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35
Term Rewriting Systems
, 1992
"... Term Rewriting Systems play an important role in various areas, such as abstract data type specifications, implementations of functional programming languages and automated deduction. In this chapter we introduce several of the basic comcepts and facts for TRS's. Specifically, we discuss Abstract Re ..."
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Cited by 567 (16 self)
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Term Rewriting Systems play an important role in various areas, such as abstract data type specifications, implementations of functional programming languages and automated deduction. In this chapter we introduce several of the basic comcepts and facts for TRS's. Specifically, we discuss Abstract Reduction Systems
Homeomorphic Embedding for Online Termination
 STATIC ANALYSIS. PROCEEDINGS OF SAS’98, LNCS 1503
, 1998
"... Recently wellquasi orders in general, and homeomorphic embedding in particular, have gained popularity to ensure the termination of program analysis, specialisation and transformation techniques. In this paper, ..."
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Cited by 61 (8 self)
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Recently wellquasi orders in general, and homeomorphic embedding in particular, have gained popularity to ensure the termination of program analysis, specialisation and transformation techniques. In this paper,
A Notation for Lambda Terms I: A Generalization of Environments
 THEORETICAL COMPUTER SCIENCE
, 1994
"... A notation for lambda terms is described that is useful in contexts where the intensions of these terms need to be manipulated. This notation uses the scheme of de Bruijn for eliminating variable names, thus obviating ffconversion in comparing terms. This notation also provides for a class of terms ..."
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Cited by 33 (12 self)
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A notation for lambda terms is described that is useful in contexts where the intensions of these terms need to be manipulated. This notation uses the scheme of de Bruijn for eliminating variable names, thus obviating ffconversion in comparing terms. This notation also provides for a class of terms that can encode other terms together with substitutions to be performed on them. The notion of an environment is used to realize this `delaying' of substitutions. The precise mechanism employed here is, however, more complex than the usual environment mechanism because it has to support the ability to examine subterms embedded under abstractions. The representation presented permits a ficontraction to be realized via an atomic step that generates a substitution and associated steps that percolate this substitution over the structure of a term. The operations on terms that are described also include ones for combining substitutions so that they might be performed simultaneously. Our notatio...
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 ...
Wellfoundedness of Term Orderings
 CTRS 94 (WORKSHOP ON CONDITIONAL AND TYPED TERM REWRITING SYSTEMS
, 1994
"... Wellfoundedness is the essential property of orderings for proving termination. We introduce a simple criterion on term orderings such that any term ordering possessing the subterm property and satisfying this criterion is wellfounded. The usual path orders fulfil this criterion, yielding a muc ..."
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Cited by 19 (5 self)
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Wellfoundedness is the essential property of orderings for proving termination. We introduce a simple criterion on term orderings such that any term ordering possessing the subterm property and satisfying this criterion is wellfounded. The usual path orders fulfil this criterion, yielding a much simpler proof of wellfoundedness than the classical proof depending on Kruskal's theorem. Even more, our approach covers nonsimplification orders like spo and gpo which can not be dealt with by Kruskal's theorem. For finite alphabets we present completeness results, i. e., a term rewriting system terminates if and only if it is compatible with an order satisfying the criterion. For infinite alphabets the same completeness results hold for a slightly different criterion.
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 16 (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...
Syntactical Analysis of Total Termination
 In Proceedings of the 4th International Conference on Algebraic and Logic Programming
, 1994
"... Termination is an important issue in the theory of term rewriting. In general termination is undecidable. There are nevertheless several methods successful in special cases. In [5] we introduced the notion of total termination: basically terms are interpreted compositionally in a total wellfounded ..."
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Cited by 15 (8 self)
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Termination is an important issue in the theory of term rewriting. In general termination is undecidable. There are nevertheless several methods successful in special cases. In [5] we introduced the notion of total termination: basically terms are interpreted compositionally in a total wellfounded order, in such a way that rewriting chains map to descending chains. Total termination is thus a semantic notion. It turns out that most of the usual techniques for proving termination fall within the scope of total termination. This paper consists of two parts. In the first part we introduce a generalization of recursive path order presenting a new proof of its wellfoundedness without using Kruskal's theorem. We also show that the notion of total termination covers this generalization. In the second part we present some syntactical characterizations of total termination that can be used to prove that many term rewriting systems are not totally terminating and hence outside the scope of the...
Algorithms for ordinal arithmetic
 In 19th International Conference on Automated Deduction (CADE
, 2003
"... Abstract. Proofs of termination are essential for establishing the correct behavior of computing systems. There are various ways of establishing termination, but the most general involves the use of ordinals. An example of a theorem proving system in which ordinals are used to prove termination is A ..."
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Cited by 11 (5 self)
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Abstract. Proofs of termination are essential for establishing the correct behavior of computing systems. There are various ways of establishing termination, but the most general involves the use of ordinals. An example of a theorem proving system in which ordinals are used to prove termination is ACL2. In ACL2, every function defined must be shown to terminate using the ordinals up to ɛ0. We use a compact notation for the ordinals up to ɛ0 (exponentially more succinct than the one used by ACL2) and define efficient algorithms for ordinal addition, subtraction, multiplication, and exponentiation. In this paper we describe our notation and algorithms, prove their correctness, and analyze their complexity. 1