states in such a way that from any two corresponding states the two ... use for identifying states of a process via abstraction homomorphisms is similar to the.
BISIMULATIONS AND ABSTRACTION HOMOMORPHISMS Ilaria
Castellani*
Computer Science Department University of Edinburgh
Abstract In t h i s
paper
to a s a m e
the
notion
of b i s i m u l a t i o n
for
a class
processes) m a y b e r e s t a t e d
of l a b e l l e d t r a n s i t i o n a s o n e of " r e d u c i b i l i t y
system" via a simple reduction relation. The reduction relation is proven to
enjoy s o m e when
we s h o w t h a t
( t h e c l a s s of no*zdet,e ~ i s i ~ , c
systems
desirable properties,
notably a Church-Rosser
restricted to finite nondeterministic
property. W e
also s h o w
processes, the relation yields unique
that,
minimal
forms for processes and can be characterised algebraically by a set of reduction rules.
I. Introduction Labelled transition systems
[K,P] are generally reeognised
as an
appropriate
model
for
nondeterministic computations. The motivation for studying such computations stems from the increasing interest in concurrent p r o g r a m m i n g . W h e n modelling c o m m u n i c a t i o n
between concurrent programs, s o m e basic difficulties have
to be faced. A concurrent p r o g r a m
is inherently part of a larger environment, with which
it interacts in the course of its computation. Therefore a simple input-output function is not an adequate about
the
behaviour from
model
internal
for such a program.
states
of a program,
as to be
should retain s o m e able to
Also, nondetermivtacy
in any interacting environment.
such parameters
The m o d e l
so
express
information
the
arises w h e n
as the relative speeds of concurrent programs:
program's abstracting
as a consequence,
we need to regard any single concurrent p r o g r a m as being itself nondeterministic. The
question
accounts
is then
to
for intermediate
find
a
model
states. On
for
the
nondeterministic
other
hand,
programs
only
should be considered which are relevant to the "interactive"
those
that
somehow
intermediate
states
(or extevmal) behaviour of
the program. N o w one can think of various criteria for selecting such significant states. In this respect labelled transition systems provide a very flexible model: definition of the transition relation one obtains a whole range going
from
a full account
of the
structure
of a p r o g r a m
by varying the
of different descriptions,
to
some
more
interesting
"abstract" descriptions. However, even these abstract descriptions still need to be factored by equivalence relations (for a. review see [B] or [DEN]). A natural notion of equivalence, bisimulation equivalence, D. Park
[Pa]
for
transition
systems:
informally
bisi~vt~late each other if a full correspondence
has b e e n recently proposed by
speaking,
two
systems
are
said
to
can be established between their sets of
*Supported by a scholarship from the Consiglio Nazionale deUe Ricerche (Italy)
224
states in such
a way
that f r o m
any two corresponding
states the two (sub)systems will
still bisimulate e a c h other. In this paper w e s h o w that the notion of bisimulation for a class of labelled transition
s y s t e m s ( t h e c l a s s of n o n d e t e r m i n i s t i c processes) m a y b e r e s t a t e d to a same enjoy
s y s t e m " via a s i m p l e r e d u c t i o n
some
desirable" properties,
when restricted
notably
relation. a
to finite nondeterministic
Church-Rosser processes,
forms for processes and can be characterised The p a p e r abstract
relation
property.
the relation
is p r o v e n to
We a l s o
yields
show that,
unique minimal
a l g e b r a i c a l l y b y a s e t of r e d u c t i o n r u l e s .
is o r g a n i s e d a s follows. In s e c t i o n 2 we p r e s e n t
c l a s s of n o n d e t e v m i n i s t i c
a s o n e of " r e d u c i b i l i t y
The r e d u c t i o n
our computational
processes. In s e c t i o n 3 we a r g u e t h a t
model, the
t h i s b a s i c m o d e l is n o t
e n o u g h , p a r t i c u l a r l y w h e n s y s t e m s a r e a l l o w e d unobsevvable t r a n s i t i o n s
o b s e r v a b l e o n e s . We t h e r e f o r e simplifying the-structure
abstraction homomorphisms
introduce
of a p r o c e s s b y m e r g i n g t o g e t h e r
a s well as
[CFM] a s a m e a n s
of
s o m e of i t s s t a t e s : t h e r e s u l t
is a p r o c e s s w i t h a s i m p l e r d e s c r i p t i o n , b u t " a b s t r a c t l y
e q u i v a l e n t " t o t h e o r i g i n a l one. We
can then infer a red~ction relation between processes
from the
homomorphisms as
invariance
reduction
e x i s t e n c e of a b s t r a c t i o n
b e t w e e n t h e m . We p r o v e s o m e s i g n i f i c a n t p r o p e r t i e s in
contexts
relation,
we
and
define
the an
announced
Church-Rosser
abstraction
equivalence
of t h i s r e l a t i o n , s u c h
property.
