Results 1  10
of
27
Mechanizing and Improving Dependency Pairs
 Journal of Automated Reasoning
, 2006
"... Abstract. The dependency pair technique [1, 11, 12] is a powerful method for automated termination and innermost termination proofs of term rewrite systems (TRSs). For any TRS, it generates inequality constraints that have to be satisfied by wellfounded orders. We improve the dependency pair techni ..."
Abstract

Cited by 105 (40 self)
 Add to MetaCart
(Show Context)
Abstract. The dependency pair technique [1, 11, 12] is a powerful method for automated termination and innermost termination proofs of term rewrite systems (TRSs). For any TRS, it generates inequality constraints that have to be satisfied by wellfounded orders. We improve the dependency pair technique by considerably reducing the number of constraints produced for (innermost) termination proofs. Moreover, we extend transformation techniques to manipulate dependency pairs which simplify (innermost) termination proofs significantly. In order to fully mechanize the approach, we show how transformations and the search for suitable orders can be mechanized efficiently. We implemented our results in the automated termination prover AProVE and evaluated them on large collections of examples.
Improving Dependency Pairs
 JOURNAL OF AUTOMATED REASONING
, 2003
"... The dependency pair approach [2, 11, 12] is one of the most powerful techniques for termination and innermost termination proofs of term rewrite systems (TRSs). For any TRS, it generates inequality constraints that have to be satisfied by weakly monotonic wellfounded orders. We improve the dependen ..."
Abstract

Cited by 23 (10 self)
 Add to MetaCart
The dependency pair approach [2, 11, 12] is one of the most powerful techniques for termination and innermost termination proofs of term rewrite systems (TRSs). For any TRS, it generates inequality constraints that have to be satisfied by weakly monotonic wellfounded orders. We improve the dependency pair approach by considerably reducing the number of constraints produced for (innermost) termination proofs. Moreover,
The Longest Perpetual Reductions in Orthogonal Expression Reduction Systems
 In: Proc. of the 3 rd International Conference on Logical Foundations of Computer Science, LFCS'94, A. Nerode and Yu.V. Matiyasevich, eds., Springer LNCS
, 1994
"... We consider reductions in Orthogonal Expression Reduction Systems (OERS), that is, Orthogonal Term Rewriting Systems with bound variables and substitutions, as in the calculus. We design a strategy that for any given term t constructs a longest reduction starting from t if t is strongly normaliza ..."
Abstract

Cited by 19 (8 self)
 Add to MetaCart
(Show Context)
We consider reductions in Orthogonal Expression Reduction Systems (OERS), that is, Orthogonal Term Rewriting Systems with bound variables and substitutions, as in the calculus. We design a strategy that for any given term t constructs a longest reduction starting from t if t is strongly normalizable, and constructs an infinite reduction otherwise. The Conservation Theorem for OERSs follows easily from the properties of the strategy. We develop a method for computing the length of a longest reduction starting from a strongly normalizable term. We study properties of pure substitutions and several kinds of similarity of redexes. We apply these results to construct an algorithm for computing lengths of longest reductions in strongly persistent OERSs that does not require actual transformation of the input term. As a corollary, we have an algorithm for computing lengths of longest developments in OERSs. 1 Introduction A strategy is perpetual if, given a term t, it constructs an infinit...
Modularity of Termination Using Dependency Pairs
 PROC. 9TH RTA
, 1997
"... The framework of dependency pairs allows automated termination and innermost termination proofs for many TRSs where such proofs were not possible before. In this paper we present a refinement of this framework in order to prove termination in a modular way. Our modularity results significantly incre ..."
Abstract

Cited by 18 (10 self)
 Add to MetaCart
The framework of dependency pairs allows automated termination and innermost termination proofs for many TRSs where such proofs were not possible before. In this paper we present a refinement of this framework in order to prove termination in a modular way. Our modularity results significantly increase the class of term rewriting systems where termination resp. innermost termination can be proved automatically. Moreover, the modular approach to dependency pairs yields new modularity criteria which extend previous results in this area considerably. In particular, existing results for modularity of innermost termination can easily be obtained as direct consequences of our new criteria.
Pushing the Frontiers of Combining Rewrite Systems Farther Outwards
 In Proceedings of the Second International Workshop on Frontiers of Combining Systems, FroCos '98, Applied Logic Series
, 1998
"... It is well known that simple termination is modular for certain kinds of combinations of term rewriting systems (TRSs). This result is of practical relevance because most techniques for (automated) termination proofs use simplification orderings, so they show in fact simple termination. On the other ..."
Abstract

Cited by 11 (3 self)
 Add to MetaCart
It is well known that simple termination is modular for certain kinds of combinations of term rewriting systems (TRSs). This result is of practical relevance because most techniques for (automated) termination proofs use simplification orderings, so they show in fact simple termination. On the other hand, in practice many systems are nonsimply terminating. In order to cope with such systems, Arts and Giesl developed the dependency pair approach. By using (quasi)simplification orderings in combination with dependency pairs, it is possible to prove termination of nonsimply terminating systems automatically. It is natural to ask whether modularity of simple termination can be extended to the class of those systems which can be handled by this technique. In this paper we show that this is indeed the case. In this way, the class of TRSs for which termination can be proved in a modular way is extended significantly. 1 Introduction Modularity is a wellknown paradigm in computer science. ...
Modular termination of contextsensitive rewriting
 IN PROC. 4TH PPDP
, 2002
"... Contextsensitive rewriting (CSR) has recently emerged as an interesting and flexible paradigm that provides a bridge between the abstract world of general rewriting and the (more) applied setting of declarative specification and programming languages such as OBJ*, CafeOBJ, ELAN, and Maude. A natura ..."
Abstract

