The Joys of Bisimulation
Colin Stirling
Department of Computer Science
University of Edinburgh
Edinburgh EH JZ UK
email cps dcs ed ac uk
Intro duction
Bisimulation is a rich concept which app ears in various areas of theoretical com
puter science Its origins lie in concurrency theory for instance see Milner
and in mo dal logic see for example van Benthem
In this pap er we review results ab out bisimulation from b oth the p oint
of view of automata and from a logical p oint of view We also consider how
bisimulation has a role in nite mo del theory and we o er a new unde nability
result
Basics
Lab elled transition systems are commonly encountered in op erational semantics
of programs and systems They are just lab elled graphs A transition system
a
is a pair T S f a Ag where S is a non empty set of states A is
a
a non empty set of lab els and for each a L is a binary relation on S
a a
We write s s instead of s s Sometimes there is extra structure in
a transition system a set of atomic colours Q such that each colour q S the
subset of states with colour q
Bisimulations were intro duced by Park as a small re nement of the b e
havioural equivalence de ned by Hennessy and Milner in b etween basic CCS
pro cesses whose b ehaviour is a transition system
De nition A binary relation R b etween states of a transition system is a
bisimulation just in case whenever s t R and a A
a a
if s s then t t for some t such that s t R and
a a
if t t then s s for some s such that s t R
In the case of an enriched transition system with colours there is an extra clause
in the de nition of a bisimulation that it preserves colours if s t R then
for all colours q s q i t q
Simple examples of bisimulations are the identity relation and the empty rela
tion Two states of a transition system s and t are bisimulation equivalent or
bisimilar written s t if there is a bisimulation relation R with s t R
One can also present bisimulation equivalence as a game G s t see for
0 0
example which is played by two participants players I and I I A play of
G s t is a nite or in nite length sequence of the form s t s t
0 0 0 0 i i
Player I attempts to show that the initial states are di erent whereas player
II wishes to establish that they are equivalent Supp ose an initial part of a
play is s t s t The next pair s t is determined by one of the
0 0 j j j +1 j +1
following two moves
a
s and then player II cho oses a trans Player I cho oses a transition s
j +1 j
a
t ition with the same lab el t
j +1 j
a
t and then player II cho oses a trans Player I cho oses a transition t
j +1 j
a
s ition with the same lab el s
j +1 j
The play continues with further moves Player I always cho oses rst and then
player I I with full knowledge of player I s selection must cho ose a corresp onding
transition of the other state
A play of a game continues until one of the players wins In a p osition s t
if one of these states has an a transition and the other do esnt then s and t
are clearly distinguishable and in the case of an enriched transition systems if
one of these states has a colour which the other do esnt have then again they
are distinguishable Consequently any p osition s t where s and t are
n n n n
distinguishable counts as a win for player I and are called I wins A play is won
by player I if the play reaches a I win p osition Any play that fails to reach such
a p osition counts as a win for player I I Consequently player II wins if the play
is in nite or if the play reaches the p osition s t and neither state has an
n n
available transition
Di erent plays of a game can have di erent winners Nevertheless for each
game one of the players is able to win any play irresp ective of what moves her
opp onent makes To make this precise the notion of strategy is essential A
strategy for a player is a family of rules which tell the player how to move
However it turns out that we only need to consider history free strategies whose
rules do not dep end on what happ ened previously in the play For player I a
rule is therefore of the form at p osition s t cho ose transition x where x is
a a
s s or t t for some a A rule for player II is at p osition s t when
a a
player I has chosen x cho ose y where x is either s s or t t and y is
a corresp onding transition of the other state A player uses the strategy in a
play if all her moves ob ey the rules in The strategy is a winning strategy if
the player wins every play in which she uses
Prop osition For any game G s t either player I or player II has a history
free winning strategy
Prop osition Player II has a winning strategy for G s t i s t
Transition systems are mo dels for basic pro cess calculi such as CCS and
CSP Mo dels for richer calculi capturing value passing mobility causality time
probability and lo cations have b een develop ed The basic notion of bisimulation
has b een generalised often in a variety of di erent ways to cover these extra
features Bisimulation also has a nice categorical representation via co algebras
due to Aczel see for example which allows a very general de nition It
