Abstract. This paper presents a technique for automated theorem proving with free variable tableaux that does not require backtracking. Most existing automated proof procedures using free variable tableaux require iterative deepening and backtracking over applied instantiations to guarantee completeness. If the correct instantiation is hard to find, this can lead to a significant amount of duplicated work. Incremental Closure is a way of organizing the search for closing instantiations that avoids this inefficiency. 1

Hyper tableau reasoning is a version of clausal form tableau reasoning where all negative literals in a clause are resolved away in a single inference step. Constrained hyper tableaux are a generalization of hyper tableaux, where branch closing substitutions, from the point of view of model generation, give rise to constraints on satisfying assignments for the branch. These variable constraints eliminate the need for the awkward `purifying substitutions' of of hyper tableaux. The paper presents a non-destructive and proof conuent calculus for constrained hyper tableaux, together with a soundness and completeness proof, with completeness based on a new way to generate models from open tableaux. Next, it is indicated how the calculus can be modi ed for minimal model generation. Finally, it is pointed out that the variable constraint approach applies to free variable tableau reasoning in general.

Several variants of a first-order simplification rule for non-normal form tableaux using syntactic constraints are presented. These can be used as a framework for porting refinements of clausal first-order proof procedures to non-normal form tableaux. Some experimental results obtained with a prototypical implementation are given.

Abstract not found

First go at a Haskell implementation of tableau theorem proving for FOL, using lazy lists of instantiating substitutions for the free tableau variables instead of bracktracking. Keywords: Tableau theorem proving, Instance Streams, leanTAP, Proof Search without Backtracking. MSC codes: : : : 1 Introduction This paper works out a suggestion from Giese [4] to do tableau proof search by merging closing substitutions for tableau branches into a closing substitution for the whole tableau. The paper contains the complete Haskell [5] code of a lazy tableau theorem prover for FOL. The boxed text constitutes the Haskell program. Instead of doing proof search by backtracking and iterative deepening, LazyTAP generates innite streams of most general closing substitutions for the branches and tries to merge them in a fair way into a closing substitution for the whole tableau. These streams are processed lazily using the lazy execution mechanism of Haskell. Use of substitution instances to guide p...

In the area of formal verification it is well known that there can be no single logic that suits all needs. This insight motivates the diversity of this dissertation: it contains contributions to SAT solving, First Order theorem proving and Model Finding, and Symbolic Model Checking. A growing number of problem domains are successfully being tackled by SAT solvers. Following the current trend of extending and adapting SAT solvers we present a detailed description of a SAT solver designed for that particular purpose. The description bridges a gap between theory and practice, serving as a tutorial on modern SAT solving algorithms. Among other things we describe how to solve a series of related SAT problems efficiently, called incremental SAT solving. For finding finite first order models, the MACE-style method that is based on SAT solving, is well-known. We improve the basic method by several techniques, that can be loosely classified as either transformations that make the reduction to SAT result in fewer clauses, or techniques that are designed to speed up the search of the SAT solver. The resulting tool, called PARADOX, performed well in the SAT division of the CASC-19 competition. Recently, there has been large interest in methods for safety property verification that are based on SAT solving. One example is temporal induction, also called k-induction. The method requires a sequence of increasingly stronger induction proofs to be performed. We show how this sequence of proofs can be solved more efficiently using incremental SAT solving.