Sparse Tiling Through Overlap Closures for Termination of String Rewriting

A strictly locally testable language is characterized by its set of admissible factors, prefixes and suffixes, called tiles. We over-approximate reachability sets in string rewriting by languages defined by sparse sets of tiles, containing only those that are reachable in derivations. Using the partial algebra defined by a tiling for semantic labeling, we obtain a transformational method for proving local termination. These algebras can be represented efficiently as finite automata of a certain shape. Using a known result on forward closures, and a new characterisation of overlap closures, we can automatically prove termination and relative termination, respectively. We report on experiments showing the strength of the method

Omega-Termination is Undecidable for Totally Terminating Term Rewriting Systems

We give a complete proof of the fact that the following problem is undecidable: Given: A term rewriting system, where the termination of its rewrite relation is provable by a total reduction order on ground terms, Wanted: Does there exist a strictly monotonic interpretation in the positive integers that proves termination? Keywords: term rewriting, termination, total termination, omega-termination, termination hierarchy, termination type 1 Introduction Termination of a term rewriting system (TRS), i.e. the nonexistence of an infinite rewrite reduction, is one of the key notions in term rewriting. It is the basis of a number of decision algorithms for properties undecidable in the general case. It is known that termination is undecidable for TRSs, even for the restricted case of TRS with only unary function symbols [7], and for the case of one-rule TRSs [2]. The following true hierarchy of TRSs is called the termination hierarchy [12]. termination ) non-self-embedding ) simple te..

Relative termination

"Relative termination" is a property that generalizes both termination and "termination modulo". In order to prove that a term rewrite system relatively terminates, one may reuse the common termination quasiorderings. Further proof methods become available by the cooperation property. Relative termination sets up new proof techniques for termination and confluence. The usefulness of the notion of relative termination is finally demonstrated by a proof of completeness for "reduced narrowing" and "normal narrowing", two attractive variants of the narrowing procedure

On a monotonic semantic path ordering

The semantic path ordering preceq_spo is an ordering that allows to prove termination of term rewriting systems. Unlike other such orderings, it is not monotonic. We construct two monotonic suborderings preceq_cspo, preceq_mspo, of preceq_spo. Both orderings rely on reasonable assumptions on the underlying semantic ordering, and mirror Kamin/L"evy"s termination proof method. Moreover, preceq_mspo is shown to cover preceq_spo up to the subterm property. In the case of the semantic ordering being a simplification quasiordering, the three orderings even coincide. Thus the Knuth/Bendix ordering turns out to be a special case of the semantic path ordering

Termination of String Rewriting Rules That Have One Pair of Overlaps

This paper presents a partial solution to the long standing open problem of termination of one-rule string rewriting. Overlaps between the two sides of the rule play a central role in existing termination criteria. Wecharacterize termination of all one-rule string rewriting systems that have one suchoverlap at either end. This both completes a result of Kurth and generalizes a result of Shikishima-Tsuji et al

Termination of single-threaded one-rule Semi-Thue systems

Abstract. This paper is a contribution to the long standing open problem of uniform termination of Semi-Thue Systems that consist of one rule s → t. McNaughton previously showed that rules incapable of (1) deleting t completely from both sides, (2) deleting t completely from the left, and (3) deleting t completely from the right, have a decidable uniform termination problem. We use a novel approach to show that Premise (2) or, symmetrically, Premise (3), is inessential. Our approach is based on derivations in which every pair of successive steps has an overlap. We call such derivations single-threaded. Key Words and Phrases: string rewriting, semi-Thue system, uniform termination, termination, one-rule, single-rule, single-threaded, well-behave

Parallelizing Functional Programs by Generalization

List homomorphisms are functions that are parallelizable using the divide-and-conquer paradigm. We study the problem of finding a homomorphic representation of a given function, based on the Bird-Meertens theory of lists. A previous work proved that to each pair of leftward and rightward sequential representations of a function, based on cons- and snoc-lists, respectively, there is also a representation as a homomorphism. Our contribution is a mechanizable method to extract the homomorphism representation from a pair of sequential representations. The method is decomposed to a generalization problem and an inductive claim, both solvable by term rewriting techniques. To solve the former we present a sound generalization procedure which yields the required representation, and terminates under reasonable assumptions. We illustrate the method and the procedure by the parallelization of the scan-function (parallel prefix). The inductive claim is provable automatically