We consider μ-calculus formulas in a normal form: after a prefix offixed-point quantifiers follows a quantifier-free expression. We are interested in theproblem of evaluating (model checking) such formulas in a powerset lattice. We assume thatthe quantifier-free part of the expression can be any monotone function given by ablack-box – we may only ask for its value for given arguments. As a first result we provethat when the lattice is fixed, the problem becomes polynomial (the assumption about thequantifier-free part strengthens this result). As a second result we show that anyalgorithm solving the problem has to ask at least about n2(namely Ω(n2/log n)) queries to the function, even when the expressionconsists of one μ and one ν (the assumption about thequantifier-free part weakens this result).

We describe a sequent calculus μLJ with primitives for inductive andcoinductive datatypes and equip it with reduction rules allowing a sound translation ofGödel’s system T. We introduce the notion of a μ-closedcategory, relying on a uniform interpretation of open μLJformulas as strong functors. We show that any μ-closed category is asound model for μLJ. We then turn to the construction of a concreteμ-closed category based on Hyland-Ong game semantics. The model relieson three main ingredients: the construction of a general class of strong functors calledopen functors acting on the category of games and strategies, thesolution of recursive arena equations by exploiting cycles in arenas, andthe adaptation of the winning conditions of parity games to build initial algebras andterminal coalgebras for many open functors. We also prove a weak completeness result forthis model, yielding a normalisation proof for μLJ.

We study iteration and recursion operators in the denotational semantics of typedλ-calculi derived from the multiset relational model of linear logic.Although these operators are defined as fixpoints of typed functionals, we prove themfinitary in the sense of Ehrhard’s finiteness spaces.

We prove that there is a constant c such that, for each positive integer k, every (2k + 1) × (2k + 1) array A on the symbols (1,. . .,2k+1) with at most c(2k+1) symbols in every cell, and each symbol repeated at most c(2k+1) times in every row and column is avoidable; that is, there is a (2k+1) × (2k+1) Latin square S on the symbols 1,. . .,2k+1 such that, for each i,j ∈ {1,. . .,2k+1}, the symbol in position (i,j) of S does not appear in the corresponding cell in A. This settles the last open case of a conjecture by Häggkvist. Using this result, we also show that there is a constant ρ, such that, for any positive integer n, if each cell in an n × n array B is assigned a set of m ≤ ρ n symbols, where each set is chosen independently and uniformly at random from {1,. . .,n}, then the probability that B is avoidable tends to 1 as n → ∞.

In the paper we develop an approach to asymptotic normality through factorial cumulants. Factorial cumulants arise in the same manner from factorial moments as do (ordinary) cumulants from (ordinary) moments. Another tool we exploit is a new identity for ‘moments’ of partitions of numbers. The general limiting result is then used to (re-)derive asymptotic normality for several models including classical discrete distributions, occupancy problems in some generalized allocation schemes and two models related to negative multinomial distribution.

We study the number of edge-disjoint Hamilton cycles one can guarantee in a sufficiently large graph G on n vertices with minimum degree δ=(1/2+α)n. For any constant α>0, we give an optimal answer in the following sense: let regeven(n,δ) denote the degree of the largest even-regular spanning subgraph one can guarantee in a graph on n vertices with minimum degree δ. Then the number of edge-disjoint Hamilton cycles we find equals regeven(n,δ)/2. The value of regeven(n,δ) is known for infinitely many values of n and δ. We also extend our results to graphs G of minimum degree δ ≥ n/2, unless G is close to the extremal constructions for Dirac's theorem. Our proof relies on a recent and very general result of Kühn and Osthus on Hamilton decomposition of robustly expanding regular graphs.

We prove the undecidability of Core XPath 1.0 (CXP) [G. Gottlob and C. Koch,inProc. of 17th Ann. IEEE Symp. on Logic in Computer Science,LICS ’02(Copenhagen,July 2002). IEEE CS Press (2002) 189–202.] extended with anInflationary Fixed Point (IFP) operator. More specifically,we provethat the satisfiability problem of this language is undecidable. In fact,the fragment ofCXP+IFP containing only the self and descendant axes is already undecidable.

In this paper, we prove several new Turán density results for 3-graphs with independent neighbourhoods. We show:

\begin{align*}\pi (K_4^-, C_5, F_{3,2})=12/49, \pi (K_4^-, F_{3,2})=5/18 \textrm {and} \pi (J_4, F_{3,2})=\pi (J_5, F_{3,2})=3/8,\end{align*}

$\binom{|A|}{2}$

\begin{align*}\pi (F_{3,2}, \textrm { induced }K_4^-)=3/8 \textrm {and} \pi (K_5, 5\textrm {-set spanning exactly 8 edges})=3/4.\end{align*}

\begin{align*}\pi (K_4, 4\textrm {-set spanning exactly 1 edge})=5/9.\end{align*}

E. Thorp introduced the following card shuffling model. Suppose the number of cards is even. Cut the deck into two equal piles, then interleave them as follows. Choose the first card from the left pile or from the right pile according to the outcome of a fair coin flip. Then choose from the other pile. Continue this way, flipping an independent coin for each pair, until both piles are empty.

