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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.
AN AUTOMATA THEORETIC APPROACH TO THE GENERALIZED WORD PROBLEM IN GRAPHS OF GROUPS
, 2009
"... We give a simpler proof using automata theory of a recent result of Kapovich, Weidmann and Myasnikov according to which socalled benign graphs of groups preserve decidability of the generalized word problem. These include graphs of groups in which edge groups are polycyclicbyfinite and vertex grou ..."
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We give a simpler proof using automata theory of a recent result of Kapovich, Weidmann and Myasnikov according to which socalled benign graphs of groups preserve decidability of the generalized word problem. These include graphs of groups in which edge groups are polycyclicbyfinite and vertex groups are either locally quasiconvex hyperbolic or polycyclicbyfinite and so in particular chordal graph groups (rightangled Artin groups).