An automaton is synchronizable if there exists an input that drives it into a definite state from any other state. This notion has been studied for various types of automata. In this work, we investigate notions of synchronizability for Parikh automata (PAs). A Parikh automaton is an automaton with counters that are checked against a semilinear set at the end of the computation. We consider several notions of synchronizability (or directability) for PAs and show that they lead to decidable and PSPACE-complete problems, even for deterministic and complete Parikh automata (DCPAs) on letters. Then, we demonstrate that for DCPAs on letters, the synchronization problems are NP-complete over a unary alphabet, and NP-complete and solvable in polynomial time over a binary and a unary alphabet, respectively, if the dimension is fixed and numbers are encoded in unary. For a unary alphabet, we also provide other restrictions that ensure polynomial-time solvability. We then show that for partially ordered Parikh automata whose semilinear constraint sets have a regular Parikh-inverse image, deciding ∃∀ - D_3 -directability is NP-complete. Additionally, we prove that deciding D_3 -directability for partially ordered nondeterministic complete word automata is in AC ^0 .

The pseudovariety 𝐃𝐒 consists of all finite monoids whose regular D-classes form subsemigroups. We exhibit a uniform winning strategy for Synchronizer in the synchronization game on every synchronizing automaton whose transition monoid lies in 𝐃𝐒, and we prove that 𝐃𝐒 is the largest pseudovariety with this property.

The literal and the initial literal shuffle have been introduced to model the behavior of two synchronized processes. However, it is not possible to describe the synchronization of multiple processes. Furthermore, both restricted forms of shuffling are not associative. Here, we extend the literal shuffle and the initial literal shuffle to multiple arguments. We also introduce iterated versions, much different from the iterated ones previously introduced for the binary literal and initial literal shuffle. We investigate formal properties, and show that in terms of expressive power, in a full trio, they coincide with the general shuffle. Furthermore, we look at closure properties with respect to the regular, context-free, context-sensitive, recursive and recursively enumerable languages for all operations introduced. Then, we investigate various decision problems motivated by analogous problems for the (ordinary) shuffle operation. Most problems we look at are tractable, but we also identify one intractable decision problem.

We consider the state complexity of projection, the shuffle operation, intersection, union, up- and downward closure and interior on commutative regular languages. Usually, state complexity bounds are given in terms of the state complexities of the input languages (the sharp bound nm for union and intersection on commutative, even unary, languages of state complexities n and m, respectively, is known in the literature). Here, we associate with every commutative regular language an index and period vector, generalizing corresponding notions for unary and periodic languages from the literature, and express sharp state complexity bounds with these novel parameters. Using relations between the state complexity of a language and the index and period vectors, we deduce the state complexity bound n|Σ| for upward closure and downward interior, and (2nm)|Σ|, (nm)|Σ| and (n+m−1)|Σ| for the shuffle of general commutative regular, commutative group and commutative aperiodic languages, respectively, with state complexities n and m over an alphabet Σ. We do not know whether these bounds are sharp. We also prove the sharp bound n for projections and downward closure and upward interior. Our results are based on a canonical automaton model for commutative languages. We also show that commutative regular languages can be written as a finite union of shuffle products of commutative finite and commutative group languages intersected with a subalphabet. Furthermore, we prove characterizations of the commutative aperiodic and commutative group languages in terms of the index and periodic vectors and investigate relations between a language and their unary projection languages.

The commutative closure operation, which corresponds to the Parikh image, is a natural operation on formal languages occurring in verification and model-checking. Commutative closures of regular languages correspond to semilinear sets and, by Parikh’s theorem, to the commutative closures of context-free languages. The commutative closure is not regularity-preserving on the class of regular languages, for example already the commutative closure of the simple language (ab)^* is not regular. Here, we show that the commutative closure of a binary regular language accepted by a circular automaton yields a regular language. Then, we deduce a sufficient condition on the cycles in automata for regularity of the commutative closure. This yields this property, for example, for the following classes of automata: automata with threshold one transformation semigroups, automata with simple idempotents and almost-group automata. The fact that the commutative closure on group languages and polynomials of group languages is regularity-preserving is known in the literature. Polynomials of group languages correspond to level one-half of the group hierarchy. We also show that on the next level in this hierarchy, i.e., level one, this property is lost and the commutative closure is no longer regularity-preserving. Lastly, we give a binary circular automaton not contained in the largest proper positive variety 𝒲 closed under shuffle and commutative closure.

We exhibit a winning strategy for Synchronizer in the synchronization game on every synchronizing automaton in whose transition monoid the regular D-classes form subsemigroups.

