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A generic tableau prover and its integration with Isabelle
 Journal of Universal Computer Science
, 1999
"... Abstract: A generic tableau prover has been implemented and integrated with Isabelle [Paulson, 1994]. Compared with classical rstorder logic provers, it has numerous extensions that allow it to reason with any supplied set of tableau rules. It has a higherorder syntax in order to support userde ne ..."
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Cited by 38 (10 self)
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Abstract: A generic tableau prover has been implemented and integrated with Isabelle [Paulson, 1994]. Compared with classical rstorder logic provers, it has numerous extensions that allow it to reason with any supplied set of tableau rules. It has a higherorder syntax in order to support userde ned binding operators, such as those of set theory. The uni cation algorithm is rstorder instead of higherorder, but it includes modi cations to handle bound variables. The proof, when found, is returned to Isabelle as a list of tactics. Because Isabelle veri es the proof, the prover can cut corners for e ciency's sake without compromising soundness. For example, the prover can use type information to guide the search without storing type information in full. Categories: F.4, I.1
Simplification  A general constraint propagation technique for propositional and modal tableaux
, 1998
"... . Tableau and sequent calculi are the basis for most popular interactive theorem provers for formal verification. Yet, when it comes to automatic proof search, tableaux are often slower than DavisPutnam, SAT procedures or other techniques. This is partly due to the absence of a bivalence principle ..."
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Cited by 24 (2 self)
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. Tableau and sequent calculi are the basis for most popular interactive theorem provers for formal verification. Yet, when it comes to automatic proof search, tableaux are often slower than DavisPutnam, SAT procedures or other techniques. This is partly due to the absence of a bivalence principle (viz. the cutrule) but there is another source of inefficiency: the lack of constraint propagation mechanisms. This paper proposes an innovation in this direction: the rule of simplification, which plays for tableaux the role of subsumption for resolution and of unit for the DavisPutnam procedure. The simplicity and generality of simplification make possible its extension in a uniform way from propositional logic to a wide range of modal logics. This technique gives an unifying view of a number of tableauxlike calculi such as DPLL, KE, HARP, hypertableaux, BCP, KSAT. We show its practical impact with experimental results for random 3SAT and the industrial IFIP benchmarks for hardware ve...
Incremental Theory Reasoning Methods for Semantic Tableaux
 In Proceedings, 5th Workshop on Theorem Proving with Analytic Tableaux and Related Methods
, 1996
"... . Theory reasoning is an important technique for increasing the efficiency of automated deduction systems. In this paper we present incremental theory reasoning, a method that improves the interaction between the foreground reasoner and the background (theory) reasoner and, thus, the efficiency of ..."
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Cited by 7 (3 self)
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. Theory reasoning is an important technique for increasing the efficiency of automated deduction systems. In this paper we present incremental theory reasoning, a method that improves the interaction between the foreground reasoner and the background (theory) reasoner and, thus, the efficiency of the combined system. The use of incremental theory reasoning in free variable semantic tableaux and the cost reduction that can be achieved are discussed; as an example, completionbased equality reasoning is presented, including experimental data obtained using an implementation. 1 Introduction Theory reasoning is an important technique for increasing the efficiency of automated deduction systems. The knowledge from a given domain (or theory) is made use of by applying efficient methods for reasoning in that domain. The general purpose foreground reasoner calls a special purpose background reasoner to handle problems from a certain theory. Following the pioneering work of Stickel [23], so...
Lean Induction Principles for Tableaux
, 1997
"... . In this paper, we deal with various induction principles incorporated in an underlying tableau calculus with equality. The induction formulae are restricted to literals. Induction is formalized as modified closure conditions which are triggered by applications of the ffirule. Examples dealing wit ..."
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Cited by 3 (1 self)
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. In this paper, we deal with various induction principles incorporated in an underlying tableau calculus with equality. The induction formulae are restricted to literals. Induction is formalized as modified closure conditions which are triggered by applications of the ffirule. Examples dealing with (weak forms of) arithmetic and strings illustrate the simplicity and usability of our induction handling. We prove the correctness of the closure conditions and discuss possibilities to strengthen the induction principles. 1 Introduction The use of induction principles in various forms certainly is an important and prominent topic in Automated Deduction, as witnessed, e.g., by [15, 6, 1] and quite recently in [17]. The complexity of the problem  both, in terms of proof search and formulation of appropriate deduction mechanisms  is well known. Our aim is to demonstrate that various forms of induction can be incorporated into classical free variable tableaux in an elegant way. We were...
