Synchronizing heuristics for weakly connected automata with various topologies

Since the problem of finding a shortest synchronizing sequence for an automaton is known to be NP-hard, heuristics algorithms are used to find synchronizing sequences. There are several heuristic algorithms in the literature for this purpose. However, even the most efficient heuristic algorithm in the literature has a quadratic complexity in terms of the number of states of the automaton, and therefore can only scale up to a couple of thousands of states. It was also shown before that if an automaton is not strongly connected, then these heuristic algorithms can be used on each strongly connected component separately. This approach speeds up these heuristic algorithms and allows them to scale to much larger number of states easily. In this paper, we investigate the effect of the topology of the automaton on the performance increase obtained by these heuristic algorithms. To this end, we consider various topologies and provide an extensive experimental study on the performance increase obtained on the existing heuristic algorithms. Depending on the size and the number of components, we obtain speed-up values as high as 10000x and more

Strong inapproximability of the shortest reset word

The \v{C}ern\'y conjecture states that every $n$-state synchronizing automaton has a reset word of length at most $(n-1)^2$. 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 $O(\log n)$ 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 $\epsilon>0$, it is NP-hard to approximate the length of the shortest reset word within a factor of $n^{1-\epsilon}$. This is essentially tight since a simple $O(n)$-approximation algorithm exists.Comment: extended abstract to appear in MFCS 201

Synchronizing Random Almost-Group Automata

In this paper we address the question of synchronizing random automata in the critical settings of almost-group automata. Group automata are automata where all letters act as permutations on the set of states, and they are not synchronizing (unless they have one state). In almost-group automata, one of the letters acts as a permutation on $n-1$ states, and the others as permutations. We prove that this small change is enough for automata to become synchronizing with high probability. More precisely, we establish that the probability that a strongly connected almost-group automaton is not synchronizing is $\frac{2^{k-1}-1}{n^{2(k-1)}}(1+o(1))$, for a $k$-letter alphabet.Comment: full version prepared for CIAA 201

Computational Complexity of Synchronization under Regular Commutative Constraints

Here we study the computational complexity of the constrained synchronization problem for the class of regular commutative constraint languages. Utilizing a vector representation of regular commutative constraint languages, we give a full classification of the computational complexity of the constraint synchronization problem. Depending on the constraint language, our problem becomes PSPACE-complete, NP-complete or polynomial time solvable. In addition, we derive a polynomial time decision procedure for the complexity of the constraint synchronization problem, given some constraint automaton accepting a commutative language as input.Comment: Published in COCOON 2020 (The 26th International Computing and Combinatorics Conference); 2nd version is update of the published version and 1st version; both contain a minor error, the assumption of maximality in the NP-c and PSPACE-c results (propositions 5 & 6) is missing, and of incomparability of the vectors in main theorem; fixed in this version. See (new) discussion after main theore

DFAs and PFAs with Long Shortest Synchronizing Word Length

It was conjectured by \v{C}ern\'y in 1964, that a synchronizing DFA on $n$ states always has a shortest synchronizing word of length at most $(n-1)^2$, and he gave a sequence of DFAs for which this bound is reached. Until now a full analysis of all DFAs reaching this bound was only given for $n \leq 4$, and with bounds on the number of symbols for $n \leq 10$. Here we give the full analysis for $n \leq 6$, without bounds on the number of symbols. For PFAs the bound is much higher. For $n \leq 6$ we do a similar analysis as for DFAs and find the maximal shortest synchronizing word lengths, exceeding $(n-1)^2$ for $n =4,5,6$. For arbitrary n we give a construction of a PFA on three symbols with exponential shortest synchronizing word length, giving significantly better bounds than earlier exponential constructions. We give a transformation of this PFA to a PFA on two symbols keeping exponential shortest synchronizing word length, yielding a better bound than applying a similar known transformation.Comment: 16 pages, 2 figures source code adde

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

Checking Whether an Automaton Is Monotonic Is NP-complete

An automaton is monotonic if its states can be arranged in a linear order that is preserved by the action of every letter. We prove that the problem of deciding whether a given automaton is monotonic is NP-complete. The same result is obtained for oriented automata, whose states can be arranged in a cyclic order. Moreover, both problems remain hard under the restriction to binary input alphabets.Comment: 13 pages, 4 figures. CIAA 2015. The final publication is available at http://link.springer.com/chapter/10.1007/978-3-319-22360-5_2

On the Number of Synchronizing Colorings of Digraphs

We deal with $k$-out-regular directed multigraphs with loops (called simply \emph{digraphs}). The edges of such a digraph can be colored by elements of some fixed $k$-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 $n$ 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 $1-1/k^d$, for every $d \ge 1$ 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 $1-1/k$ is the smallest possible fraction of synchronizing colorings, except for a single exceptional example on 6 vertices for $k=2$.Comment: CIAA 2015. The final publication is available at http://link.springer.com/chapter/10.1007/978-3-319-22360-5_1

On Varieties of Ordered Automata

The Eilenberg correspondence relates varieties of regular languages to pseudovarieties of finite monoids. Various modifications of this correspondence have been found with more general classes of regular languages on one hand and classes of more complex algebraic structures on the other hand. It is also possible to consider classes of automata instead of algebraic structures as a natural counterpart of classes of languages. Here we deal with the correspondence relating positive $\mathcal C$-varieties of languages to positive $\mathcal C$-varieties of ordered automata and we present various specific instances of this correspondence. These bring certain well-known results from a new perspective and also some new observations. Moreover, complexity aspects of the membership problem are discussed both in the particular examples and in a general setting

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 $n$-state synchronizing decoder has a reset word of length at most $O(n \log^3 n)$. In addition to that, we prove that the expected reset threshold of a uniformly random synchronizing binary $n$-state decoder is at most $O(n \log n)$. We also show that for any non-unary alphabet there exist decoders whose reset threshold is in $\varTheta(n)$. We prove the \v{C}ern\'{y} conjecture for $n$-state automata with a letter of rank at most $\sqrt[3]{6n-6}$. 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 $n$-state random synchronizing binary automaton is at most $n^{3/2+o(1)}$. 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