Specification of Reactive Hardware/Software Systems - Electronic ...

318 Behaviour-Preserving Transformations
(b) Cb
x1¥¡ ¡ ¡ xr 2 is a process class which is derived from
Cb
x1¥¡ ¡ ¡ ¥ xr 1
by replacing (in one or more of the method bodies) one or more
occurrences of ch!m(E¡1 ¥¡ ¡ ¡ ¥ E¡r) ch¡ by !m(E¡1 ¥¡ ¡ ¡ ¥ E¡r) ch¡ (where is some arbitrary
channel), and by adjusting the set of channels and the message interface.
¡ ¥¡ ¡ ¡ ¥ Er)¥ (2) for all ch b ChSort(C1(E1 Sysp § ) ChSort(Cb ¥¡ ¡ ¡ ¥ Er)¥ 2(E1 Sysp f2(ch¡
)
f1( ch) f2( ch) and f1(ch) ).
¢
(ii)
if (1) and (2) of (i) hold and if
¢
S § p£ e
1 Cb E1£ ¡ ¡ ¡£ Er¢ § f1 ¥ env¥ Sys
1 p Sys ¥
¢
(3) S p£ e
2 (S p£ e
2 ) where is defined as
¢
¡
S § p£ e
2 Cb E1£ ¡ ¡ ¡£ Er¢ § f2 ¥ env¥ Sys
2 p Sys ¥
(ch!m(E¡1 ¥¡ ¡ ¡ ¥ E¡r)) ¥¡ ¡ ¡ ¥ ch¡
(S)
!m(E¡1 E¡r)
S
¡ !m¡ ¡ (E1¡ ¡ ¥¡ ¡ ¡ ¥ Er¡ ¡ ) ch¡ ¡ !m¡ ¡ (E1¡ ¡ ¥¡ ¡ ¡ ¥ Er¡ ¡
(ch¡
¡ ?m¡ ¡ (p1¡ ¡ ¥¡ ¡ ¡ ¥ pr¡ ¡ ) ch¡ ¡ ?m¡ ¡ (p1¡ ¡ ¥¡ ¡ ¡ ¥ pr¡ ¡
(ch¡
¡ ¥¡ ¡ ¡ ¥ Em¡ ¡ )(p1¡ ¡ ¥¡ ¡ ¡ ¥ pn¡ ¡ (E1 m¡ ¡ )) ¥¡ ¡ ¡ ¥ Em¡ ¡ )(p1¡ ¡ ¥¡ ¡ ¡ ¥ pn¡ ¡ (E1 (m¡ )
( )
(compound(S p£ e
1 ¥¡ ¡ ¡ ¥ Sp£ e
i )) compound( (S p£
)¥¡ ¡ ¡ ¥
e
1 (S p£ e
i ))
Here compound(S p£ e
1 ¥¡ ¡ ¡ ¥ Sp£ e
i ) denotes any one of the compound statements of Stat p£ e .
The message-send statement printed in bold-face denotes that it is one of the replaced
occurrences. So it is assumed that different occurrences of the same message-send
statement can be told apart.
(iii) (i) and (ii) also hold if message-receive statements in stead of message-send statements are
replaced. ¨
By transformation 14 is justified to replace a process object, on which some channel
renaming is applied, by another (derived) process object, on which some different channel
renaming is applied. For example, consider a© b ¥ Sys A()§ p Sys , where A() is an
¥
instance of process class A
defined in Sys p . Assume that A() has two channels, a and
b, and suppose it can send a message m on channel a and a message n on channel b.
By Transformation 14 the specification is equivalent to ()§ Id ¥ Sys A¡ p Sys , where A¡ ()
¥
which is defined in Sys p . This class is derived from A
is an instance of class A¡
replacing each occurrence of b!n by a!n, by changing the communication channel clause
to a, and by changing the message interface to a!n. The transformation is visualised by
the following Instance Structure Diagrams:
A
a
b
a
¢
¡
A'
a
by