Decidability and Complexity in Automatic Monoids

Abstract

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...

Uniform decision problems for automatic semigroups

“…Our proof is a variation of a well-known technique for encoding a Turing machine into a string-rewriting system (see, for example, [21]). Our result improves upon a recent result of Lohrey [19], who used a similar method to show that right-reachability for automatic monoids is, in general, undecidable. We use without further comment a number of standard results from the theory of string-rewriting; these can be found in [2].…”

“…A number of authors have considered decision problems in automatic semigroups; for example, it has been established that the word problem for an automatic semigroup is always decidable in quadratic time [5] and in certain cases is P-complete [19]. In general, this research has assumed a fixed semigroup with automatic structure, and asked what computations can be performed in the semigroup.…”

Knapsack in Hyperbolic Groups

“…LogCFL has several alternative characterizations, see Section 4 for details. The LogCFL upper bound for knapsack in hyperbolic groups improves the polynomial upper bound shown in [22], and also generalizes a result from [16], stating that the word problem for a hyperbolic group is in LogCFL. For hyperbolic groups that contain a copy of a non-abelian free group (such hyperbolic groups are called non-elementary) it follows from [19] that knapsack is LogCFLcomplete.…”

“…Let G be a hyperbolic group. In [16] it is shown that the word problem for G is a growing context-sensitive language, i.e., it can be generated by a grammar where all productions are strictly length-increasing (except for the start production S → ε). In [1] it was shown that every growing context-sensitive language can be recognized by a one-way AuxPDA in logarithmic space and polynomial time.…”

Automata-based presentations of infinite structures

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