45 research outputs found
A Fast Algorithm Finding the Shortest Reset Words
In this paper we present a new fast algorithm finding minimal reset words for
finite synchronizing automata. The problem is know to be computationally hard,
and our algorithm is exponential. Yet, it is faster than the algorithms used so
far and it works well in practice. The main idea is to use a bidirectional BFS
and radix (Patricia) tries to store and compare resulted subsets. We give both
theoretical and practical arguments showing that the branching factor is
reduced efficiently. As a practical test we perform an experimental study of
the length of the shortest reset word for random automata with states and 2
input letters. We follow Skvorsov and Tipikin, who have performed such a study
using a SAT solver and considering automata up to states. With our
algorithm we are able to consider much larger sample of automata with up to
states. In particular, we obtain a new more precise estimation of the
expected length of the shortest reset word .Comment: COCOON 2013. The final publication is available at
http://link.springer.com/chapter/10.1007%2F978-3-642-38768-5_1
Algebraic synchronization criterion and computing reset words
We refine a uniform algebraic approach for deriving upper bounds on reset
thresholds of synchronizing automata. We express the condition that an
automaton is synchronizing in terms of linear algebra, and obtain upper bounds
for the reset thresholds of automata with a short word of a small rank. The
results are applied to make several improvements in the area.
We improve the best general upper bound for reset thresholds of finite prefix
codes (Huffman codes): we show that an -state synchronizing decoder has a
reset word of length at most . In addition to that, we prove
that the expected reset threshold of a uniformly random synchronizing binary
-state decoder is at most . We also show that for any non-unary
alphabet there exist decoders whose reset threshold is in .
We prove the \v{C}ern\'{y} conjecture for -state automata with a letter of
rank at most . In another corollary, based on the recent
results of Nicaud, we show that the probability that the \v{C}ern\'y conjecture
does not hold for a random synchronizing binary automaton is exponentially
small in terms of the number of states, and also that the expected value of the
reset threshold of an -state random synchronizing binary automaton is at
most .
Moreover, reset words of lengths within all of our bounds are computable in
polynomial time. We present suitable algorithms for this task for various
classes of automata, such as (quasi-)one-cluster and (quasi-)Eulerian automata,
for which our results can be applied.Comment: 18 pages, 2 figure
On Synchronizing Colorings and the Eigenvectors of Digraphs
An automaton is synchronizing if there exists a word that sends all states of the automaton to a single state. A coloring of a digraph with a fixed out-degree k is a distribution of k labels over the edges resulting in a deterministic finite automaton. The famous road coloring theorem states that every primitive digraph has a synchronizing coloring. We study recent conjectures claiming that the number of synchronizing colorings is large in the worst and average cases.
Our approach is based on the spectral properties of the adjacency matrix A(G) of a digraph G. Namely, we study the relation between the number of synchronizing colorings of G and the structure of the dominant eigenvector v of A(G). We show that a vector v has no partition of coordinates into blocks of equal sum if and only if all colorings of the digraphs associated with v are synchronizing. Furthermore, if for each b there exists at most one partition of the coordinates of v into blocks summing up to b, and the total number of partitions is equal to s, then the fraction of synchronizing colorings among all colorings of G is at least (k-s)/k. We also give a combinatorial interpretation of some known results concerning an upper bound on the minimal length of synchronizing words in terms of v
Attainable Values of Reset Thresholds
An automaton is synchronizing if there exists a word that sends all states of the automaton to a single state. The reset threshold is the length of the shortest such word. We study the set RT_n of attainable reset thresholds by automata with n states. Relying on constructions of digraphs with known local exponents we show that the intervals [1, (n^2-3n+4)/2] and
[(p-1)(q-1), p(q-2)+n-q+1], where 2 n, gcd(p,q)=1, belong to RT_n, even if restrict our attention to strongly connected automata. Moreover, we prove that in this case the smallest value that does not belong to RT_n is at least n^2 - O(n^{1.7625} log n / log log n).
This value is increased further assuming certain conjectures about the gaps between consecutive prime numbers.
We also show that any value smaller than n(n-1)/2 is attainable by an automaton with a sink state and any value smaller than n^2-O(n^{1.5}) is attainable in general case.
Furthermore, we solve the problem of existence of slowly synchronizing automata over an arbitrarily large alphabet, by presenting for every fixed size of the alphabet an infinite series of irreducibly synchronizing automata with the reset threshold n^2-O(n)
Synchronizing Strongly Connected Partial DFAs
We study synchronizing partial DFAs, which extend the classical concept of
synchronizing complete DFAs and are a special case of synchronizing unambiguous
NFAs. A partial DFA is called synchronizing if it has a word (called a reset
word) whose action brings a non-empty subset of states to a unique state and is
undefined for all other states. While in the general case the problem of
checking whether a partial DFA is synchronizing is PSPACE-complete, we show
that in the strongly connected case this problem can be efficiently reduced to
the same problem for a complete DFA. Using combinatorial, algebraic, and formal
languages methods, we develop techniques that relate main synchronization
problems for strongly connected partial DFAs with the same problems for
complete DFAs. In particular, this includes the \v{C}ern\'{y} and the rank
conjectures, the problem of finding a reset word, and upper bounds on the
length of the shortest reset words of literal automata of finite prefix codes.
We conclude that solving fundamental synchronization problems is equally hard
in both models, as an essential improvement of the results for one model
implies an improvement for the other.Comment: Full version of the paper at STACS 202
On random primitive sets, directable NDFAs and the generation of slowly synchronizing DFAs
We tackle the problem of the randomized generation of slowly synchronizing
deterministic automata (DFAs) by generating random primitive sets of matrices.
We show that when the randomized procedure is too simple the exponent of the
generated sets is O(n log n) with high probability, thus the procedure fails to
return DFAs with large reset threshold. We extend this result to random
nondeterministic automata (NDFAs) by showing, in particular, that a uniformly
sampled NDFA has both a 2-directing word and a 3-directing word of length O(n
log n) with high probability. We then present a more involved randomized
algorithm that manages to generate DFAs with large reset threshold and we
finally leverage this finding for exhibiting new families of DFAs with reset
threshold of order .Comment: 31 pages, 9 figures. arXiv admin note: text overlap with
arXiv:1805.0672
Complexity of Preimage Problems for Deterministic Finite Automata
Given a subset of states S of a deterministic finite automaton and a word w, the preimage is the subset of all states that are mapped to a state from S by the action of w. We study the computational complexity of three problems related to the existence of words yielding certain preimages, which are especially motivated by the theory of synchronizing automata. The first problem is whether, for a given subset, there exists a word extending the subset (giving a larger preimage). The second problem is whether there exists a word totally extending the subset (giving the whole set of states) - it is equivalent to the problem whether there exists an avoiding word for the complementary subset. The third problem is whether there exists a word resizing the subset (giving a preimage of a different size). We also consider the variants of the problem where an upper bound on the length of the word is given in the input. Because in most cases our problems are computationally hard, we additionally consider parametrized complexity by the size of the given subset. We focus on the most interesting cases that are the subclasses of strongly connected, synchronizing, and binary automata