Pushout

Abstract: The dynamics of reactive systems, e.g. CCS, has often been defined using a labelled transition system (LTS). More recently it has become natural in defining dynamics to use reaction rules - i.e. unlabelled transition rules - together with a structural congruence. But LTSs lead more naturally to behavioural equivalences. So one would like to derive from reaction rules a suitable LTS. This paper shows how to derive an LTS for a wide range of reactive systems. A label for an agent a is defined to be any context F which intuitively is just large enough so that the agent Fa ("a in context F") is able to perform a reaction. The key contribution of this paper is a precise definition of "just large enough", in terms of the categorical notion of relative pushout (RPO), which ensures that bisimilarity is a congruence when sufficient RPOs exist. Two examples - a simplified form of action calculi and term-rewriting - are given, for which it is shown that sufficient RPOs indeed exist. The thrust of this paper is, therefore, towards a general method for achieving useful behavioural congruence relations.

Abstract: Buffer management schemes are needed to fairly regulate the sharing of memory among different output port queues in a shared memory ATM switch. Of the conventional schemes, static threshold is simple but does not adapt to changing traffic conditions while pushout is efficient and adaptive but difficult to implement. We propose a novel scheme called dynamic threshold which combines the simplicity of static threshold and the adaptability of pushout. The key idea is that the maximum permissible length, for any individual queue at any instant of time, is proportional to the unused buffering in the switch. A queue whose length equals or exceeds the current threshold value may accept no more new cells. The dynamic threshold procedure presented improves the fairness and switch efficiency by guaranteeing access to the buffer space for all output queues. Computer simulation is used to compare the loss performance of the dynamic threshold technique with that of static threshold and pushout. The dynamic threshold scheme is shown to be a good compromise: while nearly as simple as static threshold control, it offers most of the performance benefits of pushout. Like pushout, the dynamic threshold method is adaptive, so it is more robust to uncertainties and changes in traffic conditions than, static threshold control.

Abstract: R. Stanley, in an investigation of modular flats in geometries (Algebra Universalis 1-2 (1971), 214-217), proved that the characteristic polynomial x(x) of a modular flat x divides the characteristic polynomial x(G) of a geometry G. In this paper we identify the quotient: THEOREM. If x is a modular flat of G, x(G)/x(x) = x(Tx(G))I(X 1), where Tl(G) is the complete Brown truncation of G by x. (The lattice of TX(G) consists of all flats containing x and all flats disjoint from x, with the induced order from G.) We give many characterizations of modular flats in terms of their lattice properties as well as by means of a short-circuit axiom and a modular version of the MacLane-Steinitz exchange axiom. Modular flats are shown to have many of the useful properties of points and distributive flats (separators) in addition to being much more prevalent. The theorem relating the chromatic polynomials of two graphs and the polynomial of their vertex join across a common clique generalizes to geometries: THEOREM. Given geometries G and H, if x is a modular flat of G as well as a subgeometry of H, then there exists a geometry P = PX(G, H) which is a pushout in the category of injective strong maps and such that X(P) = x(G)X(H)/x(x). The closed set structure, rank function, independent sets, and lattice properties of P are characterized. After proving a modular extension theorem we give applications of our results to Crapo's single element extension theorem, Crapo's join operation, chain groups, unimodular geometries, transversal geometries, and graphs.