Concurrent Processes and Reaction

Concurrent Processes and Reaction

John Justine Villar Jhoirene Clemente

January 10, 2013

John Justine Villar, Jhoirene Clemente () Concurrent Processes and Reaction January 10, 2013 1 / 30

Sequential Process Expression
A Process Expression carries information about both the behaviour and the
structure of the system.
Definition 1 (Sequential Process Expression)
The set Pseq of sequential process expressions is defined by the following
syntax: P ::= A < a1 , . . . , an > | i∈I αi .Pi where I is any finite indexing
set. We use P, Q, Pi , . . . to stand for process expressions.

Here are some notations defined in Section 3.
Process identifiers: A, B, . . .
Process with name parameters: A < a, b, c >
We write a to denote a sequence of names a1 , a2 , . . . , an
The notation {b/a}P means replacing ai by bi in P where 1 ≤ i ≤ n.
The notation fn(P) denotes the set of names which occur in a process
expression. (‘fn’ stands for free names)


Sequential Process Expression

We also assume that every process identiﬁer A has a deﬁning equation of the
form
def
A(a) = PA
where PA is a summation, and the names a = a1 , . . . , an include all the free
names fn(PA ) of PA .


Labels and Flowgraphs

Outline

1 Labels and Flowgraphs

2 Observation and Reactions

3 Concurrency Process Expressions

4 Structural Congruence

5 Reaction Rules



Definition 2 (Labels)
Label L is the union of the set of names and co-names.
def ¯
L = N∪N

where N is an infinite set of names,
¯
N is and infinite set of co-names, and
¯
N ∩ N = ∅.

Labels are used in Labelled Transition Systems (LTS).
Labels are used as buttons of black boxes.
We call ¯ the complement of a.
a
As an extension of complementation,

¯ def a
a=
.



Figure: Black box A with buttons a and ¯
b.

Every complementary pair (a, ¯) of labels will represent a means of
a
interaction between black boxes. So if we have another black box B with
buttons b and ¯, we have the following system.
c



This is an example of a single ﬂowgraph.

Figure: The system containing black box A and B with interaction on the
complementary pair (b, ¯
b).

Deﬁnition 3 (Flowgraph)
Flowgraph depicts the structure of a system, i.e. linkage among its
components.

Flowgraphs should not be confused with transition systems.
Flowgraphs do not depict the dynamic property of a system.


Here is another example of a ﬂowgraph

Figure: System containing a scheduler with client processes P1 , . . . , Pn .

Note that each Pi may have many other labelled ports and a port may bear any
number of arcs.

Observation and Reactions

Outline





5 Reaction Rules




The complementary label (b, ¯ is not to be thought of as a buffer or channel
b)
having some capacity. It is a means of synchronized action or handshake.
Suppose we think of black boxes A and B as separate sequential processes.
The deﬁning equations are
def
A = a.A
def
A =¯ b.A
def
B = b.B
def
B = ¯.Bc



The composite system consisting of A and B running concurrently, with no
interdependence except that any action ¯ by A must be synchronized with an
b
action b by B and conversely.

the transition a occurs, leading to states A and B holding simultaneously;
the shared transition occurs, leading to A and B ;
the transitions a and ¯ occur in either order, or simultaneously, leading to
c
A and B again;
and so on.



We shall think of the labels b and ¯ as representing observable actions, or
b
observations.

We observe b by interacting
with it, i.e. by performing
its complement ¯ and conversely.
b,
Observations = Interactions
The shared transition in the ﬁgure
above is unobservable; we can think of it as internal action or a reaction.
Deﬁnition 4 (Reaction)
Reaction is the interaction (i.e. mutual observation) between two components
of the system, which will be denoted as τ ; being unobservable, it has no
complement.




Deﬁnition 5 (Act)
def
Act = L ∪ {τ }


Concurrency Process Expressions

Outline





5 Reaction Rules



Summation
i∈I αi .Pi
Composition
P|Q
Restriction
(new a) P, where a is a bound name and fn(P) is the set of all names
occurring free, or those that are not bound to P.

