Superposition for Fixed Domains

Abstract

Superposition is an established decision procedure for a variety of first-order logic theories represented by sets of clauses. A satisfiable theory, saturated by superposition, implicitly defines a minimal term-generated model for the theory. Proving universal properties with respect to a saturated theory directly leads to a modification of the minimal model's term-generated domain, as new Skolem functions are introduced. For many applications, this is not desired.Therefore, we propose the first superposition calculus that can explicitly represent existentially quantified variables and can thus compute with respect to a given domain. This calculus is sound and refutationally complete in the limit for a first-order fixed domain semantics. For saturated Horn theories and classes of positive formulas, we can even employ the calculus to prove properties of the minimal model itself, going beyond the scope of known superposition-based approaches.

Superposition Modulo Linear Arithmetic SUP(LA)

“…Spass(LA) finitely saturates this conjecture together with the theory of the transition system without finding the empty clause in less than one second on any reasonable PC hardware. Due to completeness and the minimal model property of superposition with respect to existentially quantified conjunctions of atoms [HW08] this shows that the level of the water tank is always below 240 units. Obviously, the hierarchic superposition calculus is complete for this example, because there are no free function symbols at all.…”

Decidability Results for Saturation-Based Model Building

“…Our approach has potential for further research. We restricted our attention to a non-equational setting, whereas our initial fixed domain calculus [12] considers equations as well. It is an open problem to what extend our results also hold in an equational setting.…”

“…In [12], we introduced a superposition-based calculus to address the problem whether N |= Σ ∀ x.∃ y.φ, where N is a set of unconstrained clauses and φ a formula over Σ. There, both N and φ may contain equational atoms.…”

“…Hence its proof is exactly as in the unconstrained case or in the case of the original fixed domain calculus (cf. [1,12]): Let N 0 , N 1 Proof. Let M be a Herbrand model of N over Σ = (P, F ).…”

“…We first enhance the non-equational part of our superposition calculus for fixed domains [12] with syntactic disequations (Section 3). The result is an ordered resolution calculus for fixed domains with syntactic disequations that is sound (Proposition 1) and complete (Theorem 1).…”