1,014 research outputs found
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
Limit Synchronization in Markov Decision Processes
Markov decision processes (MDP) are finite-state systems with both strategic
and probabilistic choices. After fixing a strategy, an MDP produces a sequence
of probability distributions over states. The sequence is eventually
synchronizing if the probability mass accumulates in a single state, possibly
in the limit. Precisely, for 0 <= p <= 1 the sequence is p-synchronizing if a
probability distribution in the sequence assigns probability at least p to some
state, and we distinguish three synchronization modes: (i) sure winning if
there exists a strategy that produces a 1-synchronizing sequence; (ii)
almost-sure winning if there exists a strategy that produces a sequence that
is, for all epsilon > 0, a (1-epsilon)-synchronizing sequence; (iii) limit-sure
winning if for all epsilon > 0, there exists a strategy that produces a
(1-epsilon)-synchronizing sequence.
We consider the problem of deciding whether an MDP is sure, almost-sure,
limit-sure winning, and we establish the decidability and optimal complexity
for all modes, as well as the memory requirements for winning strategies. Our
main contributions are as follows: (a) for each winning modes we present
characterizations that give a PSPACE complexity for the decision problems, and
we establish matching PSPACE lower bounds; (b) we show that for sure winning
strategies, exponential memory is sufficient and may be necessary, and that in
general infinite memory is necessary for almost-sure winning, and unbounded
memory is necessary for limit-sure winning; (c) along with our results, we
establish new complexity results for alternating finite automata over a
one-letter alphabet
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
Strong inapproximability of the shortest reset word
The \v{C}ern\'y conjecture states that every -state synchronizing
automaton has a reset word of length at most . We study the hardness
of finding short reset words. It is known that the exact version of the
problem, i.e., finding the shortest reset word, is NP-hard and coNP-hard, and
complete for the DP class, and that approximating the length of the shortest
reset word within a factor of is NP-hard [Gerbush and Heeringa,
CIAA'10], even for the binary alphabet [Berlinkov, DLT'13]. We significantly
improve on these results by showing that, for every , it is NP-hard
to approximate the length of the shortest reset word within a factor of
. This is essentially tight since a simple -approximation
algorithm exists.Comment: extended abstract to appear in MFCS 201
On the Number of Synchronizing Colorings of Digraphs
We deal with -out-regular directed multigraphs with loops (called simply
\emph{digraphs}). The edges of such a digraph can be colored by elements of
some fixed -element set in such a way that outgoing edges of every vertex
have different colors. Such a coloring corresponds naturally to an automaton.
The road coloring theorem states that every primitive digraph has a
synchronizing coloring.
In the present paper we study how many synchronizing colorings can exist for
a digraph with vertices. We performed an extensive experimental
investigation of digraphs with small number of vertices. This was done by using
our dedicated algorithm exhaustively enumerating all small digraphs. We also
present a series of digraphs whose fraction of synchronizing colorings is equal
to , for every and the number of vertices large enough.
On the basis of our results we state several conjectures and open problems.
In particular, we conjecture that is the smallest possible fraction of
synchronizing colorings, except for a single exceptional example on 6 vertices
for .Comment: CIAA 2015. The final publication is available at
http://link.springer.com/chapter/10.1007/978-3-319-22360-5_1
On Primitivity of Sets of Matrices
A nonnegative matrix is called primitive if is positive for some
integer . A generalization of this concept to finite sets of matrices is
as follows: a set of matrices is
primitive if is positive for some indices
. The concept of primitive sets of matrices comes up in a
number of problems within the study of discrete-time switched systems. In this
paper, we analyze the computational complexity of deciding if a given set of
matrices is primitive and we derive bounds on the length of the shortest
positive product.
We show that while primitivity is algorithmically decidable, unless it
is not possible to decide primitivity of a matrix set in polynomial time.
Moreover, we show that the length of the shortest positive sequence can be
superpolynomial in the dimension of the matrices. On the other hand, defining
to be the set of matrices with no zero rows or columns, we give
a simple combinatorial proof of a previously-known characterization of
primitivity for matrices in which can be tested in polynomial
time. This latter observation is related to the well-known 1964 conjecture of
Cerny on synchronizing automata; in fact, any bound on the minimal length of a
synchronizing word for synchronizing automata immediately translates into a
bound on the length of the shortest positive product of a primitive set of
matrices in . In particular, any primitive set of
matrices in has a positive product of length
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