We approach the task of computing a carefully synchronizing word of optimum length for a given partial deterministic automaton, encoding the problem as an instance of SAT and invoking a SAT solver. Our experiments demonstrate that this approach gives satisfactory results for automata with up to 100 states even if very modest computational resources are used. We compare our results with the ones obtained by the first author for exact synchronization, which is another version of synchronization studied in the literature, and draw some theoretical conclusions.
We present a strongly exponential lower bound that applies both to the subset synchronization threshold for binary deterministic automata and to the careful synchronization threshold for binary partial automata. In the later form, the result finishes the research initiated by Martyugin (2013). Moreover, we show that both the thresholds remain strongly exponential even if restricted to strongly connected binary automata. In addition, we apply our methods to computational complexity. Existence of a subset reset word is known to be PSPACE-complete; we show that this holds even under the restricti…
Planar automata seems to be representative of the synchronizing behavior of deterministic finite state automata. We conjecture that Černy's conjecture holds true, if and only if, it holds true for planar automata. In this paper we have gathered some evidence concerning this conjecture. This evidence amounts to show that the class of planar automata is representative of the algorithmic hardness of synchronization
We approach the problem of computing a $D_{3}$-synchronizing word of minimum length for a given nondeterministic automaton via its encoding as an instance of SAT and invoking a SAT solver. We also present some experimental results.
Pin proved that every circular automaton with a prime number of states containing a non-permutation is synchronizing. In this paper, we investigate the synchronization of circular semi-flower automata. We first prove that every semi-flower automaton is a one-cluster automaton. Subsequently, we prove that every semi-flower automaton containing a 1-cycle is synchronizing. Further, we prove that every circular semi-flower automaton with an odd number of states containing a 2-cycle is synchronizing.
We approach the task of computing a carefully synchronizing word of minimum length for a given partial deterministic automaton, encoding the problem as an instance of SAT and invoking a SAT solver. Our experimental results demonstrate that this approach gives satisfactory results for automata with up to 100 states even if very modest computational resources are used.
We study the problem of synchronization of automata with random inputs. We present a series of automata such that the expected number of steps until synchronization is exponential in the number of states. At the same time, we show that the expected number of letters to synchronize any pair of the famous Cerny automata is at most cubic in the number of states.
For general input automata, there exist regular constraint languages such that asking if a given input automaton admits a synchronizing word in the constraint language is PSPACE-complete or NP-complete. Here, we investigate this problem for commutative automata over an arbitrary alphabet and automata with simple idempotents over a binary alphabet as input automata. The latter class contains, for example, the Černý family of automata. We find that for commutative input automata, the problem is always solvable in polynomial time, for every constraint language. For input automata with simple idem…
We investigate the constrained synchronization problem for weakly acyclic, or partially ordered, input automata. We show that, for input automata of this type, the problem is always in NP. Furthermore, we give a full classification of the realizable complexities for constraint automata with at most two states and over a ternary alphabet. We find that most constrained problems that are PSPACE-complete in general become NP-complete. However, there also exist constrained problems that are PSPACE-complete in the general setting but become polynomial time solvable when considered for weakly acyclic…
Berlinkov has suggested an algorithm that, given a deterministic finite automaton $\mathcal{A}$, verifies whether or not $\mathcal{A}$ is synchronizing in linear (of the number of states and letters) expected time. We present a modification of Berlinkov's algorithm which we have implemented and tested. Our experiments show that the implementation outperforms the standard quadratic algorithm even for automata of modest size and allow us to give a statistically accurate approximation of the ratio of non-synchronizing automata amongst all automata with a given number of states.
We consider the first problem that appears in any application of synchronizing automata, namely, the problem of deciding whether or not a given $n$-state $k$-letter automaton is synchronizing. First we generalize results from \cite{RandSynch},\cite{On2Problems} for the case of strongly connected partial automata. Specifically, for $k>1$ we show that an automaton is synchronizing with probability $1-O(\frac{1}{n^{0.5k}})$ and present an algorithm with linear in $n$ expected time, while the best known algorithm is quadratic on each instance. This results are interesting due to their applications…
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 pro…
In the constrained synchronization problem we ask if a given automaton admits a synchronizing word coming from a fixed regular constraint language. We show that intersecting a given constraint language with an ideal language decreases the computational complexity. Additionally, we state a theorem giving PSPACE-hardness that broadly generalizes previously used constructions and a result on how to combine languages by concatenation to get polynomial time solvable constrained synchronization problems. We use these results to give a classification of the complexity landscape for small constraint a…
A goal of this paper is to introduce the new construction of an automaton with shortest synchronizing word of length $O(d^{\frac{n}{d}})$, where $d \in \mathbb{N}$ and $n$ is the number of states for that automaton. Additionally we introduce new transformation from any synchronizable DFA or carefully synchronizable PFA of $n$ states to carefully synchronizable PFA of $d \cdot n$ states with shortest synchronizing word of length $Ω(d^{\frac{n}{d}})$.