is an interesting question whether all the di erent brands of bisimulation are
instances of this categorical account In this pap er we shall continue to examine
only the very concrete notion of bisimulation on transition systems
Bisimulation closure and invariance
It is common to identify a ro ot of a transition system as some sp ecial start state
Ab ove we de ned a bisimulation on states of the same transition graph Equally
we could have de ned it b etween states of di erent transition systems When
transition systems are ro oted we can then say that two systems are bisimilar if
their ro ots are
A family of ro oted transition graphs is said to b e closed under bisimulation
equivalence when the following holds
if T and T T then T
Given a ro oted transition system there is a smallest transition system which
is bisimilar to it this is its canonical transition graph which is the result of rst
removing any states which are not reachable from the ro ot and then identifying
bisimilar states using quotienting
An alternative p ersp ective on bisimulation closure is from the viewp oint of
prop erties of transition systems Prop erties whose transition systems are bisim
ulation closed are said to b e bisimulation invariant Over ro oted transition
graphs prop erty is bisimulation invariant provided that
if T j and T T then T j
By T j we mean that is true of the transition graph T On the whole
counting prop erties are not bisimulation invariant for example has states
or has an even numb er of states In contrast temp oral prop erties are bisimula
tion invariant for instance will eventually do an a transition or is never able
to do a b transition Other prop erties such as has an Hamiltonian circuit or
is colourable are also not bisimulation invariant Later we shall b e interested
in parameterised prop erties that is prop erties of arbitrary arity We say that
an n ary prop erty x x on transition systems is bisimulation invariant
1 n
provided that
if T j s s and t t are states of T and
1 n 1 n
t s for all i i n then T j t t
i i 1 n
By T j s s we mean that is true of the states s s of T
1 n 1 n
An example of a prop erty which is not bisimulation invariant is x x is a
1 n
cycle and an example of a bisimulation invariant prop erty is x is language
1
equivalent to x
2
The notions of bismulation closure and invariance have app eared indep end
ently in a variety of contexts see for instance
Caucal s hierarchy
Bisimulation equivalence is a very ne equivalence b etween states An interest
ing line of enquiry is to re consider classical results in automata theory replacing
language equivalence with bismulation equivalence These results concern de n
ability closure prop erties and decidability undecidability
Grammars can b e viewed as generators of transition systems Let b e a
nite family of nonterminals and assume that A is a nite set of terminals A
a
basic transition has the form where and a A A state is
then any memb er of and the transition relations on states are de ned as the
least relations closed under basic transitions and the following pre x rule
a a
PRE if then
Given a state we can de ne its ro oted transition system whose states are just
the ones reachable from
In the table b elow is a Caucal hierarchy of transition graph descriptions
according to how the family of basic transitions is sp eci ed In each case we
assume a nite family of rules Typ e captures nite state graphs Typ e cap
1
in fact tures context free grammars in Greibach normal form and Typ e
2
captures pushdown automata For Typ e and b elow this means that in each
case there are nitely many basic transitions In the other cases R and R are
1 2
a
regular expressions over the alphab et The idea is that each rule R
1
a
stands for the p ossibly in nite family of basic transitions f R g
1
a a
R stands for the family f R and R g For and R
2 1 2 1
a
instance a Typ e rule of the form X Y Y includes for each n the
a
n
basic transition X Y Y
Basic Transitions
a
R Typ e R
2 1
a
Typ e R
1
a
Typ e
a
1
Typ e where j j and j j
2
a
Typ e X
a a
Typ e X Y or X
This hierarchy is implicit in Caucal s work on understanding context free
graphs and understanding when the monadic second order theory of graphs is
decidable With resp ect to language equivalence the hierarchy collapses
to just two levels the regular and the context free The families b etween and
including Typ e and Typ e are equivalent
The standard transformation from pushdown automata to context free gram
1
mars Typ e to Typ e do es not preserve bisimulation equivalence In fact
2
with resp ect to bisimilarity pushdown automata is a richer family than context
1
free grammars For instance normed Typ e transition systems are closed un
der canonical transition systems Caucal and Monfort show that this is not
1
transition systems see for further results ab out canonical true for Typ e
2
transition graphs Caucal showed in that Typ e transition systems coincide
1
There is a strict hierarchy b etween Typ e up to isomorphism with Typ e
