Abstract
Overview. Multi agent systems (MAS) provide an important framework for formalizing various problems in computer science, artificial intelligence, game theory, social choice theory, etc. Modal logics are amongst the most suitable and versatile logical formalisms for specification and verification of computational systems. Here, we present an overview of some important developments in the area. We introduce modal logics used for specification of temporal, epistemic, and strategic properties of systems; then, we present some model checking algorithms, and discuss the computational complexity of the model checking problem. Finally, we consider symbolic (compact) representations of systems based on Binary Decision Diagrams (BDD) and propositional logic formulas, and show how the representations change the algorithmic side of model checking. We also discuss other techniques that help to reduce the complexity and make the verification feasible even for large systems.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Ågotnes, T.: Action and knowledge in alternating-time temporal logic. Synthese 149(2), 377–409 (2006); Section on Knowledge, Rationality and Action
Ågotnes, T., Goranko, V., Jamroga, W.: Alternating-time temporal logics with irrevocable strategies. In: Samet, D. (ed.) Proceedings of TARK XI, pp. 15–24 (2007)
Alur, R., Henzinger, T.A., Kupferman, O.: Alternating-time Temporal Logic. In: Proceedings of the 38th Annual Symposium on Foundations of Computer Science (FOCS), pp. 100–109. IEEE Computer Society Press (1997)
Alur, R., Henzinger, T.A., Kupferman, O.: Alternating-time Temporal Logic. Journal of the ACM 49, 672–713 (2002)
Beeri, C.: On the membership problem for functional and multivalued dependencies in relational databases. ACM Transactions on Database Systems 5(3), 241–259 (1980)
Biere, A., Cimatti, A., Clarke, E., Strichman, O., Zhu, Y.: Bounded model checking. In: Highly Dependable Software. Advances in Computers, vol. 58. Academic Press (2003) (pre-print)
Blackburn, P., de Rijke, M., Venema, Y.: Modal Logic. Cambridge Tracts in Theoretical Computer Science, vol. 53. Cambridge University Press (2001)
Bulling, N.: Model checking coalition logic on implicit models is Δ3-complete. Technical Report IfI-10-02, Clausthal University of Technology (2010)
Bulling, N., Dix, J., Jamroga, W.: Model checking logics of strategic ability: Complexity. In: Dastani, M., Hindriks, K., Meyer, J.-J. (eds.) Specification and Verification of Multi-Agent Systems, pp. 125–159. Springer (2010)
Bulling, N., Jamroga, W.: What agents can probably enforce. Fundamenta Informaticae 93(1-3), 81–96 (2009)
Bulling, N., Jamroga, W.: Alternating epistemic mu-calculus. In: Proceedings of IJCAI 2011, pp. 109–114 (2011)
Bulling, N., Jamroga, W., Dix, J.: Reasoning about temporal properties of rational play. Annals of Mathematics and Artificial Intelligence 53(1-4), 51–114 (2008)
Clarke, E., Emerson, E.A., Sistla, A.P.: Automatic verification of finite state concurrent systems using temporal logic specifications: A practical approach. ACM Transactions on Programming Languages and Systems 8(2), 244–263 (1986)
Clarke, E.M., Emerson, E.A., Sistla, A.P.: Automatic verification of finite-state concurrent systems using temporal logic specifications. ACM Transactions on Programming Languages and Systems 8(2), 244–263 (1986)
Clarke, E.M., Grumberg, O., Peled, D.: Model Checking. MIT Press (1999)
Cohen, P.R., Levesque, H.J.: Intention is choice with commitment. Artificial Intelligence 42, 213–261 (1990)
Davis, M., Logemann, G., Loveland, D.: A machine program for theorem proving. Journal of the ACM 5(7), 394–397 (1962)
