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On groups and counter automata
 Internat. J. Algebra Comput
"... Abstract. We study finitely generated groups whose word problems are accepted by counter automata. We show that a group has word problem accepted by a blind ncounter automaton in the sense of Greibach if and only if it is virtually free abelian of rank n; this result, which answers a question of Gi ..."
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Abstract. We study finitely generated groups whose word problems are accepted by counter automata. We show that a group has word problem accepted by a blind ncounter automaton in the sense of Greibach if and only if it is virtually free abelian of rank n; this result, which answers a question of Gilman, is in a very precise sense an abelian analogue of the MullerSchupp theorem. More generally, if G is a virtually abelian group then every group with word problem recognised by a Gautomaton is virtually abelian with growth class bounded above by the growth class of G. We consider also other types of counter automata. 1.
The loop problem for monoids and semigroups
 Mathematical Proceedings of the Cambridge Philosophical Society
, 2006
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THE WORD PROBLEM DISTINGUISHES COUNTER LANGUAGES
"... Abstract. Counter automata are more powerful versions of finitestate automata where addition and subtraction operations are permitted on a set of n integer registers, called counters. We show that the word problem of Z n is accepted by a nondeterministic mcounter automaton if and only if m ≥ n. 1. ..."
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Abstract. Counter automata are more powerful versions of finitestate automata where addition and subtraction operations are permitted on a set of n integer registers, called counters. We show that the word problem of Z n is accepted by a nondeterministic mcounter automaton if and only if m ≥ n. 1.
RATIONAL SUBSETS OF POLYCYCLIC MONOIDS AND VALENCE AUTOMATA
, 710
"... Abstract. We study the classes of languages defined by valence automata with rational target sets (or equivalently, regular valence grammars with rational target sets), where the valence monoid is drawn from the important class of polycyclic monoids. We show that for polycyclic monoids of rank 2 or ..."
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Abstract. We study the classes of languages defined by valence automata with rational target sets (or equivalently, regular valence grammars with rational target sets), where the valence monoid is drawn from the important class of polycyclic monoids. We show that for polycyclic monoids of rank 2 or more, such automata accept exactly the contextfree languages. For the polycyclic monoid of rank 1 (that is, the bicyclic monoid), they accept a class of languages strictly including the partially blind onecounter languages. Key to the proof is a description of the rational subsets of polycyclic and bicyclic monoids, other consequences of which include the decidability of the rational subset membership problem, and the closure of the class of rational subsets under intersection and complement. 1.
RATIONAL SEMIGROUP AUTOMATA
, 708
"... Abstract. We show that for any monoid M, the family of languages accepted by Mautomata (or equivalently, generated by regular valence grammars over M) is completely determined by that part of M which lies outside the maximal ideal. In particular, every such family arises as a family of languages ac ..."
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Abstract. We show that for any monoid M, the family of languages accepted by Mautomata (or equivalently, generated by regular valence grammars over M) is completely determined by that part of M which lies outside the maximal ideal. In particular, every such family arises as a family of languages accepted by Nautomata where N is a simple or 0simple monoid. A consequence is that every such family is either exactly the class of regular languages, contains all the blind onecounter languages, or is the family of languages accepted by Gautomata where G is an nonlocallyfinite torsion group. We then consider a natural extension of the usual definition which permits the automaton to utilise more of the structure of each monoid, and additionally allows us to define Sautomata for S an arbitrary semigroup. In the monoid case, the resulting automata are equivalent to the valence automata with rational target sets which arise in the theory of regulated rewriting systems. We study these automata in the case that the register semigroup is completely simple or completely 0simple, obtaining a complete characterisation of the classes of languages corresponding to such semigroups, in terms of their maximal subgroups. In the process, we obtain a number of results about rational subsets of Rees matrix semigroups which are likely to be of independent interest. 1.