2
and Typ e Therefore with resp ect to bisimulation equivalence there are ve
levels in the hierarchy
Baeten Bergstra and Klop proved that bisimulation equivalence is decidable
on normed Typ e transition systems The decidability result was generalized
in to encompass all Typ e graphs Gro ote and Huttel proved that other
standard equivalences traces failures simulation bisimulation etc on
Typ e graphs are all undecidable The most recent result is by S enizergues
who shows that bisimulation equivalence is decidable on Typ e transition
systems which generalises his pro of of decidability of language equivalence for
DPDA This leaves as an op en question whether it is also decidable for
Typ e systems
One can build an alternative hierarchy when a sequence is viewed as
a
a multiset In which case the rule PRE ab ove is to b e understo o d as if
a
then where is multiset union Christensen Hirshfeld and Moller
showed that bisimulation equivalence is decidable on Typ e graphs Huttel
proved that other equivalences are undecidable Typ e graphs are Petri
nets Jan car showed undecidability of bisimilarity on Petri nets Under
1
transition systems are not this commutative interpretation Typ e and Typ e
2
equivalent Hirshfeld utilizing Jan car s technique showed undecidability of
1
systems for more details see the survey bisimulation for Typ e
2
Logics
Bisimulations were indep endently intro duced in the context of mo dal logic by
van Benthem A variety of logics can b e de ned over transition graphs
Let M b e the following family of mo dal formulas where a ranges over A
tt j j j hai
1 2
The inductive stipulation b elow de nes when a state s has a mo dal prop erty
written s j however we drop the index T
T
1 0
A state t is terminal if it has no transitions A state s is normed if for all s such
w u
0 0
that s ! s for some w 2 A then there is a terminal t such that s ! t for some
u 2 A
s j tt
s j i s j
s j i s j or s j
a
s j hai i t s t and t j
This mo dal logic is known as Hennessy Milner logic In the context of an
enriched transition system one adds prop ositions q for each colour q Q to the
logic with semantic clause s j q i s q
Bisimilar states have the same mo dal prop erties Let s t just in case s
M
and t have the same mo dal prop erties
Prop osition If s t then s t
M
The converse of Prop osition holds for a restricted set of transition systems A
a
state s is immediately image nite if for each a A the set ft s tg is nite
w
And s is image nite if every memb er of ft w A s tg is immediately
image nite
Prop osition If s and t are image nite and s t then s t
M
These two results are known as the mo dal characterisation of bisimulation equi
valence due to Hennessy and Milner There is also an unrestricted charac
terisation result for in nitary mo dal logic And there are less restrictive notions
than image niteness for when characterisation holds see
The mo dal logic M is not very expressive For instance it cannot de ne safety
or liveness prop erties on transition systems which have b een found to b e very
useful when analysing the b ehaviour of concurrent systems Mo dal mu calculus
M intro duced by Kozen has the rquired extra expressive p ower The new
constructs over and ab ove those of M are
Z j j Z
where Z ranges over a family of prop ositional variables and in the case of Z
there is a restiction that all free o ccurrences of Z in are within the scop e of
an even numb er of negations to guarantee monotonicity
The semantics of M is extended to encompass the least xed p oint op erator
Z Because of free variables valuations V are used which assign to each
variable Z a subset of states in S Let V S Z b e the valuation V which agrees
with V everywhere except Z when V Z S The inductive de nition of
satisfaction stipulates when a pro cess E has the prop erty relative to V written
E j and the semantic clauses for the mo dal fragment are as b efore except
V
for the presence of V
s j Z i s V Z
V
s j Z i S S if s S then t S t S and t j
V
V [S Z ]
The stipulation for the xed p oint follows directly from the Tarski Knaster the
orem as a least xed p oint is the intersection of all pre xed p oints Again we
would add atomic formulas q if we are interested in extended transition systems
The bisimulation characterisation result ab ove Prop ositions and remain
true for closed formulas of M
Second order prop ositional mo dal logic M is de ned as an extension of M
as follows
Z j j 2 j Z
S
The mo dality 2 is the re exive and transitive closure of f a a Ag and is
included so that M includes M As with mo dal mu calculus we de ne when
s j The new clauses are
V
w
s j 2 i t w A if s t then t j
V V
s j Z i S S s j
V
V [S Z ]
The op erator Z is a set quanti er ranging over subsets of S There is a
straightforward translation of M into M Let Tr b e this translation The
imp ortant case is the xed p oint Tr Z Z 2 Tr Z Z
Formulas of M and closed formulas of M are bisimulation invariant from
Prop osition and its generalisation to M This is not true in the case of M for
