A Goal-Directed Decision Procedure for Hybrid PDL

Mark Kaminski, Gert Smolka
2013 Journal of automated reasoning  
We present the first goal-directed decision procedure for hybrid PDL. The procedure is based on a modular approach that scales from basic modal logic with eventualities to hybrid PDL. The approach is designed so that nominals and eventualities are treated orthogonally. To deal with the complex programs of PDL, the approach employs a novel disjunctive program decomposition. In arguing the correctness of our approach, we employ the novel notion of support generalizing the standard notion of
more » ... ka sets. model obtained over a finite formula universe known as the Fischer-Ladner closure. They also show that the satisfiability problem for PDL is EXPTIME-hard. Based on Fischer and Ladner's model construction, Pratt [46] gives a decision procedure for PDL that runs in deterministic exponential time, thus establishing EXPTIME-completeness of the logic (see [23, 6, 32] for variants of Pratt's procedure with correctness proofs). The procedure starts with the set H of all Hintikka sets over the Fischer-Ladner closure of the input formula and prunes H by removing unsatisfiable sets until the remaining sets form a model satisfying all of the sets. The input formula is then satisfiable if and only if it is contained in one of the remaining sets. Pratt [47] devises a more practical version of the abstract procedure in [46] that works with an AND-OR graph constructed from the input formula. While initially developed for PDL, Pratt's methods scale to temporal logics as shown by Ben-Ari et al. [5], Wolper [55], and Emerson and Halpern [15]. An alternative formulation of Pratt's procedure [47] is given by Nguyen and Szałas [44]. Although worst-case optimal, Pratt's methods are not truly practical since they always construct data structures whose size is exponential in that of the input formula, and hence take exponential time. The first decision procedure for PDL that is both practical and worst-case optimal is devised by Goré and Widmann [20, 54] . Goré and Widmann's procedure interleaves the construction of the AND-OR graph with pruning. Unlike Pratt's procedures, it is goal-directed. By following the structure of the input formula, the procedure avoids computation steps that are redundant for determining the satisfiability of the formula. This way, many formulas can be decided efficiently despite the high worst-case complexity of the logic. A different family of decision procedures for PDL is based on nondeterministic search for models as pioneered for modal logic by Kripke [40]. Such procedures stepwise grow (a syntactic representation of) a candidate model by adding states and transitions until the model satisfies the input formula. The first nondeterministic procedure for (a variant of) test-free PDL is due to Baader [4]. To check the satisfaction of eventualities, Baader develops a dedicated loop detection technique. Later, related procedures are also proposed for full PDL by De Giacomo and Massacci [13], and by Abate et al. [1] . Unlike Goré and Widmann's procedure, the procedures in [4, 13, 1] have suboptimal complexity: Baader's [4] procedure can be seen to run in NEXPTIME, the procedure by Abate et al. [1] requires 2EXPTIME, while de Giacomo and Massacci [13] claim NEXPTIME complexity for their procedure (without proof). Despite being suboptimal, the procedures improve on Pratt's procedures in terms of practical efficiency since they are goal-directed. In this paper, we devise the first goal-directed procedure for PDL extended → n → + := → • → * The transition relations for complex programs and the denotation ϕ M of a formula ϕ in M are defined by mutual induction on the structure of formulas and programs as follows: α+β → M = α → M ∪ β → M α * → M = α → * M αβ → M = α → M • β → M ϕ → M = { (w, w) | w ∈ ϕ M }
doi:10.1007/s10817-013-9294-5 fatcat:v3pqh3m6incf7kvfoo4kqo6e3e