We prove an upper bound of O(d3) for the mixing time of the Thorp shuffle with 2d cards, improving on the best known bound of O(d4). As a consequence, we obtain an improved bound on the time required to encrypt a binary message of length d using the Thorp shuffle.

A perfect Kt-matching in a graph G is a spanning subgraph consisting of vertex-disjoint copies of Kt. A classic theorem of Hajnal and Szemerédi states that if G is a graph of order n with minimum degree δ(G) ≥ (t − 1)n/t and t|n, then G contains a perfect Kt-matching. Let G be a t-partite graph with vertex classes V1, …, Vt each of size n. We show that, for any γ > 0, if every vertex x ∈ Vi is joined to at least $\bigl ((t-1)/t + \gamma \bigr )n$ vertices of Vj for each j ≠ i, then G contains a perfect Kt-matching, provided n is large enough. Thus, we verify a conjecture of Fischer [6] asymptotically. Furthermore, we consider a generalization to hypergraphs in terms of the codegree.

A Message Authentication Code (MAC) is a function that takes a message and a key asparameters and outputs an authentication of the message. MAC are used to guarantee thelegitimacy of messages exchanged through a network, since generating a correctauthentication requires the knowledge of the key defined secretly by trusted parties.However, an attacker with access to a sufficiently large number of message/authenticationpairs may use a brute force algorithm to infer the secret key: from a set containinginitially all possible key candidates, subsequently remove those that yield an incorrectauthentication, proceeding this way for each intercepted message/authentication pair untila single key remains. In this paper, we determine an exact formula for the expected numberof message/authentication pairs that must be used before such form of attack issuccessful, along with an asymptotical bound that is both simple and tight. We conclude byillustrating a modern application where this bound comes in handy, namely the estimationof security levels in reflection-based verification of software integrity.

We introduce a type of isomorphism among strategic games that we call localisomorphism. Local isomorphisms is a weaker version of the notions of strongand weak game isomorphism introduced in [J. Gabarro, A. Garcia and M. Serna,Theor. Comput. Sci. 412 (2011) 6675–6695]. In a localisomorphism it is required to preserve, for any player, the player’s preferences on thesets of strategy profiles that differ only in the action selected by this player. We showthat the game isomorphism problem for local isomorphism is equivalent to the same problemfor strong or weak isomorphism for strategic games given in: general, extensive andformula general form. As a consequence of the results in [J. Gabarro, A. Garcia and M.Serna, Theor. Comput. Sci. 412 (2011) 6675–6695] thisimplies that local isomorphism problem for strategic games is equivalent to (a) thecircuit isomorphism problem for games given in general form, (b) the boolean formulaisomorphism problem for formula games in general form, and (c) the graph isomorphismproblem for games given in explicit form.

We investigate the Sandpile Model and Chip Firing Game and an extension of these modelson cycle graphs. The extended model consists of allowing a negative number of chips ateach vertex. We give the characterization of reachable configurations and of fixed pointsof each model. At the end, we give explicit formula for the number of their fixedpoints.

We consider the problem of sums of dilates in groups of prime order. It is well known that sets with small density and small sumset in p behave like integer sets. Thus, given A ⊂ p of sufficiently small density, it is straightforward to show that

\begin{linenomath}$$| \lambda_{1}A+\lambda_{2}A+\cdots+ \lambda_{k}A | \ge\biggl(\sum_{i}|\lambda_{i}|\biggr)|A|- o(|A|).$$\end{linenomath}

A 3-graph is said to contain a generalized 4-cycle if it contains 4 edges A, B, C, D such that A ∩ B=C ∩ D =∅ and A ∪ B=C ∪ D. We show that a 3-graph in which every pair of vertices is contained in at least 4 edges must contain a generalized 4-cycle. When the number of vertices, n, is equivalent to 1 or 5 modulo 20, this result is optimum, in the sense that for such n there are 3-graphs where every pair of vertices is contained in 3 edges but which do not contain a generalized 4-cycle.

The study of extremal problems related to independent sets in hypergraphs is a problem that has generated much interest. There are a variety of types of independent sets in hypergraphs depending on the number of vertices from an independent set allowed in an edge. We say that a subset of vertices is j-independent if its intersection with any edge has size strictly less than j. The Kruskal–Katona theorem implies that in an r-uniform hypergraph with a fixed size and order, the hypergraph with the most r-independent sets is the lexicographic hypergraph. In this paper, we use a hypergraph regularity lemma, along with a technique developed by Loh, Pikhurko and Sudakov, to give an asymptotically best possible upper bound on the number of j-independent sets in an r-uniform hypergraph.

It is well known that an intersecting family of subsets of an n-element set can contain at most 2n−1 sets. It is natural to wonder how ‘close’ to intersecting a family of size greater than 2n−1 can be. Katona, Katona and Katona introduced the idea of a ‘most probably intersecting family’. Suppose that is a family and that 0 < p < 1. Let (p) be the (random) family formed by selecting each set in independently with probability p. A family is most probably intersecting if it maximizes the probability that (p) is intersecting over all families of size ||.

Katona, Katona and Katona conjectured that there is a nested sequence consisting of most probably intersecting families of every possible size. We show that this conjecture is false for every value of p provided that n is sufficiently large.