Cited by 9 (6 self)
 Add to MetaCart
Contextsensitive rewriting (CSR) has recently emerged as an interesting and flexible paradigm that provides a bridge between the abstract world of general rewriting and the (more) applied setting of declarative specification and programming languages such as OBJ*, CafeOBJ, ELAN, and Maude. A natural approach to study properties of programs written in these languages is to model them as contextsensitive rewriting systems. Here we are especially interested in proving termination of such systems, and thereby providing methods to establish termination of e.g. OBJ* programs. For proving termination of contextsensitive rewriting, there exist a few transformation methods, that reduce the problem to termination of a transformed ordinary term rewriting system (TRS). These transformations, however, have some serious drawbacks. In particular, most of them do not seem to support a modular analysis of the termination problem. In this paper we will show that a substantial part of the wellknown theory of modular term rewriting can be extended to CSR, via a thorough analysis of the additional complications arising from contextsensitivity. More precisely, we will mainly concentrate on termination (properties). The obtained modularity results correspond nicely to the fact that in the above languages the modular design of programs and specifications is explicitly promoted, since it can now also be complemented by modular analysis techniques.
Transforming Conditional Rewrite Systems with Extra Variables into Unconditional Systems
 In: Proc. LPAR '99, Tblisi
, 1999
"... . Deterministic conditional rewrite systems are interesting because they permit extra variables on the righthand sides of the rules. If such a system is quasireductive, then it is terminating and has a computable rewrite relation. It will be shown that every deterministic CTRS R can be transformed ..."
Abstract

Cited by 8 (2 self)
 Add to MetaCart
. Deterministic conditional rewrite systems are interesting because they permit extra variables on the righthand sides of the rules. If such a system is quasireductive, then it is terminating and has a computable rewrite relation. It will be shown that every deterministic CTRS R can be transformed into an unconditional TRS U(R) such that termination of U(R) implies quasireductivity of R. The main theorem states that quasireductivity of R implies innermost termination of U(R). These results have interesting applications in two different areas: modularity in term rewriting and termination proofs of wellmoded logic programs. 1 Introduction Conditional term rewriting systems (CTRSs) are the basis of functional logic programming; see [Han94] for an overview of this field. In CTRSs variables on the righthand side of a rewrite rule which do not occur on the lefthand side are often forbidden. This is because it is in general not clear how to instantiate them. On the other hand, a rest...
Perpetual Reductions in λCalculus
, 1999
"... This paper surveys a part of the theory of fireduction in λcalculus which might aptly be called perpetual reductions. The theory is concerned with perpetual reduction strategies, i.e., reduction strategies that compute infinite reduction paths from λterms (when possible), and with perpetual r ..."
Abstract

Cited by 8 (0 self)
 Add to MetaCart
This paper surveys a part of the theory of fireduction in λcalculus which might aptly be called perpetual reductions. The theory is concerned with perpetual reduction strategies, i.e., reduction strategies that compute infinite reduction paths from λterms (when possible), and with perpetual redexes, i.e., redexes whose contraction in λterms preserves the possibility (when present) of infinite reduction paths. The survey not only recasts classical theorems in a unified setting, but also offers new results, proofs, and techniques, as well as a number of applications to problems in λcalculus and type theory.
Normalisation in Weakly Orthogonal Rewriting
, 1999
"... . A rewrite sequence is said to be outermostfair if every outermost redex occurrence is eventually eliminated. Outermostfair rewriting is known to be (head)normalising for almost orthogonal rewrite systems. In this paper we study (head)normalisation for the larger class of weakly orthogonal rewr ..."
Abstract

Cited by 8 (4 self)
 Add to MetaCart
. A rewrite sequence is said to be outermostfair if every outermost redex occurrence is eventually eliminated. Outermostfair rewriting is known to be (head)normalising for almost orthogonal rewrite systems. In this paper we study (head)normalisation for the larger class of weakly orthogonal rewrite systems. Normalisation is established and a counterexample against headnormalisation is provided. Nevertheless, infinitary normalisation, which is usually obtained as a corollary of headnormalisation, is shown to hold. 1 Introduction The term f(a) in the term rewrite system fa ! a; f(x) ! bg can be rewritten to normal form b, but is also the starting point of the infinite rewrite sequence f(a) ! f(a) ! : : :. It is then of interest to design a normalising strategy, i.e. a restriction on rewriting which guarantees to reach a normal form if one can be reached. How to design a normalising strategy? Observe that in the example the normal form b was reached by contracting the redex closest...
Perpetuality and Uniform Normalization in Orthogonal Rewrite Systems
 INFORMATION AND COMPUTATION
"... We present two characterizations of perpetual redexes, which are redexes whose contractions retain the possibility of infinite reductions. These characterizations generalize and strengthen existing criteria for the perpetuality of redexes in orthogonal Term Rewriting Systems and the calculus due ..."
Abstract

Cited by 7 (2 self)
 Add to MetaCart
We present two characterizations of perpetual redexes, which are redexes whose contractions retain the possibility of infinite reductions. These characterizations generalize and strengthen existing criteria for the perpetuality of redexes in orthogonal Term Rewriting Systems and the calculus due to Bergstra and Klop, and others. To unify our results with those in the literature, we introduce Contextsensitive Conditional Expression Reduction Systems (CCERSs) and prove confluence for orthogonal CCERSs. We then define a perpetual onestep reduction strategy which enables one to construct minimal (w.r.t. Levy's permutation ordering on reductions) infinite reductions in orthogonal CCERSs. We then prove (1) perpetuality (in a specific context) of a redex whose contraction does not erase potentially infinite arguments, which are possibly finite (i.e., strongly normalizable) arguments that may become infinite after a number of outside steps, and (2) perpetuality (in every con...