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14
Modular & Incremental Proofs of ACTermination
 Journal of Symbolic Computation
, 2002
"... Recently, the framework of rewriting modules was proposed and provided modular and incremental termination criteria. In this paper, we extend these results to the important case of Associative and Commutative rewriting by means of ACdependency pairs. ..."
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Cited by 15 (3 self)
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Recently, the framework of rewriting modules was proposed and provided modular and incremental termination criteria. In this paper, we extend these results to the important case of Associative and Commutative rewriting by means of ACdependency pairs.
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
Termination by abstraction
, 2004
"... Abstract. Abstraction can be used very effectively to decompose and simplify termination arguments. If a symbolic computation is nonterminating, then there is an infinite computation with a top redex, such that all redexes are immortal, but all children of redexes are mortal. This suggests applying ..."
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Abstract. Abstraction can be used very effectively to decompose and simplify termination arguments. If a symbolic computation is nonterminating, then there is an infinite computation with a top redex, such that all redexes are immortal, but all children of redexes are mortal. This suggests applying weaklymonotonic wellfounded relations in abstractionbased termination methods, expressed here within an abstract framework for termbased proofs. Lexicographic combinations of orderings may be used to match up with multiple levels of abstraction. A small number of firms have decided to terminate their independent abstraction schemes.
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 7 (6 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
Dependency pairs for rewriting with builtin numbers and semantic data structures
, 2007
"... Abstract. This paper defines an expressive class of constrained equational rewrite systems that supports the use of semantic data structures (e.g., sets or multisets) and contains builtin numbers, thus extending our previous work presented at CADE 2007 [6]. These rewrite systems, which are based on ..."
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Abstract. This paper defines an expressive class of constrained equational rewrite systems that supports the use of semantic data structures (e.g., sets or multisets) and contains builtin numbers, thus extending our previous work presented at CADE 2007 [6]. These rewrite systems, which are based on normalized rewriting on constructor terms, allow the specification of algorithms in a natural and elegant way. Builtin numbers are helpful for this since numbers are a primitive data type in every programming language. We develop a dependency pair framework for these rewrite systems, resulting in a flexible and powerful method for showing termination that can be automated effectively. Various powerful techniques are developed within this framework, including a subterm criterion and reduction pairs that need to consider only subsets of the rules and equations. It is wellknown from the dependency pair framework for ordinary rewriting that these techniques are often crucial for a successful automatic termination proof. Termination of a large collection of examples can be established using the presented techniques. 1
Dependency Pairs for Rewriting with NonFree Constructors
"... Abstract. A method based on dependency pairs for showing termination of functional programs on data structures generated by constructors with relations is proposed. A functional program is specified as an equational rewrite system, where the rewrite system specifies the program and the equations exp ..."
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Abstract. A method based on dependency pairs for showing termination of functional programs on data structures generated by constructors with relations is proposed. A functional program is specified as an equational rewrite system, where the rewrite system specifies the program and the equations express the relations on the constructors that generate the data structures. Unlike previous approaches, relations on constructors can be collapsing, including idempotency and identity relations. Relations among constructors may be partitioned into two parts: (i) equations that cannot be oriented into terminating rewrite rules, and (ii) equations that can be oriented as terminating rewrite rules, in which case an equivalent convergent system for them is generated. The dependency pair method is extended to normalized rewriting, where constructorterms in the redex are normalized first. The method has been applied to several examples, including the Calculus of Communicating Systems and the Propositional Sequent Calculus. Various refinements, such as dependency graphs, narrowing, etc., which increase the power of the dependency pair method, are presented for normalized rewriting. 1
Termination Dependencies
"... The innovative dependencypair termination method of [1, 2] relies on two important observations: – If a rewrite system is nonterminating, then there is an infinite derivation with at least one redex at the top of a term (see, for example, [4, p. 287]). – If a rewrite system is nonterminating, then ..."
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Cited by 4 (0 self)
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The innovative dependencypair termination method of [1, 2] relies on two important observations: – If a rewrite system is nonterminating, then there is an infinite derivation with at least one redex at the top of a term (see, for example, [4, p. 287]). – If a rewrite system is nonterminating, then there is an infinite derivation in which all proper subterms of every redex are mortal (these are the “constricting ” derivations of [11]). By “mortal”, we mean that it initiates finite derivations only. Let F be some vocabulary (of constant and function symbols) and T, the set of terms constructed from it. The dependencypair method can be reformulated—and somewhat strengthened—in terms of two (related) quasiorderings, as follows: A rewrite system terminates if there are wellfounded quasiorderings � and � ′ such that: 1. (Rule) ℓ � r for all rules ℓ → r; 2. (Dependency) ℓ ≻ ′ u for all subterms u of the right side r of a rule ℓ → r that are not also subterms of the left side ℓ;
OrderSorted Dependency Pairs
, 2008
"... Types (or sorts) are pervasive in computer science and in rewritingbased programming languages, which often support subtypes (subsorts) and subtype polymorphism. Programs in these languages can be modeled as ordersorted term rewriting systems (OSTRSs). Often, termination of such programs heavily d ..."
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Cited by 4 (2 self)
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Types (or sorts) are pervasive in computer science and in rewritingbased programming languages, which often support subtypes (subsorts) and subtype polymorphism. Programs in these languages can be modeled as ordersorted term rewriting systems (OSTRSs). Often, termination of such programs heavily depends on sort information. But few techniques are currently available for proving termination of OSTRSs; and they often fail for interesting OSTRSs. In this paper we generalize the dependency pairs approach to prove termination of OSTRSs. Preliminary experiments suggest that this technique can succeed where existing ones fail, yielding easier and simpler termination proofs.
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
Effectively Checking or Disproving the Finite Variant Property
"... An equational theory decomposed into a set B of equational axioms and a set ∆ of rewrite rules has the finite variant (FV) property in the sense of ComonLundh and Delaune iff for each term t there is a finite set {t1,..., tn} of →∆,Bnormalized instances of t so that any instance of t normalizes to ..."
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An equational theory decomposed into a set B of equational axioms and a set ∆ of rewrite rules has the finite variant (FV) property in the sense of ComonLundh and Delaune iff for each term t there is a finite set {t1,..., tn} of →∆,Bnormalized instances of t so that any instance of t normalizes to an instance of some ti modulo B. This is a very useful property for cryptographic protocol analysis, and for solving both unification and disunification problems. Yet, at present the property has to be established by hand, giving a separate mathematical proof for each given theory: no checking algorithms seem to be known. In this paper we give both a necessary and a sufficient condition for FV from which we derive, both an algorithm ensuring the sufficient condition, and thus FV, and another disproving the necessary condition, and thus disproving FV. These algorithms can check automatically a number of examples and counterexamples of FV known in the literature.