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Incremental closure of free variable tableaux
 Proc. Intl. Joint Conf. on Automated Reasoning IJCAR
, 2001
"... 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 complete ..."
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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
Constrained Hyper Tableaux
, 2001
"... 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 generati ..."
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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 nondestructive 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.
A Firstorder Simplification Rule with Constraints
 3RD INT. WORKSHOP ON FIRSTORDER THEOREM PROVING (FTP
, 2000
"... Several variants of a firstorder simplification rule for nonnormal form tableaux using syntactic constraints are presented. These can be used as a framework for porting refinements of clausal firstorder proof procedures to nonnormal form tableaux. Some experimental results obtained with a protot ..."
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Cited by 3 (3 self)
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Several variants of a firstorder simplification rule for nonnormal form tableaux using syntactic constraints are presented. These can be used as a framework for porting refinements of clausal firstorder proof procedures to nonnormal form tableaux. Some experimental results obtained with a prototypical implementation are given.
LazyTAP  A Lazy Tableau Theorem Prover for FOL
, 2000
"... 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 Intro ..."
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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...
Applications of SAT solving
, 2003
"... 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 numb ..."
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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 MACEstyle method that is based on SAT solving, is wellknown. 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 CASC19 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 kinduction. 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.
Fair Constraint Merging Tableaux in Lazy Functional Programming Style
"... Constraint merging tableaux maintain a system of all closing substitutions of all subtableau up to a certain depth, which is incrementally increased. This avoids backtracking as necessary in destructive first order free variable tableaux. The first sound and complete implementation of this paradigm ..."
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Constraint merging tableaux maintain a system of all closing substitutions of all subtableau up to a certain depth, which is incrementally increased. This avoids backtracking as necessary in destructive first order free variable tableaux. The first sound and complete implementation of this paradigm was given in an objectoriented style by Giese. In this paper we analyse the reasons why lazy functional implementations so far were problematic (although appealing), and we give a solution. The resulting implementation...
Axiomatic constraint systems for proof search modulo theories
"... Abstract Goaldirected proof search in firstorder logic uses metavariables to delay the choice of witnesses; substitutions for such variables are produced when closing prooftree branches, using firstorder unification or a theoryspecific background reasoner. This paper investigates a generalisa ..."
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Abstract Goaldirected proof search in firstorder logic uses metavariables to delay the choice of witnesses; substitutions for such variables are produced when closing prooftree branches, using firstorder unification or a theoryspecific background reasoner. This paper investigates a generalisation of such mechanisms whereby theoryspecific constraints are produced instead of substitutions. In order to design modular proofsearch procedures over such mechanisms, we provide a sequent calculus with metavariables, which manipulates such constraints abstractly. Proving soundness and completeness of the calculus leads to an axiomatisation that identifies the conditions under which abstract constraints can be generated and propagated in the same way unifiers usually are. We then extract from our abstract framework a component interface and a specification for concrete implementations of background reasoners. 1