15 research outputs found
Word problems recognisable by deterministic blind monoid automata
We consider blind, deterministic, finite automata equipped with a register
which stores an element of a given monoid, and which is modified by right
multiplication by monoid elements. We show that, for monoids M drawn from a
large class including groups, such an automaton accepts the word problem of a
group H if and only if H has a finite index subgroup which embeds in the group
of units of M. In the case that M is a group, this answers a question of Elston
and Ostheimer.Comment: 8 pages, fixed some typos and clarified ambiguity in the abstract,
results unchange
The word problem distinguishes counter languages
Counter automata are more powerful versions of finite-state automata where
addition and subtraction operations are permitted on a set of n integer
registers, called counters. We show that the word problem of is accepted
by a nondeterministic -counter automaton if and only if .Comment: 8 page
On groups and counter automata
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
n-counter 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 Muller-Schupp theorem. More
generally, if G is a virtually abelian group then every group with word problem
recognised by a G-automaton is virtually abelian with growth class bounded
above by the growth class of G. We consider also other types of counter
automata.Comment: 18 page
Generalized Results on Monoids as Memory
We show that some results from the theory of group automata and monoid
automata still hold for more general classes of monoids and models. Extending
previous work for finite automata over commutative groups, we demonstrate a
context-free language that can not be recognized by any rational monoid
automaton over a finitely generated permutable monoid. We show that the class
of languages recognized by rational monoid automata over finitely generated
completely simple or completely 0-simple permutable monoids is a semi-linear
full trio. Furthermore, we investigate valence pushdown automata, and prove
that they are only as powerful as (finite) valence automata. We observe that
certain results proven for monoid automata can be easily lifted to the case of
context-free valence grammars.Comment: In Proceedings AFL 2017, arXiv:1708.0622