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Decision Problems For SemiThue Systems With A Few Rules
, 1996
"... For several decision problems about semiThue systems, we try to locate the frontier between the decidable and the undecidable from the point of view of the number of rules. We show that the the Termination Problem, the UTermination Problem, the Accessibility Problem and the CommonDescendant Probl ..."
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Cited by 56 (0 self)
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For several decision problems about semiThue systems, we try to locate the frontier between the decidable and the undecidable from the point of view of the number of rules. We show that the the Termination Problem, the UTermination Problem, the Accessibility Problem and the CommonDescendant Problem are undecidable for 3 rules STS. As a corollary we obtain the undecidability of the PostCorrespondence Problem for 7 rules.
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 ...
Total Termination of Term Rewriting is Undecidable
 Journal of Symbolic Computation
, 1995
"... Usually termination of term rewriting systems (TRS's) is proved by means of a monotonic wellfounded order. If this order is total on ground terms, the TRS is called totally terminating. In this paper we prove that total termination is an undecidable property of finite term rewriting systems. The pr ..."
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Cited by 9 (3 self)
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Usually termination of term rewriting systems (TRS's) is proved by means of a monotonic wellfounded order. If this order is total on ground terms, the TRS is called totally terminating. In this paper we prove that total termination is an undecidable property of finite term rewriting systems. The proof is given by means of Post's Correspondence Problem. 1 Introduction Termination of term rewriting systems (TRS's) is an important property. Often termination proofs are given by defining an order that is wellfounded, and proving that for every rewrite step the value of the term decreases according this order. In many cases the order is monotonic, and it suffices to prove that l oe ? r oe for all rewrite rules l ! r and all ground substitutions oe. Standard techniques following this approach include recursive path order and KnuthBendix order, see for example [17]. It is an interesting question whether these orders are total or can be extended to a total monotonic order, or are essen...
Relative Undecidability in the Termination Hierarchy of Single Rewrite Rules
 Proceedings of the Colloquium on Trees in Algebra and Programming, Lecture Notes in Computer Science
, 1997
"... . For a hierarchy of properties of term rewriting systems, related to termination, we prove relative undecidability even in the case of single rewrite rules: for implications X ) Y in the hierarchy the property X is undecidable for rewrite rules satisfying Y . 1 Introduction A fundamental problem ..."
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Cited by 3 (2 self)
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. For a hierarchy of properties of term rewriting systems, related to termination, we prove relative undecidability even in the case of single rewrite rules: for implications X ) Y in the hierarchy the property X is undecidable for rewrite rules satisfying Y . 1 Introduction A fundamental problem in the theory of term rewriting is the detection of termination: for a fixed system of rewrite rules, determine whether there are infinite rewrite sequences. Besides termination a number of related properties are of interest, linearly ordered by implication: polynomial termination ) !termination ) total termination ) simple termination ) nonselfembeddingness ) termination ) nonloopingness ) acyclicity We call this the termination hierarchy. Apart from polynomial termination, all properties in the termination hierarchy are known to be undecidable ([11, 15, 13, 18, 8, 9]). In [9] we showed the stronger result of relative undecidability : for all implications X ) Y in the termination hier...
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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Cited by 2 (1 self)
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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.