper provides a general view of characteristic formulae that are expressed in terms of logics that have a facility for the recursive definition of formulae. We show how several examples of characteristic-formula constructions in the literature can be recovered as instances of the proposed general framework, and how the framework can be used to yield novel constructions. The paper also offers general results pertaining to the definition of co-characteristic formulae and of characteristic formulae expressed in terms of infinitary modal logics.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 22 , Special Issue 2: Express '09, Ice '09 and Sos '09 , April 2012 , pp. 125 - 173
Copyright
Copyright © Cambridge University Press 2012

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References

Aceto, L. and Hennessy, M. (1992) Termination, deadlock, and divergence. Journal of the ACM 39 (1)147–187.CrossRefGoogle Scholar
Aceto, L., Ingolfsdottir, A., Larsen, K. G. and Srba, J. (2007) Reactive Systems: Modelling, Specification and Verification, Cambridge University Press.CrossRefGoogle Scholar
Aceto, L., Ingolfsdottir, A., Pedersen, M. L. and Poulsen, J. (2000) Characteristic formulae for timed automata. RAIRO, Theoretical Informatics and Applications 34 (6)565–584.CrossRefGoogle Scholar
Barwise, J. and Moss, L. (1996) Vicious Circles: On the mathematics of non-wellfounded phenomena, CSLI Lecture Notes 60, CSLI Publications.Google Scholar
Barwise, J. and Moss, L. S. (1998) Modal correspondence for models. J. Philos. Logic 27 (3)275–294.CrossRefGoogle Scholar
Bloom, B., Istrail, S. and Meyer, A. (1995) Bisimulation can't be traced. Journal of the ACM 42 (1)232–268.CrossRefGoogle Scholar
Boudol, G. and Larsen, K. G. (1992) Graphical versus logical specifications. Theoretical Computer Science 106 (1)3–20.Google Scholar
Browne, M. C., Clarke, E. M. and Grümberg, O. (1988) Characterizing finite Kripke structures in propositional temporal logic. Theoretical Computer Science 59 (1-2)115–131.Google Scholar
Clarke, E. C., Emerson, E. A. and Sistla, A. P. (1986) Automatic verification of finite-state concurrent systems using temporal logic specifications. ACM Transactions on Programming Languages and Systems 8 (2)244–263.Google Scholar
Clarke, E., Gruemberg, O. and Peled, D. (1999) Model Checking, MIT Press.Google Scholar
Clarke, E. M. and Emerson, E. A. (1981) Design and synthesis of synchronization skeletons using branching time temporal logic. In: Kozen, D. (ed.) Proceedings of the Workshop on Logics of Programs. Springer-Verlag Lecture Notes in Computer Science 131 52–71.CrossRefGoogle Scholar
Cleaveland, R. and Steffen, B. (1991) Computing behavioural relations, logically. In: Albert, J. L., Monien, B. and Rodríguez, M. (eds.) Proceedings 18th ICALP, Madrid. Springer-Verlag Lecture Notes in Computer Science 510 127–138.CrossRefGoogle Scholar
Corradini, F., De Nicola, R. and Labella, A. (1999) Graded modalities and resource bisimulation. In: Rangan, C. P., Raman, V. and Ramanujam, R. (eds.) Proceedings Foundations of Software Technology and Theoretical Computer Science, 19th Conference, Chennai, India. Springer-Verlag Lecture Notes in Computer Science 1738 381–393.Google Scholar
">De Nicola, R., Montanari, U. and Vaandrager, F. (1990) Back and forth bisimulations. In: Proceedings CONCUR' 90 (Amsterdam, 1990). Springer-Verlag Lecture Notes in Computer Science 458 152–165.CrossRefGoogle Scholar
">De Nicola, R. and Vaandrager, F. W. (1995) Three logics for branching bisimulation. Journal of the ACM 32 (2)458–487.CrossRefGoogle Scholar
">de Rijke, M. (2000) A note on graded modal logic. Studia Logica 64 (2)271–283.Google Scholar
