Process Algebra

Process Algebra
Book: Chapter 8

The Main Issue
Q: When are two models equivalent?
A: When they satisfy different
properties.

Q: Does this mean that the models
have different executions?

What is process algebra?
 An abstract description for nondeterministic
and concurrent systems.
 Focuses on the transitions observed rather
than on the states reached.
 Main correctness criterion: conformance
between two models.
 Uses: system refinement, model checking,
testing.

Different models may have
the same set of executions!

e d a a e
d a

b c b c

a-insert coin, b-press pepsi, c-press pepsi-light
d-obtain pepsi, e-obtain pepsi-light

Actions: Act={a,b,c,d}{}.
Agents: E, E’, F, F1, F2, G1, G2, …

E F
e d a a e
d a
E’ F1 F2
b c b c
G1 G2
Agent E may evolve into agent E’.
Agent F may evolve into F1 or F2.

Events.
E F
a 
 a a 

E’ F1 F2
b c b c
G1 G2

E—aE’, F—aF1, F—aF2, F1—aG1,

F2—aG2. G1—F, G1—F.

Actions and co-actions
E F
 a   a a 
E’ F1 F2
b c b c
G1 G2

For each action a, except for , there is a co-
action a. a and a interact (a input, a output).
The coaction of a is a.

Notation
E a.(b+c)
a (actually, a.((b.0)+(c.0))
F E—aF
b c F—bG
G H F—cH
0 – deadlock/termination.
a.E – execute a, then continue according to E.
E+F – execute according to E or to F.
E||F – execute E and F in parallel.

Conventions
 “.” has higher priority than “+”.
 “.0” or “.(0||0||…||0)” is omitted.

CCS - calculus of concurrent systems
[Milner]. Syntax
 a,b,c, … actions, A, B, C - agents.
 a,b,c, coactions of a,b,c. -silent action.
 nil - terminate.
 a.E - execute a, then behave like E.
 + - nondeterministic choice.
 || - parallel composition.
 L - restriction: cannot use letters of L.
 [f] - apply mapping function f between
between letters.

Semantics (proof rule and axioms).
Structural Operational Semantics SOS

 a.p –a p
 p—ap’ |-- p+q –a p’
 q—aq’ |-- p+q –a q’
 p—ap’ |-- p|q –a p’|q
 q—aq’ |-- p|q –a p|q’
 p—ap’, q—aq’ |-- p|q – p’|q’
 p—ap’ , a  R |-- pL –a p’R
 p—ap’ |-- p[m]—m(a)p’[m]

Action Prefixing
a.E—aE (Axiom)

Thus, a.(b.(c||c)+d)—a(b.(c||c)+d).

Choice
E—aE’ F—aF’
(E+F)—aE’ (E+F)—aF’

b.(c||c)—b(c||c).
Thus,
(b.(c||c)+e)—b(c||c).

If E—aE’ and F—aF’, then E+F has a
nondeterministic choice.

Concurrent Composition
E—aE’ F—aF’
E||F—aE’||F E||F—aE||F’

E—aE’, F—aF’
————————
E||F—E’||F’

c—c0, c—c0, c||c—0||0,
c||c—c0||c, c||c—cc||0.

Restriction
E—aE’, a, a R
—————————
ER –aE’R
In this case: allows only internal
interaction of c.
c||c—0||0 c||c—c0||c
c||c—cc||0
(c||c) {c}—(0||0) {c}

Relabeling
E—aE’
————
E[m] –m(a)E’[m]

No axioms/rules for agent 0.

Examples
a a.E||b.F b
E||b.F a.E||F
b a
E||F

Derivations a.(b.(c||c)+d)
a
b b.(c||c)+d
d

(c||c) c 0
c

(0||c) (c||0)
c c

(0||0)

Modeling binary variable

set_1
set_0 set_1
is_0?
C0 C1 is_1?
set_0

C0=is_0? . C0 + set_1 . C1 + set_0 . C0

C1=is_1? . C1 + set_0 . C0 + set_1 . C1

Equational Definition
E F
 a   a a 
E’ F1 F2
b c b c
G1 G2

E=a.(b..E+c..E) E—aE’, A=E
F=a.b..F+a.c..F A—aE’

Trace equivalence:
Systems have same finite sequences.

E F
a a a

b c c
b b

E=a.(b+c) F=(a.b)+a.(b+c)
Same traces

Failures: comparing also what we
cannot do after a finite sequence.

