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Equational Reasoning In SaturationBased Theorem Proving
, 1998
"... INTRODUCTION Equational reasoning is fundamental in mathematics, logics, and many applications of formal methods in computer science. In this chapter we describe the theoretical concepts and results that form the basis of stateoftheart automated theorem provers for firstorder clause logic with ..."
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Cited by 38 (2 self)
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INTRODUCTION Equational reasoning is fundamental in mathematics, logics, and many applications of formal methods in computer science. In this chapter we describe the theoretical concepts and results that form the basis of stateoftheart automated theorem provers for firstorder clause logic with equality. We mainly concentrate on refinements of paramodulation, such as the superposition calculus, that have yielded the most promising results to date in automated equational reasoning. We begin with some preliminary material in section 2 and then explain, in section 3, why resolution with the congruence axioms is an impractical theorem proving method for equational logic. In section 4 we outline the main results about paramodulationa more direct equational inference rule. This section also contains a description of the modification method, which can be used to demonstrate that the functional reflexivity axioms are redundant in the context of paramodulation. The modification
Inductionless Induction
, 1994
"... Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1 A few words explaining the title . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Some examples of the problem we are considering . . . . . . . . . . . . . . . . . . . ..."
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Cited by 23 (0 self)
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Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1 A few words explaining the title . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Some examples of the problem we are considering . . . . . . . . . . . . . . . . . . . 3 1.3 Outline of the chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2 Formal background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1 Terms and clauses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Equational deduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3 Inductive theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.4 Constructors and sufficient completeness . . . . . . . . . . . . . . . . . . . . . . . . 8 2.5 Term Rewriting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.6 Standar
Decidability and undecidability results for NelsonOppen and rewritebased decision procedures
 In Proc. IJCAR3, U. Furbach and
, 2006
"... Abstract. In the context of combinations of theories with disjoint signatures, we classify the component theories according to the decidability of constraint satisfiability problems in arbitrary and in infinite models, respectively. We exhibit a theory T1 such that satisfiability is decidable, but s ..."
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Cited by 22 (14 self)
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Abstract. In the context of combinations of theories with disjoint signatures, we classify the component theories according to the decidability of constraint satisfiability problems in arbitrary and in infinite models, respectively. We exhibit a theory T1 such that satisfiability is decidable, but satisfiability in infinite models is undecidable. It follows that satisfiability in T1 ∪ T2 is undecidable, whenever T2 has only infinite models, even if signatures are disjoint and satisfiability in T2 is decidable. In the second part of the paper we strengthen the NelsonOppen decidability transfer result, by showing that it applies to theories over disjoint signatures, whose satisfiability problem, in either arbitrary or infinite models, is decidable. We show that this result covers decision procedures based on rewriting, complementing recent work on combination of theories in the rewritebased approach to satisfiability. 1
Sat solving for argument filterings
 In Logic for Programming, Artificial Intelligence and Reasoning (LPAR
, 2006
"... Abstract. This paper introduces a propositional encoding for lexicographic path orders in connection with dependency pairs. This facilitates the application of SAT solvers for termination analysis of term rewrite systems based on the dependency pair method. We address two main interrelated issues a ..."
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Cited by 18 (10 self)
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Abstract. This paper introduces a propositional encoding for lexicographic path orders in connection with dependency pairs. This facilitates the application of SAT solvers for termination analysis of term rewrite systems based on the dependency pair method. We address two main interrelated issues and encode them as satisfiability problems of propositional formulas that can be efficiently handled by SAT solving: (1) the combined search for a lexicographic path order together with an argument filtering to orient a set of inequalities; and (2) how the choice of the argument filtering influences the set of inequalities that have to be oriented. We have implemented our contributions in the termination prover AProVE. Extensive experiments show that by our encoding and the application of SAT solvers one obtains speedups in orders of magnitude as well as increased termination proving power. 1
On Using Ground Joinable Equations in Equational Theorem Proving
 PROC. OF THE 3RD FTP, ST. ANDREWS, SCOTTLAND, FACHBERICHTE INFORMATIK. UNIVERSITAT KOBLENZLANDAU
, 2000
"... When rewriting and completion techniques are used for equational theorem proving, the axiom set is saturated with the aim to get a rewrite system that is terminating and confluent on ground terms. To reduce the computational effort it should (1) be powerful for rewriting and (2) create not too ma ..."