it is to o rich for characterising bisimulation for instance a variety of counting
prop erties are de nable such as has at least two di erent successors under an
a transition This means that two bisimilar states need not have the same M
prop erties
Besides mo dal logics we can also consider other logics over transition systems
First order logic FOL over transition systems contains binary relations E for
a
each a A and monadic predicates q x for each colour q if extended transition
systems are under consideration Formulas have the form
xE y j x y j j j x
a 1 2
A formula x x with at most free variables x x will b e true or
1 n 1 n
false of transition system T and states s s in the usual way
1 n
Richer logics include rst order logic with xed p oints FOL where there
is the extra formulas
Z x x j j Z x x y y
1 k 1 k 1 k
In the case of Z there is the same restriction as in M that all free
o ccurrences of Z in lie within the scop e of an even numb er of negations
An alternative extension of rst order logic is monadic second order logic
OL with the extra formulas
Z x j j Z
1
Van Benthem s use of bisimulation was to identify which formulas of FOL
are equivalent to mo dal formulas to M formulas see the survey A formula
x is equivalent to a mo dal formula provided that for any T and for any
state s T j s i s j T
Prop osition A FOL formula x over transition systems is bisimulation
invariant i is equivalent to an M formula
This result was generalised by Janin and Walukiewicz to OL and M as
follows
Prop osition A OL formula x over transition systems is bisimulation
invariant i it is equivalent to a closed M formula
One corollary of this result is that the bisimulation invariant closed formulas
of M coincides with closed formulas of M
An interesting question is if there is also a characterisation of the bisimulation
invariant formulas of FOL See for preliminary results but over nite
mo dels
Finite mo del theory
Finite mo del theory is concerned with relationships b etween complexity classes
and logics over nite structures It is interesting to consider bisimulation invari
ance in the context of nite mo del theory
Rosen showed that Prop osition of the previous section remains true with
the restriction to nite transition systems It is an op en question whether
Prop osition also remains true under this restriction
Part of the interest in relationships b etween M and M or OL with re
sp ect to nite transition systems is that within M and OL one can de ne
NP complete problems examples include colourability on nite connected un
directed graphs Consider such a graph If there is an edge b etween two states
a a
s and t let s t and t s So in this case A fag and colourability is
given by
X Y Z 2 X a X Y a Y Z a Z
where which says that every vertex has a unique colour is
2 X Y Z Y Z X Z X Y
In contrast M formulas over nite transition systems can only express PTIME
prop erties
An interesting op en question is whether there is a logic which captures exactly
the PTIME prop erties of transition systems Otto has shown that there is a logic
for the PTIME prop erties that are bisimulation invariant The right setting
is FOL over canonical transition systems where is and a linear ordering
on states is thereby de nable
We now consider emaciated nite transition systems whose set A is a singleton
That is now T S where S is nite We can de ne language equival
n
ence on emaciated transition systems Let s t when n if there is a
0
sequence of transitions of length n from s to t and by convention s s A
state is terminal if it has no transitions The language of state s is the set L s
i
fi s t and t is terminalg Consequently s and s are language
equivalent if L s L s The prop erty x is language equivalent to y as was
noted earlier is bisimulation invariant Notice that this is an example of a dyadic
invariant prop erty
Prop osition Language equivalence on canonical nite transition graphs is
co NP complete
Hence language equivalence over nite transition systems is de nable in
FOL i PTIME NP Dawar o ers a di erent route to this observation
A classical result due to Immermann Gurevich and Shelah in a slightly
normalised form is
Prop osition A FOL formula y y over nite transition systems is
1 n
equivalent to a formula of the form u Z x x y y ue where
1 m 1 n
is rst order and contains at most x x free
1 m
The argument places in the application from n to m are all lled by
the same element u This allows for the arity of the de ning xed p oint m to
b e larger than the arity of the FOL formula n
Consequently if one can prove that y is language equivalent to z y z is
not de nable by a FOL formula in normal form u Z x x y z ue
1 m
then this would show that PTIME is di erent from NP As a rst step we have
proved the following using tableaux
Theorem Language equivalence y z is not de nable in FOL by a normal
formula of the form u Z x x x y z u
1 2 3
Acknowledgement I would like to thank Julian Brad eld and Anuj Dawar
for help in understanding nite mo del theory
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