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Levels of Undecidability in Rewriting
, 2011
"... Undecidability of various properties of first order term rewriting systems is wellknown. An undecidable property can be classified by the complexity of the formula defining it. This classification gives rise to a hierarchy of distinct levels of undecidability, starting from the arithmetical hierarc ..."
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Undecidability of various properties of first order term rewriting systems is wellknown. An undecidable property can be classified by the complexity of the formula defining it. This classification gives rise to a hierarchy of distinct levels of undecidability, starting from the arithmetical hierarchy classifying properties using first order arithmetical formulas, and continuing into the analytic hierarchy, where quantification over function variables is allowed. In this paper we give an overview of how the main properties of first order term rewriting systems are classified in these hierarchies. We consider properties related to normalization (strong normalization, weak normalization and dependency problems) and properties related to confluence (confluence, local confluence and the unique normal form property). For all of these we distinguish between the single term version and the uniform version. Where appropriate, we also distinguish between ground and open terms. Most uniform properties are Π 0 2complete. The particular problem of local confluence turns out to be Π 0 2complete for ground terms, but only Σ 0 1complete (and thereby recursively enumerable) for open terms. The most surprising result concerns dependency pair problems without minimality flag: we prove this problem to be Π 1 1complete, hence not in the arithmetical hierarchy, but properly in the analytic hierarchy. Some of our results are new or have appeared in our earlier publications [35, 7]. Others are based on folklore constructions, and are included for completeness as their precise classifications have hardly been noticed previously.
On Efficient ReductionAlgorithms for Some Trace Rewriting Systems
, 1993
"... . We consider some basic problems on the decidability and complexity of trace rewriting systems. The new contribution of this paper is an O(n log(n)) algorithm for some computing irreducible normal forms in the case of certain onerule systems. 1 Introduction The notes of this paper are based on th ..."
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. We consider some basic problems on the decidability and complexity of trace rewriting systems. The new contribution of this paper is an O(n log(n)) algorithm for some computing irreducible normal forms in the case of certain onerule systems. 1 Introduction The notes of this paper are based on the Font Romeu Lecture and on an invited lecture at FCT93 conference [11] of the second author. In the first part of the paper we give some basic background about trace rewriting systems. There is some overlap with the published notes from FCT93. However, the second part is original and constitutes a new contribution to the theory of trace rewriting systems. The theory of rewriting over free partially commutative monoids (trace rewriting) combines combinatorial aspects from string rewriting (modulo a congruence) and graph rewriting. This restriction of graph rewriting leads to feasible algorithms, but some interesting complexity questions are still open. For example, a challenging open probl...
IOS Press On OneRule Grid SemiThue Systems
, 2012
"... Abstract. The family of onerule grid semiThue systems, introduced by Alfons Geser, is the family of onerule semiThue systems such that there exists a letter c that occurs as often in the lefthand side as the righthand side of the rewriting rule. We prove that for any onerule grid semiThue sy ..."
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Abstract. The family of onerule grid semiThue systems, introduced by Alfons Geser, is the family of onerule semiThue systems such that there exists a letter c that occurs as often in the lefthand side as the righthand side of the rewriting rule. We prove that for any onerule grid semiThue system S, the set S(w) of all words obtainable from w using repeatedly the rewriting rule of S is a constructible contextfree language. We also prove the regularity of the set Loop(S) of all words that start a loop in a onerule grid semiThue systems S. Keywords: onerule semiThue system, termination, grid semiThue system. 1.
On the Complexity of Deciding the Derivation Length in Term Rewriting Systems
"... Verification techniques based on rewriting can play an active role in many fields of software system analysis. To this respect, rewriting based tools can become complementary to modelchecking or finite state based tools. However, in order to be practical these techniques have to exhibit a reasonabl ..."
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Verification techniques based on rewriting can play an active role in many fields of software system analysis. To this respect, rewriting based tools can become complementary to modelchecking or finite state based tools. However, in order to be practical these techniques have to exhibit a reasonable degree of efficiency, therefore, according to [5], a systematic study of the quantitative aspects of term rewriting is in order. Complexity issues on term rewriting systems have been variously studied in the literature. The time complexity of determining the derivation length in a term rewriting system (T RS for short) is a wellknown and investigated issue. From a computational complexity point of view, in general this problem is intractable, since termination is even undecidable [6]. A series of papers have then shown that undecidability also holds if the T RS contains only three rules [8], only two rules [3], one rule [1] and one rule which is leftlinear, nonoverlapping and variable preserving [2]. In this paper we investigate a numerical refinement of the termination problem,