663 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
Preimage problems for deterministic finite automata
Given a subset of states of a deterministic finite automaton and a word
, the preimage is the subset of all states mapped to a state in by the
action of . We study three natural problems concerning words giving certain
preimages. The first problem is whether, for a given subset, there exists a
word \emph{extending} the subset (giving a larger preimage). The second problem
is whether there exists a \emph{totally extending} word (giving the whole set
of states as a preimage)---equivalently, whether there exists an
\emph{avoiding} word for the complementary subset. The third problem is whether
there exists a \emph{resizing} word. We also consider variants where the length
of the word is upper bounded, where the size of the given subset is restricted,
and where the automaton is strongly connected, synchronizing, or binary. We
conclude with a summary of the complexities in all combinations of the cases
Synchronizing weighted automata
We introduce two generalizations of synchronizability to automata with
transitions weighted in an arbitrary semiring K=(K,+,*,0,1). (or equivalently,
to finite sets of matrices in K^nxn.) Let us call a matrix A
location-synchronizing if there exists a column in A consisting of nonzero
entries such that all the other columns of A are filled by zeros. If
additionally all the entries of this designated column are the same, we call A
synchronizing. Note that these notions coincide for stochastic matrices and
also in the Boolean semiring. A set M of matrices in K^nxn is called
(location-)synchronizing if M generates a matrix subsemigroup containing a
(location-)synchronizing matrix. The K-(location-)synchronizability problem is
the following: given a finite set M of nxn matrices with entries in K, is it
(location-)synchronizing?
Both problems are PSPACE-hard for any nontrivial semiring. We give sufficient
conditions for the semiring K when the problems are PSPACE-complete and show
several undecidability results as well, e.g. synchronizability is undecidable
if 1 has infinite order in (K,+,0) or when the free semigroup on two generators
can be embedded into (K,*,1).Comment: In Proceedings AFL 2014, arXiv:1405.527
Synchronization Problems in Automata without Non-trivial Cycles
We study the computational complexity of various problems related to
synchronization of weakly acyclic automata, a subclass of widely studied
aperiodic automata. We provide upper and lower bounds on the length of a
shortest word synchronizing a weakly acyclic automaton or, more generally, a
subset of its states, and show that the problem of approximating this length is
hard. We investigate the complexity of finding a synchronizing set of states of
maximum size. We also show inapproximability of the problem of computing the
rank of a subset of states in a binary weakly acyclic automaton and prove that
several problems related to recognizing a synchronizing subset of states in
such automata are NP-complete.Comment: Extended and corrected version, including arXiv:1608.00889.
Conference version was published at CIAA 2017, LNCS vol. 10329, pages
188-200, 201
How Much Lookahead is Needed to Win Infinite Games?
Delay games are two-player games of infinite duration in which one player may
delay her moves to obtain a lookahead on her opponent's moves. For
-regular winning conditions it is known that such games can be solved
in doubly-exponential time and that doubly-exponential lookahead is sufficient.
We improve upon both results by giving an exponential time algorithm and an
exponential upper bound on the necessary lookahead. This is complemented by
showing EXPTIME-hardness of the solution problem and tight exponential lower
bounds on the lookahead. Both lower bounds already hold for safety conditions.
Furthermore, solving delay games with reachability conditions is shown to be
PSPACE-complete.
This is a corrected version of the paper https://arxiv.org/abs/1412.3701v4
published originally on August 26, 2016
Degree of Sequentiality of Weighted Automata
Weighted automata (WA) are an important formalism to describe quantitative properties. Obtaining equivalent deterministic machines is a longstanding research problem. In this paper we consider WA with a set semantics, meaning that the semantics is given by the set of weights of accepting runs. We focus on multi-sequential WA that are defined as finite unions of sequential WA. The problem we address is to minimize the size of this union. We call this minimum the degree of sequentiality of (the relation realized by) the WA.
For a given positive integer k, we provide multiple characterizations of relations realized by a union of k sequential WA over an infinitary finitely generated group: a Lipschitz-like machine independent property, a pattern on the automaton (a new twinning property) and a subclass of cost register automata. When possible, we effectively translate a WA into an equivalent union of k sequential WA. We also provide a decision procedure for our twinning property for commutative computable groups thus allowing to compute the degree of sequentiality. Last, we show that these results also hold for word transducers and that the associated decision problem is PSPACE
-complete
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