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PDL with intersection and converse: satisfiability and infinite-state model checking

Published online by Cambridge University Press:  12 March 2014

Stefan Göller ,
Markus Lohrey  and
Carsten Lutz
Stefan Göller
Affiliation:
Universität Leipzig, Institut für Informatik, Abteilung Algebraische und Logische Grundlagen der Informatik, Johannisgasse 26, 04103 Leipzig, Germany, E-mail: goeller@informatik.uni-leipzig.de
Markus Lohrey
Affiliation:
Universität Leipzig, Institut für Informatik, Abteilung Algebraische und Logische Grundlagen der Informatik, Johannisgasse 26, 04103 Leipzig, Germany, E-mail: lohrey@informatik.uni-stuttgart.de
Carsten Lutz
Affiliation:
Dresden University of Technology, Institute for Theoretical Computer Science, Department of Computer Science, Nöthnitzer Str. 46, 01062 Dresden, Germany, E-mail: lutz@tcs.inf.tu-dresden.de
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Abstract

We study satisfiability and infinite-state model checking in ICPDL, which extends Propositional Dynamic Logic (PDL) with intersection and converse operators on programs. The two main results of this paper are that (i) satisfiability is in 2ΕΧΡΤΙΜΕ, thus 2ΕΧΡΤΙΜΕ-complete by an existing lower bound, and (ii) infinite-state model checking of basic process algebras and pushdown systems is also 2ΕΧΡΤΙΜΕ-complete. Both upper bounds are obtained by polynomial time computable reductions to ω-regular tree satisfiability in ICPDL, a reasoning problem that we introduce specifically for this purpose. This problem is then reduced to the emptiness problem for alternating two-way automata on infinite trees. Our approach to (i) also provides a shorter and more elegant proof of Danecki's difficult result that satisfiability in IPDL is in 2ΕΧΡΤΙΜΕ. We prove the lower bound(s) for infinite-state model checking using an encoding of alternating Turing machines.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 74 , Issue 1 , March 2009 , pp. 279 - 314
Copyright
Copyright © Association for Symbolic Logic 2009