We study extremal and algorithmic questions of subset and careful synchronization in monotonic automata. We show that several synchronization problems that are hard in general automata can be solved in polynomial time in monotonic automata, even without knowing a linear order of the states preserved by the transitions. We provide asymptotically tight bounds on the maximum length of a shortest word synchronizing a subset of states in a monotonic automaton and a shortest word carefully synchronizing a partial monotonic automaton. We provide a complexity framework for dealing with problems for mo…
A synchronizing word for an automaton is a word that brings that automaton into one and the same state, regardless of the starting position. Cerny conjectured in 1964 that if a n-state deterministic automaton has a synchronizing word, then it has a synchronizing word of size at most (n-1)^2. Berlinkov recently made a breakthrough in the probabilistic analysis of synchronization by proving that with high probability, an automaton has a synchronizing word. In this article, we prove that with high probability an automaton admits a synchronizing word of length smaller than n^(1+ε), and therefore t…
The class of known constraint automata for which the constrained synchronization problem is in NP all admit a special form. In this work, we take a closer look at them. We characterize a wider class of constraint automata that give constrained synchronization problems in NP, which encompasses all known problems in NP. We call these automata polycyclic automata. The corresponding language class of polycyclic languages is introduced. We show various characterizations and closure properties for this new language class. We then give a criterion for NP-completeness and a criterion for polynomial ti…
Register automata (RAs) are finite automata extended with a finite set of registers to store and compare data from an infinite domain. We study the concept of synchronizing data words in RAs: does there exist a data word that sends all states of the RA to a single state? For deterministic RAs with k registers (k-DRAs), we prove that inputting data words with 2k+1 distinct data from the infinite data domain is sufficient to synchronize. We show that the synchronization problem for DRAs is in general PSPACE-complete, and it is NLOGSPACE-complete for 1-DRAs. For nondeterministic RAs (NRAs), we sh…
We generalize the concept of synchronizing words for finite automata, which map all states of the automata to the same state, to deterministic visibly push-down automata. Here, a synchronizing word w does not only map all states to the same state but also fulfills some conditions on the stack content of each run after reading w. We consider three types of these stack constraints: after reading w, the stack (1) is empty in each run, (2) contains the same sequence of stack symbols in each run, or (3) contains an arbitrary sequence which is independent of the other runs. We show that in contrast …
We study synchronizing automata with the shortest reset words of relatively large length. First, we refine the Frankl-Pin result on the length of the shortest words of rank $m$, and the Béal, Berlinkov, Perrin, and Steinberg results on the length of the shortest reset words in one-cluster automata. The obtained results are useful in computation aimed in extending the class of small automata for which the Černý conjecture is verified and discovering new automata with special properties regarding synchronization.
The synchronization problem is investigated for the class of locally strongly transitive automata introduced in a previous work of the authors. Some extensions of this problem related to the notions of stable set and word of minimal rank of an automaton are studied. An application to synchronizing colorings of aperiodic graphs with a Hamiltonian path is also considered.
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 th…
Complex systems in a wide variety of areas such as biological modeling, image processing, and language recognition can be modeled using networks of very simple machines called finite automata. Connecting subsystems modeled using finite automata into a network allows for more computational power. One such network, called a cellular automaton, consists of an n-dimensional array for n > 1 with a single finite automaton located at each point of the array. One of the oldest problems associated with cellular automata is the firing synchronization problem, originally proposed by John Myhill in 1957. …
The firing squad synchronization problem (FSSP) on cellular automata has been studied extensively for more than forty years, and a rich variety of synchronization algorithms have been proposed for not only one-dimensional arrays but two-dimensional arrays. In the present paper, we propose a simple recursive-halving based optimum-time synchronization algorithm that can synchronize any rectangle arrays of size m*n with a general at one corner in m+n+max(m, n)-3 steps. The algorithm is a natural expansion of the well-known FSSP algorithm proposed by Balzer [1967], Gerken [1987], and Waksman [1966…
Instead of looking at the lengths of synchronizing words as in Černý's conjecture, we look at the switch count of such words, that is, we only count the switches from one letter to another. Where the synchronizing words of the Černý automata $\mathcal{C}_n$ have switch count linear in $n$, we wonder whether synchronizing automata exist for which every synchronizing word has quadratic switch count. The answer is positive: we prove that switch count has the same complexity as synchronizing word length. We give some series of synchronizing automata yielding quadratic switch count, the best one re…
We present a few classes of synchronizing automata exhibiting certain extremal properties with regard to synchronization. The first is a series of automata with subsets whose shortest extending words are of length $\varTheta(n^2)$, where $n$ is the number of states of the automaton. This disproves a conjecture that every subset in a strongly connected synchronizing automaton is $cn$-extendable, for some constant $c$, and in particular, shows that the cubic upper bound on the length of the shortest reset words cannot be improved generally by means of the extension method. A detailed analysis sh…
We present a detailled timed automata model of the clock synchronization algorithm that is currently being used in a wireless sensor network (WSN) that has been developed by the Dutch company Chess. Using the Uppaal model checker, we establish that in certain cases a static, fully synchronized network may eventually become unsynchronized if the current algorithm is used, even in a setting with infinitesimal clock drifts.