Davis, M., Putnam, H.: A computing procedure for quantification theory. Journal of the ACM 7(3), 201–215 (1960)
de Alfaro, L., Faella, M., Henzinger, T.A., Majumdar, R., Stoelinga, M.: Model checking discounted temporal properties. Theoretical Computer Science 345, 139–170 (2005)
Dembinski, P., Janowska, A., Janowski, P., Penczek, W., Półrola, A., Szreter, M., Woźna, B., Zbrzezny, A.: \(\surd\)erics: A Tool for Verifying Timed Automata and Estelle Specifications. In: Garavel, H., Hatcliff, J. (eds.) TACAS 2003. LNCS, vol. 2619, pp. 278–283. Springer, Heidelberg (2003)
Dima, C., Tiplea, F.L.: Model-checking atl under imperfect information and perfect recall semantics is undecidable. CoRR, abs/1102.4225 (2011)
Doyen, L., Raskin, J.-F.: Games with imperfect information: Theory and algorithms. Lecture Notes in Game Theory for Computer Scientists, pp. 185–212. Cambridge University Press (2011)
Emerson, E.A.: Temporal and modal logic. In: van Leeuwen, J. (ed.) Handbook of Theoretical Computer Science, vol. B, pp. 995–1072. Elsevier Science Publishers (1990)
Emerson, E.A., Lei, C.-L.: Efficient model checking in fragments of the propositional mu-calculus. In: Proc. of the 1st Symp. on Logic in Computer Science (LICS 1986), pp. 267–278. IEEE Computer Society (1986)
Emerson, E.A., Lei, C.-L.: Modalities for model checking: Branching time logic strikes back. Science of Computer Programming 8(3), 275–306 (1987)
Emerson, E.A., Sistla, A.P.: Deciding branching time logic. In: STOC 1984: Proceedings of the Sixteenth Annual ACM Symposium on Theory of Computing, pp. 14–24. ACM, New York (1984)
Emerson, E.A., Halpern, J.Y.: “sometimes” and ”not never” revisited: On branching versus linear time temporal logic. Journal of the ACM 33(1), 151–178 (1986)
Fagin, R., Halpern, J.Y., Moses, Y., Vardi, M.Y.: Reasoning about Knowledge. MIT Press (1995)
Fisher, M.: Temporal Logics. Kluwer (2006)
Fitting, M.: First-Order Logic and Automated Theorem Proving. Springer (1990)
Furst, M., Saxe, J.B., Sipser, M.: Parity, circuits, and the polynomial-time hierarchy. Math. Systems Theory 17, 13–27 (1984)
Goranko, V., Jamroga, W.: State and path effectivity models for logics of multi-player games. In: Proceedings of AAMAS (to appear, 2012)
Goranko, V., Jamroga, W., Turrini, P.: Strategic games and truly playable effectivity functions. In: Proceedings of AAMAS 2011, pp. 727–734 (2011)
Goranko, V., van Drimmelen, G.: Complete axiomatization and decidability of alternating-time temporal logic. Theoretical Computer Science 353(1), 93–117 (2006)
Guelev, D.P., Dima, C., Enea, C.: An alternating-time temporal logic with knowledge, perfect recall and past: axiomatisation and model-checking. Journal of Applied Non-Classical Logics 21(1), 93–131 (2011)
Halpern, J.Y.: Reasoning about knowledge: a survey. In: Gabbay, D.M., Hogger, C.J., Robinson, J.A. (eds.) The Handbook of Logic in Artificial Intelligence and Logic Programming, vol. IV, pp. 1–34. Oxford University Press (1995)
Harel, D., Kozen, D., Tiuryn, J.: Dynamic Logic. MIT Press (2000)
Harrenstein, B.P., van der Hoek, W., Meyer, J.-J., Witteveen, C.: A modal characterization of Nash equilibrium. Fundamenta Informaticae 57(2-4), 281–321 (2003)
Harrenstein, P., van der Hoek, W., Meijer, J.-J., Witteveen, C.: Subgame-perfect Nash equilibria in dynamic logic. In: Pauly, M., Baltag, A. (eds.) Proceedings of the ILLC Workshop on Logic and Games, vol. 25, pp. 29–30. University of Amsterdam (2002); Tech. Report PP-1999-25
Hawke, P.: Coordination, almost perfect information and strategic ability. In: Proceedings of LAMAS (2010)
