What Makes Atl* Decidable? A Decidable Fragment of Strategy Logic
Strategy Logic (Sl, for short) has been recently introduced by Mogavero, Murano, and Vardi as a formalism for reasoning explicitly about strategies, as first-order objects, in multi-agent concurrent games. This logic turns out to be very powerful, strictly subsuming all major previously studied modal logics for strategic reasoning, including Atl, Atl*, and the like. The price that one has to pay for the expressiveness of Sl is the lack of important model-theoretic properties and an increased complexity of decision problems. In particular, Sl does not have the bounded-tree model property and the related satisfiability problem is highly undecidable while for Atl* it is 2ExpTime-complete. An obvious question that arises is then what makes Atl* decidable. Understanding this should enable us to identify decidable fragments of Sl. We focus, in this work, on the limitation of Atl* to allow only one temporal goal for each strategic assertion and study the fragment of Sl with the same restriction. Specifically, we introduce and study the syntactic fragment One-Goal Strategy Logic (Sl[1g], for short), which consists of formulas in prenex normal form having a single temporal goal at a time for every strategy quantification of agents. We show that Sl[1g] is strictly more expressive than Atl*. Our main result is that Sl[1g] has the bounded tree-model property and its satisfiability problem is 2ExpTime-complete, as it is for Atl*.
KeywordsModal Logic Atomic Proposition Strategy Logic Tree Automaton Universal Variable
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- 1.Albert, M.H., Nowakowski, R.J., Wolfe, D.: Lessons in Play: An Introduction to Combinatorial Game Theory. AK Peters (2007)Google Scholar
- 4.Clarke, E.M., Grumberg, O., Peled, D.A.: Model Checking. MIT Press (2002)Google Scholar
- 5.Da Costa, A., Laroussinie, F., Markey, N.: ATL with Strategy Contexts: Expressiveness and Model Checking. In: FSTTCS 2010. LIPIcs, vol. 8, pp. 120–132 (2010)Google Scholar
- 10.Hodges, W.: Model theory. Encyclopedia of Mathematics and its Applications. Cambridge University Press (1993)Google Scholar
- 16.Mogavero, F., Murano, A., Perelli, G., Vardi, M.Y.: Reasoning About Strategies: On the Model-Checking Problem. Technical Report 1112.6275, arXiv (December 2011)Google Scholar
- 17.Mogavero, F., Murano, A., Perelli, G., Vardi, M.Y.: A Decidable Fragment of Strategy Logic. Technical Report 1202.1309, arXiv (February 2012)Google Scholar
- 18.Mogavero, F., Murano, A., Vardi, M.Y.: Reasoning About Strategies. In: FSTTCS 2010. LIPIcs, vol. 8, pp. 133–144 (2010)Google Scholar
- 24.Pnueli, A.: The Temporal Logic of Programs. In: FOCS 1977, pp. 46–57 (1977)Google Scholar
- 27.Vardi, M.Y.: Why is Modal Logic So Robustly Decidable? In: DCFM 1996, pp. 149–184. American Mathematical Society (1996)Google Scholar
- 28.Vardi, M.Y., Wolper, P.: An Automata-Theoretic Approach to Automatic Program Verification. In: LICS 1986, pp. 332–344. IEEE Computer Society (1986)Google Scholar