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52
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 ..."
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Cited by 105 (40 self)
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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.
Modular Properties of Composable Term Rewriting Systems
 Journal of Symbolic Computation
, 1995
"... this paper we prove several new modularity results for unconditional and conditional term rewriting systems. Most of the known modularity results for the former systems hold for disjoint or constructorsharing combinations. Here we focus on a more general kind of combination: socalled composable sy ..."
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Cited by 56 (6 self)
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this paper we prove several new modularity results for unconditional and conditional term rewriting systems. Most of the known modularity results for the former systems hold for disjoint or constructorsharing combinations. Here we focus on a more general kind of combination: socalled composable systems. As far as conditional term rewriting systems are concerned, all known modularity result but one apply only to disjoint systems. Here we investigate conditional systems which may share constructors. Furthermore, we refute a conjecture of Middeldorp (1990, 1993). 1. Introduction
Complete Monotonic Semantic Path Orderings
 In Proc. 17th CADE, LNAI 1831
, 2000
"... Although theoretically it is very powerful, the semantic path ordering (SPO) is not so useful in practice, since its monotonicity has to be proved by hand for each concrete term rewrite system (TRS). In this paper we present a monotonic variation of SPO, called MSPO. It characterizes termination ..."
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Cited by 34 (9 self)
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Although theoretically it is very powerful, the semantic path ordering (SPO) is not so useful in practice, since its monotonicity has to be proved by hand for each concrete term rewrite system (TRS). In this paper we present a monotonic variation of SPO, called MSPO. It characterizes termination, i.e., a TRS is terminating if and only if its rules are included in some MSPO. Hence MSPO is a complete termination method. On the practical side, it can be easily automated using as ingredients standard interpretations and generalpurpose orderings like RPO. This is shown to be a sufficiently powerful way to handle several nontrivial examples and to obtain methods like dummy elimination or dependency pairs as particular cases. Finally, we obtain some positive modularity results for termination based on MSPO. 1 Introduction Rewrite systems are sets of rules (directed equations) used to compute by repeatedly replacing parts of a given formula with equal ones until the simplest po...
Modular & Incremental Automated Termination Proofs
 Journal of Automated Reasoning
, 2004
"... We propose a modular approach of term rewriting systems, making the best of their hierarchical structure. We define rewriting modules and then provide a new method to prove termination incrementally. We obtain new and powerful termination criteria for standard rewriting. Our policy of restraining ..."
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Cited by 32 (6 self)
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We propose a modular approach of term rewriting systems, making the best of their hierarchical structure. We define rewriting modules and then provide a new method to prove termination incrementally. We obtain new and powerful termination criteria for standard rewriting. Our policy of restraining termination itself (thus relaxing constraints over hierarchies components) together with the generality of the module approach are sufficient to express previous results and methods the premisses of which either include restrictions over unions or make a particular reduction strategy compulsory.
Unravelings and Ultraproperties
 In Proceedings of the Fifth International Conference on Algebraic and Logic Programming (ALP'96), volume 1139 of LNCS
, 1996
"... Conditional rewriting is universally recognized as being much more complicated than unconditional rewriting. In this paper we study how much of conditional rewriting can be automatically inferred from the simpler theory of unconditional rewriting. We introduce a new tool, called unraveling, to autom ..."
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Cited by 30 (3 self)
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Conditional rewriting is universally recognized as being much more complicated than unconditional rewriting. In this paper we study how much of conditional rewriting can be automatically inferred from the simpler theory of unconditional rewriting. We introduce a new tool, called unraveling, to automatically translate a conditional term rewriting system (CTRS) into a term rewriting system (TRS). An unraveling enables to infer properties of a CTRS by studying the corresponding ultraproperty on the corresponding TRS. We show how to rediscover properties like decreasingness, and to give easy proofs of some existing results on CTRSs. Moreover, we show how unravelings provide a valuable tool to study modularity of CTRSs, automatically giving a multitude of new results.
Improved modular termination proofs using dependency pairs
 In Proc. 2nd IJCAR, volume 3097 of LNAI
, 2004
"... Abstract. The dependency pair approach is one of the most powerful techniques for automated (innermost) termination proofs of term rewrite systems (TRSs). For any TRS, it generates inequality constraints that have to be satisfied by wellfounded orders. However, proving innermost termination is cons ..."
