Constraint manipulation in SGGS

SGGS (Semantically-Guided Goal-Sensitive theorem proving) is a clausal theorem-proving method, with a seemingly rare combination of properties: it is first order, DPLL-style model based, semantically guided, goal sensitive, and proof confluent. SGGS works with constrained clauses, and uses a sequence of constrained clauses to represent a tentative model of the given set of clauses.A basic building block in SGGS inferences is splitting, which partitions a clause into clauses that have the same set of ground instances. Splitting introduces constraints and their manipulation, which is the subject of this paper. Specifically, splitting a clause with respect to another clause requires to compute their difference, which captures the ground instances of one that are not ground instances of the other. We give a set of inference rules to compute clause difference, and reduce SGGS constraints to standard form, and we prove that it is guaranteed to terminate, provided the standardization rules are applied only within the clause difference computation

Arithmetic Classification of Perfect Models of Stratified Programs (Addendum)

RECURSION-FREE PROGRAMS The following section completes the analysis of arithmetic complexity of perfect models and has been inadvertently omitted in the previous version of the paper. We say that a general program P is recursion-free if in its dependency graph Dp there is no cycle. Clearly recursion-free programs form a subclass of stratified programs. Recursion-free programs form a very simple generalization of the class of hierarchical programs introduced in [C78]. Hierarchical programs satisfy an additional condition on variable occurrences in clauses that prevents floundering, i.e. a forced selection of a non-ground negative literal in an SLDNF- derivation. In this section we study the complexity of perfect models of recursion-free programs

A Denotational Semantics for First-Order Logic

In Apt and Bezem [AB99] (see cs.LO/9811017) we provided a computational interpretation of first-order formulas over arbitrary interpretations. Here we complement this work by introducing a denotational semantics for first-order logic. Additionally, by allowing an assignment of a non-ground term to a variable we introduce in this framework logical variables. The semantics combines a number of well-known ideas from the areas of semantics of imperative programming languages and logic programming. In the resulting computational view conjunction corresponds to sequential composition, disjunction to ``don't know'' nondeterminism, existential quantification to declaration of a local variable, and negation to the ``negation as finite failure'' rule. The soundness result shows correctness of the semantics with respect to the notion of truth. The proof resembles in some aspects the proof of the soundness of the SLDNF-resolution.Comment: 17 pages. Invited talk at the Computational Logic Conference (CL 2000). To appear in Springer-Verlag Lecture Notes in Computer Scienc

An Inference Rule for the Acyclicity Property of Term Algebras

Term algebras are important structures in many areas of mathematics and computer science. Reasoning about their theories in superposition-based first-order theorem provers is made difficult by the acyclicity property of terms, which is not finitely axiomatizable. We present an inference rule that extends the superposition calculus and allows reasoning about term algebras without axioms to describe the acyclicity property. We detail an indexing technique to efficiently apply this rule in problems containing a large number of clauses. Finally we experimentally evaluate an implementation of this extended calculus in the first-order theorem prover Vampire. The results show that this technique is able to find proofs for difficult problems that existing SMT solvers and first-order theorem provers are unable to solve