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Folding variant narrowing and optimal variant termination
 In WRLA 2010, LNCS 6381:52–68
, 2010
"... Abstract. If a set of equations E∪Ax is such that E is confluent, terminating, and coherent modulo Ax, narrowing with E modulo Ax provides a complete E ∪Axunification algorithm. However, except for the hopelessly inefficient case of full narrowing, nothing seems to be known about effective narrowin ..."
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Abstract. If a set of equations E∪Ax is such that E is confluent, terminating, and coherent modulo Ax, narrowing with E modulo Ax provides a complete E ∪Axunification algorithm. However, except for the hopelessly inefficient case of full narrowing, nothing seems to be known about effective narrowing strategies in the general modulo case beyond the quite depressing observation that basic narrowing is incomplete modulo AC. In this work we propose an effective strategy based on the idea of the E ∪Axvariants of a term that we call folding variant narrowing. This strategy is complete, both for computing E ∪Axunifiers and for computing a minimal complete set of variants for any input term. And it is optimally variant terminating in the sense of terminating for an input term t iff t has a finite, complete set of variants. The applications of folding variant narrowing go beyond providing a complete E ∪ Axunification algorithm: computing the E ∪Axvariants of a term may be just as important as computing E∪Axunifiers in recent applications of folding variant narrowing such as termination methods modulo axioms, and checking confluence and coherence of rules modulo axioms. 1
C.L.: Variants, unification, narrowing, and symbolic reachability in Maude 2.6
 Schloss Dagstuhl  LeibnizZentrum fuer Informatik (2011
"... This paper introduces some novel features of Maude 2.6 focusing on the variants of a term. Given an equational theory (Σ,Ax ∪ E), the E,Axvariants of a term t are understood as the set of all pairs consisting of a substitution σ and the E,Axcanonical form of tσ. The equational theory (Ax ∪ E) has ..."
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This paper introduces some novel features of Maude 2.6 focusing on the variants of a term. Given an equational theory (Σ,Ax ∪ E), the E,Axvariants of a term t are understood as the set of all pairs consisting of a substitution σ and the E,Axcanonical form of tσ. The equational theory (Ax ∪ E) has the finite variant property iff there is a finite set of most general variants. We have added support in Maude 2.6 for: (i) ordersorted unification modulo associativity, commutativity and identity, (ii) variant generation, (iii) ordersorted unification modulo finite variant theories, and (iv) narrowingbased symbolic reachability modulo finite variant theories. We also explain how these features have a number of interesting applications in areas such as unification theory, cryptographic protocol verification, business processes, and proofs of termination, confluence and coherence.
A Dependency Pair Framework for A∨CTermination ⋆
"... Abstract. The development of powerful techniques for proving termination of rewriting modulo a set of equations is essential when dealing with rewriting logicbased programming languages like CafeOBJ, Maude, OBJ, etc. One of the most important techniques for proving termination over a wide range of ..."
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Abstract. The development of powerful techniques for proving termination of rewriting modulo a set of equations is essential when dealing with rewriting logicbased programming languages like CafeOBJ, Maude, OBJ, etc. One of the most important techniques for proving termination over a wide range of variants of rewriting (strategies) is the dependency pair approach. Several works have tried to adapt it to rewriting modulo associative and commutative (AC) equational theories, and even to more general theories. However, as we discuss in this paper, no appropriate notion of minimality (and minimal chain of dependency pairs) which is wellsuited to develop a dependency pair framework has been proposed to date. In this paper we carefully analyze the structure of infinite rewrite sequences for rewrite theories whose equational part is a (free) combination of associative and commutative axioms which we call A∨Crewrite theories. Our analysis leads to a more accurate and optimized notion of dependency pairs through the new notion of stably minimal term. Then, we have developed a suitable dependency pair framework for proving termination of A∨Crewrite theories. Key words: equational rewriting, termination, dependency pairs 1
OrderSorted Equality Enrichments Modulo Axioms
"... Abstract. Builtin equality and inequality predicates based on comparison of canonical forms in algebraic specifications are frequently used because they are handy and efficient. However, their use places algebraic specifications with initial algebra semantics beyond the pale of theorem proving tool ..."
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Abstract. Builtin equality and inequality predicates based on comparison of canonical forms in algebraic specifications are frequently used because they are handy and efficient. However, their use places algebraic specifications with initial algebra semantics beyond the pale of theorem proving tools based, for example, on explicit or inductionless induction techniques, and of other formal tools for checking key properties such as confluence, termination, and sufficient completeness. Such specifications would instead be amenable to formal analysis if an equationallydefined equality predicate enriching the algebraic data types were to be added to them. Furthermore, having an equationallydefined equality predicate is very useful in its own right, particularly in inductive theorem proving. Is it possible to effectively define a theory transformation E ↦ → E ≃ that extends an algebraic specification E to a specification E ≃ where equationallydefined equality predicates have been added? This paper answers this question in the affirmative for a broad class of ordersorted conditional specifications E that are sortdecreasing, ground confluent, and operationally terminating modulo axioms B and have subsignature of constructors. The axioms B can consist of associativity, or commutativity, or associativitycommutativity axioms, so that the constructors are free modulo B. We prove that the transformation E ↦ → E ≃ preserves all the justmentioned properties of E. The transformation has been automated in Maude using reflection and it is used in Maude formal tools. 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.
Computing finite variants for subterm convergent rewrite systems ⋆
"... Abstract. Driven by an application in the verification of security protocols, we introduce the strong finite variant property, an extention of the finite variant property defined in [1] and we show that subterm convergent rewrite systems enjoy the strong finite variant property modulo the empty equa ..."
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Abstract. Driven by an application in the verification of security protocols, we introduce the strong finite variant property, an extention of the finite variant property defined in [1] and we show that subterm convergent rewrite systems enjoy the strong finite variant property modulo the empty equational theory. We argue that the strong finite variant property is more natural and more useful in practice than the finite variant property. We also compare the two properties and we provide a prototype implementation of an algorithm that computes a finite strongly complete set of variants for any term t with respect to a subterm convergent rewrite system. 1