relation
Based
on
on
the
processes:
two
p r o c e s s e s a r e e q u i v a l e n t iff t h e y a r e b o t h r e d u c i b l e to a s a m e ( s i m p l e r ) p r o c e s s . In s e c t i o n s
4
abstraction
and the
and
5 we s t u d y notion
use for identifying states one
underlying
homomorphism
the is
relationship
of a p r o c e s s
definition a
the
of b i s i m u l a t i o n of
single-valued
between
our
notions
between transition
via a b s t r a c t i o n
bisimulation:
we
bisimulation.
homomorphisms
show
in
We f i n a l l y
of r e d u c t i o n
and
s y s t e m s . The c r i t e r i o n we is s i m i l a r to t h e
fact
that
any
abstraction
prove
that
the
abstraction
e q u i v a l e n c e is s u b s t i t u t i v e in c o n t e x t s a n d t h a t it c o i n c i d e s w i t h t h e largest ( s u b s t i t u t i v e )
bisimulation.
Our e q u i v a l e n c e
can then
be regarded
as a simple alternative
formulation
for bisimulation equivalence. In s e c t i o n 6 we c o n s i d e r a s m a l l language f o r d e f i n i n g f i n i t e n o n d e t e r m i n i s t i c e s s e n t i a l l y a s u b s e t of R. M i l n e r ' s CCS ( C a l c u l u s of C o m m u n i c a t i n g that
our results
language
c o m b i n e n e a t l y w i t h s o m e e s t a b l i s h e d f a c t s a b o u t t h e l a n g u a g e . On t h i s
o u r e q u i v a l e n c e is j u s t M i l n e r ' s observational
finite a x i o m a t i s a t i o n algebraic
congruence, f o r w h i c h a c o m p l e t e
h a s b e e n g i v e n in [HM]. So, o n t h e o n e h a n d , we g e t a r e a d y - m a d e
characterisation
for
the
abstraction
equivalence;
charaeterisation
p r o v e s h e l p f u l in w o r k i n g o u t a c o m p l e t e
that
We c o n c l u d e
language.
processes:
S y s t e m s ) [M1]. We find
by proposing
w h i c h is i s o m o r p h i c t o t h e t e r m - m o d e l
a
on
system
denotational
the
other
hand,
our
of r e d u c t i o n rules f o r
tree-model
for
the
language,
in [HM].
Most of t h e r e s u l t s will b e s t a t e d w i t h o u t p r o o f . F o r t h e p r o o f s we r e f e r t o t h e c o m p l e t e v e r s i o n of t h e p a p e r [C].
2. N o n d e t e r m i n i s t i c S y s t e m s In t h i s s e c t i o n we i n t r o d u c e systems.
Nondeterministic
i n i t i a l state.
our basic computational systems
are
essentially
m o d e l , t h e c l a s s of nondeter',ninistie labelled
transition
systems
with
an
225
L e t A be, a s e t
a n d a, b . . .
actions or transitions, containing
of e l e m e n t a r y
which denotes
o r unobser"vable t r a n s i t i o n .
a hidden
to range
Qu~rl is the set of states
> c_c_ [ ( Q u l r ~ )
the
o v e r Q u t r I, a n d w r i t e q~U_>q, f o r
According
to
> whenever
our
evolving through each state
(q, ~, q ' ) e
q' v i a a t r a n s i t i o n
and reflexive closure
an explicit reference
definition, successive
fact,
of S m a y b e t h o u g h t
if we
described
as
---->s b e
whenever
consider
the
a transition the
(Qu~r~, A,
class
S
than of
all
NDS's,
derivation
not
---->* of ---->
, w h i c h we c a l l Qs ' r s ' - - - > s
in some
transitions.
definite
derivative
between
corresponding
the states to
the
we
successive
may
relation:
we
say
notice that
that
S'
q
and
by
qs'
S
itself
is
state
regard
can
be
rooted).
derivative
a
of
S
correspondence
of S. We s h a l l d e n o t e
the
and
states.
a n NDS, s i n c e S i s o b v i o u s l y n o t
and the derivatives
state
state
On t h e o t h e r h a n d ,
of s o m e NDS: t h e n we m i g h t
going through
(although
. We i n t e r p r e t
>), w e w i l l u s e
starting
of e l e m e n t a r y
of a s t h e i n i t i a l s t a t e rather
system
associated
by means
--->
~.
