Union-Complexities of Kleene Plus Operation

Union-free expressions are regular expressions without using the union operation. Consequently, (nondeterministic) union-free languages are described by regular expressions using only concatenation and Kleene star. The language class is also characterised by a special class of finite automata: 1CFPAs have exactly one cycle-free accepting path from each of their states. Obviously such an automaton has exactly one accepting state. The deterministic counterpart of such class of automata defines the deterministic union-free (d-union-free, for short) languages. In this paper [Formula: see text]-free nondeterministic variants of 1CFPAs are used to define n-union-free languages. The defined language class is shown to be properly between the classes of (nondeterministic) union-free and d-union-free languages (in case of at least binary alphabet). In case of unary alphabet the class of n-union-free languages coincides with the class of union-free languages. Some properties of the new subregular class of languages are discussed, e.g., closure properties. On the other hand, a regular expression is in union normal form if it is a finite union of union-free expressions. It is well known that every regular expression can be written in union normal form, i.e., all regular languages can be described as finite unions of (nondeterministic) union-free languages. It is also known that the same fact does not hold for deterministic union-free languages, that is, there are regular languages that cannot be written as finite unions of d-union-free languages. As an important result here we show that every regular language can be defined by a finite union of n-union-free languages. This fact also allows to define n-union-complexity of regular languages.

Every infinite regular language contains a regular subset of the formuv+w for some words u, v, w, where v is not the empty word. The regularsubsets of the above form are called regular components. Some well-knowncontext-free languages, such as the Dyck language and the balanced language(over an alphabet X with |X| = 2), are decomposed as disjointunions of disjunctive languages. In this paper, we investigate thedecompositions of some of the regular languages and the context-freelanguages as disjoint unions of regular components. An infinite language iscalled regular component splittable if it can be expressed as a disjointunion of regular components and a finite set. We show that the Dycklanguage, the balanced language and some disjunctive context-free splittablelanguages are regular component splittable. We also present an example toshow that there is a disjunctive context-free language which is not regularcomponent splittable.

Special section on Descriptional Complexity of Formal Systems

Descriptional Complexity of Formal Systems

A regular language L is non-returning if in the minimal deterministic finite automaton accepting it there are no transitions into the initial state. Eom, Han and Jiraskova derived upper bounds on the state complexity of boolean operations and Kleene star, and proved that these bounds are tight using two different binary witnesses. They derived upper bounds for concatenation and reversal using three different ternary witnesses. These five witnesses use a total of six different transformations. We show that for each \(n \geqslant 4\) there exists a ternary witness of state complexity n that meets the bound for reversal and that at least three letters are needed to meet this bound. Moreover, the restrictions of this witness to binary alphabets meet the bounds for product, star, and boolean operations. We also derive tight upper bounds on the state complexity of binary operations that take arguments with different alphabets. We prove that the maximal syntactic semigroup of a non-returning language has \((n-1)^n\) elements and requires at least \(\left( {\begin{array}{c}n 2\end{array}}\right) \) generators. We find the maximal state complexities of atoms of non-returning languages. Finally, we show that there exists a most complex non-returning language that meets the bounds for all these complexity measures.

A regular language [Formula: see text] is non-returning if in the minimal deterministic finite automaton accepting it there are no transitions into the initial state. Eom, Han and Jirásková derived upper bounds on the state complexity of boolean operations and Kleene star, and proved that these bounds are tight using two different binary witnesses. They derived tight upper bounds for concatenation and reversal using three different ternary witnesses. These five witnesses use a total of six different transformations. We show that for each [Formula: see text], there exists a ternary witness of state complexity [Formula: see text] that meets the bound for reversal, and restrictions of this witness to binary alphabets meet the bounds for star, product, and boolean operations. Hence all of these operations can be handled simultaneously with a single witness, using only three different transformations. We also derive tight upper bounds on the state complexity of binary operations that take arguments with different alphabets. We prove that the maximal syntactic semigroup of a non-returning language has [Formula: see text] elements and requires at least [Formula: see text] generators. We find the maximal state complexities of atoms of non-returning languages. We show that there exists a most complex sequence of non-returning languages that meet the bounds for all of these complexity measures. Furthermore, we prove there is a most complex sequence that meets all the bounds using alphabets of minimal size.

