We introduce adhesive categories, which are categories with structure ensuring that pushouts along monomorphisms are well-behaved, as well as quasiadhesive categories which restrict attention to regular monomorphisms. Many examples of graphical structures used in computer science are shown to be examples of adhesive and quasiadhesive categories. Double-pushout graph rewriting generalizes well to rewriting on arbitrary adhesive and quasiadhesive categories.

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 .

In this paper, the identification of stochastic regular languages is addressed. For this purpose, we propose a class of algorithms which allow for the identification of the structure of the minimal stochastic automaton generating the language. It is shown that the time needed grows only linearly with the size of the sample set and a measure of the complexity of the task is provided. Experimentally, our implementation proves very fast for application purposes.

Considering each occurrence of a word w in a recurrentinfinite word, we define the setof return words of w to be the set of all distinct words beginningwith an occurrence of wand ending exactly just before the next occurrence of w in the infiniteword. We give a simpler proof of therecent result (of the second author) that an infinite word is Sturmianif and only if each of its factors has exactly two return words in it.Then, considering episturmian infinite words, which are a naturalgeneralization of Sturmian words,we study the position of the occurrences of any factorin such infinite wordsand we determinate the return words. At last, we apply these results inorder to get a kind of balance property ofepisturmian words and to calculate the recurrence function of thesewords.

This paper gives a semantical underpinning for a many-sorted modal logic associated with certain dynamical systems, like transition systems, automata or classes in object-oriented languages. These systems will be described as coalgebras of so-called polynomial functors, built up from constants and identities, using products, coproducts and powersets. The semantical account involves Boolean algebras with operators indexed by polynomial functors, called MBAOs, for Many-sorted Boolean Algebras with Operators, combining standard (categorical) models of modal logic and of many-sorted predicate logic. In this setting we will see Lindenbaum MBAO models as initial objects, and canonical coalgebraic models of maximally consistent sets of formulas as final objects. They will be used to (re)prove completeness results, and Hennessey-Milner style characterisation results for the modal logic, first established by Rößiger.

This paper establishes computational equivalence of two seemingly unrelated concepts: linear conjunctive grammars and trellis automata. Trellis automata, also studied under the name of one-way real-time cellular automata, have been known since early 1980s as a purely abstract model of parallel computers, while linear conjunctive grammars, introduced a few years ago, are linear context-free grammars extended with an explicit intersection operation. Their equivalence implies the equivalence of several other formal systems, including a certain restricted class of Turing machines and a certain type of language equations, thus giving further evidence for the importance of the language family they all generate.

Beame, Cook and Hoover were the first to exhibita log-depth, polynomial size circuit family for integerdivision. However, the family was not logspace-uniform.In this paper we describe log-depth, polynomialsize, logspace-uniform, i.e., NC 1 circuit family forinteger division. In particular, by a well-known result this showsthat division is in logspace. We also refine the methodof the paper to show that division is in dlogtime-uniformNC 1.

Probabilistic operational semantics for a nondeterministic extension of pure λ-calculus is studied. In this semantics, a term evaluates to a (finite or infinite) distribution of values. Small-step and big-step semantics, inductively and coinductively defined, are given. Moreover, small-step and big-step semantics are shown to produce identical outcomes, both in call-by-value and in call-by-name. Plotkin’s CPS translation is extended to accommodate the choice operator and shown correct with respect to the operational semantics. Finally, the expressive power of the obtained system is studied: the calculus is shown to be sound and complete with respect to computable probability distributions.

In this paper, we survey the rich theory of infinite episturmian words which generalize to any finite alphabet, in a rather resembling way, the well-known family of Sturmian words on two letters. After recalling definitions and basic properties, we consider episturmian morphisms that allow for a deeper study of these words. Some properties of factors are described, including factor complexity, palindromes, fractional powers, frequencies, and return words. We also consider lexicographical properties of episturmian words, as well as their connection to the balance property, and related notions such as finite episturmian words, Arnoux-Rauzy sequences, and “episkew words” that generalize the skew words of Morse and Hedlund.

In this paper we give a simple proof that the alternation-depth hierarchy of the μ-calculus for binary trees is strict. The witnesses for this strictness are the automata that determine whether there is a winning strategy for the parity game played on a tree.

We propose a new way of characterizing the complexity of online problems. Instead of measuring the degradation of the output quality caused by the ignorance of the future we choose to quantify the amount of additional global information needed for an online algorithm to solve the problem optimally. In our model, the algorithm cooperates with an oracle that can see the whole input. We define the advice complexity of the problem to be the minimal number of bits (normalized per input request, and minimized over all algorithm-oracle pairs) communicated by the algorithm to the oracle in order to solve the problem optimally. Hence, the advice complexity measures the amount of problem-relevant information contained in the input. We introduce two modes of communication between the algorithm and the oracle based on whether the oracle offers an advice spontaneously (helper) or on request (answerer). We analyze the Paging and DiffServ problems in terms of advice complexity and deliver upper and lower bounds in both communication modes; in the case of DiffServ problem in helper mode the bounds are tight.