Dechesne, F., Mousavi, M. R. and Orzan, S. (2007) Operational and epistemic approaches to protocol anlaysis: Bridging the gap. In: Proceedings of the 14th International Conference on Logic for Programming Artificial Intelligence and Reasoning (LPAR'07). Springer-Verlag Lecture Notes in Artificial Intelligence 4790 226–241.Google Scholar
Ebbinghaus, H.-D., Flum, J. and Thomas, W. (1994) Mathematical Logic, second edition, Undergraduate Texts in Mathematics, Springer-Verlag. (Translated from the German by Margit Meßmer.)CrossRefGoogle Scholar
Emerson, E. A. (1990) Temporal and modal logic. In: Handbook of Theoretical Computer Science, Vol. B, Elsevier 995–1072.Google Scholar
Fábregas, I., de Frutos-Escrig, D. and Palomino, M. (2009) Non-strongly stable orders also define interesting simulation relations. In: Kurz, A., Lenisa, M. and Tarlecki, A. (eds.) Proceedings: Algebra and Coalgebra in Computer Science, Third International Conference, CALCO 2009, Udine, Italy. Springer-Verlag Lecture Notes in Computer Science 5728 221–235.CrossRefGoogle Scholar
Fecher, H. and Steffen, M. (2005) Characteristic μ-calculus formula for an underspecified transition system. In: EXPRESS'04. Electronic Notes in Theoretical Computer Science 128 103–116.CrossRefGoogle Scholar
van Glabbeek, R. (1987) Bounded nondeterminism and the approximation induction principle in process algebra. In: Brandenburg, F.-J., Vidal-Naquet, G. and Wirsing, M. (eds.) Proceedings: STACS 87, 4th Annual Symposium on Theoretical Aspects of Computer Science, Passau, Germany. Springer-Verlag Lecture Notes in Computer Science 247 336–347.CrossRefGoogle Scholar
van Glabbeek, R. (2001) The linear time–branching time spectrum. I. The semantics of concrete, sequential processes. In: Bergstra, J., Ponse, A. and Smolka, S. A. (eds.) Handbook of Process Algebra, Elsevier 3–99.CrossRefGoogle Scholar
van Glabbeek, R. and Weijland, W. P. (1996) Branching time and abstraction in bisimulation semantics. Journal of the ACM 43 (3)555–600.CrossRefGoogle Scholar
Graf, S. and Sifakis, J. (1986) A modal characterization of observational congruence on finite terms of CCS. Information and Control 68 (1-3)125–145.Google Scholar
Groote, J. F. and Vaandrager, F. W. (1992) Structured operational semantics and bisimulation as a congruence. Information and Computation 100 (2)202–260.Google Scholar
Hennessy, M. and Milner, R. (1985) Algebraic laws for nondeterminism and concurrency. Journal of the ACM 32 (1)137–161.Google Scholar
Ingolfsdottir, A., Godskesen, J. C. and Zeeberg, M. (1987) Fra Hennessy–Milner logik til CCS-processer. Master's thesis, Department of Computer Science, Aalborg University. (In Danish.)Google Scholar
Keller, R. M. (1976) Formal verification of parallel programs. Communications of the ACM 19 (7)371–384.CrossRefGoogle Scholar
Korver, H. (1991) Computing distinguishing formulas for branching bisimulation. In: Larsen, K. G. and Skou, A. (eds.) Proceedings of the Third Workshop on Computer Aided Verification, Aalborg, Denmark. Springer-Verlag Lecture Notes in Computer Science 575 13–23.CrossRefGoogle Scholar
Kozen, D. (1983) Results on the propositional mu-calculus. Theoretical Computer Science 27 333–354.CrossRefGoogle Scholar
Kucera, A. and Schnoebelen, Ph. (2006) A general approach to comparing infinite-state systems with their finite-state specifications. Theoretical Computer Science 358 (2-3)315–333.CrossRefGoogle Scholar
Laroussinie, F., Larsen, K. G. and Weise, C. (1995) From timed automata to logic – and back. In: Wiedermann, J. and Hájek, P. (eds.) Mathematical Foundations of Computer Science 1995, 20th International Symposium, Prague, Czech Republic. Springer-Verlag Lecture Notes in Computer Science 969 529–539.Google Scholar
