2 - Behavioural Equivalence

Summary

2.1 Strong barbed congruence

This section begins to develop the theory of behavioural equivalence by introducing strong barbed bisimilaritv and strong barbed congruence. The former is defined via a kind of bisimulation involving internal action and a notion of observation. Two terms are strong barbed congruent if the processes obtained by placing them into an arbitrary context are strong barbed bisimilar. The barbed bisimilaritv and congruence are strong in that they take processes to evolve only in single Ƭ transitions: there is no provision for abstraction from the number of Ƭ transitions that comprise an evolution. When such provision is made, we obtain (weak) barbed bisimilaritv and barbed congruence, the latter being the relation we adopt as the principal behavioural equivalence for π-terms throughout the book. These relations will be introduced in Section 2.4.

We begin by introducing strong barbed bisimilaritv. Consider first the following two-plaver game on the directed graph whose nodes are the processes and whose arrows are given by the Ƭ-transition relation. The players move alternately. A play either is infinite, or in its final position the player whose turn it is cannot move. A play begins with two nodes occupied by tokens. The first player can move either of the tokens from the node it is on along an outgoing edge to a neighbouring node. The second player can respond only by moving the other token from the node it is on along an outgoing edge to a neighbouring node. If a play is infinite then the second player wins. If after some finite number of moves the player whose turn it is cannot move, then that player loses.

For given starting processes, the second player has a winning strategy for the game iff the processes are reduction bisimilar, that is, they are related by a bisimulation on the graph.