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11
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 49 (7 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...
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 following res ..."
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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 27 (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...
A Simple Proof of Sufficient Conditions for the Termination of the Disjoint Union of Term Rewriting Systems
 Bulletin of the European Association for Theoretical Computer Science 49
, 1993
"... this paper. For a comprehensive survey of more results in this field of term rewriting we refer to [Mid90]. Additionally, [Gra92] discusses more recent results. 2 Preliminaries ..."
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Cited by 11 (4 self)
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this paper. For a comprehensive survey of more results in this field of term rewriting we refer to [Mid90]. Additionally, [Gra92] discusses more recent results. 2 Preliminaries
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
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 ..."
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Cited by 9 (6 self)
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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.
Combinations of Simplifying Conditional Term Rewriting Systems
 In Proceedings of the 3rd International Workshop on Conditional Term Rewriting Systems
, 1992
"... . A conditional term rewriting system (CTRS) is called simplifying if there exists a simplification ordering ? on terms such that the lefthand side of any rewrite rule is greater than the righthand side and the terms occurring in the conditions of that rule. If a simplifying join CTRS consists of ..."
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Cited by 4 (2 self)
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. A conditional term rewriting system (CTRS) is called simplifying if there exists a simplification ordering ? on terms such that the lefthand side of any rewrite rule is greater than the righthand side and the terms occurring in the conditions of that rule. If a simplifying join CTRS consists of finitely many rules, it is terminating and the applicability of a rewrite rule is decidable by recursively reducing the terms in the conditions. Consider two finite CTRSs R1 and R2 which may share constructors (symbols that do not occur at the root position of the lefthand side of any rewrite rule) but no other function symbols. It will be shown that the combined CTRS R = R1[R2 is simplifying if and only if R1 and R2 are simplifying. Moreover, confluence is a modular property of finite simplifying join CTRSs. 1 Introduction During the past decade, term rewriting has gained an enormous importance in fields of computer science concerned with symbolic manipulation. Among others, it may be vie...
Termination is not Modular for Confluent VariablePreserving Term Rewriting Systems
, 1993
"... Introduction A term rewriting system (TRS for short) is a pair (F ; R) consisting of a signature F and a set R of rewrite rules [1, 4]. Every rewrite rule l ! r 2 R, where l; r are terms from T (F ; V), must satisfy the following two constraints: (i) l is not a variable, and (ii) variables occurr ..."
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Cited by 4 (2 self)
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Introduction A term rewriting system (TRS for short) is a pair (F ; R) consisting of a signature F and a set R of rewrite rules [1, 4]. Every rewrite rule l ! r 2 R, where l; r are terms from T (F ; V), must satisfy the following two constraints: (i) l is not a variable, and (ii) variables occurring in r also occur in l. Two TRSs are disjoint if their signatures are disjoint. A property P of TRSs is called modular, if for all disjoint TRSs (F 1 ; R 1 ) and (F 2 ; R 2 ) their disjoint union (F 1 ]F 2 ; R 1 ] R 2 ) has the property P if and only if both (F 1 ;<F30
Modular termination of rconsistent and leftlinear term rewriting systems
 Theoretical Computer Science
, 1995
"... A modular property of term rewriting systems is one that holds for the direct sum of two disjoint term rewriting systems, iff it holds for every involved term rewriting system. A term rewriting system is rconsistent, iff there is no term that can be rewritten to two different variables. We show tha ..."
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Cited by 3 (0 self)
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A modular property of term rewriting systems is one that holds for the direct sum of two disjoint term rewriting systems, iff it holds for every involved term rewriting system. A term rewriting system is rconsistent, iff there is no term that can be rewritten to two different variables. We show that the subclass of leftlinear and rconsistent term rewriting systems has the modular termination property. This subclass may also contain nonconfluent term rewriting systems. Since confluence implies rconsistency, this constitutes a generalisation of the theorem of Toyama, Klop, and Barendregt on the modularity of termination for confluent and leftlinear term rewriting systems. Acknowledgements.
On Properties of Monoids That Are Modular for Free Products and for Certain Free Products With Amalgamated Submonoids
, 1997
"... A property P of stringrewriting systems is called modular if the disjoint union R 1 [ R 2 of two stringrewriting systems R 1 and R 2 has this property if and only if R 1 and R 2 both have this property. Analogously, a property P of monoids is modular if the free product M 1 M 2 of two monoids M ..."
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Cited by 2 (2 self)
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A property P of stringrewriting systems is called modular if the disjoint union R 1 [ R 2 of two stringrewriting systems R 1 and R 2 has this property if and only if R 1 and R 2 both have this property. Analogously, a property P of monoids is modular if the free product M 1 M 2 of two monoids M 1 and M 2 has this property if and only if M 1 and M 2 both have this property. Since the stringrewriting systems form a subclass of the linear termrewriting systems, it follows that many properties are modular for stringrewriting systems. Here we give a summary of these modularity results, providing fairly simple proofs for them. In addition, we show that the property of having finite derivation type (FDT) is modular for finitely presented monoids. In a second part we consider the algebraic operation of forming a free product with amalgamating certain submonoids, which corresponds to a nondisjoint union of stringrewriting systems. Following Toyama and Aoto (1996) we show that certain ...