Günther Wirsching

Cohomology of small categories

In this paper we introduce and study the cohomology of a small category with coefficients in a natural system. This generalizes the known concepts of Watts [23] (resp. of Mitchell [17]) which use modules (resp. bimodules) as coefficients. We were led to consider natural systems since they arise in numerous examples of linear extensions of categories; in Section 3 four examples are discussed explicitly which indicate deep connection with algebraic and topological problems: (1) The category of Z//p2-modules, p prime. (2) The homotopy category of Moore spaces in degree n, n22. (3) The category of group rings of cyclic groups. (4) The homotopy category of Eilenberg-MacLane fibrations. We prove the following results on the cohomology with coefficients in a natural system: (5) An equivalence of small categories induces an isomorphism in cohomology. (6) Linear extensions of categories are classified by the second cohomology group HZ. (7) The group H’ can be described in terms of derivations. (8) Free categories have cohomological dimension 5 1, and category of fractions preserve dimension one. (9) A double cochain complex associated to a cover yields a method of computation for the cohomology; two examples are given. The results (7) and (8) correspond to known properties of the Hochschild-Mitchell cohomology, see [7] and [ 171. In the final section we discuss the various notions of cohomology of small categories, and we show that all these can be described in terms of Ext functors studied in the classical paper [l l] of Grothendieck.

Semantic dialogue modeling

This paper describes an abstract model for the semantic level of a dialogue system. We introduce mathematical structures which make it possible to design a semantic-driven dialogue system. We describe essential parts of such a system, which comprise the construction of feature-values relations representing meaning from a given world model, the modeling of the flow of information between the dialogue strategy controller and speech recogniser by a horizon of comprehension and the horizon of recognition results, the connection of these horizons to wordings via utterance-meaning pairs, and the incorporation of new horizons into a state of information. Finally, the connection to dialogue strategy controlling is sketched.

On Weighted Petri Net Transducers

In this paper we present a basic framework for weighted Petri net transducers (PNTs) for the translation of partial languages (consisting of partial words) as a natural generalisation of finite state transducers (FSTs).

On the combinatorial structure of 3 N + 1 predecessor sets

The 3n + 1 predecessor set P(a) of an integer a consists of all integers n whose iterates Ti(n) eventually hit a, where T is the 3n + 1 function defined by T(n) = n/2 if n is even, T(n) = (3n + 1)/2 if n is odd. This paper gives a representation of the sets P(a) by certain sets of finite integer sequences and studies their combinatorial structure. The notion of small sequences is introduced and explored in connection with the sets P(a). We also investigate an averaging and approximation heuristics based on small sequences.

Computational Aspects of Ordered Integer Partition with Upper Bounds

We propose a novel algorithm for computing the number of ordered integer partitions with upper bounds. This problem’s task is to compute the number of distributions of z balls into n urns with constrained capacities \(i_1,\hdots,i_n\) (see [10]). Besides the fact that this elementary urn problem has no known combinatoric solution, it is interesting because of its applications in the theory of database preferences as described in [3] and [9]. The running time of our algorithm depends only on the number of urns and not on their capacities as in other previously known algorithms.

Prime covers and periodic patterns

The work of I. Korec on irreducible (prime) disjoint covering systems of residue classes is continued. We exploit combinatorial methods to obtain two new existence theorems. First, one can prescribe the number s of occurrences of a certain modulus in a prime cover, provided s is small enough. Second, there are prime covers without supremum, provided the order is divisible by sufficiently many different primes.