Unprovability results for clause set cycles

Abstract

The notion of clause set cycle abstracts a family of methods for automated inductive theorem proving based on the detection of cyclic dependencies between clause sets. By discerning the underlying logical features of clause set cycles, we are able to characterize clause set cycles by a logical theory. We make use of this characterization to provide practically relevant unprovability results for clause set cycles that exploit different logical features.We will also be interested in sets of formulas that contain at most one free variable.Definition 3. Let Γ be a set of formulas, then by Γ − we denote the set of formulas of Γ that have at most one free variable.Clause sets are an alternative representation of ∀ 1 formulas, that is preferred by automated theorem provers because of its uniformity.Definition 4 (Literals, clauses, clause sets). Let L be a first-order language. By an L literal we understand an L atom or the negation of an L atom. An L clause is a finite set of L literals. An L clause set is a set of L clauses. Whenever the language L is clear from the context, we simply speak of atoms, literals, clauses and clause sets.We will now recall some basic model-theoretic concepts.Definition 5. Let L be a first-order language, then L structures and the first-order satisfaction relation |= are defined as usual. Let L ′ ⊆ L be a first-order language and M an L structure, then by M | L ′ we denote the L ′ reduct of M . Let M be an L structure, then we write b ∈ M to express that b is an element of the domain of M . Formulas and clauses are interpreted as usual. In particular, a clause is interpreted as the universal closure of the disjunction of its literals. Let ∆ be a set of L formulas and L clauses, then M |= ∆ if M |= δ for each δ ∈ ∆.Let us conclude this section by introducing some notation to manipulate clauses and clause sets.