A Framework for Using Knowledge in Tableau Proofs
 Proc. International Conference on Automated Reasoning with Analytic Tableaux and Related Methods, PontMousson
, 1997
"... . The problem of automatically reasoning using a knowledge base containing axioms, definitions and theorems from a firstorder theory is recurrent in automated reasoning research. Here we present a sound and complete method for reasoning over an arbitrary firstorder theory using the tableau cal ..."
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Cited by 2 (1 self)
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. The problem of automatically reasoning using a knowledge base containing axioms, definitions and theorems from a firstorder theory is recurrent in automated reasoning research. Here we present a sound and complete method for reasoning over an arbitrary firstorder theory using the tableau calculus. A natural, wellmotivated and simple restriction (implemented in IPR) to the method provides a powerful framework for the automation of the selection of theorems from a knowledge base for use in theorem proving. The restrictions are related to semantic resolution restrictions and the setofsupport restriction in resolution, and to hypertableaux and the weak connection condition in tableaux. We also present additional tableau rules used by the IPR prover for handling some equality which is not complete but is sufficient for handling the problems in its intended domain of problem solving. 1 Introduction The rules presented in this paper allow an automatic theorem proving pro...
A Further and Effective Liberalization of the deltaRule in Free Variable Semantic Tableaux
"... In this paper, we present a further liberalization of the ffirule in free variable semantic tableaux. It is effective in that (1) it is both a natural and intuitive liberalization, and (2) can reduce the proof size exponentially as compared to previous versions of the ffirule. 1 Introduction Proo ..."
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Cited by 2 (1 self)
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In this paper, we present a further liberalization of the ffirule in free variable semantic tableaux. It is effective in that (1) it is both a natural and intuitive liberalization, and (2) can reduce the proof size exponentially as compared to previous versions of the ffirule. 1 Introduction Proof procedures for first order predicate logic such as semantic tableaux need means to deal with existential quantifiers. In general there are two different ways to do this. One way is to Skolemize the formula to be proven in a preprocessing step, obtaining a purely universal formula at the expense of a richer signature. The other approach is not to use a preliminary Skolemization but to add a tableau expansion rule for treating the essentially existential formulae, so that Skolemization is performed during the proof construction when existential formulae are encountered on tableau branches. In substance, there is no difference in applying either of the two methods, but we believe that adding ...
Full FirstOrder Sequent and Tableau Calculi With Preservation of Solutions and the Liberalized deltaRule but Without Skolemization
, 1998
"... . We present a combination of raising, explicit variable dependency representation, the liberalized ffirule, and preservation of solutions for firstorder deductive theorem proving. Our main motivation is to provide the foundation for our work on inductive theorem proving. 1 Introduction 1.1 W ..."
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Cited by 2 (2 self)
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. We present a combination of raising, explicit variable dependency representation, the liberalized ffirule, and preservation of solutions for firstorder deductive theorem proving. Our main motivation is to provide the foundation for our work on inductive theorem proving. 1 Introduction 1.1 Without Skolemization We discuss how to analytically prove firstorder theorems in contexts where Skolemization is not appropriate. It has at least three problematic aspects. 1. Skolemization enrichs the signature or introduces higherorder variables. Unless special care is taken, this may introduce objects into empty universes and change the notion of termgeneratedness or Herbrand models. Above that, the Skolem functions occur in answers to goals or solutions of constraints which in general cannot be translated into the original signature. For a detailed discussion of these problems cf. Miller (1992). 2. Skolemization results in the following simplified quantification structure: "For all...
Discoveries and Experiments in the Automation of Mathematical Reasoning
, 2002
"... vii List of Figures xii Chapter 1. ..."
Skolem functions and Hilbert’s ǫterms in Free Variable Tableau Systems
"... Abstract. We apply the framework of the generic δrule presented in [4] to show how to map the δ ǫrule, which uses ǫterms as syntactical objects to expand existentially quantified formulae, in the context of standard δrules based on Skolem terms. Structural results coming out from such mapping pr ..."
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Cited by 1 (1 self)
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Abstract. We apply the framework of the generic δrule presented in [4] to show how to map the δ ǫrule, which uses ǫterms as syntactical objects to expand existentially quantified formulae, in the context of standard δrules based on Skolem terms. Structural results coming out from such mapping process are discussed. 1