S ---->s S'. Now i t is e a s y t o s e e t h a t , f o r a n y SE S, a o n e - t o - o n e
can be established
>),
to S is r e q u i r e d .
a n NDS S is a m a c h i n e states
S as giving rise to new systems,
Let
q to state
u s e of t h e t r a n s i t i v e
of Q, r,
S =
is the initial state (or root) of S, and
d e r i v a t i o n r e l a t i o n o n S. F o r a n NDS S = ( Q u t r I, A,
instead
In
(NDS) o v e r A i s a t r i p l e
of S, r ~ Q
S may evolve from state
We w i l l a l s o m a k e
o v e r A,
i s t h e trar~sitio~% rel=~io~% o n S.
x A x (Qutr~) ]
We w i l l u s e q, q' t o r a n g e q_a__> q, a s :
s y m b o l -r
to range
o v e r A - IT t.
D e f i n i t i o n B.I: k nondetev~r~i~is~ic s y s t e m
where
a distinguished
We w i l l u s e ~, u . . .
by S
corresponding
the
q
to
the
d e r i v a t i v e S'.
In
the
following
we
will
often
avail
of
this
correspondence
between
s o m e s i m p l e operators:
a nullary
states
and
operator
NIL,
(sub)systems. We a s s u m e a set
t h e c l a s s S to be c l o s e d w.r.t,
of u n a r y
meaning operator,
operators
of t h e s e
~.
operators
(one for each
#EA), a n d
a binary
a simple form
+. The i n t e n d e d
t e r m i n a t i o n , + is a f r e e - c h o i c e
i s t h e f o l l o w i n g : NIL r e p r e s e n t s
and the ~'s provide
operator
called prefixing
of s e q u e n t i a l i s a t i o n ,
by the
a c t i o n p~. The t r a n s i t i o n by means i) ii) The
~s
relation
of a c o m p o u n d
NDS m a y b e i n f e r r e d
of t h e c o m p o n e n t s
~u_> s
s J~---> s '
implies
operators
will
S + S" be
given
~ > S' , S" + S ~ a
precise
2.1 N o n d e t e r m i n i s t i c
the
and
other
concentrate
arcs
hand,
for
a
subclass
of
S,
the
class
of
in the next section.
processes
As t h e y a r e , NDS's h a v e a n i s o m o r p h i c whose nodes
S'
definition
nondefe'rrr~inistic p r o c e s s e s t h a t we w i l l i n t r o d u c e
On
from those
of t h e r u l e s :
represent any
NDS
representation
respectively may
be
the
unfolded
as (rooted)
states into
and an
the
labelled directed graphs, transitions
acyclic
graph.
of a s y s t e m . We s h a l l
here
o n a c l a s s of a c y c l i c NDS's t h a t we c a l l n o n d e t e r m i n i s t i c p r o c e s s e s (NDP's).
226
---->" is a p a r t i a l ordering. E a c h s t a t e
B a s i c a l l y , NDP's a r e NDS's w h o s e d e r i v a t i o n r e l a t i o n of
a
process
is
assigned
leading from the
root
to that
state.
To m a k e
to
The l a b e l l i n g is s u b j e c t t o t h e
there
at
most this
graph
represents
sequence. are
in t h e
that
two p a t h s
subsequently,
join
label,
a
finitely amounts
if t h e y
many to
correspond
sequence
to
the
following f u r t h e r
states
impose
the
such a labelling
a
of
observable
consistent,
same
observable
restriction:
labelled
by
general
image-finiteness
a.
