Matthias Horbach

Superposition for fixed domains

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 models 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.

Deciding the inductive validity of ∀∃* queries

We present a new saturation-based decidability result for inductive validity. Let Σ be a finite signature in which all function symbols are at most unary and let N be a satisfiable Horn clause set without equality in which all positive literals are linear. If N ∪ {A1, ... , An →} belongs to a class that can be finitely saturated by ordered resolution modulo variants, then it is decidable whether a sentence of the form ¬x.∃y→.A1 ∧ ... ∧ An is valid in the minimal model of N.

Superposition for Fixed Domains

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 perfect term-generated model for the theory. Proving universal properties with respect to a saturated theory directly leads to a modification of the perfect models 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 complete for a first-order fixed domain semantics. For some classes of formulas and theories, we can even employ the calculus to prove properties of the perfect model itself, going beyond the scope of known superposition based approaches.

Obtaining Finite Local Theory Axiomatizations via Saturation

In this paper we present a method for obtaining local sets of clauses from possibly non-local ones. For this, we follow the work of Basin and Ganzinger and use saturation under a version of ordered resolution. In order to address the fact that saturation can generate infinite sets of clauses, we use constrained clauses and show that a link can be established between saturation and locality also for constrained clauses: This often allows us to give a finite representation of possibly infinite saturated sets of clauses.

On the Combination of the Bernays–Schönfinkel–Ramsey Fragment with Simple Linear Integer Arithmetic

In general, first-order predicate logic extended with linear integer arithmetic is undecidable. We show that the Bernays-Schonfinkel-Ramsey fragment (\(\exists ^* \forall ^*\)-sentences) extended with a restricted form of linear integer arithmetic is decidable via finite ground instantiation. The identified ground instances can be employed to restrict the search space of existing automated reasoning procedures considerably, e.g., when reasoning about quantified properties of array data structures formalized in Bradley, Manna, and Sipma’s array property fragment. Typically, decision procedures for the array property fragment are based on an exhaustive instantiation of universally quantified array indices with all the ground index terms that occur in the formula at hand. Our results reveal that one can get along with significantly fewer instances.

Disunification for ultimately periodic interpretations

Disunification is an extension of unification to first-order formulae over syntactic equality atoms. Instead of considering only syntactic equality, I extend a disunification algorithm by Comon and Delor to ultimately periodic interpretations, i.e. minimal many-sorted Herbrand models of predicative Horn clauses and, for some sorts, equations of the form sl(x)≃sk(x). The extended algorithm is terminating and correct for ultimately periodic interpretations over a finite signature and gives rise to a decision procedure for the satisfiability of equational formulae in ultimately periodic interpretations. As an application, I show how to apply disunification to compute the completion of predicates with respect to an ultimately periodic interpretation. Such completions are a key ingredient to several inductionless induction methods.

Locality Transfer: From Constrained Axiomatizations to Reachability Predicates

In this paper, we build upon our previous work in which we used constrained clauses in order to finitely represent infinite sets of clauses and proved that constrained axiomatizations are local if they are saturated under a version of resolution. We extend this result by identifying situations in which locality of saturated axiomatizations is maintained if we enrich the base theory by introducing new predicates (often reachability predicates) instead of using constraints for these properties.

Saturation-based Decision Procedures for Fixed Domain and Minimal Model Semantics

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 Herbrand model for the theory. This raises the question in how far superposition calculi can be employed for reasoning about such minimal models. This is indeed often possible when existential properties are considered. However, proving universal properties directly leads to a modification of the minimal model’s termgenerated domain, as new Skolem functions are introduced. For many applications, this is not desired because it changes the problem. In this thesis, I propose the first superposition calculus that can explicitly represent existentially quantified variables and can thus compute with respect to a given fixed domain. It does not eliminate existential variables by Skolemization, but handles them using additional constraints with which each clause is annotated. This calculus is sound and refutationally complete in the limit for a fixed domain semantics. For saturated Horn theories and classes of positive formulas, the calculus is even complete for proving properties of the minimal model itself, going beyond the scope of known superpositionbased approaches. The calculus is applicable to every set of clauses with equality and does not rely on any syntactic restrictions of the input. Extensions of the calculus lead to various new decision procedures for minimal model validity. A main feature of these decision procedures is that even the validity of queries containing one quantifier alternation can be decided. In particular, I prove that the validity of any formula with at most one quantifier alternation is decidable in models represented by a finite set of atoms and that the validity of several classes of such formulas is decidable in models represented by so-called disjunctions of implicit generalizations. Moreover, I show that the decision of minimal model validity can be reduced to the superposition-based decision of first-order validity for models of a class of predicative Horn clauses where all function symbols are at most unary.