International Journal of Foundations of Computer Science

The set of all regular languages is closed under concatenation and forms a monoid known as the monoid of regular languages. In this paper the structure of finitely generated subsemigroups of this monoid in case of one letter alphabet is investigated. We prove that finitely generated semigroups of regular languages over a one letter alphabet are Kleene, rational and thus automatic. It is already known that not all of finitely generated commutative semigroups are automatic, thus we may conclude that semigroups of unary regular languages have more rigid structure.

Fine and Wilf's well-known theorem states that any word having periods p,q and length at least p+q-gcd(p,q) also has gcd(p,q) as a period. Moreover, the length p+q-gcd(p,q) is critical since counterexamples can be provided for shorter words. This result has since been extended to partial words, or finite sequences that may contain some "holes." More precisely, any partial word u with H holes having weak periods p,q and length at least the so-denoted l_H(p,q) also has strong period gcd(p,q) provided u is not (H,(p,q))-special. This extension was done for one hole by Berstel and Boasson (where the class of (1,(p,q))-special partial words is empty), and for an arbitrary number of holes by Blanchet-Sadri. In this paper, we further extend these results, allowing an arbitrary number of weak periods. In addition to speciality, the concepts of intractable period sets and interference between periods play a role.

Tree controlled grammars are context-free grammars where the associated language only contains those terminal words which have a derivation where the word of any level of the corresponding derivation tree belongs to a given regular language. We present some results on the power of such grammars where we restrict the regular languages to some known subclasses of the family of regular languages.

In this paper we extend finite-memory automata with non-deterministic reassignment that allows an automaton to "guess" the future content of its registers, and introduce the corresponding notion of a regular expression over an infinite alphabet.

A language L ⊆ A* is literally idempotent in case that ua²v ∈ L if and only if uav ∈ L, for each u, v ∈ A*, a ∈ A. We already studied classes of such languages closely related to the (positive) varieties of the famous Straubing-Thérien hierarchy. In the present paper we start a systematic study of literal varieties of literally idempotent languages, namely we deal with the case of two letter alphabet. First, we consider natural canonical expressions for such languages. Secondly, we describe all possible classes of the form where is a literal variety of literally idempotent languages. At the end we consider also positive literal varieties of literally idempotent languages.

Restarting automata and two-pushdown automata are investigated that have a single internal state only. As such an automaton must always stay in the same state, this state is of no importance for the behaviour of the automaton. Accordingly, these automata are called stateless. We consider various types of stateless two-pushdown automata and restarting automata. We investigate their expressive power, comparing them in particular to each other and to the corresponding types of automata with states.

It is proved that the language family generated by Boolean grammars is effectively closed under injective gsm mappings and inverse gsm mappings (where gsm stands for a generalized sequential machine). The same results hold for conjunctive grammars, unambiguous Boolean grammars and unambiguous conjunctive grammars.

Membership problems for compressed strings in regular languages are investigated. Strings are represented by straight-line programs, i.e., context-free grammars that generate exactly one string. For the representation of regular languages, various formalisms with different degrees of succinctness (e.g., suitably extended regular expressions, hierarchical automata) are considered. Precise complexity bounds are derived. Among other results, it is shown that the compressed membership problem for regular expressions with intersection is PSPACE-complete. This solves an open problem of Plandowski and Rytter.

Iterative arrays (IAs) are a parallel computational model with a sequential processing of the input. They are one-dimensional arrays of interacting identical deterministic finite automata. In this paper, realtime-IAs with sublinear space bounds are used to recognize formal languages. The existence of an infinite proper hierarchy of space complexity classes between logarithmic and linear space bounds is proved. Some decidability questions on logarithmically space bounded realtime-IAs are investigated, and an optimal logarithmic space lower bound for non-regular language recognition on realtime-IAs is shown. Finally, some non-recursive trade-offs between space bounded realtime-IAs are emphasized.

We consider several problems regarding the iterated or non-iterated hairpin completion of some subclasses of regular languages. Thus we obtain a characterization of the class of regular languages as the weak-code images of the k-hairpin completion of center-disjoint k-locally testable languages in the strict sense. This result completes two results from [3] and [11]. Then we investigate some decision problems and closure properties of the family of the iterated hairpin completion of singleton languages. Finally, we discuss some algorithms regarding the possibility of computing the values of k such that the non-iterated or iterated k-hairpin completion of a given regular language does not produce new words.