For a finite permutation group on n elements we show the following (and variants thereof) equivalences: (1) the permutation group is primitive, (2) in the transformation monoid generated by the group and any rank n−1 mapping there exists, for every non-empty subset, an element mapping the whole permutation domain onto this subset, (3) in the transformation monoid generated by the group and any rank n−1 mapping there exists, for every two distinct subsets, an element mapping precisely one to a singleton set. We also investigate further properties related to the reachability of subsets. Lastly, we apply our results to automata and show that automata whose transformation monoids contain a primitive permutation group and a mapping that excludes precisely one state from its image are completely reachable and have the property that a minimal automaton for the set of synchronizing words has the maximal possible number of states.

We investigate the state complexity of the permutation operation, or the commutative closure, on Alphabetical Pattern Constraints (APCs). This class corresponds to level [Formula: see text] of the Straubing-Thérien hierarchy and includes the finite, the piecewise testable, or [Formula: see text]-trivial, and the [Formula: see text]-trivial and [Formula: see text]-trivial languages. We give a sharp state complexity bound expressed in terms of the longest strings in the unary projection languages of an associated finite language. Additionally, for a subclass, we give sharp bounds expressed in terms of the size of a recognizing input automaton and the size of the alphabet. We also state a related state complexity bound for the commutative closure on finite languages. Lastly, we investigate the language inclusion, equivalence and universality problems on APCs up to permutational, or Parikh, equivalence. These problems are known to be [Formula: see text]-complete on APCs in general, even for fixed alphabets. We show them to be decidable in polynomial time for fixed alphabets if we only want to solve them up to Parikh equivalence. We also correct a mistake from the conference version in a bound on the size of recognizing automata for the commutative closure.

An n -state automaton is synchronizing if it can be reset, or synchronized, into a definite state. The set of input words that synchronize the automaton is acceptable by an automaton of size 2 n - n . Shortest paths in such an accepting automaton correspond to shortest synchronizing words. Here, we introduce completely distinguishable automata, a subclass of the synchronizing automata. Being completely distinguishable is a necessary condition for a minimal automaton for the set of synchronizing words to have size 2 n - n . In fact, as we show, it has size 2 n - n if and only if the automaton is completely distinguishable and has a completely reachable subautomaton that only missed at most one state. We give different characterizations of completely distinguishable automata. Then we relate these notions to graph-theoretical constructions and investigate the subclass of automata with simple idempotents (SI-automata). We show that for these automata the properties of synchronizability, complete distinguishability and complete reachability and the minimal automaton for the set of synchronizing word having 2 n - n states are equivalent when the transformation monoid contains a transitive permutation group. A related result from the literature about SI-automata is wrong, we discuss and correct that mistake here. Lastly, using the results on SI-automata, we show that deciding complete reachability, complete distinguishability and whether the minimal automaton for the set of synchronizing words has 2 n - n states are all NL -hard problems.

Synchronizability of automata has been investigated and extended to various models. Here, we investigate notions of synchronizability for Parikh automata, i.e., automata with counters that are checked against a semilinear set at the end of the computation. We consider various notions of synchronizability (or directability) for Parikh automata and show that they give decidable and $$\textsf {PSPACE} $$ -complete problems. We then show that for deterministic and complete Parikh automata on letters the synchronization problems are $$\textsf{NP} $$ -complete over a binary alphabet and solvable in polynomial time for a unary alphabet when the dimension is fixed and the semilinear set is encoded in unary. For a binary encoding the problem remains $$\textsf{NP} $$ -hard even for unary two-state deterministic and complete Parikh automata of dimension one.

We present new automata-theoretical characterizations for the regularity of the iterated shuffle on commutative regular languages. Using these characterizations we show that, for a fixed alphabet, it is tractable to decide whether the iterated shuffle of a regular commutative language is itself regular when the input language is given by a deterministic automaton. Additionally, we introduce two new subclasses of commutative regular languages, called Type I and Type II languages, on which the iterated shuffle is regularity-preserving and show that the iterated shuffle of a commutative language is a Type I language if it is regular. Additionally, we establish various closure properties and show that we can decide if a language given by deterministic automaton is in one of these classes in polynomial time.

We introduce a subclass of the commutative regular languages that is characterized by the property that the state set of the minimal deterministic automaton can be written as a certain Cartesian product. This class behaves much better with respect to the state complexity of the shuffle, for which we find the bound~$2nm$ if the input languages have state complexities $n$ and $m$, and the upward and downward closure and interior operations, for which we find the bound~$n$. In general, only the bounds $(2nm)^{|\Sigma|}$ and $n^{|\Sigma|}$ are known for these operations in the commutative case. We prove different characterizations of this class and present results to construct languages from this class. Lastly, in a slightly more general setting of partial commutativity, we introduce other, related, language classes and investigate the relations between them.