As
it
actions
we only allow derivation
f o r a n y l a b e l a,
will
be
made
condition
clear on
the
systems. In t h e
formal
sequences
definition,
over
we will u s e
A, w i t h
the
usual
the
following n o t a t i o n :
prefix-ordering,
and
with
A* is empty
s i m p l i c i t y t h e s t r i n g will b e d e n o t e d b y ~. The covering r e l a t i o n partial ordering
< is g i v e n by: x--Cy iff x < y a n d ~ z s u c h t h a t
following c o n v e n t i o n : T a c t s
the
set
of f i n i t e
sequence
e
. For
--C a s s o c i a t e d t o a
x < z < y . Also, we m a k e t h e
as t h e i d e n t i t y o v e r A a n d will t h u s
be r e p l a c e d
by e when
o c c u r r i n g in s t r i n g s . D e f i n i t i o n 2.1.1:
A nondeterrn, i n i s t i c
process
(NDP) o v e r
A is
a triple
P =
( Q u l r ], P2
Ps ~
P2
equivalence
abk~ g i v e s u s a c r i t e r i o n
to r e g a r d
However, being e s s e n t i a l l y a simplification,
~
two p r o c e s s e s
as " a b s t r a c t l y
is n o t s y m m e t r i c
and therefore
the same". does not,
f o r e x a m p l e , r e l a t e t h e two p r o c e s s e s : S
a
a
b
a
b
b
or the processes:
B a s e d o n ~ - ~ , we will t h e n d e f i n e o n NDP's a m o r e g e n e r a l r e l a t i o n ~ a b s ' of r e d u c i b i l i t y to
a same process: D e f i n i t i o n 3.1.1 :
~
abs
=
~-~.~-
def
We c a n i m m e d i a t e l y p r o v e a few p r o p e r t i e s f o r Property
I:
~bs
is a n e q u i v a l e n c e .
Proof: T r a n s i t i v i t y f e l l o w s f r o m t h e f a c t t h a t as:
abs"
a~
is C h u r c h - R o s s e r , w h i c h c a n be r e s t a t e d
[~bbu~b~ ]'= " % . ~ 2:
Property
"~abs is p r e s e r v e d b y t h e o p e r a t o r s Fz. a n d +.
P r o o f : Consequence To s u m
of ab~ and ~-k-~-I invariance in ~. and ÷ contexts.
[3
up, w e have n o w a sz~bstit~ive equivalence ~ab~ for NDP*s that can be split, w h e n
required,
in
equivalence. bisimulation systems.
o
two
reduction
In the coming equivalence,
a
halves.
The
equivalence
section we will study h o w notion
introduced
by
~abs will be
called
abstraction equivalence
D. Park
[Pa]
for
general
abstraction relates to transition
231
4. B i s i m u l a t i o n A natural
relations
method
for comparing
b e h a v e li,Ee e a c h o t h e r , Now, w h a t
different
according
is t o b e t a k e n
systems
as the
behavio~r
of a s y s t e m
can always, in fact, having fixed a criterion a system be reeursively Based
on
such
equivalence bisimulate
For
an
notion
behaviour,
or
each
recursively
other
'NDS S, S': from
transition
internal
they can
need not
b e k n o w n a p r i o r i . One
one gets
between
let the behaviour
of
of i t s s u b s y s t e m s . an
(equally implicit) notion
systems:
two
systems
are
said
of to
of e i t h e r of t h e t w o, s e l e c t e d w i t h s o m e c r i t e r i o n ,
of t h e o t h e r , s e l e c t e d w i t h t h e s a m e c r i t e r i o n .
relation
a ~-subsystem
of
transitions,
following weak transition n
of b e h a v i o u r ,
bisimulation,
a subsystem
the
S' i s
to w h i c h e x t e n t
for deriving subsystems,
of t h e b e h a v i o u r s
iff a n y s u b s y s t e m
bisimulates
subsystem abstract
defined in terms
an implicit
of
i s to c h e c k
t o s o m e d e f i n i t i o n of b e h a v i o u r .
provides
S iff
a weaker
relations ~
an
S ~ >S'
obvious
for
criterion
some
criterion ~.
for
However,
w i l l be n e e d e d .
deriving
if we
are
To t h i s p u r p o s e
a to the
are introduced:
m
=: !__~_~__>.L__~
n, m > O
n
= 2--> S' is c a l l e d
n>_O
a ~-derivative
of S iff S ~ S ' .
We c a n
then
formally
define bisimulations
on
NDS's a s f o l l o w s : Definition
A (weak)
4.1:
bisimulalion
relation
is
a relation
Rc_(S ×S)
such
that
RcF(R),
w h e r e (S1,Ss) E F(R) iff ¥ p~ E A:
i) S 1 ~=> S'1
implies
3 S'2
s,t.
