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

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Extra variables in a clause are variables which occur in the body but not in the head. It has been argued that extra variables are necessary and contribute to the expressive power of logic languages. In the first part of this paper, we show that this is not true in general. For this purpose, we provide a simple syntactic transformation of each logic program into a logic program without extra variables. Moreover, we show a strong correspondence between the original and the transformed program with respect to the declarative and the operational semantics. In the second part of this paper, we use a similar technique to provide new completeness results for equational logic programs with extra variables. In equational logic programming it is well known that extra variables cause problems since narrowing, the standard operational semantics for equational logic programming, may become incomplete in the presence of extra variables. Since extra variables are useful from a programmin...

Introduction Interest in the theory and applications of rewriting has been growing rapidly, as evidenced in part by four conference proceedings #including this one# #15, 26, 41,66#; three workshop proceedings #33, 47, 77#; #ve special journal issues #5,88, 24, 40, 67#; more than ten surveys #2,7,27, 28, 44, 56,57,76, 82, 81#; one edited collection of papers #1#; four monographs #3, 12,55,65#; and seven books #four of them still in progress# #8,9, 35, 54, 60,75, 84#. To encourage and stimulate continued progress in this area, wehave collected #with the help of colleagues# a number of problems that appear to us to be of interest and regarding whichwe do not know the answer. Questions on rewriting and other equational paradigms have been included; manyhave not aged su#ciently to be accorded the appellation #open problem". Wehave limited ourselves to theoretical questions, though there are certainly many additional interesting questions relating to applications and implementation

. This work investigates inductive theorem proving techniques for first-order functions whose meaning and domains can be specified by Horn Clauses built up from the equality and finitely many unary membership predicates. In contrast with other works in the area, constructors are not assumed to be free. Techniques originating from tree automata are used to describe ground constructor terms in normal form, on which the induction proofs are built up. Validity of (free) constructor clauses is checked by an original technique relying on the recent discovery of a complete axiomatisation of finite trees and their rational subsets. Validity of clauses with defined symbols or non-free constructor terms is reduced to the latter case by appropriate inference rules using a notion of ground reducibility for these symbols. We show how to check this property by generating proof obligations which can be passed over to the inductive prover. 1 Introduction The need for large formal proofs has lead to t...

Erlang is a functional programming language developed by Ericsson Telecom, which is particularly well suited for implementing concurrent processes. In this paper we show how methods from the area of term rewriting are presently used at Ericsson. To verify properties of processes, such a property is transformed into a termination problem of a conditional term rewriting system (CTRS). Subsequently, this termination proof can be performed automatically using dependency pairs. The paper illustrates how the dependency pair technique can be applied for termination proofs of conditional TRSs. Secondly, we present three refinements of this technique, viz. narrowing, rewriting, and instantiating dependency pairs. These refinements are not only of use in the industrial applications sketched in this paper, but they are generally applicable to arbitrary (C)TRSs. Thus, in this way dependency pairs can be used to prove termination of even more (C)TRSs automatically.

. Recently we have shown the following abstract result for unconditional term rewriting systems (TRSs). Whenever the disjoint union R1 \Phi R2 of two (finite) terminating TRSs R1 , R2 is non-terminating, then one of the systems, say R1 , enjoys an interesting property, namely it is not termination preserving under non-deterministic collapses, i.e. R1 \Phi fG(x; y) ! x; G(x; y) ! yg is non-terminating, and the other system R2 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. Here we extend this result and derived sufficient criteria for modularity of termination to the case of conditional term rewriting systems (CTRSs). Moreover we relate various definitions of notions related to termination of CTRSs to each other and discuss some subtleties and problems concerning extra variables in the rules. 1 Introduction From a theoretical point of view and also for efficiency ...

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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 non-terminating, then one of the systems, say R 1, enjoys an interesting (undecidable) property, namely it is not termination preserving under non-deterministic collapses, i.e. R 1 \Phi fG(x; y) ! x; G(x; y) ! yg is non-terminating, 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

We prove constructively that the class of ground-confluent and decreasing conditional term rewriting systems (CTRSs) (without extra variables) coincides with the class of orthogonal and terminating, unconditional term rewriting systems (TRSs). TRSs being included in CTRSs, this result follows from a transformation from any ground-confluent and decreasing CTRS specifying a computable function f into a TRS with the mentioned properties for f . The generated TRS is ordersorted, but we outline a similar transformation yielding an unsorted TRS.