PSPACE Bounds for Rank-1 Modal Logics

Modal logics are attractive from a computational point of view, as they often combine expressiveness with decidability. For many modal logics not involving dynamic features, satisfiability is known to be in PSPACE . This is typically proved for one logic at a time, e.g. by modifications of the witness algorithm for the modal logic K [Ladner 1977;Blackburn et al. 2001], but also using markedly different methods such as the constraint-based PSPACE -algorithm for graded modal logic [Tobies 2001]. Vardi [1989] gives a first glimpse of a generalisable method, equipping various epistemic logics with a neighbourhood frame semantics and showing them to be in NP and PSPACE , respectively (with the K axiom being responsible for PSPACE -hardness; recent work by Halpern and Rêgo [2007] shows that negative introspection brings the complexity back down to NP). Nevertheless, there is to date no generally applicable theorem that allows establishing PSPACE -bounds for large classes of modal logics in a uniform way.

Here, we generalise the methods of [Vardi 1989] to obtain PSPACE bounds for rank-1 modal logics, i.e. logics axiomatisable by formulas whose modal depth uniformly equals one, in a systematic way. Although presently limited to rank 1, our approach covers numerous relevant and non-trivial examples. We recover known PSPACE bounds not only for normal modal logics such as K and KD, but most notably also for a range of non-normal modal logics such as graded modal logic [Fine 1972], coalition logic [Pauly 2002], and probabilistic modal logic [Larsen and Skou 1991;Heifetz and Mongin 2001]. Moreover, our methods lead to a previously unknown PSPACE upper bound for majority logic [Pacuit and Salame 2004] that was independently discovered by Demri and Lugiez [2006] at the same time. These logics are far from exotic: graded modal logic plays a role e.g. in decision support and knowledge representation [van der Hoek and Meyer 1992;Ohlbach and Koehler 1999], and probabilistic modal logic has appeared in connection with model checking [Larsen and Skou 1991] and in modelling economic behaviour [Heifetz and Mongin 2001].

The key to such a degree of generality is to parametrise the theory over the type of systems defining the semantics, using coalgebraic methods. Coalgebra conveniently abstracts from the details of a concrete class of models as it encapsulates the precise nature of models in an endofunctor on the category of sets. As specific instances, one obtains e.g. (serial) Kripke frames, (monotone) neighbourhood frames [Hansen and Kupke 2004], game frames [Pauly 2002], probabilistic transition systems and automata [Rabin 1963;Bartels et al. 2004], weighted automata, linear automata [Carlyle and Paz 1971], and multigraphs [D’ Agostino and Visser 2002]. Despite the broad range of systems covered by the coalgebraic approach, a substantial body of concepts and non-trivial results has emerged, encompassing e.g. generic notions of bisimilarity and coinduction [Bartels 2003], corecursion [Turi and Plotkin 1997], duality, and ultrafilter extensions [Kupke et al. 2005]. On the applications side, coalgebraic modal logic features in actual specification languages such as the object oriented specification language CCSL [Rothe et al. 2001] and CoCasl [Mossakowski et al. 2006].

The coalgebraic study of computational aspects of modal logic was initiated in [Schröder 2007], where the finite model property and associated NEXPTIMEbounds were proved. Here, we push these results further and present a shallow model property based on coalgebraic semantics. This leads to a generic PSPACEalgorithm for deciding satisfiability that traverses a shallow model and strips off one layer of modalities in every step. Alternatively, our algorithm may be seen as computing a shallow proof that enjoys a number of pleasant proof-theoretic properties, including a weak subformula property (i.e. it mentions only propositional combinations of subformulas of the goal).

The model construction relies on extending the axiomatisation of a given logic to a set of rules which is closed under rule resolution, i.e. every resolvent of two substituted rule conclusions can also be derived directly using a third rule. This process typically results in an infinite but recursive set of rules. Resolution closedness then enables us to build the shallow model using induction on the modal depth of formulas. Since we are working with an infinite set of rules, we have to impose a second condition to ensure that we can decide satisfiability: a rule set is closed under contraction if every substituted rule conclusion with duplicate literals can be derived using a substitution instance of a second rule in whose conclusion all literals remain distinct. The decision procedure will run in PSPACE if both closure under resolution and closure under contraction can be controlled, i.e. there is a polynomial bound on the size of rules that are applicable at every step of the deductive process. T

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