S s ~=> S'~ , S'1 R S~
ii) S e ~
impnes
3 S'
s.t.
S
S~
Now we k n o w t h a t
~
S'l ,
F has a maximal fixed-point
g i v e n b y u c_r(R)~R~. We will d e n o t e
this largest
S ' R S'z (which is also its m a x i m a l
bisimulation
by ,
and,
postfixed-point) since turns
o u t t o b e a n e q u i v a l e n c e , r e f e r t o i t a s the b i s i m u l a t i o n e q u i v a l e n c e . Unfortunately,
< ~ > is n o t p r e s e r v e d
by the operator
by all the operators.
P r e c i s e l y , < ~ > is n o t p r e s e r v e d
+, a s s h o w n b y t h e e x a m p l e :
NIL
, but
a
On t h e o t h e r
hand
S 1 + S~ can
be
shown
equivalence
the relation iff Y S:
to
be
contained
a
+ seems
We will
in
equivalence
~bs"
the
next
by c l o s i n g w.r.t, t h e o p e r a t o r
+:
S + S 1 S + Ss
substitutive
equivalence,
i n . ( F o r m o r e
To c o n c l u d e , see
+, o b t a i n e d
a convenient section
that
and
in
fact
to
be
the
largest
such
d e t a i l s o n < ~ > a n d < ~ > + w e r e f e r t o [MS]). restriction +
o n t o a d o p t
coincides,
on
NDP's,
w h e n m o d e l l i n g NDS's. with
our
abstraction
232
5. Relating Bisimulations to Abstraction H o m o m o r p h i s m s Looking back
at out relations
a~
and
~abs' we notice that they rely on a notion of
of states which, like bisimulations, is rec~rsive.
eq~$v~le~ce
Moreover, the recursion
builds u p on the basis of a similarity requirement (equality of t~bels) that reminds of the criterion (equality of obse~vmble
c~eriv~f~ovt sequences) used in bisimulations to derive
"bisimilar"
indicates
subsystems.
All this
there
might
be
a
close
analogy
between
abstraction equivalence and bisimulation equivalence. In fact, since we substitutive
know
bisimulation
that ~bs
is substitutive, w e
equivalence
+.
To
shall try to relate
this purpose,
we
it with the
will need
a
direct
(recursive) definition for +. Note that < ~ > + cmi~c~cifies a
only differs f r o m
system
capacities depend speaking, s o m e
can
develop
when
in that it takes into account placed
on the system having s o m e
of the "alternatives"
in
a
sum-context.
the p~-eerrtpt~xe Such
preemptive
silently reachable state where, informally
offered by the sum-context are no m o r e
available.
This suggests that w e should adopt, w h e n looking for a direct definition of +, the m o r e ~'est~,ct~.ve t r a n s i t i o n
re~at~o~s ~==>: m
n,m>O
= z___> ~ > _z__> In particular, we will have ~ be: ~
=
. > , n>O. Note on the other h a n d that, for aEA, it will
= ~.
However, + is ~-est~ct~e with respect to < ~ >
steps are concerned:
at further
only as far as the first = ~
derivation
s t e p s + b e h a v e s like , a s i t c a n b e s e e n f r o m t h e
example:
So, if w e are to recursively define < ~ > + in terms of the transitions ~=>, w e will have to somehow
counteract the strengthening effect of the ~ ' s
To this end, for any relation RC_ (S x S), a relation R E R
at steps other than the first.
("almost" R) is introduced:
(S I, $2)
iff (St,S2) E R, or (7Si,S2) E R, or (SI,7S~) E R
Then w e can define a-bisimulation ("almost" bisimulation) relations on NDS's as follows: Definition 5.1: A (~ea/c) a-b£s~m~a~io~ relation is a relation R c (Sx S) such that RcFa(R ), where (SI,Sz) E Fa(R ) iff V /z E A:
i) s~ ~=~ s'~ implies 3 S~
s.t. s~ e=~ s~. s'~ R s~
ii) S 2 ~=> S~ i m p l i e s
s.t.
Again, F
has a m a x i m a l
3 S',
S1 ~
S'1,
S'1 R S~
(post)fized-paint which is an equivalence, and which we will Both the
denote by a. The equivalence " has been proven to coivtcide with +,
233
d e f i n i t i o n of a a n d t h e p r o o f t h a t
< ~ > a = < ~ > + a r e d u e t o M. H e n n e s s y .
