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17
Theorem Proving Modulo
 Journal of Automated Reasoning
"... Abstract. Deduction modulo is a way to remove computational arguments from proofs by reasoning modulo a congruence on propositions. Such a technique, issued from automated theorem proving, is of much wider interest because it permits to separate computations and deductions in a clean way. The first ..."
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Cited by 75 (14 self)
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Abstract. Deduction modulo is a way to remove computational arguments from proofs by reasoning modulo a congruence on propositions. Such a technique, issued from automated theorem proving, is of much wider interest because it permits to separate computations and deductions in a clean way. The first contribution of this paper is to define a sequent calculus modulo that gives a proof theoretic account of the combination of computations and deductions. The congruence on propositions is handled via rewrite rules and equational axioms. Rewrite rules apply to terms and also directly to atomic propositions. The second contribution is to give a complete proof search method, called Extended Narrowing and Resolution (ENAR), for theorem proving modulo such congruences. The completeness of this method is proved with respect to provability in sequent calculus modulo. An important application is that higherorder logic can be presented as a theory modulo. Applying the Extended Narrowing and Resolution method to this presentation of higherorder logic subsumes full higherorder resolution.
Rewriting Modulo a Rewrite System
, 1995
"... . We introduce rewriting with two sets of rules, the first interpreted equationally and the second not. A semantic view considers equational rules as defining an equational theory and reduction rules as defining a rewrite relation modulo this theory. An operational view considers both sets of rules ..."
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Cited by 13 (3 self)
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. We introduce rewriting with two sets of rules, the first interpreted equationally and the second not. A semantic view considers equational rules as defining an equational theory and reduction rules as defining a rewrite relation modulo this theory. An operational view considers both sets of rules as similar. We introduce sufficient properties for these two views to be equivalent (up to different notions of equivalence). The paper ends with a collection of example showing the effectiveness of this approach. Rewriting can be viewed simultaneously as the most basic symbolmanipulating method, and as a very expressive specification framework, given the expressive power of rewriting modulo equations. It is a primary candidate to the role of a general logical framework [Mes92, MOM93]. Historically, rewriting has been given an equational semantics, saying that a rewrite rule u \Gamma! v is interpreted as u is equal to v. This is the case for instance when defining functions or solving the w...
Decidability and Complexity Analysis by Basic Paramodulation
, 1998
"... It is shown that for sets of Horn clauses saturated under basic paramodulation, the word and unifiability problems are in NP, and the number of minimal unifiers is simply exponential (i). For Horn sets saturated wrt. a special ordering under the more restrictive inference rule of basic superpositio ..."
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Cited by 12 (7 self)
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It is shown that for sets of Horn clauses saturated under basic paramodulation, the word and unifiability problems are in NP, and the number of minimal unifiers is simply exponential (i). For Horn sets saturated wrt. a special ordering under the more restrictive inference rule of basic superposition, the word and unifiability problems are still decidable and unification is finitary (ii). These two results are applied to the following languages. For shallow presentations (equations with variables at depth at most one) we show that the closure under paramodulation can be computed in polynomial time. Applying result (i), it follows that shallow unifiability is in NP, which is optimal since unifiability in ground theories is already NPhard. The shallow word problem is even shown to be polynomial. Generalizing shallow theories to the Horn case, we obtain (two versions of) a language we call Catalog, a natural extension of Datalog to include functions and equality. The closure under paramo...
Paramodulation with BuiltIn Abelian Groups
 in `15th IEEE Symposium on Logic in Computer Science (LICS
, 2000
"... A new technique is presented for superposition with firstorder clauses with builtin abelian groups (AG). Compared with previous approaches, it is simpler, and no inferences with the AG axioms or abstraction rules are needed. Furthermore, AGunification is used instead of the computationally more ex ..."
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Cited by 6 (4 self)
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A new technique is presented for superposition with firstorder clauses with builtin abelian groups (AG). Compared with previous approaches, it is simpler, and no inferences with the AG axioms or abstraction rules are needed. Furthermore, AGunification is used instead of the computationally more expensive unification modulo associativity and commutativity. Due to the simplicity and restrictiveness of our inference system, its compatibility with redundancy notions and constraints, and the fact that standard term orderings like RPO can be used, we believe that our technique will become the method of choice for practice, as well as a basis for new theoretical developments like logicbased complexity and decidability analysis. Keywords: term rewriting, automated deduction. 1 Introduction It is crucial for the performance of a deduction system that it incorporates specialized techniques to work efficiently with standard algebraic theories, since a nave handling of some axioms (like assoc...
Termination Modulo Combinations of Equational Theories
"... Abstract. Rewriting with rules R modulo axioms E is a widely used technique in both rulebased programming languages and in automated deduction. Termination methods for rewriting systems modulo specific axioms E (e.g., associativitycommutativity) are known. However, much less seems to be known abou ..."