It is proposed to compare strategies in a parity game by comparing the sets of behaviours they allow. For such a game, there may be no winning strategy that encompasses all the behaviours of all winning strategies. It is shown, however, that there always exists a permissive strategy that encompasses all the behaviours of all memoryless strategies. An algorithm for finding such a permissive strategy is presented. Its complexity matches currently known upper bounds for the simpler problem of finding the set of winning positions in a parity game. The algorithm can be seen as a reduction of a parity to a safety game and computation of the set of winning positions in the resulting game.

We characterize conjugation classes of Christoffel words (equivalently of standard words) by the number of factors. We give several geometric proofs of classical results on these words and sturmian words.

This paper provides a framework to address termination problems in term rewriting by using orderings induced by algebras over the reals. The generation of such orderings is parameterized by concrete monotonicity requirements which are connected with different classes of termination problems: termination of rewriting, termination of rewriting by using dependency pairs, termination of innermost rewriting, top-termination of infinitary rewriting, termination of context-sensitive rewriting, etc. We show how to define term orderings based on algebraic interpretations over the real numbers which can be used for these purposes. From a practical point of view, we show how to automatically generate polynomial algebras over the reals by using constraint-solving systems to obtain the coefficients of a polynomial in the domain of the real or rational numbers. Moreover, as a consequence of our work, we argue that software systems which are able to generate constraints for obtaining polynomial interpretations over the naturals which prove termination of rewriting (e.g., AProVE, CiME, and TTT), are potentially able to obtain suitable interpretations over the reals by just solving the constraints in the domain of the real or rational numbers.

We present an algorithm which produces, in some cases, infinite words avoiding both large fractional repetitions and a given set of finite words. We use this method to show that all the ternary patterns whose avoidability index was left open in Cassaigne's thesis are 2-avoidable. We also prove that there exist exponentially many $\frac{7}{4}^+$-free ternary words and $\frac{7}{5}^+$-free 4-ary words. Finally we give small morphisms for binary words containing only the squares 2, 12 and (01)² and for binary words avoiding large squares and fractional repetitions.

The standard procedure to transform a regular expression of size n to an ϵ-free nondeterministic finite automaton yields automata with O(n) states and O(n2) transitions. For a long time this was supposed to be also the lower bound, but a result by Hromkovic et al. showed how to build an ϵ-free NFA with only O(n log2(n)) transitions. The current lower bound on the number of transitions is Ω(n log(n)). A rough running time estimation for the common follow sets (CFS) construction proposed by Hromkovič et al. yields a cubic algorithm. In this paper we present a sequential algorithm for the CFS construction which works in time O(n log(n) + size of the output). On a CREW PRAM the CFS construction can be performed in time O(log(n)) using O(n + (size of the output)/log(n)) processors. We also present a simpler proof of the lower bound on the number of transitions.

Recently, a new measurement – the advice complexity – was introduced for measuring the information content of online problems. The aim is to measure the bitwise information that online algorithms lack, causing them to perform worse than offline algorithms. Among a large number of problems, a well-known scheduling problem, job shop scheduling with unit length tasks, and the paging problem were analyzed within this model. We observe some connections between advice complexity and randomization. Our special focus goes to barely random algorithms, i.e., randomized algorithms that use only a constant number of random bits, regardless of the input size. We adapt the results on advice complexity to obtain efficient barely random algorithms for both the job shop scheduling and the paging problem. Furthermore, so far, it has not yet been investigated for job shop scheduling how good an online algorithm may perform when only using a very small (e.g., constant) number of advice bits. In this paper, we answer this question by giving both lower and upper bounds, and also improve the best known upper bound for optimal algorithms.

Fine and Wilf's theorem has recently been extended to words having threeperiods. Following the method of the authors we extend it to anarbitrary number of periods and deduce from that a characterization ofgeneralized Arnoux-Rauzy sequences or episturmian infinite words.

We associate with a word w on a finite alphabet A an episturmian (or Arnoux-Rauzy) morphism and a palindrome. We study their relations with the similar ones for the reversal of w. Then when |A|=2 we deduce, using the Sturmian words that are the fixed points of the two morphisms, a proof of a Galois theorem on purely periodic continued fractions whose periods are the reversal of each other.

The cyclic shift of a language L, defined as SHIFT(L) = {vu | uv ∈ L}, is an operation known to preserve both regularity and context-freeness. Its descriptional complexity has been addressed in Maslov's pioneering paper on the state complexity of regular language operations [Soviet Math. Dokl.11 (1970) 1373–1375], where a high lower bound for partial DFAs using a growing alphabet was given. We improve this result by using a fixed 4-letter alphabet, obtaining a lower bound (n-1)! . 2(n-1)(n-2), which shows that the state complexity of cyclic shift is 2n2+ nlogn - O(n) for alphabets with at least 4 letters. For 2- and 3-letter alphabets, we prove $2^{\Theta(n^2)}$ state complexity. We also establish a tight 2n2+1 lower bound for the nondeterministic state complexity of this operation using a binary alphabet.