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Cited by 29 (8 self)
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Abstract. The dependency pair approach is one of the most powerful techniques for automated (innermost) termination proofs of term rewrite systems (TRSs). For any TRS, it generates inequality constraints that have to be satisfied by wellfounded orders. However, proving innermost termination is considerably easier than termination, since the constraints for innermost termination are a subset of those for termination. We show that surprisingly, the dependency pair approach for termination can be improved by only generating the same constraints as for innermost termination. In other words, proving full termination becomes virtually as easy as proving innermost termination. Our results are based on splitting the termination proof into several modular independent subproofs. We implemented our contributions in the automated termination prover AProVE and evaluated them on large collections of examples. These experiments show that our improvements increase the power and efficiency of automated termination proving substantially. 1
On the Modularity of Termination of Term Rewriting Systems
 Theoretical Computer Science
, 1993
"... It is wellknown that termination is not a modular property of term rewriting systems, i.e., it is not preserved under disjoint union. The objective of this paper is to provide a "uniform framework" for sufficient conditions which ensure the modularity of termination. We will prove the fol ..."
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Cited by 29 (3 self)
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It is wellknown that termination is not a modular property of term rewriting systems, i.e., it is not preserved under disjoint union. The objective of this paper is to provide a "uniform framework" for sufficient conditions which ensure the modularity of termination. We will prove the following result. Whenever the disjoint union of two terminating term rewriting systems is nonterminating, then one of the systems is not C E terminating (i.e., it looses its termination property when extended with the rules Cons(x; y) ! x and Cons(x; y) ! y) and the other is collapsing. This result has already been achieved by Gramlich [7] for finitely branching term rewriting systems. A more sophisticated approach is necessary, however, to prove it in full generality. Most of the known sufficient criteria for the preservation of termination [24, 15, 13, 7] follow as corollaries from our result, and new criteria are derived. This paper particularly settles the open question whether simple termination ...
Relating Innermost, Weak, Uniform and Modular Termination of Term Rewriting Systems
, 1993
"... We investigate restricted termination and confluence properties of term rewriting systems, in particular weak termination and innermost termination, and their interrelation. New criteria are provided which are sufficient for the equivalence of innermost / weak termination and uniform termination of ..."
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Cited by 28 (5 self)
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We investigate restricted termination and confluence properties of term rewriting systems, in particular weak termination and innermost termination, and their interrelation. New criteria are provided which are sufficient for the equivalence of innermost / weak termination and uniform termination of term rewriting systems. These criteria provide interesting possibilities to infer completeness, i.e. termination plus confluence, from restricted termination and confluence properties. Using these basic results we are also able to prove some new results about modular termination of rewriting. In particular, we show that termination is modular for some classes of innermost terminating and locally confluent term rewriting systems, namely for nonoverlapping and even for overlay systems. As an easy consequence this latter result also entails a simplified proof of the fact that completeness is a decomposable property of socalled constructor systems. Furthermore we show how to obtain similar re...
Simple Termination of Rewrite Systems
 Theoretical Computer Science
, 1997
"... In this paper we investigate the concept of simple termination. A term rewriting system is called simply terminating if its termination can be proved by means of a simplification order. The basic ingredient of a simplification order is the subterm property, but in the literature two different defini ..."
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Cited by 21 (2 self)
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In this paper we investigate the concept of simple termination. A term rewriting system is called simply terminating if its termination can be proved by means of a simplification order. The basic ingredient of a simplification order is the subterm property, but in the literature two different definitions are given: one based on (strict) partial orders and another one based on preorders (or quasiorders). We argue that there is no reason to choose the second one, while the first one has certain advantages. Simplification orders are known to be wellfounded orders on terms over a finite signature. This important result no longer holds if we consider infinite signatures. Nevertheless, wellknown simplification orders like the recursive path order are also wellfounded on terms over infinite signatures, provided the underlying precedence is wellfounded. We propose a new definition of simplification order, which coincides with the old one (based on partial orders) in case of finite signatu...
Monads and Modular Term Rewriting
, 1997
"... . Monads can be used to model term rewriting systems by generalising the wellknown equivalence between universal algebra and monads on the category Set. In [Lu96], the usefulness of this semantics was demonstrated by giving a purely categorical proof of the modularity of confluence for the disjoint ..."
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Cited by 19 (14 self)
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. Monads can be used to model term rewriting systems by generalising the wellknown equivalence between universal algebra and monads on the category Set. In [Lu96], the usefulness of this semantics was demonstrated by giving a purely categorical proof of the modularity of confluence for the disjoint union of term rewriting systems (Toyama's theorem). This paper provides further support for the use of monads in term rewriting by giving a categorical proof of the most general theorem concerning the modularity of strong normalisation. In the process, we also improve upon some technical aspects of the earlier work. 1 Introduction Term rewriting systems (TRSs) are widely used throughout computer science as they provide an abstract model of computation while retaining a relatively simple syntax and semantics. Reasoning about large term rewriting systems can be very difficult and an alternative is to define structuring operations which build large term rewriting systems from smaller ones. Of...