# 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.