van der Hoek, W., Wooldridge, M.: Model Checking Knowledge and Time. In: Bošnački, D., Leue, S. (eds.) SPIN 2002. LNCS, vol. 2318, pp. 95–111. Springer, Heidelberg (2002)
Huang, X., Su, K., Zhang, C.: Probabilistic alternating-time temporal logic of incomplete information and synchronous perfect recall. In: Proceedings of AAAI 2012 (to appear, 2012)
Huth, M., Ryan, M.: Logic in Computer Science: Modelling and Reasoning about Systems. Cambridge University Press (2004)
Immerman, N.: Number of quantifiers is better than number of tape cells. Journal of Computer and System Sciences 22(3), 384–406 (1981)
Jamroga, W.: Some remarks on alternating temporal epistemic logic. In: Dunin-Keplicz, B., Verbrugge, R. (eds.) Proceedings of Formal Approaches to Multi-Agent Systems (FAMAS 2003), pp. 133–140 (2003)
Jamroga, W.: A Temporal Logic for Stochastic Multi-Agent Systems. In: Bui, T.D., Ho, T.V., Ha, Q.T. (eds.) PRIMA 2008. LNCS (LNAI), vol. 5357, pp. 239–250. Springer, Heidelberg (2008)
Jamroga, W.: Easy Yet Hard: Model Checking Strategies of Agents. In: Fisher, M., Sadri, F., Thielscher, M. (eds.) CLIMA IX. LNCS, vol. 5405, pp. 1–12. Springer, Heidelberg (2009)
Jamroga, W., Ågotnes, T.: Modular interpreted systems: A preliminary report. Technical Report IfI-06-15, Clausthal University of Technology (2006)
Jamroga, W., Ågotnes, T.: Constructive knowledge: What agents can achieve under incomplete information. Journal of Applied Non-Classical Logics 17(4), 423–475 (2007)
Jamroga, W., Ågotnes, T.: Modular interpreted systems. In: Proceedings of AAMAS 2007, pp. 892–899 (2007)
Jamroga, W., Dix, J.: Do Agents Make Model Checking Explode (Computationally)? In: Pěchouček, M., Petta, P., Varga, L.Z. (eds.) CEEMAS 2005. LNCS (LNAI), vol. 3690, pp. 398–407. Springer, Heidelberg (2005)
Jamroga, W., Dix, J.: Model checking ATL ir is indeed \(\Delta_2^P\)-complete. In: Proceedings of EUMAS 2006 (2006)
Jamroga, W., Dix, J.: Model checking abilities of agents: A closer look. Theory of Computing Systems 42(3), 366–410 (2008)
Jamroga, W., van der Hoek, W.: Agents that know how to play. Fundamenta Informaticae 63(2-3), 185–219 (2004)
Jonker, G.: Feasible strategies in Alternating-time Temporal Epistemic Logic. Master thesis, University of Utrecht (2003)
Kacprzak, M., Lomuscio, A., Łasica, T., Penczek, W., Szreter, M.: Verifying Multi-agent Systems via Unbounded Model Checking. In: Hinchey, M.G., Rash, J.L., Truszkowski, W.F., Rouff, C.A. (eds.) FAABS 2004. LNCS (LNAI), vol. 3228, pp. 189–212. Springer, Heidelberg (2004)
Kacprzak, M., Lomuscio, A., Niewiadomski, A., Penczek, W., Raimondi, F., Szreter, M.: Comparing BDD and SAT based techniques for model checking Chaum’s dining cryptographers protocol. Fundamenta Informaticae 72(1-2), 215–234 (2006)
Kacprzak, M., Lomuscio, A., Penczek, W.: From bounded to unbounded model checking for temporal epistemic logic. Fundamenta Informaticae 63(2-3), 221–240 (2004)
Kacprzak, M., Lomuscio, A., Penczek, W.: From bounded to unbounded model checking for temporal epistemic logic. Fundam. Inform. 63(2-3), 221–240 (2004)
Kacprzak, M., Nabialek, W., Niewiadomski, A., Penczek, W., Pólrola, A., Szreter, M., Woźna, B., Zbrzezny, A.: VerICS 2007 - a model checker for knowledge and real-time. Fundamenta Informaticae 85(1-4), 313–328 (2008)
Kacprzak, M., Penczek, W.: Fully symbolic unbounded model checking for alternating-time temporal logic. Autonomous Agents and Multi-Agent Systems 11(1), 69–89 (2005)
Kozen, D.: Results on the propositional mu-calculus. Theoretical Computer Science 27, 333–354 (1983)
Kupferman, O., Vardi, M.Y., Wolper, P.: An automata-theoretic approach to branching-time model checking. Journal of the ACM 47(2), 312–360 (2000)