Definition 6 (Concurrent Process Expression)
The set P of (concurrent) process expressions is defined by the following
syntax:

P := A < a1 , . . . , an > | αi .Pi | P1 |P2 | (new a) P
i∈I

where I is any finite indexing set.



Changing a bound name into a fresh name us called alpha-conversion. Two
terms are structurally congruent if one is derived from the other by
alpha-conversion.
Example:

(new b)a.b = (new b )a.b
Alpha-conversion is also used to perform substitution
Example:

P = (new b)a.b

{b/a}P = (new b )b.b




To illustrate a reaction,
let P = A |B, where A = ¯ and B = b.B . Thus
b.A

P ≡ ¯ |b.B ,
b.A

so reaction between b and ¯ will occur.
b
We have the reaction
P → A|B




To illustrate an alternative reaction,
let P = new a ((a.Q1 + b.Q2 )|¯.0) | (¯ 1 + ¯.R2 )
a b.R a

P → new a Q1 |(¯ 1 + ¯.R2 )
b.R a
and
P → new a (Q2 |¯)|R1
a
Note that, a’s in a.Q1 and ¯.R2 are different. Therefore,
a

P new a(Q1 |¯)|R2
a

¯ b.R
P ≡ new a ((a .Q1 + b.Q2 )|a |(¯ 1 + ¯.R2 ))
a


Structural Congruence

Outline





5 Reaction Rules




Deﬁnition 7 (Process Context)
A process context C is, informally speaking, a process expression containing a
hole, represented by [ ]. Formally, process context are given by the syntax

C := [ ] | α.C + M |new a C | C|P | P|C
C[Q] denotes the results of ﬁlling the hole in the context C by the process Q.
The elementary contexts are α.[ ] + M, new a [ ], [ ]|P, P|[ ].
Note in particular that C = [ ] is the identify context; in this case C[Q] = Q.




Deﬁnition 8 (Process Congruence)
∼ ∼
Let = be an equivalence relation over P, i.e. it is reﬂexive (P = P),
∼ ∼ ∼ ∼
symmetric (if P = Q then Q = P) and transitive (if P = Q and Q = R then
∼ ∼
P = R). Then = is said to be a process congruence if it is preserved by all
∼
elementary contexts; that is, if P = Q then
∼
α.P + M = α.Q + M
∼
new a P = new a Q
∼
P|R = Q|R
∼
R|P = R|Q




Proposition 4.1
∼
An arbitrary equivalence relation = is a process congruence if and only if ,
for all contexts C,
∼ ∼
P = Q implies C[P] = C[Q].




Deﬁnition 9 (Structural Congruence)
Structural Congruence, written ≡, is the process congruence over P
determined by the following equations:
1 Change of bound name (alpha-conversion)
2 Reordering of terms in a summation
3 P|0 ≡ P, P|Q ≡ Q|P, P|(Q|R) ≡ (P|Q)|R
4 new a (P|Q) ≡ P|new a Q if a ∈ fn(P)
/
new a 0 ≡ 0, new ab P ≡ new ba P
def
5 A(b) ≡ {b/a}PA if A(a) = PA




Deﬁnition 10 (Standard Form)
A process expression new a (M1 | . . . |Mn ), where each Mi is a non-empty sum,
is said to be in standard form. (If n = 0 we take M1 | . . . | Mn to mean 0, If a is
empty then there is no restriction.)

Theorem 11
Every process is structurally congruent to a standard form.



Linking

Binary linking operator
def
P Q = new m({m/r}P | {m/l}Q)




Linking in a more general case

The linking operator can be generalised thus:
def
P Q = new m({m/r}P) | {m/l}Q)


Reaction Rules

Outline





5 Reaction Rules


Reaction Rules

Reaction Rules

REACT Rule
TAU Rule


Reaction Rules

Reaction Rules

Example 1:

Q = ¯.(b.B|¯
a bC)
Example 2:

a.A|Q


Reaction Rules

Reaction Rules

Example 3:

P = new a((a.Q1 + b.Q2 )|¯.0)|(¯ 1 + ¯.R2 )
a b.R a


Concurrent Processes and Reaction

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