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References

REFERENCES

[1]Afanasiev, Loredana, Blackburn, Patrick, Dlmitriou, Ioanna, Gaiffe, Evan Goris Bertrand, Marx, Maarten, and de Rijke, Maarten, PDLfor ordered trees, Journal of Applied Non-Classical Logics, vol. 15 (2005), no. 2, pp. 115–135.CrossRefGoogle Scholar
[2]Alechina, Natasha, Demri, Stéphane, and de Rijke, Maarten, A modal perspective on path constraints, Journal of Logic and Computation, vol. 13 (2003), no. 6, pp. 939–956.CrossRefGoogle Scholar
[3]Alur, Rajeev, Benedikt, Michael, Etessami, Kousha, Godefroid, Patrice, Reps, Thomas W., and Yannakakis, Mihalis, Analysis of recursive state machines, ACM Transactions on Programming Languages and Systems (TOPLAS), vol. 27 (2005), no. 4, pp. 786–818.CrossRefGoogle Scholar
[4]Baader, Franz, McGuiness, Deborah L., Nardi, Daniele, and Patel-Schneider, Peter, The description logic handbook: Theory, implementation and applications, Cambridge University Press, 2003.Google Scholar
[5]Cachat, Thierry, Uniform solution of parity games on prefix-recognizable graphs, Electronic Notes in Theoretical Computer Science, vol. 68 (2002), no. 6.Google Scholar
[6]Caucal, Didier, On infinite transition graphs having a decidable monadic theory, Theoretical Computer Science, vol. 290 (2002), no. 1, pp. 79–115.CrossRefGoogle Scholar
[7]Chandra, Ashok K., Kozen, Dexter C., and Stockmeyer, Larry J., Alternation, Journal of the Association for Computing Machinery, vol. 28 (1981), no. 1, pp. 114–133.CrossRefGoogle Scholar
[8]Clarke, Edmund M., Grumberg, Orna, and Peled, Doron A., Model checking, MIT Press, 2000.Google Scholar
[9]Danecki, Ryszard, Nondeterministic propositional dynamic logic with intersection is decidable. Proceedings of the 5th Symposium on Computation Theory (Zaborow, Poland), Lecture Notes in Computer Science, no. 208, 1984, pp. 34–53.Google Scholar
[10]Danecki, Ryszard, Propositional dynamic logic with strong loop predicate, Proceedings of Mathematical Foundations of Computer Science 1984 (MFCS 1984), Lecture Notes in Computer Science, no. 176, 1984, pp. 573–581.Google Scholar
[11]Esparza, J., On the decidabilty of model checking for several mu-calculi andpetri nets, Proceedings of the 19th international Colloquium on Trees in Algebra and Programming (CAAP '94), Edinburgh, U.K. (Tison, S., editor), Lecture Notes in Computer Science, no. 787, Springer, 1994, pp. 115–129.Google Scholar
[12]Esparza, Javier, Kucera, Antonín, and Schwoon, Stefan, Model checking LTL with regular valuations for pushdown systems, Information and Computation, vol. 186 (2003), no. 2, pp. 355–376.CrossRefGoogle Scholar
[13]Fagin, R., Halpern, J. Y., Moses, Y., and Vardi, M. Y., Reasoning about knowledge, MIT Press, 1995.Google Scholar
[14]Cerro, L. Farinas del and Orlowska, E., DAL—a logic for data analysis, Theoretical Computer Science, vol. 36 (1985), no. 2–3, pp. 251–264.CrossRefGoogle Scholar
[15]Fischer, Michael J. and Ladner, Richard E., Propositional dynamic logic of regular programs, Journal of Computer and System Sciences, vol. 18 (1979), no. 2, pp. 194–211.CrossRefGoogle Scholar
[16]Flum, J., On the (infinite) model theory of fixed point logics, Models, algebras, and proofs: selected papers of the X Latin American symposium on mathematical logic held in Bogota (Caicedo, X. and Montenegro, C. H., editors), Lecture Notes in Pure and Applied Mathematics, vol. 203, Marcel Dekker, Inc., 1999, pp. 67–75.Google Scholar
[17]de giacomo, Giuseppe and Lenzerini, Maurizio, Boosting the correspondence between description logics and propositional dynamic logics, Proceedings of the 12th national conference on Artifical Intelligence (AAAI '94), 1994, pp. 205–212.Google Scholar
[18]Göller, Stefan and Lohrey, Markus, Infinite state model-checking of propositional dynamic logics, Technical Report 2006/04, Universität Stuttgart, FMI, 02 2006.CrossRefGoogle Scholar
[19]Göller, Stefan and Lohrey, Markus, Infinite state model-checking of propositional dynamic logics, Proceedings of the 20th international conference on Computer Science Logic (CSL 2006, Szeged, Hungary, Lecture Notes in Computer Science, no. 4207, Springer, 2006, pp. 349–364.Google Scholar
[20]Göller, Stefan, Lohrey, Markus, and Lutz, Casten, PDL with intersection and converse is 2 EXP-complete, Proceedings of the 10th international conference on Foundations of Software Science and Computational Structures (FoSSaCS 2007), Lecture Notes in Computer Science, no. 4423, Springer, 2007, pp. 198–212.Google Scholar
[21]Grädel, E., Guarded fixed point logics and the monadic theory of countable trees, Theoretical Computer Science, vol. 288 (2002), no. 1, pp. 129–152.CrossRefGoogle Scholar
[22]Harel, D., Recurring dominoes: making the highly undecidable highly understandable, Annals of Discrete Mathematics, vol. 24 (1985), pp. 51–72.Google Scholar
[23]Harel, David, Kozen, Dexter, and Tiuryn, Jerzy, Dynamic logic, Foundations of computing, The MIT Press, 2000.CrossRefGoogle Scholar
[24]Kozen, D., Results on the propositional μ-calculus, Automata, Languages and Programming, 9th colloquium (Nielsen, Mogens and Schmidt, Erik Meineche, editors), Lecture Notes in Computer Science, vol. 140, Springer-Verlag, 1982, pp. 348–359.CrossRefGoogle Scholar
