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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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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.
Modular ChurchRosser Modulo: The Full Picture
"... Abstract. In [17], Toyama proved that the union of two confluent termrewriting systems that share absolutely no function symbols or constants is likewise confluent, a property called modularity. The proof of this beautiful modularity result, technically based on slicing terms into an homogeneous ca ..."
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Abstract. In [17], Toyama proved that the union of two confluent termrewriting systems that share absolutely no function symbols or constants is likewise confluent, a property called modularity. The proof of this beautiful modularity result, technically based on slicing terms into an homogeneous cap and a so called alien, possibly heterogeneous substitution, was later substantially simplified in [7, 9]. In this paper 3 we present a further simplification of the proof of Toyama’s result for confluence, which shows that the crux of the problem lies in two different properties: a cleaning lemma, whose goal is to anticipate the application of collapsing reductions; a modularity property of ordered completion, that allows to pairwise match the caps and alien substitutions of two equivalent terms obtained from the cleaning lemma. The approach allows for rules with extra variables on the right and scales up to rewriting modulo arbitrary sets of equations. 1
AssociativeCommutative Reduction Orderings via HeadPreserving Interpretations
, 1995
"... We introduce a generic definition of reduction orderings on term algebras containing associativecommutative (hereafter denoted AC) operators. These orderings are compatible with the AC theory hence makes them suitable for use in deduction systems where AC operators are builtin. Furthermore, they ..."
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We introduce a generic definition of reduction orderings on term algebras containing associativecommutative (hereafter denoted AC) operators. These orderings are compatible with the AC theory hence makes them suitable for use in deduction systems where AC operators are builtin. Furthermore, they have the nice property of being total on AC classes of ground terms, a required property for example to avoid failure in ACcompletion, or to insure completeness of ordered strategies in firstorder theorem proving with builtin AC operators. We show that the two definitions already known of such total and ACcompatible orderings [24, 25] are actually instances of our definition. Finally, we find new such orderings which have more properties, first an ordering based on an integer polynomial interpretation, answering positively to a question left open by Narendran and Rusinowitch, and second an ordering which allow to orient the distributivity axiom in the usual way, answering positively to a ...
Superposition with Completely Builtin Abelian Groups
"... A new technique is presented for superposition with firstorder clauses with builtin abelian groups (AG). Compared with previous approaches, it is simpler, and AGunification is used instead of the computationally more expensive unification modulo associativity and commutativity. Furthermore, n ..."
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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 AGunification is used instead of the computationally more expensive unification modulo associativity and commutativity. Furthermore, no inferences with the AG axioms or abstraction rules are needed; in this sense this is the first approach where AG is completely built in. 1.
Symmetrization based completion
 SYMBOLIC REWRITING TECHNIQUES
, 1998
"... We argue that most completion procedures for finitely presented algebras can be simulated by term completion procedures based on a generalized symmetrization process. Therefore we present three different constructive definitions of symmetrization procedures that can take the role of the orientation ..."
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We argue that most completion procedures for finitely presented algebras can be simulated by term completion procedures based on a generalized symmetrization process. Therefore we present three different constructive definitions of symmetrization procedures that can take the role of the orientation step in a symmetrization based completion procedure. We investigate confluence and compatibility properties of the symmetrized rules computed by the different symmetrization procedures. Based on semicompatibility properties we can present a generic version of the critical pair theorem that specializes to the critical pair theorems of KnuthBendix completion procedures and algebraic completion procedures like Buchberger's algorithm respectively. This critical pair theorem also applies to symmetrization based completion procedures using a normalized reduction relation if the result of the symmetrization is both semicompatible and semistable. We conclude our paper showing how a generic Buchberger algorithm for polynomials over arbitrary finitely presented rings can be formulated as a symmetrization based completion procedure.
Modular ChurchRosser Modulo: The Complete Picture
"... Abstract. In [19], Toyama proved that the union of two confluent termrewriting systems that share absolutely no function symbols or constants is likewise confluent, a property called modularity. The proof of this beautiful modularity result, technically based on slicing terms into an homogeneous ca ..."
