Complete Monotonic Semantic Path Orderings

In this paper we present some original variations of the recursive path ordering. Additionally we de ne a restricted semantic path ordering which, in general, does not include the subterm relation, but is shown to be monotonic. By combining both kind of orderings we can prove (automatically) the termination of several (non-simply terminating) examples.

Lecture Notes in Computer Science, 2001

We present a new approach to termination analysis of logic programs. The essence of the approach is that we make use of general term-orderings (instead of level mappings), like it is done in transformational approaches to logic program termination analysis, but that we apply these orderings directly to the logic program and not to the term-rewrite system obtained through some transformation. We de ne a variant of acceptability, based on general term-orderings, and show how it is equivalent to LD-termination for well-moded, simply moded programs. We develop a demand driven, constraintbased approach to verify this acceptability-variant.

Conditional and Typed Rewriting Systems, 1991

It is well known that the disjoint union of terminating term rewriting systems does not yield a terminating system in general. We show that this undesirable phenomenon vanishes if one implements term rewriting by graph reduction: given two terminating term rewrite systems 7~o and 7~1, the graph reduction system implementing 7~0-F T~1 is terminating. In fact, we prove the stronger result that the graph reduction system for the union ~o U T~ 1 is terminating provided that the left-hand sides of ~i have no common function symbols with the right-hand sides of 7~1-i (i = O, 1). The implementation is complete in the sense that it computes a normal form for each term over the signature of R0 u 7~1. 1 I n t r o d u c t i o n The operational semantics of algebraic specification languages are usually based on term rewriting (see, e.g., [BCV 85], [BHK 89], [EM 85], [FG3M 85], [GH 86]). In this context, confluence and termination are particularly relevant properties of term rewriting systems. Hence, when specifications are structured as combinations of smaller subspecifications, the question arises whether these properties are preserved by a given combination mechanism. Recently, research in this direction has been started by,considering the disjoint union T£0 +7~1 of two term rewriting systems 7"£o and 7~1: the rule set of 7£o ~-7~1 is the union of the rules of T~ and T£1 where the function symbols occurring in T~ and 7~1 are disjoint (or are made disjoint by renaming). Toyama [Toy 87a] proves that 7~o + 7~1 is confluent *Work supported by ESPRIT project #390, PROSPECTRA, and by ESPRIT Basic Research Working Group #3264, COMPASS.

Proceedings. Thirteenth Annual IEEE Symposium on Logic in Computer Science (Cat. No.98CB36226)

A term rewrite system (TRS) terminates i its rules are contained in a reduction ordering >. In order to deal with any set of equations, including inherently non-terminating ones (like commutativity), TRS have been generalised to ordered TRS (E; >), where equations of E are applied in whatever direction agrees with >. The con uence of terminating TRS is well-known to be decidable, but for ordered TRS the decidability of con uence has been open. Here we show that the con uence of ordered TRS is decidable if ordering constraints for > can be solved in an adequate way, which holds in particular for the class of LPO orderings. For sets E of constrained equations, con uence is shown to be undecidable. Finally, ground reducibility is proved undecidable for ordered TRS.

We propose proving left-termination of well-moded logic programs by transforming them into term rewrite systems (TRSs). We introduce the new notions single-redex reduction, a reduction in which exactly one redex occurs in each term, and single-redex normalising, normalising with respect to the single-redex reduction. We describe a transformation of wellmoded logic programs into TRSs for which termination of the logic program follows from single-redex normalisation of the TRS, which is far stronger than previous results, since termination, innermost normalisation etc., all imply single-redex normalisation. A powerful tool for proving termination of TRSs that are obtained by the proposed transformation is semantic labelling. We use it for proving termination of implementations of quick-sort and generation of permutations.

Lecture Notes in Computer Science, 2009

Current techniques and tools for automated termination analysis of term rewrite systems (TRSs) are already very powerful. However, they fail for algorithms whose termination is essentially due to an inductive argument. Therefore, we show how to couple the dependency pair method for TRS termination with inductive theorem proving. As confirmed by the implementation of our new approach in the tool AProVE, now TRS termination techniques are also successful on this important class of algorithms.

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 definitions are given: one based on (strict) partial orders and another one based on preorders (or quasi-orders). We argue that there is no reason to choose the second one, while the first one has certain advantages.

Journal of Automated Reasoning, 2012

This paper introduces a propositional encoding for recursive path orders (RPO), in connection with dependency pairs. Hence, we capture in a uniform setting all common instances of RPO, i.e., lexicographic path orders (LPO), multiset path orders (MPO), and lexicographic path orders with status (LPOS). This facilitates the application of SAT solvers for termination analysis of term rewrite systems (TRSs).

Theoretical Computer Science, 1995

It is shown that a termination proof for a term rewriting system using a lexicographic path ordering yields a multiply recursive bound on the length of derivations, measured in the depth of the starting term. This result is essentially optimal since for every multiply recursive function f a rewrite system (which reduces under the lexicographic path ordering) can be found such that its derivation length cannot be bounded by f.

1995

We propose a new semantics for rewrite systems ba.~ed on interpreting rewrite rules as in equatioIlB between terms in an ordered algebra. In part.icular, we show thai the algebra. of normal forms in a terminating system is a uniqnely minimal covering of the term algebra. In the non-terminating ca..~e, the existence of this minimal covering is established in the comple tion of an ordered algebra formed by rewrit.ing sequences. We thus generalize the properties of normal forms far: non-terminating systelil~ to this minimal covering. ThesE' include the exi~tence of normal forms for arbitrary rewrite ~ystems, and their uniqueness for conBue-nt ~ystems, in which Ca<le the algebra of normal forms i~ isomorphic to the canonical quotient. algebra associated with the rule~ when seen as eqnations. This extend!> the benefits of alge braic semantics to systems with non-determinist.ic and non-t.erminating computations. V•le first study properties of abstract. order~, and then instantiat.e the~e to term rewriting sy~tems.