Mainstream object-oriented languages often fail to provide complete powerful features altogether, such as, multiple inheritance, dynamic overloading and copy semantics of inheritance. In this paper we present a core object-oriented imperative language that integrates all these features in a formal framework. We define a static type system and a translation of the language into the meta-language λ_object,, in order to account for semantic issues and prove type safety of our proposal.

The algebraic counterpart of the Wagner hierarchy consists of a well-founded and decidable classification of finite pointed ω-semigroups of width 2 and height ωω . This paper completes the description of this algebraic hierarchy. We first give a purely algebraic decidability procedure of this partial ordering by introducing a graph representation of finite pointed ω-semigroups allowing to compute their precise Wagner degrees.The Wagner degree of any ω-rational language can therefore be computed directly on its syntactic image. We then show how to build a finite pointed ω-semigroup of any given Wagner degree. We finally describe the algebraic invariants characterizing every degree of this hierarchy.

In the XML standard, data are represented as unranked labeledordered trees. Regular unranked tree automata provide a usefulformalism for the validation of schemas enforcing regular structuralconstraints on XML documents. However some concrete applicationcontexts need the expression of more general constraints than theregular ones. In this paper we propose a new framework in whichcontext-free style structural constraints can be expressed andvalidated. This framework is characterized by: (i) the introductionof a new notion of trees, the so-called typed unranked labeledtrees (tulab trees for short) in which each node receivesone of three possible types (up, down or fix), and (ii) thedefinition of a new notion of tree automata, the so-callednested sibling tulab tree automata, able to enforcecontext-free style structural constraints on tulab tree languages.During their structural control process, such automata are usingvisibly pushdown languages of words [R. Alur and P. Madhusudan, Visibly pushdown languages, 36th ACM symposium on Theory of Computing, Chicago, USA (2004) 202–211] on theiralphabet of states. We show that the resulting class NSTL oftulab tree languages recognized by nested sibling tulab treeautomata is robust, i.e. closed under Boolean operations and withdecision procedures for the classical membership, emptiness andinclusion problems. We then give three characterizations of NSTL :a logical characterization by defining an adequate logic in whichNSTL happens to coincide with the models of monadic second ordersentences; the two other characterizations are using adequateencodings and map together languages of NSTL with some regularsets of 3-ary trees or with particular sets of binary trees.

This paper discusses context-free rewriting systems inwhich there exist two disjoint finite sets of rules, and a symbol, referred to as a condition of applicability, is attached to each rule in either of these two sets. In one set, a rule with a symbol attached to it is applicable if the attached symbol occurs in the current rewritten string while in the other set, such a rule is applicable if the attached symbol does not occur there. The present paper demonstrates that these rewriting systems are computationally complete. From this main result, the paper derives several consequences and solves several open problems.

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.

The algebraic study of formal languages shows that ω-rational sets correspond precisely to the ω-languages recognizable by finite ω-semigroups. Within this framework, we provide a construction of the algebraic counterpart of the Wagner hierarchy. We adopt a hierarchical game approach, by translating the Wadge theory from the ω-rational language to the ω-semigroup context. More precisely, we first show that the Wagner degree is indeed a syntactic invariant. We then define a reduction relation on finite pointed ω-semigroups by means of a Wadge-like infinite two-player game. The collection of these algebraic structures ordered by this reduction is then proven to be isomorphic to the Wagner hierarchy, namely a well-founded and decidable partial ordering of width 2 and height ωω .

We prove that every Sturmian word ω has infinitely many prefixes ofthe form UnVn3 , where |Un| < 2.855|Vn| andlimn→∞|Vn| = ∞. In passing, we give a very simple proof of theknown fact that every Sturmian word begins in arbitrarily long squares.

Consider the following one-player game. Starting with the empty graph on n vertices, in every step a new edge is drawn uniformly at random and inserted into the current graph. This edge has to be coloured immediately with one of r available colours. The player's goal is to avoid creating a monochromatic copy of some fixed graph F for as long as possible. We prove a lower bound of nβ(F,r) on the typical duration of this game, where β(F,r) is a function that is strictly increasing in r and satisfies limr→∞ β(F,r) = 2 − 1/m2(F), where n2−1/m2(F) is the threshold of the corresponding offline colouring problem.

