Emanuela Fachini

Locally Thin Set Families

A family of subsets of an n-set is k-locally thin if, for every k of its member sets, the ground set has at least one element contained in exactly 1 of them. We derive new asymptotic upper bounds for the maximum cardinality of locally thin set families for every even k. This improves on previous results of two of the authors with Monti.

Permutation Capacities of Families of Oriented Infinite Paths

Korner and Malvenuto asked whether one can find

C-tree systolic automata

\binom{n}{\lfloor n/2\rfloor}

Nonacceptability criteria and closure properties for the class of languages accepted by binary systolic tree automata

linear orderings (i.e., permutations) of the first

A Better Bound for Locally Thin Set Families

n

Colour Number, Capacity, and Perfectness of Directed Graphs

natural numbers such that any pair of them places two consecutive integers somewhere in the same position. This led to the notion of graph-different permutations. We extend this concept to directed graphs, focusing on orientations of the semi-infinite path whose edges connect consecutive natural numbers. Our main result shows that the maximum number of permutations satisfying all the pairwise conditions associated with all of the various orientations of this path is exponentially smaller, for any single orientation, than the maximum number of those permutations which satisfy the corresponding pairwise relationship. This is in sharp contrast to a result of Gargano, Korner, and Vaccaro concerning the analogous notion of Sperner capacity of families of finite graphs. We improve the exponential lower bound for the original problem and list a number of open questions.

A kleene-like characterization of languages accepted by systolic tree automata

Abstract A new type of systolic automaton is introduced, its structural properties, such as homogeneity and stability, are investigated and the class of languages accepted by these automata is studied. This class of languages, in the nondeterministic case, contains a large subclass of the Lindenmayer EPT0L languages. A characterization of the defined model is also given in terms of sequential machines.

Path Separation by Short Cycles

Abstract In this paper a contribution is given to the solution of the problem of finding an inductive characterization of the class of languages accepted by binary systolic tree automata, L (BSTA), in terms of the closure of a class of languages with respect to certain operations. It is shown that L (BSTA) is closed with respect to some new operations: selective concatenation, restricted concatenation and restricted iteration. The known nonclosure of L (BSTA) with respect to classical language operations, like concatenation and Kleene iteration is proved here by using a new nonacceptability criterion.

Tight Packings of Hamming Spheres

A family of subsets of an n-set is 4-locally thin if for every quadruple of its members the ground set has at least one element contained in exactly 1 of them. We show that such a family has at most 20.4561n members. This improves on our previous results with Noga Alon. The new proof is based on a more careful analysis of the self-similarity of the graph associated with such set families by the graph entropy bounding technique.

Classes of systolic Y-tree automata and a comparison with systolic trellis automata

Abstract. We introduce a new concept of chromatic number for directed graphs, called the colour number and use it to upper bound the transitive clique number and the Sperner capacity of arbitrary directed graphs. Our results represent a common generalization of previous bounds of Alon and the second author and lead to a concept of perfectness for directed graphs.