The question if a deterministic finite automaton admits a software reset in the form of a so-called synchronizing word can be answered in polynomial time. In this paper, we extend this algorithmic question to deterministic automata beyond finite automata. We prove that the question of synchronizability becomes undecidable even when looking at deterministic one-counter automata. This is also true for another classical mild extension of regularity, namely that of deterministic one-turn push-down automata. However, when we combine both restrictions, we arrive at scenarios with a PSPACE-complete (…
Two topics are presented: synchronization games and synchronization costs. In a synchronization game on a deterministic finite automaton, there are two players, Alice and Bob, whose moves alternate. Alice wants to synchronize the given automaton, while Bob aims to make her task as hard as possible. We answer a few natural questions related to such games. Speaking about synchronization costs, we consider deterministic automata in which each transition has a certain price. The problem is whether or not a given automaton can be synchronized within a given budget. We determine the complexity of th…
It was conjectured by Černý in 1964, that a synchronizing DFA on $n$ states always has a 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 $n \leq 5$, and with bounds on the number of symbols for $n \leq 12$. Here we give the full analysis for $n \leq 7$, without bounds on the number of symbols. For PFAs (partial automata) on $\leq 7$ states we do a similar analysis as for DFAs and find the maximal shortest synchronizing word lengths, exceeding $(n-1)^2$ fo…
This paper situates itself in the theory of variable length codes and of finite automata where the concepts of completeness and synchronization play a central role. In this theoretical setting, we investigate the problem of finding upper bounds to the minimal length of synchronizing words and incompletable words of a finite language X in terms of the length of the words of X. This problem is related to two well-known conjectures formulated by Cerny and Restivo, respectively. In particular, if Restivo's conjecture is true, our main result provides a quadratic bound for the minimal length of a s…
In an asynchronous cooperative editing workflow of a structured document, each of the co-authors receives in the different phases of the editing process, a copy of the document to insert its contribution. For confidentiality reasons, this copy may be only a partial replica containing only parts of the (global) document which are of demonstrated interest for the considered co-author. Note that some parts may be a demonstrated interest over a co-author; they will therefore be accessible concurrently. When it's synchronization time (e.g. at the end of an asynchronous editing phase of the process)…
The set of synchronizing words of a given $n$-state automaton forms a regular language recognizable by an automaton with $2^n - n$ states. The size of a recognizing automaton for the set of synchronizing words is linked to computational problems related to synchronization and to the length of synchronizing words. Hence, it is natural to investigate synchronizing automata extremal with this property, i.e., such that the minimal deterministic automaton for the set of synchronizing words has $2^n - n$ states. The sync-maximal permutation groups have been introduced in [{\sc S. Hoffmann}, Complete…
A set of nonnegative matrices $\mathcal{M}=\{M_1, M_2, \ldots, M_k\}$ is called primitive if there exist indices $i_1, i_2, \ldots, i_m$ such that $M_{i_1} M_{i_2} \ldots M_{i_m}$ is positive (i.e. has all its entries $>0$). The length of the shortest such product is called the exponent of $\mathcal{M}$. The concept of primitive sets of matrices comes up in a number of problems within control theory, non-homogeneous Markov chains, automata theory etc. Recently, connections between synchronizing automata and primitive sets of matrices were established. In the present paper, we significantly str…
We present Woorpje, a string solver for bounded word equations (i.e., equations where the length of each variable is upper bounded by a given integer). Our algorithm works by reformulating the satisfiability of bounded word equations as a reachability problem for nondeterministic finite automata, and then carefully encoding this as a propositional satisfiability problem, which we then solve using the well-known Glucose SAT-solver. This approach has the advantage of allowing for the natural inclusion of additional linear length constraints. Our solver obtains reliable and competitive results an…
Imagine an assembly line where a box with a lid and liquid in it enters in some unknown orientation. The box should leave the line with the open lid facing upwards with the liquid still in it. To save costs there are no complex sensors or image recognition software available on the assembly line, so a reset sequence needs to be computed. But how can the dependencies of the deforming impact of a transformation of the box, such as 'do not tilt the box over when the lid is open' or 'open the lid again each time it gets closed' be modeled? We present three attempts to model constraints of these ki…