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Abstract. In [19], Toyama proved that the union of two confluent termrewriting systems that share absolutely no function symbols or constants is likewise confluent, a property called modularity. The proof of this beautiful modularity result, technically based on slicing terms into an homogeneous cap and a so called alien, possibly heterogeneous substitution, was later substantially simplified in [8, 12]. In this paper 3, we present a further simplification of the proof of Toyama’s result for confluence, which shows that the crux of the problem lies on two different properties: a cleaning lemma, whose goal is to anticipate the application of collapsing reductions and a modularity property of ordered completion that allows to pairwise match the caps and alien substitutions of two equivalent terms obtained from the cleaning lemma. The approach allows for arbitrary kinds of rules, and scales up to rewriting modulo arbitrary sets of equations. 1
Incremental Checking of WellFounded Recursive Specifications Modulo Axioms
, 2011
"... We introduce the notion of wellfounded recursive ordersorted equational logic (OS) theories modulo axioms. Such theories define functions by wellfounded recursion and are inherently terminating. Moreover, for wellfounded recursive theories important properties such as confluence and sufficient c ..."
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We introduce the notion of wellfounded recursive ordersorted equational logic (OS) theories modulo axioms. Such theories define functions by wellfounded recursion and are inherently terminating. Moreover, for wellfounded recursive theories important properties such as confluence and sufficient completeness are modular for socalled fair extensions. This enables us to incrementally check these properties for hierarchies of such theories that occur naturally in modular rulebased functional programs. Wellfounded recursive OS theories modulo axioms contain only commutativity and associativitycommutativity axioms. In order to support arbitrary combinations of associativity, commutativity and identity axioms, we show how to eliminate identity and (under certain conditions) associativity (without commutativity) axioms by theory transformations in the last part of the paper.
Specification and Proof in Membership Equational Logic1
"... Abstract: This paper is part of a longterm effort to increase expressiveness of algebraic specification languages while at the same time having a simple semantic foundation on which efficient execution by rewriting and powerful theoremproving tools can be based. In particular, our rewriting techni ..."
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Abstract: This paper is part of a longterm effort to increase expressiveness of algebraic specification languages while at the same time having a simple semantic foundation on which efficient execution by rewriting and powerful theoremproving tools can be based. In particular, our rewriting techniques provide semantic foundations for Maude's functional sublanguage, where they have been efficiently implemented. This effort started in the late seventies, led by the ADJ group, who promoted equational logic and universal algebra as the semantic basis of program specification languages. An important later milestone was the work around ordersorted algebras and the OBJ family of languages developed at SRIInternational in the eighties. This effort has been substantially advanced in the midnineties with the development of Maude, a language based on membership equational logic. Membership equational logic is quite simple, and yet quite powerful. Its atomic formulae are equations and sort membership assertions, and its sentences are Horn clauses. It extends in a conservative way both (a version of) ordersorted equational logic and partial algebra approaches, while Horn logic with equality can be very easily encoded.
Rewrite Orderings for HigherOrder Terms in ηLong βNormal Form and the Recursive Path Ordering
"... : This paper extends the termination proof techniques based on rewrite orderings to a higherorder setting, by defining a recursive path ordering for simply typed higherorder terms in jlong finormal form. This ordering is powerful enough to show termination of several complex examples. Key words ..."
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: This paper extends the termination proof techniques based on rewrite orderings to a higherorder setting, by defining a recursive path ordering for simply typed higherorder terms in jlong finormal form. This ordering is powerful enough to show termination of several complex examples. Key words: higherorder rewriting ; typed lambda calculus; termination orderings. 1 Introduction Higherorder rewrite rules are used in various programming languages and logical systems with two different meanings. Some functional languages like ML or type theories like the calculus of inductive constructions use higherorder rewrite rules to define functions or recursors by firstorder pattern matching. In this setting termination is known to be satisfied when all rules follow a generalized form of a primitive recursive schema of higher type [9, 1, 10]. In functional languages like Elf, or theorem provers like Isabelle, higherorder rewrite rules define functions by higherorder pattern matching. ...