Automata-Based Computation of Temporal Equilibrium Models

Abstract

Abstract. Temporal Equilibrium Logic (TEL) is a formalism for temporal logic programming that generalizes the paradigm of Answer Set Programming (ASP) introducing modal temporal operators from standard Linear-time Temporal Logic (LTL). In this paper we solve some open problems that remained open for TEL like decidability, complexity assessment and computation of temporal equilibrium models for arbitrary theories. We propose a method for the latter that consists in building a Büchi automaton that accepts exactly the temporal equilibrium models of a given theory, providing an automata-based decision procedure and illustrating the ω-regularity of such sets. We are able to show that TEL satisfiability can be solved in exponential space and it is hard for polynomial space. Finally, given two theories, we provide a decision procedure to check whether they have the same temporal equilibrium models.

Temporal Answer Set Programming on Finite Traces

“…For instance, we may conclude that satisfiability for both THT f and TEL f are PSpace-hard, since LTL f , proved to be PSpace-complete (De Giacomo and Vardi 2013), can be easily reduced to THT f or TEL f by adding the excluded middle axiom (EM). In fact, THT f is also PSpace-complete: its membership can be proved by encoding THT f into LTL f as in the reduction from THT ω to LTL ω made in (Cabalar and Demri 2011). In the case of TEL f satisfiability, we conjecture its ExpSpace membership by using a translation into TEL ω , which is ExpSpace-complete (Bozzelli and Pearce 2015), similar to the one from LTL f into LTL ω in (De Giacomo and Vardi 2013).…”

Temporal equilibrium logic: a survey

“…Proposition 5 (from Cabalar & Demri, 2011). Given a temporal formula ϕ for a propositional signature At, for every THT-model H, T , the propositions below are equivalent: This gives us a direct way of encoding LTL in THT, since LTL-models of ϕ coincide with its total THT-models.…”

“…In this way, the constraint ¬ ♦ p → ⊥ would be ruling out all models where p does not occur infinitely often. Note also that all LTL is embeddable both in THT (Proposition 5) and in fact in TEL too (for details, see Cabalar & Demri, 2011).…”

“…{ p, q}, { p , p, q}, {q , q}) is an LTL-model of Corollary 8 (from Cabalar & Demri, 2011). Let ϕ be the formula ϕ * ∧ p∈At ( p → p).…”

“…Cabalar & Demri, 2011). Let ϕ be a temporal formula built over the connectives ∨, ∧, →, and U and such that → occurs only in subformulae of the form p →⊥ with p ∈ At.…”