International Journal of Foundations of Computer Science

In this paper we study the non-deterministic communication complexity of regular languages. We show that a regular language has either constant or at least logarithmic non-deterministic communication complexity. We prove several linear lower bounds which we know cover a wide range of regular languages with linear complexity. Furthermore we find evidence that previous techniques (Tesson and Thérien 2005) for proving linear lower bounds, for instance in deterministic and probabilistic models, do not work in the non-deterministic setting.

Considering the uniform distribution on sets of m non-empty words whose sum of lengths is n, we establish that the average state complexities of the rational operations are asymptotically linear.

The classes of languages which are Boolean combinations of languages of the form

Here we deal with similar questions concerning the finite unions and positive Boolean combinations of the languages of the form above. In the first case the corresponding pseudovarieties are given by a single identity, in the second case there are finite bases for k equals to 1 and 2 and there is no finite basis for k ≥ 4 (the case k = 3 remains open). All the pseudovarieties are generated by a single algebraic structure.

We show a Θ(n log n) bound on the maximal number of occurrences of primitively-rooted k-th powers occurring in a string of length n for any integer k, k ≥ 2. We also show a Θ(n²) bound on the maximal number of primitively-rooted powers with fractional exponent e, 1 < e < 2, occurring in a string of length n. This result holds obviously for their maximal number of occurrences. The first result contrasts with the linear number of occurrences of maximal repetitions of exponent at least 2.

In this paper we introduce and study some new cooperation protocols for cooperating distributed (CD) grammar systems. These derivation modes depend on the number of different nonterminals present in the sentential form obtained when a component finished a derivation phase. This measure describes the competence of the grammar on the string (the competence is high if the number of the different nonterminals is small). It is also a measure of the efficiency of the grammar on the given string (a component is more efficient than another one if it is able to decrease the number of nonterminals in the string to a greater extent). We prove that if the underlying derivation mode is the t-mode derivation, then some variants of these systems determine the class of random context ET0L languages. If these CD grammar systems use the k step limited derivations as underlying derivation mode, then they are able to generate any recursively enumerable language.

We define tree automata with global equality and disequality constraints (TAGED). TAGEDs can test (dis)equalities between subtrees which may be arbitrarily faraway. In particular, they are equipped with an equality relation and a disequality relation on states, so that whenever two subtrees t and t′ evaluate (in an accepting run) to two states which are in the (dis)equality relation, they must be (dis)equal. We study several properties of TAGEDs, and prove emptiness decidability of for several expressive subclasses of TAGEDs.

We give an O(n + t) time algorithm to determine whether an NFA with n states and t transitions accepts a language of polynomial or exponential growth. Given an NFA accepting a language of polynomial growth, we can also determine the order of polynomial growth in O(n+t) time. We also give polynomial time algorithms to solve these problems for context-free grammars.

We investigate deterministic forgetting automata, i.e., deterministic linear bounded automata which can only use the operations 'move', 'erase' (rewrite with a blank symbol) and 'delete' (remove completely). We give a taxonomy of deterministic forgetting automata and draw comparisons to other kinds of automata (namely deterministic one-turn pushdown automata and one-way one-counter automata).

Finite words and their overlap properties are considered in this paper. Let w be a finite word of length n with period p and where the maximal length of its unbordered factors equals k. A word is called unbordered if it possesses no nonempty proper prefix which is also a suffix of that word. Suppose k < p in w. It is known that n ≤ 2k - 2 if w has an unbordered prefix u of length k. We show that if n = 2k - 2, then u ends in abⁱ, with two different letters a and b and i ≥ 1, and bⁱ occurs exactly once in w. This answers a conjecture by Harju and the second author of this paper about a structural property of maximal Duval extensions. Moreover, we show here that i < k/3, which in turn leads us to the solution of a special case of a problem raised by Ehrenfeucht and Silberger in 1979.

We study balance properties of circular words over alphabets of size greater than two. We give some new characterizations of balanced words connected to the Kawasaki-Ising model and to the notion of derivative of a word. Moreover we consider two different generalizations of the notion of balance, and we find some relations between them. Some of our results can be generalized to non periodic infinite words as well.

Shared-memory systems appear as a generalization of asynchronous cellular automata. In this paper we relate the partial-order semantics of shared-memory systems to Mazurkiewicz trace languages by means of a new refinement construction. We show that a set of labeled partial orders is recognized by some unambiguous shared-memory system if and only if it is definable in monadic second-order logic and media-bounded.