It c a n b e easily' s h o w n t h a t , if R is a n a - b i s i m u l a t i o n , In p a r t i e u l a r ,
for the maximal
a-bisimulation
a
t h e n R a is a n o r d i n a r y
i t is t h e e a s e t h a t
bisimulation.
~a = < ~ > .
Now, it c a n b e p r o v e d t h a t : Theorem
5.1:
abs) is a n a - b i s i m u l a t i o n .
T h e p r o o f r e l i e s o n t h e two f o l l o w i n g l e m m a ' s : L e m m a 5.1:
If P i a-k-~P2 t h e n :
Pl~=~,P'l
implies
3 P'2
s.t.
P2~=>P'2 w h e r e
eith.er P'i~b--k~P' or P'iab--k~P'2. L e m m a 5.2:
If P i a-k~P2 t h e n :
P2[~=~P'2 implies either
Note
that
reason
3 P' s.t. Pi~=>P'i where
P~bSP' s
in lemma's
or
5.1
P'i~-k~'rP'.
and
5.2 we do
C o r o l l a r y 5.1:
sb.>
not
need
consider
the
case
TP'1 ab.) P'2" T h e
a.h.'s are single-valued relations.
t h i s c a s e d o e s n o t a r i s e is t h a t c_ < ~ > s
Proof: a is t h e m a x i m a l
[]
a-bisimulation
Moreover, we have the following eharacterisation for a.h.'s: Terrni~zolo.gy:
F o r a n y NDP P, l e t Sp = Sp = IP' I P
(a-bisimulation) Theorem
)* P'] . We s a y t h a t
R is between Pi a n d P2 iff (Pl' P2 ) E R
relation
5.2: An a b s t r a c t i o n
homomorphisrn
from
Pi
to
and
P2 is
a bisimulation
Rc__ (SPiX Sp2 ).
a single-valued relation
w h i c h is b o t h a b i s i m u l a t i o n a n d a n a - b i s i m u l a t i o n b e t w e e n P l a n d P2" We n o w
come
equivalence
to
our
~sbs a n d
t h e s e two e q u i v a l e n c e s
Theorem
main the
result,
substitutive
concerning
the
bisimulation
relationship equivalence
between a
the
abstraction
. It t u r n s
out
that
coincide:
5.3: ~ ~ b s : < ~ > a
Proof of &-: F r o m c o r o l l a r y is s y m m e t r i c a l l y
5.1 we c a n i n f e r t h a t
and transitively
~ab~ :
[ "-~
Proof of -~: S u p p o s e P1 < ~ > a p~ . We w a n t to s h o w t h a t 3 P3 s.t. Let R be an a-bisimulation
._~-l]
Pl
between Pi and P . Then R can be written
R = (P. P~) - R~ [(s h- P1) x ( s p - P~)] Now c o n s i d e r :
R' : (Pl' P~) u R [ [ ( S p 1
C_ < ~ > " , s i n c e ~
closed.
P1) × (Spa- P2) ]
abs) P3 xah~- Pz " as:
234
It is e a s y to s e e t h a t However
R'
R' is b o t h a b i s i m u l a t i o n a n d a n a - b i s i m u l a t i o n b e t w e e n P1 a n d P2"
will n o t , in g e n e r a l , be s i n g l e - v a l u e d . Let t h e n ~ be t h e
equivalence induced
b y R' o n t h e s t a t e s of P2: qph
~
qp~
iff 3 P'l 6 Spi s.t. b o t h (P'f P~) a n d (P'i' P~) e R' . It c a n b e s h o w n t h a t a.h.. So Pz a ~
~ is a c o n g r u e n c e
o n Pz a n d t h e r e f o r e
Also, b y t h e o r e m
5.2, h
now the c o m p o s i t i o n
h : P z - - - > P s is a n
c a n be r e g a r d e d as a b i s i m u l a t i o n R" b e t w e e n Ps a n d Ps . C o n s i d e r
R"R":
a single-'ualzLed r e l a t i o n
t h i s is b y c o n s t r u c t i o n
(Sp x Sp ) a n d c o n t a i n i n g ( P f Ps).
up, we have Pi ~-~
Ps ~
In view of the last theorem,
c o n t a i n e d in
M o r e o v e r R"R" is a b i s i m u l a t i o n a n d a n a - b i s i m u l a t i o n ,
b e c i a u s e S b o t h R' a n d R" a r e . So, b y t h e o r e m
Summing
3 P3 s.t.
Ps"
5.2 a g a i n , P i a.bs> p .