For two graphs S and T, the constrained Ramsey number f(S, T) is the minimum n such that every edge colouring of the complete graph on n vertices (with any number of colours) has a monochromatic subgraph isomorphic to S or a rainbow subgraph isomorphic to T. Here, a subgraph is said to be rainbow if all of its edges have different colours. It is an immediate consequence of the Erdős–Rado Canonical Ramsey Theorem that f(S, T) exists if and only if S is a star or T is acyclic. Much work has been done to determine the rate of growth of f(S, T) for various types of parameters. When S and T are both trees having s and t edges respectively, Jamison, Jiang and Ling showed that f(S, T) ≤ O(st2) and conjectured that it is always at most O(st). They also mentioned that one of the most interesting open special cases is when T is a path. In this paper, we study this case and show that f(S, Pt) = O(st log t), which differs only by a logarithmic factor from the conjecture. This substantially improves the previous bounds for most values of s and t.

This special issue is devoted to papers from the meeting on Combinatorics, Probability and Computing, held at the Mathematisches Forschungsinstitut in Oberwolfach from 29 October 2006 to 4 November 2006. Like most conferences at Oberwolfach, this was an exciting and stimulating occasion; we were treated to numerous excellent talks, many of which provoked a great deal of interest and discussion among the participants.

We study relaxations of proper two-colourings, such that the order of the induced monochromatic components in one (or both) of the colour classes is bounded by a constant. A colouring of a graph G is called (C1, C2)-relaxed if every monochromatic component induced by vertices of the first (second) colour is of order at most C1 (C2, resp.). We prove that the decision problem ‘Is there a (1, C)-relaxed colouring of a given graph G of maximum degree 3?’ exhibits a hardness jump in the component order C. In other words, there exists an integer f(3) such that the decision problem is NP-hard for every 2 ≤ C < f(3), while every graph of maximum degree 3 is (1, f(3))-relaxed colourable. We also show f(3) ≤ 22 by way of a quasilinear time algorithm, which finds a (1, 22)-relaxed colouring of any graph of maximum degree 3. Both the bound on f(3) and the running time greatly improve earlier results. We also study the symmetric version, that is, when C1 = C2, of the relaxed colouring problem and make the first steps towards establishing a similar hardness jump.

A family of subsets of {1, . . ., n} is called a j-junta if there exists J ⊆ {1, . . ., n}, with |J| = j, such that the membership of a set S in depends only on S ∩ J.

In this paper we provide a simple description of intersecting families of sets. Let n and k be positive integers with k < n/2, and let be a family of pairwise intersecting subsets of {1, . . ., n}, all of size k. We show that such a family is essentially contained in a j-junta , where j does not depend on n but only on the ratio k/n and on the interpretation of ‘essentially’.

When k = o(n) we prove that every intersecting family of k-sets is almost contained in a dictatorship, a 1-junta (which by the Erdős–Ko–Rado theorem is a maximal intersecting family): for any such intersecting family there exists an element i ∈ {1, . . ., n} such that the number of sets in that do not contain i is of order (which is approximately times the size of a maximal intersecting family).

Our methods combine traditional combinatorics with results stemming from the theory of Boolean functions and discrete Fourier analysis.

Consider the following one-player game. Starting with the empty graph on n vertices, in every step a new edge is drawn uniformly at random and inserted into the current graph. This edge has to be coloured immediately with one of r available colours. The player's goal is to avoid creating a monochromatic copy of some fixed graph F for as long as possible. We prove an upper bound on the typical duration of this game if F is from a large class of graphs including cliques and cycles of arbitrary size. Together with lower bounds published elsewhere, explicit threshold functions follow.

