64,142 research outputs found
On the Complexity of the Word Problem for Automaton Semigroups and Automaton Groups
In this paper, we study the word problem for automaton semigroups and
automaton groups from a complexity point of view. As an intermediate concept
between automaton semigroups and automaton groups, we introduce
automaton-inverse semigroups, which are generated by partial, yet invertible
automata. We show that there is an automaton-inverse semigroup and, thus, an
automaton semigroup with a PSPACE-complete word problem. We also show that
there is an automaton group for which the word problem with a single rational
constraint is PSPACE-complete. Additionally, we provide simpler constructions
for the uniform word problems of these classes. For the uniform word problem
for automaton groups (without rational constraints), we show NL-hardness.
Finally, we investigate a question asked by Cain about a better upper bound for
the length of a word on which two distinct elements of an automaton semigroup
must act differently
Automaton semigroup constructions
The aim of this paper is to investigate whether the class of automaton
semigroups is closed under certain semigroup constructions. We prove that the
free product of two automaton semigroups that contain left identities is again
an automaton semigroup. We also show that the class of automaton semigroups is
closed under the combined operation of 'free product followed by adjoining an
identity'. We present an example of a free product of finite semigroups that we
conjecture is not an automaton semigroup. Turning to wreath products, we
consider two slight generalizations of the concept of an automaton semigroup,
and show that a wreath product of an automaton monoid and a finite monoid
arises as a generalized automaton semigroup in both senses. We also suggest a
potential counterexample that would show that a wreath product of an automaton
monoid and a finite monoid is not a necessarily an automaton monoid in the
usual sense.Comment: 13 pages; 2 figure
Semigroups Arising From Asynchronous Automata
We introduce a new class of semigroups arising from a restricted class of
asynchronous automata. We call these semigroups "expanding automaton
semigroups." We show that the class of synchronous automaton semigroups is
strictly contained in the class of expanding automaton semigroups, and that the
class of expanding automaton semigroups is strictly contained in the class of
asynchronous automaton semigroups. We investigate the dynamics of expanding
automaton semigroups acting on regular rooted trees, and show that
undecidability arises in these actions. We show that this class is not closed
under taking normal ideal extensions, but the class of asynchronous automaton
semigroups is closed under taking these extensions. We construct every free
partially commutative monoid as a synchronous automaton semigroup.Comment: 31 pages, 4 figure
Automaton semigroups: new construction results and examples of non-automaton semigroups
This paper studies the class of automaton semigroups from two perspectives:
closure under constructions, and examples of semigroups that are not automaton
semigroups. We prove that (semigroup) free products of finite semigroups always
arise as automaton semigroups, and that the class of automaton monoids is
closed under forming wreath products with finite monoids. We also consider
closure under certain kinds of Rees matrix constructions, strong semilattices,
and small extensions. Finally, we prove that no subsemigroup of arises as an automaton semigroup. (Previously, itself was
the unique example of a finitely generated residually finite semigroup that was
known not to arise as an automaton semigroup.)Comment: 27 pages, 6 figures; substantially revise
Truly On-The-Fly LTL Model Checking
We propose a novel algorithm for automata-based LTL model checking that
interleaves the construction of the generalized B\"{u}chi automaton for the
negation of the formula and the emptiness check. Our algorithm first converts
the LTL formula into a linear weak alternating automaton; configurations of the
alternating automaton correspond to the locations of a generalized B\"{u}chi
automaton, and a variant of Tarjan's algorithm is used to decide the existence
of an accepting run of the product of the transition system and the automaton.
Because we avoid an explicit construction of the B\"{u}chi automaton, our
approach can yield significant improvements in runtime and memory, for large
LTL formulas. The algorithm has been implemented within the SPIN model checker,
and we present experimental results for some benchmark examples
Fuzzy !-automata and its relationships
A notion of finite ! - automata with single initial state is proposed.
The concept of fuzzy deterministic Buchi automaton and Muller automaton
with full acceptance component which is recognize the same fuzzy language are
studied. We also establish the relationship between fuzzy deterministic Rabin
automaton and Muller automaton. Further, we define the transition fuzzy !
- automata and show that these automata recognize the same fuzzy language
as in the fuzzy ! - automata. Finally, we give some closure properties of fuzzy
deterministic ! - automata
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