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Cited by 12 (2 self)
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When rewriting and completion techniques are used for equational theorem proving, the axiom set is saturated with the aim to get a rewrite system that is terminating and confluent on ground terms. To reduce the computational effort it should (1) be powerful for rewriting and (2) create not too many critical pairs. These problems become especially important if some operators are associative and commutative (AC ). We show in this paper how these two goals can be reached to some extent by using ground joinable equations for simplification purposes and omitting them from the generation of new facts. For the special case of AC operators we present a simple redundancy criterion which is easy to implement, efficient, and effective in practice, leading to significant speedups.
Solved Forms for Path Ordering Constraints
 in `In Proc. 10th International Conference on Rewriting Techniques and Applications (RTA
, 1999
"... . A usual technique in symbolic constraint solving is to apply transformation rules until a solved form is reached for which the problem becomes simple. Ordering constraints are wellknown to be reducible to (a disjunction of) solved forms, but unfortunately no polynomial algorithm deciding the ..."
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Cited by 11 (4 self)
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. A usual technique in symbolic constraint solving is to apply transformation rules until a solved form is reached for which the problem becomes simple. Ordering constraints are wellknown to be reducible to (a disjunction of) solved forms, but unfortunately no polynomial algorithm deciding the satisfiability of these solved forms is known. Here we deal with a different notion of solved form, where fundamental properties of orderings like transitivity and monotonicity are taken into account. This leads to a new family of constraint solving algorithms for the full recursive path ordering with status (RPOS), and hence as well for other path orderings like LPO, MPO, KNS and RDO, and for all possible total precedences and signatures. Apart from simplicity and elegance from the theoretical point of view, the main contribution of these algorithms is on efficiency in practice. Since guessing is minimized, and, in particular, no linear orderings between the subterms are guessed, ...
Induction = IAxiomatization + FirstOrder Consistency
 Information and Computation
, 1998
"... In the early 80's, there was a number of papers on what should be called proofs by consistency. They describe how to perform inductive proofs, without using an explicit induction scheme, in the context of equational specifications and groundconvergent rewrite systems. The method was explici ..."
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Cited by 10 (0 self)
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In the early 80's, there was a number of papers on what should be called proofs by consistency. They describe how to perform inductive proofs, without using an explicit induction scheme, in the context of equational specifications and groundconvergent rewrite systems. The method was explicitly stated as a firstorder consistency proof in case of pure equational, constructor based, specifications. In this paper, we show how, in general, inductive proofs can be reduced to firstorder consistency and hence be performed by a firstorder theorem prover. Moreover, we extend previous methods, allowing nonequational specifications (even nonHorn specifications), designing some specific strategies. Finally, we also show how to drop the ground convergence requirement (which is called saturatedness for general clauses). 1 http://www.lsv.enscachan.fr/Publis/ Research Report LSV989, Lab. Spcification et Vrification, CNRS & ENS de Cachan, France, Oct. 1998 This paper was presente...
Citius altius fortius: Lessons learned from the Theorem Prover Waldmeister
 Proceedings of the 4th International Workshop on FirstOrder Theorem Proving, number 86.1 in Electronic Notes in Theoretical Computer Science
, 2003
"... In the last years, the development of automated theorem provers has been advancing in a so to speak Olympic spirit, following the motto "faster, higher, stronger"; and the Waldmeister system has been a part of that endeavour. We will survey the concepts underlying this prover, which implem ..."
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Cited by 9 (0 self)
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In the last years, the development of automated theorem provers has been advancing in a so to speak Olympic spirit, following the motto "faster, higher, stronger"; and the Waldmeister system has been a part of that endeavour. We will survey the concepts underlying this prover, which implements KnuthBendix completion in its unfailing variant. The system architecture is based on a strict separation of active and passive facts, and is realized via speci cally tailored representations for each of the central data structures: indexing for the active facts, setbased compression for the passive facts, successor sets for the conjectures. In order to cope with large search spaces, specialized redundancy criteria are employed, and the empirically gained control knowledge is integrated to ease the use of the system. We conclude with a discussion of strengths and weaknesses, and a view of future prospects.