The Converse of a Stochastic Relation Ernst-Erich Doberkat Chair for Software Technology University of Dortmund doberkat@acm.org Abstract. Transition probabilities are proposed as the stochastic coun- terparts to set-based relations. We propose the construction of the con- verse of a stochastic relation. It is shown that two of the most useful properties carry over: the converse is idempotent as well as anticom- mutative. The nondeterminism associated with a stochastic relation is defined and briefly investigated. We define a bisimulation relation, and indicate conditions under which this relation is transitive; moreover it is shown that bisimulation and converse are compatible. Keywords: Stochastic relations, concurrency, bisimulation, converse, re- lational calculi, nondeterminism. 1 Introduction The use of relations is ubiquitous in Mathematics, Logic and Computer Science, their systematic study goes back as far as Schr¨ oder’s seminal work. Ongoing research with a focus on program specification may be witnessed from the wealth of material collected in [18, 4]. The map calculus [5] shows that these methods determine an active line of research in Logic. This paper deals with stochastic rather than set-valued relations, it studies the converse of such a relation. It investigates furthermore some similarities between forming the converse for set-theoretic relations and for their stochastic cousins. For introducing into the problem, let R be a relation, i.e., a set of pairs of, say, states. If 〈x, y〉∈ R, then this is written as x → R y and interpreted as a state transition from x to y. The converse R ⌣ shifts attention to the goal of the transition: y → R ⌣ x is interpreted as y being the goal of a transition from x. Now let p(x, y) be the probability that there is a transition from x to y, and the question arises with which probability state y is the goal of a transition from x. This question cannot be answered unless we know the initial probabilities for the states. Then we can calculate p ⌣ μ (y,x) as the probability to make a transition from x to y weighted by the probability to start from x conditional to the event to reach y at all, i.e. p ⌣ μ (y,x) := μ(x) · p(x, y) ∑ t μ(t) · p(t, y) . Consider as an example the simple transition system p on three states given in the left hand side of Fig. 1. The converse p ⌣ μ for the initial probability μ := [1/21/41/4] is given on the right hand side. A.D. Gordon (Ed.): FOSSACS 2003, LNCS 2620, pp. 233–249, 2003. c  Springer-Verlag Berlin Heidelberg 2003