An Approximative Inference Method for Solving ∃ ∀SO Satisfiability Problems

Abstract

The fragment ∃∀SO(ID) of second order logic extended with inductive definitions is expressive, and many interesting problems, such as conformant planning, can be naturally expressed as finite domain satisfiability problems of this logic. Such satisfiability problems are computationally hard (Σ P 2 ). In this paper, we develop an approximate, sound but incomplete method for solving such problems that transforms a ∃∀SO(ID) to a ∃SO(ID) problem. The finite domain satisfiability problem for the latter language is in NP and can be handled by several existing solvers. We show that this provides an effective method for solving practically useful problems, such as common examples of conformant planning. We also propose a more complete translation to ∃SO(F P ), existential SO extended with nested inductive and coinductive definitions.

Constraint Propagation for First-Order Logic and Inductive Definitions

“…Recently, Vlaeminck et al [2010] showed how to represent the propagator lim I (V ) by a nested fixpoint definition [Hou 2010]. Methods to evaluate such nested definitions are currently being investigated [Hou et al 2010].…”

“…There exists conformant planning problems where determining whether a conformant plan of length less than a given length l exists is Σ P 2 -hard, even if l is polynomial in the size of the problem. Vlaeminck et al [2010] show how to approximate an MX(∀SO) problem by an MX(FO(ID)) problem, in the sense that solutions of the latter are solutions of the former (but not necessarily vice versa). The representation of FO propagation by a rule set is crucial in the approximation.…”

“…This paper is an extended and improved presentation of [Wittocx et al 2008a]. It describes (part of) the theoretical foundation for applications presented in [Wittocx et al 2010;Wittocx et al 2009;Vlaeminck et al 2010;Vlaeminck et al 2009]. A less densely written version of this paper is part of the PhD thesis of the first author [Wittocx 2010].…”

A General Representation and Approximate Inference Algorithm for Sensing Actions

“…However, it is possible to efficiently approximate the set of all models of a theory. We will use here a general approximation method for FO(ID), that efficiently allows us to detect that a formula holds in all models of a theory [17,16]. While we lack space to recall this method in detail, the essence is as follows.…”

“…An FO(ID) theory T over a vocabulary Σ is syntactically transformed into an inductive definition Approx(T ) over a new vocabulary that contains, for every predicate P/n ∈ Σ, two predicates P ct /n and P cf /n that represent whether P is certainly true (i.e., true in all models of T ) or certainly false (i.e., false in all models of T ). The definition Appox(T ) has the property that if Approx(T ) |= P ct (d), then T |= P (d) (see [16], proposition 5.4) and similar for P cf . In addition, Approx(T ) also defines a number of predicates A ct ϕ and A cf ϕ that tell us whether certain non-atomic formulas ϕ (or their negations, respectively) are entailed by T .…”