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model checking abstractions of the full AVI and/or FML layers against reference
processes satisfying the properties. Instead, the abstraction principles were ap-
plied to smaller sub-systems { for example to each of the process sub-systems
shown in Figure 1 { in order to prove a modied local verication goal. The
new goals dened for the sub-systems were elaborated in such a way that the
original correctness requirements could be deduced from the new goals by com-
positional reasoning over the parallel combination of sub-systems. To increase
the eciency of the compositional reasoning process, it was tried to identify
whether sub-systems could be regarded as (renements of) instances of generic
process classes with known compositional properties.
Example 2. In the compositional reasoning process performed to prove dead-
lock freedom of the FML, it could be shown by model checking that each of
the process sub-systems depicted in Figure 1 is a renement of a process of
type \multiplexer/concentrator" specied as MUXCON below. The denition
of MUXCON is generic in the number N specifying how many outputs must
be produced before the next input can be accepted and the number, names and
alphabet of input channels
.
Observe that an instance of MUXCON dened with N = 0 never refuses an
input.
f
in 1 ;:::;
in n g
and output channels
f
out 1 ;:::;
out m g
MUXCON [ N
; f
in 1 ;:::;
in n g; f
out 1 ;:::;
out m g
]=
MC [ N
; f
in 1 ;:::;
in ` g; f
out 1 ;:::;
out m g
](0)
MC [ N
; f
in 1 ;:::;
in ` g; f
out 1 ;:::;
out m g
]( n )=
if ( n =0)
then ( GET [ N
; f
in 1 ;:::;
in ` g; f
out 1 ;:::;
out m g
]
( STOP
u
PUT [ N
; f
in 1 ;:::;
in ` g; f
out 1 ;:::;
out m g
](1)))
else PUT [ N
; f
in 1 ;:::;
in ` g; f
out 1 ;:::;
out m g
]( n )
GET [ N
; f
in 1 ;:::;
in ` g; f
out 1 ;:::;
out m g
]=
( e :
fj
in 1 ;:::;
in `
jg
e
!
MC [ N
; f
in 1 ;:::;
in ` g; f
out 1 ;:::;
out m g
]( N ))
PUT [ N
; f
in 1 ;:::;
in ` g; f
out 1 ;:::;
out m g
]( n )=
(
u
e :
fj
out 1 ;:::;
out m jg
e
!
MC [ N
; f
in 1 ;:::;
in ` g; f
out 1 ;:::;
out m g
]( n
1))
The following generic theory is associated with the above process class:
Theorem 1. A network of process instances P 1 ;:::;
P q from class
MUXCON [ N
; f
in 1 ;:::;
in n g; f
out 1 ;:::;
out m g
] is free of deadlocks, if every com-
munication cycle
c j 1
−!
c j 2
−! :::
c j k 1
−!
c j k
−!
P j 1
P j 2
P j k
P j 1
contains at least one process instance P j `
denedwithN =0 .
t
 
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