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 the following classes of input automata: monotonic automata, solvable automata and automata with simple idempotents over a binary alphabet. The latter class contains, for example, the Černý family of automata, an infinite family of n -state automata whose shortest synchronizing words have length ( n - 1 ) 2 . Solvable automata generalize both commutative automata and weakly acyclic automata. We find that for monotonic automata, the problem is solvable in nondeterministic logarithmic space, for every regular constraint language. We identify a subclass of solvable automata for which the problem is in NP. This subclass strictly contains the weakly acyclic automata. In the course of our investigation we derive a sharp linear upper bound for the length of a shortest synchronizing word in a given constraint language, improving previous quadratic bounds for weakly acyclic and commutative automata. We also give structural characterizations of solvable automata and their recognized languages that imply that they recognize languages for which partial commutation closures give regular languages. Lastly, we show that for input automata with simple idempotents over a binary alphabet and with a constraint language given by a partial automaton with up to three states, the constrained synchronization problem is in P.

The notion of synchronization of finite automata is connected to one of the long-standing open problems in combinatorial automata theory, which is Černý’s Conjecture. In this paper, we focus on so-called synchronization games. We will discuss how to present synchronization questions in a playful way. This leads us to study related complexity questions on certain classes of finite automata. More precisely, we consider weakly acyclic, commutative and k -simple idempotent automata. We encounter a number of complexity classes, ranging from L up to PSPACE .

We present new automata-theoretical characterizations for the regularity of the iterated shuffle on commutative regular languages. Using these characterizations we show that, for a fixed alphabet, it is tractable to decide whether the iterated shuffle of a regular commutative language is itself regular when the input language is given by a deterministic automaton. Additionally, we introduce two new subclasses of commutative regular languages, called Type I and Type II languages, on which the iterated shuffle is regularity-preserving and show that the iterated shuffle of a commutative language is a Type I language if it is regular. Additionally, we establish various closure properties and show that we can decide if a language given by deterministic automaton is in one of these classes in polynomial time.

We investigate the state complexity of the upward and downward closure and interior operations on commutative regular languages. Then, we systematically study the state complexity of these operations and of the shuffle operation on commutative group languages and commutative aperiodic (or star-free) languages.

Every regular ideal language is the set of synchronizing words of some automaton. The reset complexity of a regular ideal language is the size of such an automaton with the minimal number of states. The state complexity is the size of a minimal automaton recognizing a regular language in the usual sense. There exist regular ideal languages whose state complexity is exponentially larger than its reset complexity. We call an automaton sync-maximal, if the reset complexity of the ideal language induced by its set of synchronizing words equals the number of states of the automaton and the gap between the reset complexity and the state complexity of this language is maximal possible. An automaton is completely reachable, if we can map the whole state set to any nonempty subset of states (for synchronizing automata, it is only required that the whole state set can be mapped to a singleton set). We first state a general structural result for sync-maximal automata. This shows that sync-maximal automata are closely related to completely reachable automata. We then investigate automata with simple idempotents and show that for these automata complete reachability and sync-maximality are equivalent. Lastly, we find that for automata with simple idempotents over a binary alphabet, subset reachability problems that are PSPACE-complete in general are solvable in polynomial time.

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 [S. Hoffmann, Completely Reachable Automata, Primitive Groups and the State Complexity of the Set of Synchronizing Words, LATA 2021] by stipulating that an associated automaton to the group and a non-permutation has this extremal property. The definition is in analogy with the synchronizing groups and analog to a characterization of primitivity obtained in the mentioned work. The precise relation to other classes of groups was mentioned as an open problem. Here, we solve this open problem by showing that the sync-maximal groups are precisely the primitive groups. Our result gives a new characterization of the primitive groups. Lastly, we explore an alternative and stronger definition than sync-maximality.

In the Intersection Non-Emptiness problem, we are given a list of finite automata $A_1,A_2,\dots,A_m$ over a common alphabet $\Sigma$ as input, and the goal is to determine whether some string $w\in \Sigma^*$ lies in the intersection of the languages accepted by the automata in the list. We analyze the complexity of the Intersection Non-Emptiness problem under the promise that all input automata accept a language in some level of the dot-depth hierarchy, or some level of the Straubing-Thérien hierarchy. Automata accepting languages from the lowest levels of these hierarchies arise naturally in the context of model checking. We identify a dichotomy in the dot-depth hierarchy by showing that the problem is already NP-complete when all input automata accept languages of the levels zero or one half and already PSPACE-hard when all automata accept a language from the level one. Conversely, we identify a tetrachotomy in the Straubing-Thérien hierarchy. More precisely, we show that the problem is in AC$^0$ when restricted to level zero; complete for LOGSPACE or NLOGSPACE, depending on the input representation, when restricted to languages in the level one half; NP-complete when the input is given as DFAs accepting a language in from level one or three half; and finally, PSPACE-complete when the input automata accept languages in level two or higher. Moreover, we show that the proof technique used to show containment in NP for DFAs accepting languages in the Straubing-Thérien hierarchy levels one ore three half does not generalize to the context of NFAs. To prove this, we identify a family of languages that provide an exponential separation between the state complexity of general NFAs and that of partially ordered NFAs. To the best of our knowledge, this is the first superpolynomial separation between these two models of computation.