This paper has two main focal points. We first consider an important class of machine learning algorithms: large margin classifiers, such as Support Vector Machines. The notion of margin complexity quantifies the extent to which a given class of functions can be learned by large margin classifiers. We prove that up to a small multiplicative constant, margin complexity is equal to the inverse of discrepancy. This establishes a strong tie between seemingly very different notions from two distinct areas.

In the same way that matrix rigidity is related to rank, we introduce the notion of rigidity of margin complexity. We prove that sign matrices with small margin complexity rigidity are very rare. This leads to the question of proving lower bounds on the rigidity of margin complexity. Quite surprisingly, this question turns out to be closely related to basic open problems in communication complexity, e.g., whether PSPACE can be separated from the polynomial hierarchy in communication complexity.

Communication is a key ingredient in many types of learning. This explains the relations between the field of learning theory and that of communication complexity [6, l0, 16, 26]. The results of this paper constitute another link in this rich web of relations. These new results have already been applied toward the solution of several open problems in communication complexity [18, 20, 29].

Let C(3)n denote the 3-uniform tight cycle, that is, the hypergraph with vertices v1, .–.–., vn and edges v1v2v3, v2v3v4, .–.–., vn−1vnv1, vnv1v2. We prove that the smallest integer N = N(n) for which every red–blue colouring of the edges of the complete 3-uniform hypergraph with N vertices contains a monochromatic copy of C(3)n is asymptotically equal to 4n/3 if n is divisible by 3, and 2n otherwise. The proof uses the regularity lemma for hypergraphs of Frankl and Rödl.

Given ω ≥ 1, let be the graph with vertex set in which two vertices are joined if they agree in one coordinate and differ by at most ω in the other. (Thus is precisely .) Let pc(ω) be the critical probability for site percolation on . Extending recent results of Frieze, Kleinberg, Ravi and Debany, we show that limω→∞ωpc(ω)=log(3/2). We also prove analogues of this result for the n-by-n grid and in higher dimensions, the latter involving interesting connections to Gilbert's continuum percolation model. To prove our results, we explore the component of the origin in a certain non-standard way, and show that this exploration is well approximated by a certain branching random walk.

Consider a random multigraph G* with given vertex degrees d1,. . .,dn, constructed by the configuration model. We show that, asymptotically for a sequence of such multigraphs with the number of edges , the probability that the multigraph is simple stays away from 0 if and only if . This was previously known only under extra assumptions on the maximum degree maxidi. We also give an asymptotic formula for this probability, extending previous results by several authors.

Jim Propp's rotor–router model is a deterministic analogue of a random walk on a graph. Instead of distributing chips randomly, each vertex serves its neighbours in a fixed order. We analyse the difference between the Propp machine and random walk on the infinite two-dimensional grid. It is known that, apart from a technicality, independent of the starting configuration, at each time the number of chips on each vertex in the Propp model deviates from the expected number of chips in the random walk model by at most a constant. We show that this constant is approximately 7.8 if all vertices serve their neighbours in clockwise or order, and 7.3 otherwise. This result in particular shows that the order in which the neighbours are served makes a difference. Our analysis also reveals a number of further unexpected properties of the two-dimensional Propp machine.

In majority bootstrap percolation on a graph G, an infection spreads according to the following deterministic rule: if at least half of the neighbours of a vertex v are already infected, then v is also infected, and infected vertices remain infected forever. We say that percolation occurs if eventually every vertex is infected.

The elements of the set of initially infected vertices, A ⊂ V(G), are normally chosen independently at random, each with probability p, say. This process has been extensively studied on the sequence of torus graphs [n]d, for n = 1,2, . . ., where d = d(n) is either fixed or a very slowly growing function of n. For example, Cerf and Manzo [17] showed that the critical probability is o(1) if d(n) ≤ log*n, i.e., if p = p(n) is bounded away from zero then the probability of percolation on [n]d tends to one as n → ∞.

In this paper we study the case when the growth of d to ∞ is not excessively slow; in particular, we show that the critical probability is 1/2 + o(1) if d ≥ (log log n)2 log log log n, and give much stronger bounds in the case that G is the hypercube, [2]d.