Génération aléatoire uniforme de mots de langages rationnels

Multidimensional generalized automatic sequences and shape-symmetric morphic words

Varietes de langages et operations

We consider in this paper the approach to description of algorithms for minimizing nondeterministic finite automata. For them, we consider algorithms that are general for both exact and heuristic algorithms, as well as general for options for minimizing automata according to three different criteria. The complexity of the resulting automaton, and, if necessary, the complexity of the algorithm for constructing it, we consider depending on some characteristics of the original automaton, and not the original regular language. The problem of constructing a minimal automaton can be reduced to enumerating subsets of the set of grids, i.e. to a variant of the brute force method. But, of course, such an approach is unacceptable for practical algorithms, and other solutions to the tasks are required. Common for the algorithms considered in this paper are such auxiliary algorithms that primarily use the basic finite automata, and instead of the universal automata we use another (however, incomplete) invariant of a regular language, i.e. a special binary relation on the pairs of states set of two canonical finite automata, one of which defines the given regular language. The auxiliary algorithms considered by us are desirable to be applied together, i.e. in a complex; it is this application that gives the greatest improvement in heuristic algorithms based on the theory presented in the paper. We also give a specific use case of the local search algorithm, made using the basic automaton, and present the results of computational experiments showing an improvement in the operating time of algorithms about using this algorithm #COMESYSO1120.

The synchronized alternating devices are introduced as a generalization of alternating devices. The synchronization enables the communication among parallel processes in alternating computations. A new complexity measure — synchronization complexity is introduced to measure the amount of communication among processes. The basic results covering the reduction theorem for synchronization complexity and simulations among nondeterministic, alternating and synchronized alternating devices are established. There are found very simple devices that, adding synchronized alternation, are as powerful as Turing machines. As opposed to the fact that two-way alternating finite automata recognize only regular languages it is proved that NLOG is exactly the family of languages recognized by two-way synchronized alternating finite automata with parallel complexity 1. Already one-way synchronized alternating finite automata are more powerful than finite automata. However, the log n synchronization adds nothing to any multihead alternating device. Several further results for different types of devices are established. Using simulation through synchronized alternating devices a new simulation result of nondeterministic devices by alternating ones are obtained.

This article shows a general result about finite monoids and weight reducing string rewriting systems. As a consequence it proves a long standing conjecture in formal language theory: All regular languages are Church-Rosser congruential. The class of Church-Rosser congruential languages was introduced by McNaughton, Narendran, and Otto in 1988. A language L is Church-Rosser congruential if there exists a finite, confluent, and length-reducing semi-Thue system S such that L is a finite union of congruence classes modulo S . It was known that there are deterministic linear context-free languages which are not Church-Rosser congruential, but the conjecture was that all regular languages are of this form. The article offers a stronger statement: A language is regular if and only if it is strongly Church-Rosser congruential. It is the journal version of the conference abstract which was presented at ICALP 2012.

Minimization of nondeterministic finite automata (NFA) is a hard problem (PSPACE-complete). Bisimulations are then an attractive alternative for reducing the size of NFAs, as even bisimilarity (the largest bisimulation) is almost linear using the Paige and Tarjan algorithm. NFAs obtained from regular expressions (REs) can have the number of states linear with respect to the size of the REs and conversion methods from REs to equivalent NFAs can produce NFAs without or with transitions labelled with the empty word (e-NFA). The standard conversion without e-transitions is the position automaton, \(\mathcal{A}_{pos}\). Other conversions, such as partial derivative automata (\(\mathcal{A}_{pd}\)) or follow automata (\(\mathcal{A}_{f}\)), were proven to be quotients of the position automata (by some bisimulations). Recent experimental results suggested that for REs in (normalized) star normal form the position bisimilarity almost coincide with the \(\mathcal{A}_{pd}\) automaton. Our goal is to have a better characterization of \(\mathcal{A}_{pd}\) automata and their relation with the bisimilarity of the position automata. In this paper, we consider \(\mathcal{A}_{pd}\) automata for regular expressions without Kleene star and establish under which conditions they are isomorphic to the bisimilarity of \(\mathcal{A}_{pos}\).

We investigate the descriptional complexity of the subregular language classes of (strongly) bounded regular languages. In the first part, we study the costs for the determinization of nondeterministic finite automata accepting strongly bounded regular languages. The upper bound for the costs is larger than the costs for determinizing unary regular languages, but lower than the costs for determinizing arbitrary regular languages. In the second part, we study for (strongly) bounded languages the deterministic operational state complexity of the Boolean operations as well as the operations reversal, concatenation, and iteration. In detail, we present upper and lower bounds and we develop for the proof of the lower bounds a tool that exploits the number of different colorings of cycles occurring in deterministic finite automata accepting bounded languages.

Inductive [formula omitted]-semirings