[25]Kupferman, O., Piterman, N., and Vardi, M., Model checking linear properties of prefix-recognizable systems, Proceedings of the 14th international conference on Computer Aided Verification CAV 2002, Copenhagen, Denmark (Brinksma, Ed and Larsen, Kim Guldstrand, editors), Lecture Notes in Computer Science, vol. 2404, 2002, pp. 371–385.Google Scholar
[26]Kupferman, Orna and Vardi, Moshe Y., An automata-theoretic approach to reasoning about infinite-state systems, Proceedings of the 12th international conference on Computer Aided Verification (CAV 2000), Chicago, USA (Emerson, E. Allen and Sistla, A. Prasad, editors), Lecture Notes in Computer Science, no. 1855, Springer, 2000, pp. 36–52.Google Scholar
[27]Lange, M., Model checking propositional dynamic logic with all extras, Journal of Applied Logic, vol. 4 (2005), no. 1, pp. 39–49.CrossRefGoogle Scholar
[28]Lange, M. and Lutz, C., 2-Exp Time lower bounds for propositional dynamic logics with intersection, this Journal, vol. 70 (2005), no. 4, pp. 1072–1086.Google Scholar
[29]Lugiez, Denis and Schnoebelen, Ph., The regular viewpoint on PA-processes, Theoretical Computer Science, vol. 274 (2002), no. 1-2, pp. 89–115.CrossRefGoogle Scholar
[30]Lutz, Carsten, PDL with intersection and converse is decidable, Proceedings of the 19th international workshop on Computer Science Logic (CSL 2005), Oxford, UK (Ong, C.-H. Luke, editor), Lecture Notes in Computer Science, no. 3634, Springer, 2005, pp. 413–427.Google Scholar
[31]Lutz, Carsten and Walther, Dirk, PDL with negation of atomic programs, Journal of Applied Non-Classical Logics, vol. 15 (2005), no. 2, pp. 189–213.CrossRefGoogle Scholar
[32]Mayr, Richard, Decidability of model checking with the temporal logic EF, Theoretical Computer Science, vol. 256 (2001), no. 1-2, pp. 31–62.CrossRefGoogle Scholar
[33]Meyer, John-Jules Ch., Dynamic logic for reasoning about actions and agents, (Minker, Jack, editor), Logic-based artificial intelligence, Kluwer Academic Publishers, 2000, pp. 281–311.Google Scholar
[34]Muller, David and Schupp, Paul, Alternating automata on infinite trees, Theoretical Computer Science, vol. 54 (1987), no. 2–3, pp. 267–276.CrossRefGoogle Scholar
[35]Pratt, V. R., A near-optimal method for reasoning about action, Journal of Computer and System Sciences, vol. 20 (1980), pp. 231–254.CrossRefGoogle Scholar
[36]Rogers, H., Theory of recursive functions and effective computability, McGraw-Hill, 1968.Google Scholar
[37]Cate, B. Ten and Lutz, C., The complexity of query containment in expressive fragments of XPath 2.0, Proceedings of the 26th ACM symposium on Principles of Database Systems (PODS 2007), ACM Press, 2007.Google Scholar
[38]cate, Balder ten, The expressivity of XPath with transitive closure, Proceedings of the twenty-fifth ACM SIGACT-S1GMOD-SIGART symposium on Principles of Database Systems [PODS 2006), ACM Press, 2006, pp. 328–337.CrossRefGoogle Scholar
[39]Thomas, Wolfgang, Some perspectives of infinite-state verification, Proceedings of the 3rd international symposium on Automated Technology for Verification and Analysis (ATVA 2005), Taipei, Taiwan (Peled, Doron and Tsay, Yih-Kuen, editors), Lecture Notes in Computer Science, no. 3707, Springer, 2005, pp. 3–10.CrossRefGoogle Scholar
[40]van der Hoek, W. and Meyer, J., A complete epistemic logic for multiple agents—combining distributed and common knowledge, 1997.Google Scholar
[41]van Ditmarsch, Hans P., van der Hoek, Wiebe, and Kooi, Barteld P., Concurrent dynamic epistemic logic for MAS, Proceedings of the 2nd international joint conference on Autonomous Agents and Multiagent Systems (AAMAS 2002), ACM Press, 2003, pp. 201–208.CrossRefGoogle Scholar
[42]Vardi, M. Y. and Wolper, P., Automata theoretic techniques for modal logics of programs, Journal of Computer and System Sciences, vol. 32 (1986), no. 2, pp. 183–221.CrossRefGoogle Scholar
[43]Vardi, Moshe Y., The taming of converse: Reasoning about two-way computations, Proceedings of logics of programs, Lecture Notes in Computer Science, no. 193, Springer, 1985, pp. 413–423.CrossRefGoogle Scholar
[44]Vardi, Moshe Y., Reasoning about the past with two-way automata, Proceedings of the 25th International Colloquium on Automata, Languages and Programming (1CALP '98), Aalborg, Denmark (Larsen, Kim Guldstrand, Skyum, Sven, and Winskel, Glynn, editors), Lecture Notes in Computer Science, no. 1443, Springer, 1998, pp. 628–641.Google Scholar
[45]Walukiewicz, Igor, Model checking CTL properties of pushdown systems. Proceedings of the 20th conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2000), New Delhi, India (Kapoor, Sanjiv and Prasad, Sanjiva, editors), Lecture Notes in Computer Science, no. 1974, Springer, 2000, pp. 127–138.Google Scholar
[46]Walukiewicz, Igor, Pushdown processes: Games and model-checking, Information and Computation, vol. 164 (2001), no. 2, pp. 234–263.CrossRefGoogle Scholar
[47]Wöhrle, Stefan, Decision problems over infinite graphs: Higher-order pushdown systems and synchronized products, Dissertation, RWTH, Aachen, 2005.Google Scholar