Fall 2001, CSE 520: Lecture 20

An equational theory for bisimulation

CCS is provided with equational theories that help reasoning about bisimulation. The idea is to have a set of equality axioms from which we can prove, by using the laws of equality, that two processes are bisimilar.

The axioms of CCS are divided into four main categories:

  1. the static laws, which involve only the static operators, namely those operators which persists during transitions: pefix, summation and constants
  2. the dynamic laws, which involve only the dynamic operators, namely those operators which change during the transitions: pefix, summation and constants,
  3. the laws which relate one group to the other. This group consists only of the so-called expansion law.
  4. the laws for recursion

Dynamic laws

Axioms for choice

CCS is a commutative monoid wrt +: 
   1) P + 0 = P  
   2) P + (Q + R) = (P + Q) + R
   3) P + Q = Q + P
Furthermore, + is idempotent: 
   4) P + P = P

Expansion law

This is an axiom schema which expresses the behaviour of parallel process in terms of nondeterministic choice between interleavings and synchronization: 
   5) (P | Q)\L  =  Sum { a.(P'| Q) | P -a-> P' and a, ^a not in L}
                    + 
                    Sum { a.(P'| Q') | P -a-> P' and Q -^a-> Q' }
                    +
                    Sum { a.(P | Q') | Q -a-> Q' and a, ^a not in L}

Static laws

Axioms for parallel

CCS is a commutative monoid wrt |: 
   6) P | 0 = P  
   7) P | (Q | R) = (P | Q) | R
   8) P | Q = Q | P
Note that | is not idempotent.

Axioms for restriction

   9) P\L = P   if none of the names in L occurs in P
  10) P\K\L = P \(K union L)
  11) P[f]\L = P\f-1(L)[f]
  12) (P | Q)\L = P\L | Q\L   if whenever a occurs in P and ^a occurs in Q, then a and ^a are not in L

Axioms for relabeling

  13) P[Id] = P   where Id is the Identity
  14) P[f][g] = P[g o f]   where g o f represents the functional composition of f and g
  15) P[f] = P[g]   if f and g coincide on the names occurring in P
  16) (P | Q)[f] = P[f] | Q[f]   if f is one-to-one on L, 
      where L is the set of names occuring in P union the set of names occurring in Q

Recursion laws

The first law says that a definition can be interpred as an equality. In the following, E[] represents a context (a term with some "holes") and E[P] represents the term obtained by inserting P in the holes of E[]. Please note that although contexts and the relabeling operation use the same notation, they are very different concepts. 
  17) If P =def= E[P]  then  P = E[P]
The following law says that in certain cases, equations have unique solutions: 
  18) Assume that E[] is weakly guarded, namely that all the "holes" come after at least one action. Then:
      if P = E[P],  and Q = E[Q], then P = Q.

Relation with strong bisimulation

Let Ax be the set of the above axioms (1)-(18).

Proposition All the above axioms (1)-(18) are sound wrt to strong bisimulation, namely: If P = Q is in Ax, then P is strongly bisimilar to Q.

The above properties, together with the fact that bisimulation is a congruence relation, allow to derive that the equational theory based on the above axioms is sound wrt bisimulation:

Proposition Let |- denote the standard inference relation in equational reasoning, namely the derivability relation between formulae based on the equality laws. Then we have that

if Ax |- P = Q, then P is strongly bisimilar to Q.

Additional Axioms for weak bisimulation

The above axioms are sound also wrt to weak bisimulation, because strong bisimulation implies weak bisimulation. However, when we need to prove a weak bisimulation, the above laws alone are not very helpful, because they don't say anything about the invisible actions (they treat tau as any aother action). In order to have a more powerful theory, we then consider also the following laws, called tau-laws. Note that they are dynamic laws.

Tau laws

  19) a.tau.P = a.P  
  20) P + tau.P = tau.P
  21) a.(P + tau.Q) + a.Q = a.(P + tau.Q)

Relation with weak bisimulation

Let Ax' be the set of the above axioms (1)-(21). We have the following propositions:

Proposition If P = Q is in Ax', then P is weakly bisimilar to Q.

Since also weak bisimulation is a congruence relation, allow to derive that the equational theory based on the above axioms is sound wrt bisimulation:

Proposition If Ax' |- P = Q, then P is weakly bisimilar to Q.