<i>G</i>-automata, counter languages and the Chomsky hierarchy

Elder, Murray

doi:10.1017/cbo9780511721212.023

Cited by 4 publications

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References 8 publications

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“…There are many variants of counter automata and languages in the literature, see for example [2,8,9,11,15,17,18,22,36]. In this article we define a counter automaton as follows.…”

Section: Counter Automatamentioning

confidence: 99%

C-graph automatic groups

Elder

Taback

2014

Journal of Algebra

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Abstract. We generalize the notion of a graph automatic group introduced by Kharlampovich, Khoussainov and Miasnikov by replacing the regular languages in their definition with more powerful language classes. For a fixed language class C, we call the resulting groups C-graph automatic. We prove that the class of C-graph automatic groups is closed under change of generating set, direct and free product for certain classes C. We show that for quasirealtime counter-graph automatic groups where normal forms have length that is linear in the geodesic length, there is an algorithm to compute normal forms (and therefore solve the word problem) in polynomial time. The class of quasirealtime counter-graph automatic groups includes all Baumslag-Solitar groups, and the free group of countably infinite rank. Context-sensitive-graph automatic groups are shown to be a very large class, which encompasses, for example, groups with unsolvable conjugacy problem, the Grigorchuk group, and Thompson's groups F, T and V .

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“…There are many variants of counter automata and languages in the literature, see for example [2,8,9,11,15,17,18,22,36]. In this article we define a counter automaton as follows.…”

Section: Counter Automatamentioning

confidence: 99%

C-graph automatic groups

Elder

Taback

2014

Journal of Algebra

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“…The following result, which has been observed independently by several authors [3,6], is an immediate corollary of Proposition 2 together with the fact that rational transductions are closed under composition [1, Theorem III.4.4].…”

Section: -Automata and G-automatamentioning

confidence: 64%

Formal Languages and Groups as Memory

Kambites

2009

Communications in Algebra

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“…One way of increasing the accepting power of finite automata that is currently getting lots of consideration is by adding a counter that behaves like a given group or monoid M ; see, for example, [6,8,10,13,14,16]. Such a machine is called an extended finite automaton with register M or, for short, an M -automaton (see Sec.…”

Section: Introductionmentioning

confidence: 99%

Automata with Counters that Recognize Word Problems of Free Products

Corson

Ross

2015

Int. J. Found. Comput. Sci.

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An M -automaton is a finite automaton with a blind counter that mimics a monoid M . The finitely generated groups whose word problems (when viewed as formal languages) are accepted by M -automata play a central role in understanding the family L(M ) of all languages accepted by M -automata. If G 1 and G 2 are finitely generated groups whose word problems are languages in L(M ), in general, the word problem of the free product G 1 * G 2 is not necessarily in L(M ). However, we show that if M is enlarged to the free product M * P 2 , where P 2 is the polycyclic monoid of rank two, then this closure property holds. In fact, we show more generally that the special word problem of M 1 * M 2 lies in L(M * P 2 ) whenever M 1 and M 2 are finitely generated monoids with special word problems in L(M * P 2 ). We also observe that there is a monoid without zero, denoted by CF 2 , that can be used in place of P 2 for this purpose. The monoid CF 2 is the rank two case of what we call a monoid with right invertible basis and its Rees quotient by its maximal ideal is P 2 . The fundamental theory of monoids with right invertible bases is completely analogous to that of free groups, and thus they are very convenient to use. We also investigate the questions of whether there is a group that can be used instead of the monoid P 2 in the above result and under what circumstances P 1 (or the bicyclic monoid) is enough to do the job of P 2 .

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G-automata, counter languages and the Chomsky hierarchy

Cited by 4 publications

References 8 publications

C-graph automatic groups

C-graph automatic groups

Formal Languages and Groups as Memory

Automata with Counters that Recognize Word Problems of Free Products

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