Circulations, Fuzzy Relations and Semirings

Abstract

Circulations are similar to flows in capacity-constrained networks, with the difference that they also observe lower bounds and, unlike flows, are not directed from a source to a sink. We give a new description of circulations in networks using a technique introduced by Kawahara; he applied the same methods to network flows. We show the power and flexibility of his approach in a new application, refining it at the same time by introducing the concept of test relations. Furthermore we will give algebraic formulations of a generic algorithm for computing a flow in a network with lower bounds and a sufficient and necessary criterion for the existence of a circulation.

A Semiring Approach to Equivalences, Bisimulations and Control

“…In our example we have | |T id [7,8] = id [4,5] and [8,9[ . Corresponding box operators can be defined as standard de Morgan duals of the diamonds, but we do not need them for the present paper.…”

“…Semirings have turned out to be useful for algebraic reasoning about relations and graphs, for example in [3]. Even edge-weighted graphs were successfully treated in this setting by means of fuzzy relations, as shown in [5] and [6]. Hence it is surprising that up to now no treatment of equivalence relations and bisimulations in this area has taken place, although a relational-algebraic approach was given in [11].…”