Schöning [14] introduced a notion of helping and suggested the study of the class ${\rm P}_{\rm help}({\cal C})$ of the languages that can be helped by oracles in a given class ${\cal C}$. Later, Ko [12], in order to study the connections between helping and "witness searching" , introduced the notion of self-helping for languages. We extend this notion to classes of languages and show that there exists a self-helping class that we call SH which contains all the self-helping classes. We introduce the Helping hierarchy whose levels are obtained applying a constant number of times the operator ${\rm P}_{\rm help}(\cdot)$ to the set of all the languages. We show that the Helping hierarchy collapses to the k-th level if and only if SH is equal to the k-th level. We give characterizations of all the levels and use these to construct a relativized world in which the Helping hierarchy is infinite.

We prove that for each positive integer n, the finite commutative languageEn = c(a 1 a 2...an ) possesses a test set of size at most 5n. Moreover, it is shown that each test set for E n has at least n-1 elements. The result is then generalized to commutative languages L containing a word w such that (i) alph(w) = alph}(L); and (ii) each symbol a ∈ alph}(L) occurs at least twice in w if it occurs at least twice in some word of L: each such L possesses a test set of size 11n, where n = Card(alph(L)). The considerations rest on the analysis of some basic types of word equations.

This contribution extends the notions of roots and periodicity to strings of transfinite lengths. It shows that given a transfinite string, either it possesses a unique root or the set of its roots are equivalent in a strong way.

Coalgebras for endofunctors ${\mathcal C}\rightarrow{\mathcal C}$ can be used to model classes of object-oriented languages. However, binary methods do not fit directly into this approach. This paper proposes an extension of the coalgebraic framework, namely the use of extended polynomial functors${\mathcal C}^{op} \times {\mathcal C}\rightarrow{\mathcal C}$. This extension allows the incorporation of binary methods into coalgebraic class specifications. The paper also discusses how to define bisimulation and invariants for coalgebras of extended polynomial functors and proves many standard results.

We investigate the complexity of languages described by some expressions containing shuffle operator and intersection. We show that deciding whether the shuffle of two words has a nonempty intersection with a regular set (or fulfills some regular pattern) is NL-complete. Furthermore we show that the class of languages of the form $L\cap R$, with a shuffle language L and a regular language R, contains non-semilinear languages and does not form a family of mildly context- sensitive languages.

We say that two languages X and Y are conjugates if they satisfy the conjugacy equationXZ = ZY for some language Z. We study several problems associated with this equation. For example, we characterize all sets which are conjugated via a two-element biprefix set Z, as well as all two-element sets which are conjugates.

We investigate the succinctness of several kinds of unary automata by studying their state complexity in accepting the family {Lm } of cyclic languages, where Lm = akm | k ∈ N. In particular, we show that, for any m, the number of states necessary and sufficient for accepting the unary language L m with isolated cut point on one-way probabilistic finite automata is $p_1^{\alpha_1}+ p_2^{\alpha_2} +\cdots +p_s^{\alpha_s}$ , with $p_1^{\alpha_1}p_2^{\alpha_2} \cdots p_s^{\alpha_s}$ being the factorization of m. To prove this result, we give a general state lower bound for accepting unary languages with isolated cut point on the one-way probabilistic model. Moreover, we exhibit one-way quantum finite automata that, for any m, accept L m with isolated cut point and only two states. These results are settled within a survey on unary automata aiming to compare the descriptional power of deterministic, nondeterministic, probabilistic and quantum paradigms.

The depth hierarchy results for monotone circuits of Raz and McKenzie [5] are extended to the case of monotone circuits of semi-unbounded fan-in. It follows that the inclusions NCi ⊆ SACi ⊆ ACi are proper in the monotone setting, for every i ≥ 1.

We investigate the number of iterations needed by an addition algorithm due to Burks et al. if the input is random. Several authors have obtained results on the average case behaviour, mainly using analytic techniques based on generating functions. Here we take a more probabilistic view which leads to a limit theorem for the distribution of the random number of steps required by the algorithm and also helps to explain the limiting logarithmic periodicity as a simple discretization phenomenon.

In this paper we introduce a sharpening of the Parikh mapping and investigate its basic properties. The new mapping is based on square matrices of a certain form. The classical Parikh vector appears in such a matrix as the second diagonal. However, the matrix product gives more information about a word than the Parikh vector. We characterize the matrix products and establish also an interesting interconnection between mirror images of words and inverses of .

We study hardness of approximating several minimaximal and maximinimal NP-optimizationproblems related to the minimum linear ordering problem (MINLOP). MINLOP is to find a minimum weight acyclic tournament in a given arc-weighted completedigraph. MINLOP is APX-hard but its unweighted version is polynomial timesolvable. We prove that MIN-MAX-SUBDAG problem, which is a generalization ofMINLOP and requires to find a minimum cardinality maximal acyclic subdigraphof a given digraph, is, however, APX-hard. Using results of Håstad concerning hardness of approximating independence number of a graph we then prove similar results concerning MAX-MIN-VC (respectively, MAX-MIN-FVS) which requires to find a maximum cardinality minimal vertex cover in a given graph (respectively,a maximum cardinality minimal feedback vertex set in a given digraph). We alsoprove APX-hardness of these and several related problems on various degree bounded graphs and digraphs.

We determine minimal elements, i.e., atoms, in certain partial orders of factor closed languages under ⊆. This is in analogy to structural Ramsey theory which determines minimal structures in partial orders under embedding.

In this paper methods and results related to the notion of minimal forbidden words are applied to the fragment assembly problem. The fragment assembly problem can be formulated, in its simplest form, as follows: reconstruct a word w from a given set I of substrings (fragments) of a word w. We introduce an hypothesis involving the set of fragments I and the maximal length m(w) of the minimal forbidden factors of w. Such hypothesis allows us to reconstruct uniquely the word w from the set I in linear time. We prove also that, if w is a word randomly generated by a memoryless source with identical symbol probabilities, m(w) is logarithmic with respect to the size of w. This result shows that our reconstruction algorithm is suited to approach the above problem in several practical applications e.g. in the case of DNA sequences.

This paper surveys the area of Free Burnside Semigroups. The theory of these semigroups, as is the case for groups, is far from being completely known. For semigroups, the most impressive results were obtained in the last 10 years. In this paper we give priority to the mathematical treatment of the problem and do not stress too much neither motivation nor the historical aspects. No proofs are presented in this paper, but we tried to give as many examples as was possible.

In a previous paper, the authors studied the polynomial closure of a variety of languages and gave an algebraic counterpart, in terms of Mal'cev products, of this operation. They also formulated a conjecture about the algebraic counterpart of the boolean closure of the polynomial closure – this operation corresponds to passing to the upper level in any concatenation hierarchy. Although this conjecture is probably true in some particular cases, we give a counterexample in the general case. Another counterexample, of a different nature, was independently given recently by Steinberg. Taking these two counterexamples into account, we propose a modified version of our conjecture and some supporting evidence for that new formulation. We show in particular that a solution to our new conjecture would give a solution of the decidability of the levels 2 of the Straubing–Thérien hierarchy and of the dot-depth hierarchy. Consequences for the other levels are also discussed.