P2 "
0
~,bs can be regarded
as a n alternative definition for < ~ > a
+. In the next section, w e will see h o w this n e w eharacterisation
=
can be used to derive
a set of reduction rules for + o n finite processes.
6. A l a n g u a g e f o r f i n i t e p r o c e s s e s In this section, w e model terms The
language
is
Systems[M]]). equivalence here
that
study the subclass
of finite NDP's, a n d
show
how
it c a n be used to
of a simple language L. essentially
In [HM]
a the
a
subset
a set of a x i o m s
(and
therefore
reduction
a_~
of
R. Milner's
is presented
Nabs) o n
CCS
(Calculus
the corresponding
itself c a n
be
of
Communicating
for L that exactly characterises
characterised
transition systems. algebraically,
by
We a
the show
set
of
reductio~z rules. T h e s e r u l e s yield n o r m a l f o r m s w h i c h c o i n c i d e w i t h t h e o n e s s u g g e s t e d in [HM]. Finally, we e s t a b l i s h a n o t i o n of m i n i m a l i t y f o r NDP's a n d u s e it t o d e f i n e a d e n o t a t i o n a l m o d e l f o r L, a c l a s s of NDP's t h a t we call R e p r e s e n t a t i o n
Trees. The m o d e l is s h o w n to be
i s o m o r p h i c with H e n n e s s y and Milner's t e r m - m o d e l . We s h a l l n o w i n t r o d u c e
t h e l a n g u a g e L. Following t h e a p p r o a c h
of [HM], we d e f i n e L as t h e
t e r m a l g e b r a Tz o v e r t h e s i g n a t u r e : Z = If w e a s s u m e
A u ~ NIL, + ]
the operators in E to denote the corresponding
denote the set of u n a r y
operators
operators ~.), w e can use finite NDP's to m o d e l
a t e r m t, w e will use Pt for the corresponding
o n NDP's ( A will t e r m s in T E. For
NDP.
W e shall point out, however, that the denotations for t e r m s of T~ in P will always be trees, i.e. NDP's P = (Q~.~rl,
t' would
Pt' '
as reference,
we are
able
to
derive
a new
system
relations
~
on
terms
o f Tz:
Y /~ E A*, l t ~
is
the
least
rules:
/~t ~t_÷ t
it)
t -L>
t'
implies
t + t"
~
of
c sbs>.
oharaeterises
define
rule
= ~-(aNIL+bNIL).
axiomatisation
which
We f i r s t
Pt'
for a
aNIL+r(aNIL+bNIL), whereas
i)
~abs
analogue
viewed
not b e a l l o w e d ,
reduction
Pt
U
, t " + t _a__> t '
*Our restrictionon the labellingfor NDP's corresponds to the g~n~ra~image-finitenexscondition: Vq,V/*, ~q' I q ~ q q is finite
236
The weak
relations
Let now
~
---->¢ b e
R e (where
(
are
the
derived
reduction
> stands
for (
from
the
relation
~'s
just
generated
as in section
by the
4.
following
set
of r e d u c t i o n
rules
> (~___>-l)): R¢
-
sumlaws
- 1st r-law - generalised absorption
6.1:
Corollary We c a n terms
x + x'
Re.
(x+x')+x"
R3.
x + NIL
R4.
/z'rx
R5.
x + /~x'
~
>
x' + x
< --->
> x+
(x'+x")
x
#x ---->
x
, whenever
x ~
x'
[C] t h a t :
t ____>c t '
6.1:
iff
Pt ~
Rc i s a r e w r i t i n g
make
use
of
our
i n T~. We s a y t h a t
can be applied Theorem
t;
forms,
we
E c.
charaeterise
normal
reduction
forms
for
(R3, R4 o r R5)
that:
i s of t h e f o r m ti
theory
if n o p r o p e r
t = ~ i /~itf i s a n o r m a l f o r m i f f
that
a process
Pv
(Hennessy-Milner
characterisation):
xt' form Y t:j
s.t.
have
/zjtj ~
a notion
or m i n i m a l iff P
tI
of m i n i m a l i t y
ab5 P'
implies
for
processes.