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Cited by 6 (5 self)
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Abstract. Rewriting with rules R modulo axioms E is a widely used technique in both rulebased programming languages and in automated deduction. Termination methods for rewriting systems modulo specific axioms E (e.g., associativitycommutativity) are known. However, much less seems to be known about termination methods that can be modular in the set E of axioms. In fact, current termination tools and proof methods cannot be applied to commonly occurring combinations of axioms that fall outside their scope. This work proposes a modular termination proof method based on semantics and terminationpreserving transformations that can reduce the proof of termination of rules R modulo E to an equivalent proof of termination of the transformed rules modulo a typically much simpler set B of axioms. Our method is based on the notion of variants of a term recently proposed by Comon and Delaune. We illustrate its practical usefulness by considering the very common case in which E is an arbitrary combination of associativity, commutativity, left and rightidentity axioms for various function symbols. 1
Basic Paramodulation and Decidable Theories (Extended Abstract)
 in `Proceedings 11th IEEE Symposium on Logic in Computer Science, LICS'96', IEEE Computer
, 1996
"... ) Robert Nieuwenhuis Technical University of Catalonia Pau Gargallo 5, 08028 Barcelona, Spain Email: roberto@lsi.upc.es. Abstract We prove that for sets of Horn clauses saturated under basic paramodulation, the word and unifiability problems are in NP, and the number of minimal unifiers is simpl ..."
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Cited by 6 (0 self)
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) Robert Nieuwenhuis Technical University of Catalonia Pau Gargallo 5, 08028 Barcelona, Spain Email: roberto@lsi.upc.es. Abstract We prove that for sets of Horn clauses saturated under basic paramodulation, the word and unifiability problems are in NP, and the number of minimal unifiers is simply exponential (i). For Horn sets saturated wrt. a special ordering under the more restrictive inference rule of basic superposition, the word and unifiability problems are still decidable and unification is finitary (ii). We define standard theories, which include and significantly extend shallow theories. Standard presentations can be finitely closed under superposition and result (ii) applies. Generalizing shallow theories to the Horn case, we obtain (two versions of) a language we call Catalog, a natural extension of Datalog to include functions and equality. The closure under paramodulation is finite for Catalog sets, hence (i) applies. Since for shallow sets this closure is even polynom...
ACcomplete Unification and its Application to Theorem Proving
 In Proceedings of the 7th International Conference on Rewriting Techniques and Applications
, 1996
"... . The inefficiency of ACcompletion is mainly due to the doubly exponential number of ACunifiers and thereby of critical pairs generated. We present ACcomplete Eunification, a new technique whose goal is to reduce the number of ACcritical pairs inferred by performing unification in a extension E ..."
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Cited by 5 (0 self)
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. The inefficiency of ACcompletion is mainly due to the doubly exponential number of ACunifiers and thereby of critical pairs generated. We present ACcomplete Eunification, a new technique whose goal is to reduce the number of ACcritical pairs inferred by performing unification in a extension E of AC (e.g. ACU, Abelian groups, Boolean rings, ...) in the process of normalized completion [21]. The idea is to represent complete sets of ACunifiers by (smaller) sets of Eunifiers. Not only the theories E used for unification have exponentially fewer most general unifiers than AC, but one can remove from a complete set of Eunifiers those solutions which have no Einstance which is an ACunifier. First, we define ACcomplete Eunification and describe its fundamental properties. We show how ACcomplete Eunification can be done in the elementary case, and how the known combination techniques for unification algorithms can be reused for our purposes. Finally, we give some evidence of t...
Experiments with Partial Evaluation Domains for Rewrite Specifications
 In Magne Haveraaen, Olaf Owe, and OleJohan Dahl, editors, Recent Trends in Data Type Specifications (LNCS 1130
, 1995
"... . We describe a method to improve the efficiency of normalization procedures for term rewriting systems. This improvement does not restrict the semantics of the term rewriting specification in any respect. In particular, the expressive power of term rewriting systems as a programming language for ge ..."
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Cited by 3 (2 self)
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. We describe a method to improve the efficiency of normalization procedures for term rewriting systems. This improvement does not restrict the semantics of the term rewriting specification in any respect. In particular, the expressive power of term rewriting systems as a programming language for generic programs and as a theorem prover has been preserved. Our method is basedon the following observation. Many rewrite specifications are instances of theories for which efficient data structures exist. In that case we can exploit the canonical representation of objects of such a data structure by translating terms to corresponding objects, and retranslating these objects to terms (in normal form). We will call an implementation of a data structure that allows for this kind of transformations for all (not necessarily ground) terms an evaluation domain. This is then extended to the case where only part of a rewrite specification can directly be transformed using an evaluation domain. We dev...
Superposition modulo a Shostak theory
 AUTOMATED DEDUCTION (CADE19), VOLUME 2741 OF LNAI
, 2003
"... We investigate superposition modulo a Shostak theory T in order to facilitate reasoning in the amalgamation of T and a free theory F. Free operators occur naturally for instance in program verication problems when abstracting over subroutines. If their behaviour in addition can be specied axiomatic ..."
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Cited by 2 (0 self)
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We investigate superposition modulo a Shostak theory T in order to facilitate reasoning in the amalgamation of T and a free theory F. Free operators occur naturally for instance in program verication problems when abstracting over subroutines. If their behaviour in addition can be specied axiomatically, much more of the program semantics can be captured. Combining the Shostakstyle components for deciding the clausal validity problem with the ordering and saturation techniques developed for equational reasoning, we derive a refutationally complete calculus on mixed ground clauses which result for example from CNF transforming arbitrary universally quantied formulae. The calculus works modulo a Shostak theory in the sense that it operates on canonizer normalforms. For the Shostak solvers that we study, coherence comes for free: no coherence pairs need to be considered.