E F
a a
a

b c c
b b

Failure of agent E: (σ, X), where after executing
σ from E, none of the events in X is enabled.
Agent F has failure (a, {c}), which is not a failure
of E.

Simulation equivalence
E F
a a a
b b b b

c d c d

 Relation over set of agents S. RSS.
 ERF
 If E’ R F’ and E’—aE’’, then there exists F’’,
F’—aF’’, and E’’ R F’’.

Simulation equivalence
E F
a a a

b b b b

c d c d

 ERF
F’—aF’’, and E’’ R F’’.

Here, simulation works only in
one direction. No equivalence!
want to establish
E F
a a a
symmetrically
b b b b
necessarily

c d c d

problem!!!
 ERF
F’—aF’’, and E’’ R F’’.

Simulation equivalent
but not failure equivalent

E F
a a
a

b b

Left agent a.b+a has a failure (a,{b}).

Bisimulation: same relation
simulates in both directions

E F
a a
a

b b

Not in this case: different simulation relations.

Hierarchy of equivalences

Bisimulation

Simulation Failure

Trace

Example:
a s1 b s2
s0 c
d s3 b
a t1 t2
t0 a
t4 c
A=a.((b.nil)+(c.d.A)) d t3
B=(a.(b.nil))+(a.c.d.B)

Bisimulation between G1 and G2

 Let N= N1 U N2
 A relation R : N1 x N2 is a bisumulation if
If (m,n) in R then
1. If m—am’ then n’:n—an’
and (m’,n’) in R
2. If n—an’ then m’:m—am’
and (m’,n’) in R.
 Other simulation relations are possible, I.e.,
m=a=> m’ when m—…—a—m’.

Algorithm for bisimulation:

 Partition N into blocks B1B2…Bn=N.
 Initially: one block, containing all of N.
 Repeat until no change:
Choose a block Bi and a letter a.
If some of the transitions of Bi move to
some block Bj and some not, partition
Bi accordingly.
 At the end: Structures bisimilar if initial states
of two structures are in same blocks.

Correctness of algorithm

 Invariant: if (m,n) in R then m and n remain
in the same block throughout the algorithm.
 Termination: can split only a finite number of
times.

Example:
a s1 b s2
s0 c
d s3 b
a t1 t2
t0 a
t4 c
d t3
{s0,s1,s2,s3,t0,t1,t2,t3,t4}

Example:
a s1 b s2
s0 c
d s3 b
a t1 t2
t0 a
t4 c
d t3
{s0,s1,s2,s3,t0,t1,t2,t3,t4} split on a.
{s0,t0},{s1,s2,s3,t1,t2,t3,t4}

Example:
a s1 b s2
s0 c
d s3 b
a t1 t2
t0 a
t4 c
d t3
{s0,t0},{s1,s2,s3,t1,t2,t3,t4} split on b
{s0,t0},{s1,t1},{s0,s2,s3,t2,t3,t4}

Example:
a s1 b s2
s0 c
d s3 b
a t1 t2
t0 a
t4 c
d t3
{s0,t0},{s1,t1},{s2,s3,t2,t3,t4} split on c
{s0,t0},{s1},{t1},{s2,s3,t2,t3,t4}

Example:
a s1 b s2
s0 c
d s3 b
a t1 t2
t0 a
t4 c
d t3
{s0,t0},{s1},{t1},{s2,s3,t2,t3,t4} split on c

{s0,t0},{s1},{t1},{t4},{s2,s3,t2,t3}

Example:
a s1 b s2
s0 c
d s3 b
a t1 t2
t0 a
t4 c
d t3
{s0,t0},{s1},{t1},{t4},{s2,s3,t2,t3} split on d
{s0,t0},{s1},{t1},{t4},{s3, t3},{s2,t2}

Example:
a s1 b s2
s0 c
d s3 b
a t1 t2
t0 a
t4 c
d t3
{s0,t0},{s1},{t1},{t4},{s2,t2},{s3,t3} split on a
{s0},{t0},{s1},{t1},{t4},{s3, t3},{s2,t2}

Example:
a s1 b s2
s0 c
d s3 b
a t1 t2
t0 a
t4 c
d t3
{s0},{t0},{s1},{t1},{t4},{s2,s3,t2,t3} split on d
{s0},{t0},{s1},{t1},{t4},{s3},{t3},{s2,t2}

Process Algebra

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