Automatic semigroups

Campbell, C. M.; Robertson, E. F.; Ruškuc, Nik; Thomas, Richard M.

doi:10.1016/s0304-3975(99)00151-6

Cited by 86 publications

(117 citation statements)

References 14 publications

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“…Automatic monoids were investigated for instance in [10,18,20,22,36]. They generalize automatic groups, see [15].…”

Section: Automatic Monoidsmentioning

confidence: 99%

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Decidability and Complexity in Automatic Monoids

Lohrey

2004

Developments in Language Theory

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Several complexity and decidability results for automatic monoids are shown: (i) there exists an automatic monoid with a P-complete word problem, (ii) there exists an automatic monoid such that the first-order theory of the corresponding Cayley-graph is not elementary decidable, and (iii) there exists an automatic monoid such that reachability in the corresponding Cayley-graph is undecidable. Moreover, it is shown that for every hyperbolic group the word problem belongs to LOGCFL, which improves a result of Cai [8].Keywords: automatic monoids, hyperbolic groups, word problems, Cayley-graphs, complexity, decidability IntroductionAutomatic groups attracted a lot of attention in combinatorial group theory during the last 15 years, see e.g. the textbook [15]. Roughly speaking, a finitely generated group G, generated by the finite set Γ, is automatic, if the elements of G can be represented by words from a regular language over Γ, and the multiplication with a generator can be recognized by a synchronized 2-tape automaton. This concept easily yields a quadratic time algorithm for the word problem of an automatic group.It is straight forward to extend the definition of an automatic group to the monoid case; this leads to the class of automatic monoids, see e.g. [10,18,22,36]. In the present paper, we study the complexity and decidability of basic algorithmic questions in automatic monoids. In Section 4 we consider the complexity of the word problem for automatic monoids. Analogously to the group case, it is easy to show that for every automatic monoid the word problem can be solved in quadratic time. Here, we prove that there exists a fixed automatic monoid with a P-complete word problem. Thus, unless P = NC, where NC is the class of all problems that can be solved in polylogarithmic time using a polynomial amount of hardware, there exist * This work was partly done while the author was at RWTH Aachen, Germany. 1 automatic monoids for which the word problem cannot be efficiently parallelized. Whether there exists an automatic group with a P-complete word problem was asked for the first time by Cai [8]. This problem remains open.An important subclass of the class of automatic groups is the class of hyperbolic groups, which are defined via a geometric hyperbolicity condition on the Cayleygraph [16]. In [8], Cai has shown that for every hyperbolic group the word problem belongs to the parallel complexity class NC 2 . Cai also asked, whether the upper bound of NC 2 can be improved. Using known results from formal language theory, we show in Section 4 that the word problem for every hyperbolic group belongs to the complexity class LOGCFL ⊆ NC 2 . LOGCFL is the class of all problems that are logspace reducible to a context-free language [44]. We also present a class of automatic monoids, namely monoids that can be presented by finite, terminating, confluent, and left-basic semi-Thue systems [41], for which the complexity of the word problem captures the class LOGDCFL (the logspace closure of the deterministic contex...

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“…Automatic monoids were investigated for instance in [10,18,20,22,36]. They generalize automatic groups, see [15].…”

Section: Automatic Monoidsmentioning

confidence: 99%

“…[10,18,22,36]. In the present paper, we study the complexity and decidability of basic algorithmic questions in automatic monoids.…”

Section: Introductionmentioning

confidence: 99%

Decidability and Complexity in Automatic Monoids

Lohrey

2004

Developments in Language Theory

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“…More recently, this notion has been extended to semigroups [29,30,84,28] and monoids [83,129,102] and has subsequently caught considerable attention.…”

Section: Automatic Groupsmentioning

confidence: 99%

Automata-based presentations of infinite structures

Bárány¹,

Grädel²,

Rubin³

2011

Finite and Algorithmic Model Theory

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The model theory of finite structures is intimately connected to various fields in computer science, including complexity theory, databases, and verification. In particular, there is a close relationship between complexity classes and the expressive power of logical languages, as witnessed by the fundamental theorems of descriptive complexity theory, such as Fagin's Theorem and the ImmermanVardi Theorem (see [78, Chapter 3] for a survey).However, for many applications, the strict limitation to finite structures has turned out to be too restrictive, and there have been considerable efforts to extend the relevant logical and algorithmic methodologies from finite structures to suitable classes of infinite ones. In particular this is the case for databases and verification where infinite structures are of crucial importance [130]. Algorithmic model theory aims to extend in a systematic fashion the approach and methods of finite model theory, and its interactions with computer science, from finite structures to finitely-presentable infinite ones.There are many possibilities to present infinite structures in a finite manner. A classical approach in model theory concerns the class of computable structures; these are countable structures, on the domain of natural numbers, say, with a finite collection of computable functions and relations. Such structures can be finitely presented by a collection of algorithms, and they have been intensively studied in model theory since the 1960s. However, from the point of view of algorithmic model theory the class of computable structures is problematic. Indeed, one of the central issues in algorithmic model theory is the effective evaluation of logical formulae, from a suitable logic such as, for instance, first-order logic (FO), monadic second-order logic (MSO), or a fixed point logic like LFP or the modal µ-calculus. But on computable structures, only the quantifier-free formulae generally admit effective evaluation, and already the existential fragment of first-order logic is undecidable, for instance on the computable structure (N, +, · ).This leads us to the central requirement that for a suitable logic L (depending on the intended application) the model-checking problem for the class C of finitely presented structures should be algorithmically solvable. At the very 1

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“…Since s is accessible and coaccessible, there are words u, v e A* such that F(i, u) = s and V(s, v) e T. Since the two cycles on s are distinct, there are distinct words 101, w 2 € A + (which are not prefixes of one another) which label the edges of the cycles, such that all words determined by the regular expression u{w\, w 2 }*v are accepted by F. Let / = lcm(|tui|, \w 2 \) and fix p\,p 2 € N such that / = |u>ilpi = \u)2\p 2 . Then the number of words accepted by F of length m = |M| + \v\ + Ik is at least 2*.…”

Section: Word W € A* Is Said To Be Accepted By F If F(/ W) G T a Stmentioning

confidence: 99%

“…Let a = u\ u 2 • • • M (| V" and once again choose b\ and b 2 so that W\ = ab t and w 2 = ab 2 . We claim that rex(tui) = rex(u;2) = rex(a).…”

Section: And So Thatend(a) Is Within N Of the Lower Extremum Of W's Pmentioning

confidence: 99%

Remarks concerning finitely generated semigroups having regular sets of unique normal forms

Cutting

Solomon

2001

J. Aust. Math. Soc.

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Properties such as automaticity, growth and decidability are investigated for the class of finitely generated semigroups which have regular sets of unique normal forms. Knowledge obtained is then applied to the task of demonstrating that a class of semigroups derived from free inverse semigroups under certain closure operations is not automatic.2000 Mathematics subject classification: primary 20M18; secondary 20M05, 20F10.

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Automatic semigroups

Cited by 86 publications

References 14 publications

Decidability and Complexity in Automatic Monoids

Decidability and Complexity in Automatic Monoids

Automata-based presentations of infinite structures

Remarks concerning finitely generated semigroups having regular sets of unique normal forms

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