Finite complete rewriting systems and the complexity of the word problem

Bauer, Gertrud; Otto, Friedrich

doi:10.1007/bf00271645

Cited by 55 publications

(37 citation statements)

References 12 publications

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“…On the other hand, this algorithm does not yield any upper bound on the computational complexity of the word problem [5]. Complexity results on word problems for restricted classes of finitely presented monoids can be found for instance in [10,41,42].…”

Section: Complexity Theorymentioning

confidence: 96%

“…final) state, and δ : Q \ {q f } × Σ → Q × Σ × {left, right} is the transition function) such that the question whether a word w ∈ Σ * is accepted by A is PSPACE-complete. Such a linear bounded automaton exists, see, e.g., [5]. The one-step transition relation between configurations of A is denoted by ⇒ A .…”

Section: Compressed Word Problems In Pspacementioning

confidence: 99%

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Word Problems and Membership Problems on Compressed Words

Lohrey¹

2006

SIAM J. Comput.

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Abstract. We consider a compressed form of the word problem for finitely presented monoids, where the input consists of two compressed representations of words over the generators of a monoid M, and we ask whether these two words represent the same monoid element of M. Words are compressed using straight-line programs, i.e., context-free grammars that generate exactly one word. For several classes of finitely presented monoids we obtain completeness results for complexity classes in the range from P to EXPSPACE. As a by-product of our results on compressed word problems we obtain a fixed deterministic context-free language with a PSPACE-complete compressed membership problem. The existence of such a language was open so far. Finally, we will investigate the complexity of the compressed membership problem for various circuit complexity classes.Key words. grammar-based compression, word problems for monoids, context-free languages, complexity AMS subject classifications. 20F10, 68Q17, 68Q421. Introduction. During the last decade, the massive increase in the volume of data has motivated the investigation of algorithms on compressed data, like for instance compressed strings, trees, or pictures. The general goal is to develop algorithms that directly work on compressed data without prior decompression. Let us mention here the work on compressed pattern matching, see, e.g., [19,23,49,60].In this paper we investigate two classes of computational problems on compressed data that are of central importance in theoretical computer science since its very beginning: the word problem and the membership problem.In its most general form, the word problem asks whether two terms over an algebraic structure represent the same element of the structure. Here, we restrict to the word problem for finitely presented monoids, i.e., monoids that are given by a finite set of generators and defining relations. In this case the input consists of two finite words over the set of generators and we ask whether these two words represent the same monoid element. The undecidability results concerning the word problem for finitely presented monoids [47,56] and finitely presented groups [12,51] are among the first undecidability results that touched "real mathematics". Moreover, these negative results motivated a still ongoing investigation of decidable subclasses of word problems and their computational complexity. In particular, monoids that can be presented by terminating and confluent semi-Thue systems (i.e., string rewriting systems where every word can be rewritten in a finite number of steps to a unique irreducible word), see [11,33], received a lot of attention: these monoids have decidable word problems, and if the restriction to terminating systems is suitably sharpened, then precise complexity bounds can be deduced [10,41,42]. All relevant definitions concerning semi-Thue systems and finitely presented monoids are collected in Section 3.3.In its compressed variant, the input to the word problem for a (finitely presented) monoid consists...

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Section: Complexity Theorymentioning

confidence: 96%

Section: Compressed Word Problems In Pspacementioning

confidence: 99%

Word Problems and Membership Problems on Compressed Words

Lohrey¹

2006

SIAM J. Comput.

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“…For term rewriting, both confluence [3] and termination [27] are, in general, undecidable. However, for systems known to be terminating, confluence is decidable.…”

Section: Introductionmentioning

confidence: 99%

Confluence of Graph Rewriting with Interfaces

Bonchi

Gadducci

Kissinger

et al. 2017

Programming Languages and Systems

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Abstract. For terminating double-pushout (DPO) graph rewriting systems confluence is, in general, undecidable. We show that confluence is decidable for an extension of DPO rewriting to graphs with interfaces. This variant is important due to it being closely related to rewriting of string diagrams. We show that our result extends, under mild conditions, to decidability of confluence for terminating rewriting systems of string diagrams in symmetric monoidal categories.

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“…Despite this result, there are plenty of rewriting systems with a decidable word problem, the most famous class being that of confluent and terminating systems [32]. In general, also confluence and termination are undecidable properties of a semi-Thue system, see [28] for termination and [1] for confluence. A large amount of research tries to identify sufficient conditions for confluence/termination of rewriting systems (cf.…”

Section: Introductionmentioning

confidence: 99%

Decidable First-Order Theories of One-Step Rewriting in Trace Monoids

Kuske

Lohrey²

2004

Theory Comput Systems

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Abstract. We prove that the first-order theory of the one-step rewriting relation associated with a trace rewriting system is decidable but in general not elementary. This extends known results on semi-Thue systems but our proofs use new methods; these new methods yield the decidability of local properties expressed in first-order logic augmented by modulo-counting quantifiers. Using the main decidability result, we define several subclasses of trace rewriting systems for which the confluence problem is decidable.

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Finite complete rewriting systems and the complexity of the word problem

Cited by 55 publications

References 12 publications

Word Problems and Membership Problems on Compressed Words

Word Problems and Membership Problems on Compressed Words

Confluence of Graph Rewriting with Interfaces

Decidable First-Order Theories of One-Step Rewriting in Trace Monoids

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