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Decidable Approximations of Sets of Descendants and Sets of Normal Forms
, 1997
"... We present here decidable approximations of sets of descendants and sets of normal forms of Term Rewriting Systems, based on specific tree automata techniques. In the context of rewriting logic, a Term Rewriting System is a program, and a normal form is a result of the program. Thus, approximations ..."
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Cited by 54 (18 self)
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We present here decidable approximations of sets of descendants and sets of normal forms of Term Rewriting Systems, based on specific tree automata techniques. In the context of rewriting logic, a Term Rewriting System is a program, and a normal form is a result of the program. Thus, approximations of sets of descendants and sets of normal forms provide tools for analysing a few properties of programs: we show how to compute a superset of results, to prove the sufficient completeness property, or to find a criterion for proving termination under a specific strategy, the sequential reduction strategy.
Generalized Sufficient Conditions for Modular Termination of Rewriting
 IN ENGINEERING, COMMUNICATION AND COMPUTING
, 1992
"... Modular properties of term rewriting systems, i.e. properties which are preserved under disjoint unions, have attracted an increasing attention within the last few years. Whereas confluence is modular this does not hold true in general for termination. By means of a careful analysis of potential cou ..."
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Cited by 51 (8 self)
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Modular properties of term rewriting systems, i.e. properties which are preserved under disjoint unions, have attracted an increasing attention within the last few years. Whereas confluence is modular this does not hold true in general for termination. By means of a careful analysis of potential counterexamples we prove the following abstract result. Whenever the disjoint union R1 \Phi R2 of two (finitely branching) terminating term rewriting systems R1 , R2 is nonterminating, then one of the systems, say R1 , enjoys an interesting (undecidable) property, namely it is not termination preserving under nondeterministic collapses, i.e. R1 \Phi fG(x; y) ! x; G(x; y) ! yg is nonterminating, and the other system R2 is collapsing, i.e. contains a rule with a variable right hand side. This result generalizes known sufficient criteria for modular termination of rewriting and provides the basis for a couple of derived modularity results. Furthermore, we prove that the minimal rank of pote...
Completeness of Combinations of Constructor Systems
 Journal of Symbolic Computation
, 1993
"... this paper we show that it is sufficient to impose the constructor discipline for obtaining the modularity of completeness. This result is a simple consequence of a quite powerful divide and conquer technique for establishing completeness of such constructor systems. Our approach is not limited to s ..."
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Cited by 31 (2 self)
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this paper we show that it is sufficient to impose the constructor discipline for obtaining the modularity of completeness. This result is a simple consequence of a quite powerful divide and conquer technique for establishing completeness of such constructor systems. Our approach is not limited to systems which are composed of disjoint parts. The importance of our method is that we may decompose a given constructor system into parts which possibly share function symbols and rewrite rules in order to infer completeness. We obtain a similar technique for semicompleteness, i.e. the combination of confluence and weak normalisation. 1. Introduction
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 29 (6 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...
On the Modularity of Confluence of ConstructorSharing Term Rewriting Systems
 In Proceedings of the 19th colloquium on Trees in Algebra and Programming, LNCS 787
, 1994
"... . Toyama's Theorem states that confluence is a modular property of disjoint term rewriting systems. This theorem does not generalize to combined systems with shared constructors. Thus the question arises naturally whether there are sufficient conditions which ensure the modularity of confluence ..."
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Cited by 13 (1 self)
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. Toyama's Theorem states that confluence is a modular property of disjoint term rewriting systems. This theorem does not generalize to combined systems with shared constructors. Thus the question arises naturally whether there are sufficient conditions which ensure the modularity of confluence in the presence of shared constructors. In particular, Kurihara and Krishna Rao posed the problem whether there are interesting sufficient conditions independent of termination. This question appeared as Problem 59 in the list of open problems in the theory of rewriting published recently [DJK93]. The present paper gives an affirmative answer to that question. Among other sufficient criteria, it is shown that confluence is preserved under the combination of constructorsharing systems if the systems are also normalizing. This in conjunction with the fact that normalization is modular for those systems implies the modularity of semicompleteness. 1 Introduction It is wellknown from software engi...