P = P'. T h e n t h e
We s a y following
is trivial: Theorem
F o r a n y f i n i t e NDP P, 3 ! m i n i m a l NDP P' s . t .
6.3:
Proof: f o r u n i q u e n e s s , We s h a l l Corollary
denote 6.2:
~'s unique
earlier, the ^ Pt might
denotation
of' T z, w h i c h
Church-Rosser minimal
P''
property
process
O
corresponding
to the
NDP P.
h A P ~~bs P' iff P = P',
As w e m e n t i o n e d "abstract"
use
h by P the
P ~bs
is isomorphic
[]
denotation not
P t of
be a tree.
to the term-model
Note first
that
unwinding
of a n NDP P ( w h i c h i s n o t
a n y NDP w h i c h
is not
a tree defined
a term
t
is
always
We s h a l l n o w p r o p o s e
a
tree.
However
its
a tree-model f o r t e r m s
Tz/=c. has
a unique unwinding
formally
here)
into
will be denoted
a tree.
The tree-
by U(P).
237
Let n o w R T ( r e p r e s e n t a t i o n d e n o t a t i o n T t of a t e r m tE T
t r e e s ) b e t h e c l a s s : RT = f U(P) I P is a m i n i m a l NDP ] . T h e h in RT is d e f i n e d by: Tt = U(Pt) .
It c a n b e s h o w n t h a t : T h e o r e m 6.4:
t =c t ~ iff
Tt = Tt,
We s h a l l f i n a l l y a r g u e t h a t o u r m o d e l R T is i s o m o r p h i c to t h e t e r m - m o d e l RT is a ~ . - a l g e b r a s a t i s f y i n g t h e a x i o m s E c (by t h e o r e m
Tr/= c :
6.4), w i t h t h e o p e r a t o r s
d e f i n e d by:
A
~v(p) = v(~p) v(pl) + u(P 2) = V ( ~ + ~ ) T h e r e f o r e , s i n c e T / = ¢ is t h e i n i t i a l ~ - a l g e b r a s a t i s f y i n g t h e a x i o m s E¢, we k n o w t h a t : 3 !
~-homomorphism
~ : T J = ¢ ---> R T
A It is e a s i l y s e e n t h a t x~ is g i v e n by: ~I,([t]) = U(Pt) = Tt. Also, b y t h e o r e m
6.4 a g a i n , • is a
b i j e c t i o n b e t w e e n T~ a n d RT.
Conclusion We h a v e p r o p o s e d
an alternative
< ~ > + f o r a c l a s s of t r a n s i t i o n could
be
characterised
homomorphism: should
be
just
as
Note t h a t
easily,
by
(substitutive) bisimulation
equivalence
the ordinary
equivalence
slightly
bisimulation
changing
the
in f a c t it w o u l d be e n o u g h to d r o p t h e r e q u i r e m e n t
preserved.
denotational
definition for the
systems.
Also,
using
our
definition,
we
have
definition
of
that proper states
been
able
to
derive
a
m o d e l f o r t h e l a n g u a g e L, w h i c h is i s o m o r p h i c to H e n n e s s y a n d M i l n e r ' s t e r m
model for the same language. Our a p p r o a c h
is i n t e n d e d
nondeterministic
to
extend
and concurrent
nondeterministically).
to
richer
languages,
(meaning that
for
programs
the actual concurrency
which are
both
is n o t i n t e r p r e t e d
Some simple results have already been reached in that direction.
Acknowledgements The
definition
of
abstraction
homomorphism
and
the
idea
M i l n e r ' s n o t i o n s of o b s e r v a t i o n a l e q u i v a l e n c e a n d c o n g r u e n c e U. M o n t a n a r i
at
Pisa
University.
I would
like
I would
a l s o like
to t h a n k
to
thank my
of u s i n g
it
to
characterise
s t e m s f r o m a joint work with him
for
supervisor
inspiration
subsequent
discussions.
substantial
h e l p h e g a v e m e all a l o n g , a n d R. Milner f o r h e l p f u l s u g g e s t i o n s .
and
for
M. H e n n e s s y f o r t h e Many t h a n k s
to m y c o l l e a g u e s F r a n c i s Wai a n d T a t s u y a H a g i n o f o r h e l p i n g m e w i t h t h e w o r d p r o c e s s i n g of t h e p a p e r .
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[BR]
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