Operational union-complexity
Introduction
Descriptional complexity of formal systems, especially of formal languages is an interesting and fruitful branch of theoretical computer science. Regular languages are the most studied, as they are very well known and applied in various places. The family of regular languages is the smallest, the simplest class of the Chomsky-hierarchy. Their descriptions by regular expressions are widely used. They are generated by regular, by left-linear and also by right-linear grammars. They are accepted by finite state automata: both nondeterministic and deterministic variants characterize this class of languages. Recently various classes of subregular languages play also importance [11]. The most known measure of the descriptional complexity of regular languages is the number of states of the minimal deterministic finite automaton accepting them (in several cases, completely defined finite automata are used, which may contain a sink state) [5]. A similar measure based on nondeterministic finite automata is also studied [9]. Also, another measure that could be more interesting if the finite automata are only partially defined, namely, the transition-complexity [8], [16]. These measures are somehow connected to some global properties of the finite automata. A more recent measure is based on the number of final (or accepting) states of the automata [7], [13], comparing it to the previous measure, it seems to be a measure which reflects only part of the automata, not the whole, but a crucial part. Descriptional complexity of regular languages may also be based on regular grammars, however, since the concept of grammars and automata are closely related to each other with very simple constructions from one to the other, these measures do not seem really different than the previously mentioned ones, e.g., the state complexity is closely related to the number of nonterminals needed in the grammar, while the transition complexity is closely related to the number of rewriting rules applied in the grammar.
Measures of regular languages can also be defined based on their regular expressions. A kind of “global measure” could be the length or the number of regular operators used in an expression, and for each regular language there is a regular expression (or there are some regular expressions) having the smallest such value, which can be assigned to the described language. However, we may also give some measures which are not measuring the whole expression, but some, yet significant, details of them. For example, such a measure is the star-height, the number of nested Kleene-stars needed to describe the language [10]. Here, another, a relatively new measure is studied which is also based on the possible regular expressions of a regular language. The union-free languages are defined by regular expressions without the union. They were first mentioned as star-dot regular languages in [3]. Later on, in [6], their description by equations were examined, and it was shown that this class cannot be axiomatized by a finite set of equations. Automata theoretical characterisation of this language class was given in [18]: nondeterministic finite automata with the property that there is exactly one cycle-free accepting path from each of their states accept these languages. This class of automata allowed to define the deterministic counterpart of the class, the family of deterministic union-free languages [4], [14], [15]. On the other hand, every regular language is a finite union of union-free languages [3], [17], [21]. The union-complexity of the regular languages is defined subsequently based on minimal decompositions [17]. In this paper, we use union-complexity to measure for the complexity of regular languages.
While the invited talk and the paper appeared in the proceedings of the DCFS [20] were more about a general view about union-free languages, the corresponding 1-cycle-free-path automata and some known results about union-complexity, here, in this paper, we focus (see Section 3) on the operational union-complexity of regular languages. More specifically, we present and prove theorems about the possible values of the union-complexity of the resulting languages after applying some basic (regular, set theoretical and other) operations on the languages with known union-complexity. As far as we know, this is the first study of union-complexity from the view of descriptional complexity. In the next section we formally define the union-complexity and we give some preliminaries. In Section 3, we present our main results, while some further thought about deterministic union-complexity in Section 4. Finally, summary and conclusions close the paper.
Section snippets
Preliminaries
In this section first we recall the definition of union-complexity of regular languages [17], [19] and we also recall some known results. We assume that the reader is familiar with the basic concepts of formal languages and regular expressions, thus for each unexplained concepts she/he is referred to any standard textbook on the topic, e.g., to [12] or the Handbook chapter [22]. We also fix our notation here. The empty word is denoted by λ, V is a finite alphabet, while are the regular
Main results
As the name of the operation union is involved in the term union-complexity, we start our studies by analysing how the union-complexity may change if the union of two languages is considered.
Theorem 3.1 Let and be two regular languages with union-complexities n and m, respectively. Then the union of them, i.e., could have the union-complexity at most . Moreover, this bound is tight, i.e., for any two positive integers , there are languages and with union-complexities n and m, such
On deterministic union-complexity
We start this section by recalling the definition of deterministic union-free languages, see, e.g., [15], [20]. Since this language class is defined by a specific class of automata, we present definitions of automata first.
Definition 4.1 A 5-tuple is a non-deterministic finite automaton, with the finite set of states Q. Further, is the initial state, V is the (input) alphabet and is the set of final (or accepting) states. The function is the transition function. A path
Summary and conclusions
Every regular language is a finite union of union-free languages, that type of description of a regular language is called its union normal-form [17]. Based on the union normal-form of regular expressions, union-complexity of languages has been defined (see, e.g., [17], [19], [20]). Consequently, the class of union-free regular languages plays an important role studying union-complexity. In this paper, the union-complexity as a complexity measure of the regular languages was considered.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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