A structural analysis of modular termination of term rewriting systems
, 1991
"... Modular properties of term rewriting systems, i.e. properties which are preserved under disjoint unions, have attracted an increasing attention within the last few years. Whereas confluence is modular this does not hold true in general for termination. By result. Whenever the disjoint union R 1 \Phi ..."
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Cited by 9 (4 self)
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Modular properties of term rewriting systems, i.e. properties which are preserved under disjoint unions, have attracted an increasing attention within the last few years. Whereas confluence is modular this does not hold true in general for termination. By result. Whenever the disjoint union R 1 \Phi R 2 of two (finite) terminating term rewriting systems R 1, R 2 is nonterminating, then one of the systems, say R 1, enjoys an interesting (undecidable) property, namely it is not termination preserving under nondeterministic collapses, i.e. R 1 \Phi fG(x; y) ! x; G(x; y) ! yg is nonterminating, and the other system R 2 is collapsing, i.e. contains a rule with a variable right hand side. This result generalizes known sufficient syntactical criteria for modular termination of rewriting. Then we develop a specialized version of the `increasing interpretation method' for proving termination of combinations of term rewriting systems. This method is applied to establish modularity of termination for certain classes of term rewriting systems. In particular, termination turns out to be modular for the class of
Abstract Modularity
"... Modular rewriting seeks criteria under which rewrite systems inherit properties from their smaller subsystems. This divide and conquer methodology is particularly useful for reasoning about large systems where other techniques fail to scale adequately. Research has typically focused on reasoning ab ..."
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Cited by 1 (0 self)
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Modular rewriting seeks criteria under which rewrite systems inherit properties from their smaller subsystems. This divide and conquer methodology is particularly useful for reasoning about large systems where other techniques fail to scale adequately. Research has typically focused on reasoning about the modularity of specific properties for specific ways of combining specific forms of rewriting. This paper is, we believe, the first to ask a much more general question. Namely, what can be said about modularity independently of the specific form of rewriting, combination and property at hand. A priori there is no reason to believe that anything can actually be said about modularity without reference to the specifics of the particular systems etc. However, this paper shows that, quite surprisingly, much can indeed be said.
Proving Termination of Sequential Reduction Relation using Tree Automata
, 1997
"... The sequential reduction relation is a specific rewrite relation, rewriting terms by separate normalisations w.r.t. several systems, or modules. When considering simple combinations of systems, like disjoint or constructor sharing, termination (or weak termination) of each system implies termination ..."
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Cited by 1 (1 self)
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The sequential reduction relation is a specific rewrite relation, rewriting terms by separate normalisations w.r.t. several systems, or modules. When considering simple combinations of systems, like disjoint or constructor sharing, termination (or weak termination) of each system implies termination of the sequential reduction relation. In the case of more complex combinations, like hierarchical or general combination, there is no general result for termination of the sequential reduction relation. We propose here a criterion for termination of the sequential reduction relation in the hierarchical (and general) case. This criterion is based on decidable approximations of sets of normal forms, computed thanks to specific tree automata techniques. Some examples of application to modular termination proof of programs are also presented. Keywords: Term Rewriting, Termination, Modularity, Sequential Reduction Relation, Tree Automata, Approximation. 1 Introduction In the context of rewriti...
HIGHERORDER (NON)MODULARITY
"... Abstract. We show that, contrary to the situation in firstorder term rewriting, almost none of the usual properties of rewriting are modular for higherorder rewriting, irrespective of the higherorder rewriting format. For the particular format of simply typed applicative term rewriting systems we ..."
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Abstract. We show that, contrary to the situation in firstorder term rewriting, almost none of the usual properties of rewriting are modular for higherorder rewriting, irrespective of the higherorder rewriting format. For the particular format of simply typed applicative term rewriting systems we prove that modularity of confluence, normalization, and termination can be recovered by imposing suitable linearity constraints.