Laroussinie, F., Markey, N., Oreiby, G.: On the expressiveness and complexity of ATL. Logical Methods in Computer Science 4, 7 (2008)
Laroussinie, F., Markey, N., Schnoebelen, P.: Model Checking CTL + and FCTL is Hard. In: Honsell, F., Miculan, M. (eds.) FOSSACS 2001. LNCS, vol. 2030, pp. 318–331. Springer, Heidelberg (2001)
Leyton-Brown, K., Shoham, Y.: Essentials of Game Theory: A Concise, Multidisciplinary Introduction. Morgan & Claypool (2008)
Lichtenstein, O., Pnueli, A.: Checking that finite state concurrent programs satisfy their linear specification. In: POPL 1985: Proceedings of the 12th ACM SIGACT-SIGPLAN Symposium on Principles of Programming Languages, pp. 97–107. ACM, New York (1985)
Lomuscio, A., Penczek, W.: Logic column 19: Symbolic model checking for temporal-epistemic logics. CoRR, abs/0709.0446 (2007)
Lomuscio, A., Qu, H., Raimondi, F.: MCMAS: A Model Checker for the Verification of Multi-Agent Systems. In: Bouajjani, A., Maler, O. (eds.) CAV 2009. LNCS, vol. 5643, pp. 682–688. Springer, Heidelberg (2009)
Moskewicz, M., Madigan, C., Zhao, Y., Zhang, L., Malik, S.: Chaff: Engineering an efficient SAT solver. In: Proc. of the 38th Design Automation Conference (DAC 2001), pp. 530–535 (June 2001)
Osborne, M., Rubinstein, A.: A Course in Game Theory. MIT Press (1994)
Papadimitriou, C.H.: Computational Complexity. Addison Wesley, Reading (1994)
Pauly, M.: Logic for Social Software. PhD thesis, University of Amsterdam (2001)
Penczek, W., Lomuscio, A.: Verifying epistemic properties of multi-agent systems via bounded model checking. In: Proc. of the 2nd Int. Conf. on Autonomous Agents and Multi-Agent Systems (AAMAS 2003), pp. 209–216. ACM (2003)
Penczek, W., Lomuscio, A.: Verifying epistemic properties of multi-agent systems via bounded model checking. Fundamenta Informaticae 55(2), 167–185 (2003)
Penczek, W., Polrola, A.: Advances in Verification of Time Petri Nets and Timed Automata: A Temporal Logic Approach. SCI, vol. 20. Springer (2006)
Peterson, G., Reif, J.: Multiple-person alternation. In: Proceedings of the 20th Annual Symposium on Foundations of Computer Science (FOCS), pp. 348–363. IEEE Computer Society Press (1979)
Peterson, G., Reif, J., Azhar, S.: Lower bounds for multiplayer noncooperative games of incomplete information. Computers and Mathematics with Applications 41(7), 957–992
Pnueli, A., Rosner, R.: On the synthesis of a reactive module. In: POPL 1989: Proceedings of the 16th ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, pp. 179–190. ACM, New York (1989)
Pnueli, A., Rosner, R.: Distributed reactive systems are hard to synthesize. In: Proceedings of the 31th Annual Symposium on Foundations of Computer Science (FOCS), pp. 746–757. IEEE Computer Society Press (1990)
Raimondi, F.: Model Checking Multi-Agent Systems. PhD thesis, University College London (2006)
Raimondi, F., Lomuscio, A.: Automatic verification of deontic interpreted systems by model checking via OBDD’s. In: de Mántaras, R.L., Saitta, L. (eds.) Proceedings of ECAI, pp. 53–57 (2004)
Raimondi, F., Lomuscio, A.: Verification of multiagent systems via ordered binary decision diagrams: An algorithm and its implementation. In: AAMAS, pp. 630–637 (2004)
Rao, A.S., Georgeff, M.P.: Modeling rational agents within a BDI-architecture. In: Proceedings of the 2nd International Conference on Principles of Knowledge Representation and Reasoning, pp. 473–484 (1991)
Rosner, R.: Modular Synthesis of Reactive Systems. PhD thesis, Weizmann Institute of Science (1992)
Schnoebelen, P.: The complexity of temporal model checking. In: Advances in Modal Logics, Proceedings of AiML 2002. World Scientific (2003)
