The axioms of CCS are divided into four main categories:

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

1) P + 0 = P 2) P + (Q + R) = (P + Q) + R 3) P + Q = Q + PFurthermore, + is idempotent:

4) P + P = P

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}

6) P | 0 = P 7) P | (Q | R) = (P | Q) | R 8) P | Q = Q | PNote that | is not idempotent.

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

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

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.

**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.

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

**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.