Larsen, K. G. (1990) Proof systems for satisfiability in Hennessy–Milner logic with recursion. Theoretical Computer Science 72 (2-3)265–288.CrossRefGoogle Scholar
Larsen, K. G. and Skou, A. (1991) Bisimulation through probabilistic testing. Information and Computation 94 (1)1–28.Google Scholar
Larsen, K. G. and Skou, A. (1992) Compositional verification of probabilistic processes. In: Cleaveland, R. (ed.) Proceedings CONCUR 92, Stony Brook, N.Y., U.S.A. Springer-Verlag Lecture Notes in Computer Science 630 456–471.Google Scholar
Lassen, S. B. (1998) Relational Reasoning about Functions and Nondeterminism, Ph.D. thesis, University of Aarhus.Google Scholar
Levy, P. B. (2009) Boolean precongruences.Google Scholar
Margaria, T. and Steffen, B. (1993) Distinguishing formulas for free. In: Proceedings EDAC–EUROASIC'93: IEEE European Design Automation Conference, Paris, France, IEEE Computer Society Press.Google Scholar
Milner, R. (1981) A modal characterisation of observable machine behaviour. In: Astesiano, E. and Böhm, C. (eds.) CAAP '81: Trees in Algebra and Programming, 6th Colloquium. Springer-Verlag Lecture Notes in Computer Science 112 25–34.CrossRefGoogle Scholar
Milner, R. (1989) Communication and Concurrency, Prentice-Hall International.Google Scholar
Moran, A. K. (1998) Call-by-name, Call-by-need, and McCarthy's Amb, Ph.D. thesis, Department of Computing Science, Chalmers University of Technology and University of Gothenburg, Gothenburg, Sweden.Google Scholar
Moss, L. S. (2007) Finite models constructed from canonical formulas. J. Philos. Logic 36 (6)605–640.CrossRefGoogle Scholar
Müller-Olm, M. (1998) Derivation of characteristic formulae. In: MFCS'98 Workshop on Concurrency, Brno, Czech Republic. Electronic Notes in Theoretical Computer Science 18.Google Scholar
Park, D. (1981) Concurrency and automata on infinite sequences. In: Deussen, P. (ed.) 5th GI Conference, Karlsruhe, Germany. Springer-Verlag Lecture Notes in Computer Science 104 167–183.CrossRefGoogle Scholar
Pitcher, C. (2001) Functional Programming and Erratic Non-Determinism, Ph.D. thesis, Oxford University.Google Scholar
Plotkin, G. D. (2004) A structural approach to operational semantics. Journal of Logic and Algebraic Programming (60-61) 17–139.Google Scholar
Pnueli, A. (1997) The temporal logic of programs. In: Proceedings 18th Annual Symposium on Foundations of Computer Science, IEEE 46–57.Google Scholar
Queille, J. P. and Sifakis, J. (1981) Specification and verification of concurrent systems in Cesar. In: Proceedings of the 5th International Symposium on Programming. Springer-Verlag Lecture Notes in Computer Science 137 337–351.Google Scholar
Steffen, B. and Ingolfsdottir, A. (1994) Characteristic formulae for processes with divergence. Information and Computation 110 (1)149–163.Google Scholar
Stirling, C. (1987) Modal logics for communicating systems. In: Twelfth international colloquium on automata, languages and programming. Theoretical Computer Science 49 (2-3)311–347.Google Scholar
Tarski, A. (1955) A lattice-theoretical fixpoint theorem and its applications. Pacific Journal of Mathematics 5 285–309.Google Scholar
Thomas, W. (1993) On the Ehrenfeucht–Fraïssé game in theoretical computer science. In: Gaudel, M.-C. and Jouannaud, J.-P. (eds.) Proceedings TAPSOFT. Springer-Verlag Lecture Notes in Computer Science 668 559–568.Google Scholar
Thomsen, B. (1987) An extended bisimulation induced by a preorder on actions. Master's thesis, Aalborg University Centre.Google Scholar
Ulidowski, I. (1992) Equivalences on observable processes. In: Scedrov, A. (ed.) Proceedings of the 7th Annual IEEE Symposium on Logic in Computer Science, IEEE Computer Society Press 148–161.Google Scholar