Schnoor, H.: Strategic planning for probabilistic games with incomplete information. In: Proceedings of AAMAS 2010, pp. 1057–1064 (2010)
Schobbens, P.Y.: Alternating-time logic with imperfect recall. Electronic Notes in Theoretical Computer Science 85(2), 82–93 (2004)
Sheeran, M., Singh, S., Stålmarck, G.: Checking Safety Properties Using Induction and a SAT-Solver. In: Johnson, S.D., Hunt Jr., W.A. (eds.) FMCAD 2000. LNCS, vol. 1954, pp. 108–125. Springer, Heidelberg (2000)
Shilov, N.V., Garanina, N.O., Choe, K.-M.: Update and abstraction in model checking of knowledge and branching time. Fundamenta Informaticae 72(1-3), 347–361 (2006)
Shilov, N.V., Garanina, N.O., Kalinina, N.A.: Model checking knowledge, actions and fixpoints. In: Proceedings of CS&P 2004, pp. 351–357 (2004)
Sistla, A.P., Clarke, E.M.: The complexity of propositional linear temporal logics. Journal of ACM 32(3), 733–749 (1985)
Stirling, C.: Modal and Temporal Properties of Processes. Springer (2001)
Tarski, A.: A lattice-theoretical fixpoint theorem and its applications. Pacific Journal of Mathematics 5, 285–309 (1955)
van der Hoek, W., Jamroga, W., Wooldridge, M.: A logic for strategic reasoning. In: Proceedings of AAMAS 2005, pp. 157–164 (2005)
van der Hoek, W., Lomuscio, A., Wooldridge, M.: On the complexity of practical ATL model checking. In: Stone, P., Weiss, G. (eds.) Proceedings of AAMAS 2006, pp. 201–208 (2006)
van der Hoek, W., Verbrugge, R.: Epistemic logic: A survey. Game Theory and Applications 8, 53–94 (2002)
van der Hoek, W., Wooldridge, M.: Tractable multiagent planning for epistemic goals. In: Castelfranchi, C., Johnson, W.L. (eds.) Proceedings of the First International Joint Conference on Autonomous Agents and Multi-Agent Systems (AAMAS 2002), pp. 1167–1174. ACM Press, New York (2002)
van der Hoek, W., Wooldridge, M.: Cooperation, knowledge and time: Alternating-time Temporal Epistemic Logic and its applications. Studia Logica 75(1), 125–157 (2003)
van der Meyden, R., Shilov, N.V.: Model Checking Knowledge and Time in Systems with Perfect Recall (Extended Abstract). In: Pandu Rangan, C., Raman, V., Sarukkai, S. (eds.) FST TCS 1999. LNCS, vol. 1738, pp. 432–445. Springer, Heidelberg (1999)
van Otterloo, S., Jonker, G.: On Epistemic Temporal Strategic Logic. Electronic Notes in Theoretical Computer Science XX, 35–45 (2004); Proceedings of LCMAS 2004
Vardi, M.Y., Wolper, P.: An automata-theoretic approach to automatic program verification (preliminary report). In: Proceedings of the First Annual IEEE Symposium on Logic in Computer Science (LICS 1986), pp. 332–344. IEEE Computer Society Press (1986)
Walther, D., van der Hoek, W., Wooldridge, M.: Alternating-time temporal logic with explicit strategies. In: Samet, D. (ed.) Proceedings TARK XI, pp. 269–278. Presses Universitaires de Louvain (2007)
Weiss, G. (ed.): Multiagent Systems. A Modern Approach to Distributed Artificial Intelligence. MIT Press, Cambridge (1999)
Wooldridge, M.: An Introduction to Multi Agent Systems. John Wiley & Sons (2002)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Jamroga, W., Penczek, W. (2012). Specification and Verification of Multi-Agent Systems. In: Bezhanishvili, N., Goranko, V. (eds) Lectures on Logic and Computation. ESSLLI ESSLLI 2011 2010. Lecture Notes in Computer Science, vol 7388. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31485-8_6
Download citation
DOI: https://doi.org/10.1007/978-3-642-31485-8_6
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-31484-1
Online ISBN: 978-3-642-31485-8
eBook Packages: Computer ScienceComputer Science (R0)