Computational Complexity of Synchronization under Regular Commutative Constraints

正则集交换闭包的自动机

Automata for the commutative closure of regular sets

针对有限字母表生成的自由幺半群中单词交换像的概念，提出一种代数方法的简单算法：给定正则集$L$，若其交换闭包${C}(L)$正则，可生成接受该闭包的有限状态自动机。

Ideal Separation and General Theorems for Constrained Synchronization and their Application to Small Constraint Automata

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…

Commutative Regular Languages with Product-Form Minimal Automata

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 $(2nm)^{|Σ|}$ and $n^{|Σ|}$ are known for these operations in the commutative case. We prove dif…

Characterizing classes of regular languages using prefix codes of bounded synchronization delay

In this paper we continue a classical work of Schützenberger on codes with bounded synchronization delay. He was interested to characterize those regular languages where the groups in the syntactic monoid belong to a variety $H$. He allowed operations on the language side which are union, intersection, concatenation and modified Kleene-star involving a mapping of a prefix code of bounded synchronization delay to a group $G\in H$, but no complementation. In our notation this leads to the language classes $SD_G(A^\infty)$ and $SD_H(A^\infty$). Our main result shows that $SD_H(A^\infty)$ always c…

State Complexity Investigations on Commutative Languages -- The Upward and Downward Closure, Commutative Aperiodic and Commutative Group Languages

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.

Commutative Languages and their Composition by Consensual Methods

Commutative languages with the semilinear property (SLIP) can be naturally recognized by real-time NLOG-SPACE multi-counter machines. We show that unions and concatenations of such languages can be similarly recognized, relying on -- and further developing, our recent results on the family of consensually regular (CREG) languages. A CREG language is defined by a regular language on the alphabet that includes the terminal alphabet and its marked copy. New conditions, for ensuring that the union or concatenation of CREG languages is closed, are presented and applied to the commutative SLIP langu…

State Complexity Bounds for the Commutative Closure of Group Languages

In this work we construct an automaton for the commutative closure of a given regular group language. The number of states of the resulting automaton is bounded by the number of states of the original automaton, raised to the power of the alphabet size, times the product of the order of the letters, viewed as permutations of the state set. This gives the asymptotic state bound $O((n\exp(\sqrt{n\ln n}))^{|Σ|})$, if the original regular language is accepted by an automaton with $n$ states. Depending on the automaton in question, we label points of $\mathbb N_0^{|Σ|}$ by subsets of states and int…

Tightening the Complexity of Equivalence Problems for Commutative Grammars

We show that the language equivalence problem for regular and context-free commutative grammars is coNEXP-complete. In addition, our lower bound immediately yields further coNEXP-completeness results for equivalence problems for communication-free Petri nets and reversal-bounded counter automata. Moreover, we improve both lower and upper bounds for language equivalence for exponent-sensitive commutative grammars.

有限自动机子类上的同步游戏的统一胜利策略

Uniform winning strategies for the synchronization games on subclasses of finite automata

本文提出了一种适用于特定有限自动机子类的同步游戏统一胜利策略，并证明了DS伪变种是具有该性质的最大伪变种。

Uniformization Problems for Synchronizations of Automatic Relations on Words

A uniformization of a binary relation is a function that is contained in the relation and has the same domain as the relation. The synthesis problem asks for effective uniformization for classes of relations and functions that can be implemented in a specific way. We consider the synthesis problem for automatic relations over finite words (also called regular or synchronized rational relations) by functions implemented by specific classes of sequential transducers. It is known that the problem "Given an automatic relation, does it have a uniformization by a subsequential transducer?" is decida…

Partially-commutative context-free languages

The paper is about a class of languages that extends context-free languages (CFL) and is stable under shuffle. Specifically, we investigate the class of partially-commutative context-free languages (PCCFL), where non-terminal symbols are commutative according to a binary independence relation, very much like in trace theory. The class has been recently proposed as a robust class subsuming CFL and commutative CFL. This paper surveys properties of PCCFL. We identify a natural corresponding automaton model: stateless multi-pushdown automata. We show stability of the class under natural operations…