Bisimulat ion Collapse and the Process Taxonomy

Olaf Burkart .1, Didier Caucal **1 and Bernhard Steffen 2

1 IRISA, Campus de Beaulieu, 35042 Rennes, France. ({burkart, caucal}@irisa.fr) 2 Fakult~it fiir Mathematik und Informatik, UniversitKt Passau, Innstrafle 33,

94032 Passau, Germany. (steffen@fmi.uni-passau.de)

Abstract . We consider the factorization (collapse) of infinite transition graphs wrt. bisimulation equivalence. It turns out that almost none of the more complex classes of the process taxonomy, which has been established in the last years, are preserved by this operation. However, for the class of BPA graphs (i.e. prefix transition graphs of context-free grammars) we can show that the factorization is effectively a regular graph, i.e. finitely representable by means of a deterministic hypergraph grammar. Since finiteness of regular graphs is decidable, this yields, as a corollary, a decision procedure for the finiteness problem of context-free processes wrt. bisimulation equivalence.

1 Introduction

In concurrency theory, process calculi are widely accepted as algebraic descrip- tion languages for concurrent systems. Their semantics are usually formulated in terms of labelled transition graphs which model the dynamic behaviour together with some notion of behavioural equivalence. Since there is a great choice in the point of view for observing processes, in the last decade a plethora of behavioural equivalences have been suggested in order to capture the various underlying no- tions of concurrency. However, to be of practical interest decidability of the equivalence at hand is a main concern. Therefore, much research has focused on decision questions for different classes of processes (or transition graphs) with respect to a given equivalence relation.

It is folklore that any reasonable equivalence is decidable for finite-state sys- tems. Moreover, newer results show that notably strong bisimulation equivalence is decidable even for certain classes of infinite-state systems [BBK87, CHS92, CHM93a, Sti96].

In this paper we investigate the structure of transition graphs factorized (collapsed) wrt. bisimulation equivalence, i.e. transition graphs where bisimilar

* Supported by the European Community under HCM grant ERBCHBGCT 920017. Current address: LFCS, University of Edinburgh, JCMB, King's Buildings, Edin- burgh EH9 3JZ, UK. (ola]@dcs.ed.ac.uk)

** Supported by Esprit BRA 6317 (ASMICS).

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states are identified, for different classes of infinite-state systems. It turns out that almost none of the more complex classes, like e.g. BPP and BPA pro- cesses, of the process taxonomy, which has been established in the last years, are preserved by this operation. Whereas it is comparatively straightforward to establish that the class of normed BPP graphs is closed by factorization, while this property does not hold for the whole class 1, the treatment of the much more intensely studied class of BPA processes is quite intricate.

Here the normed case has already been solved in [Cau90] showing that also the class of normed BPA processes is closed under bisimulation collapse. We show that this is not true for BPA in general. Rather, the bisimulation collapse of BPA graphs (i.e. prefix transition graphs of context-free grammars) is shown to be a regular graph, i.e. finitely representable by means of a deterministic hypergraph grammar. As our corresponding construction is effective, the well- known decidability of the finiteness of regular graphs yields, as a corollary, the decidability of the finiteness of context-free processes wrt. the bisimulation se- mantics. This reduction solves the open problem first considered in [MM94], and subsequently in [BG96] where decidability was proved only for certain subclasses of BPA.

The remainder of the paper is organised as follows. Section 2 presents the basic notions and the process taxonomy, which has been established in the last years. In particular, it provides examples closing the remaining separation gaps between the taxonomic classes. The same examples will also be used in the subsequent sections in order to illustrate the power of the bisimulation collapse. Section 3 considers the factorization problem for BPP graphs, while Section 4 presents our main result, the existence of an effective procedure yielding a regular graph for each collapsed BPA graph. Finally, Section 5 gives our conclusions and directions for further research.

2 A T a x o n o m y o f I n f i n i t e - S t a t e T r a n s i t i o n G r a p h s

In process theory the fundamental notion of transition graph is often used to model the behaviour of concurrent programs or processes. However, beyond the finite-state case, only few classes of transition graphs have been considered in greater detail. In this section we summarise known results by presenting a taxonomy for these classes according to their expressive power.

Defini t ion 2.1. A labelled transition graph is a triple 7" = (S, •, ---~) where S is the set of vertices (or states), ~ is the set of transition labels (or actions), and --~ C S • 27 • S is the transition relation. If the transition graph 7" possesses a terminal coroot e E S, i.e. e has no out- going transitions and e is accessible from every vertex v E S, we say that 7" is norraed. The norm of any vertex v E S is then defined as the length of the shortest path from v to e.

i Nevertheless, Jan~ar and Esparza [JE96] proved recently that finiteness up to bisim- ulation is decidable even for the strictly larger class of Petri net transition graphs.

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In particular, we are interested in classes of infinite transition graphs which can be finitely represented by labelled rewrite systems.

D e f i n i t i o n 2.2. A labelled rewrite system is a triple ~ = (V, 27, R) where V is an alphabet (or set of nonterminals), 27 is a set of labels, and R C V* x 27 x V* is a finite set of rewrite rules. If the rewrite rules R are of the form V • 27 x V* the rewrite system is called alphabetic.

In the remainder of the paper, a rewrite rule (u, a, v) E R is also written as u -% v. Moreover, we will denote a rewrite system simply by R if V and 27 are clear from the context. Different classes of labelled transition graphs arise now by considering for rewrite systems the following forms of rewritings.

D e f i n i t i o n 2.3. Let 7~ = (V, ,U, R) be a labelled rewrite system. The (unres- tricted) rewriting relation of R is then defined by

= , t { (wuw', a, w w ' ) I (u -% v) �9 R, w, w' �9 V* }, R

while the prefix rewriting relation of R is given by

~ - + =~,{(uw, a, vw) ] (u-%v)�9 w �9 R

For technical reasons we will consider in our Main Lemma 4.1 also the suffix rewriting relation of R defined by

=~,{(wu, a, wv) l (u-%v)�9 w � 9 R

as well as the prefix reduction relation of R which is defined by

~=~ =~,{(u~,a,v)I ( ~ v ) � 9 ~ � 9 v*}. R

Any of the relations given above can inductively be extended to words over 27. Moreover, in the case of prefix rewritings we have the following lemma due to Biichi [Biic64].

P r o p o s i t i o n 2.4. The set { v [ u ~ v } of words reachable by prefix deriva.

tion from a given axiom u is regular, and a corresponding finite automaton is effectively constructible from R 2

Labelled rewrite systems R are a convenient formalism to finitely represent infinite-state processes since each of the rewrite relations "-*R C V* x 27 • V* given in Definition 2.3 may also be interpreted as a labelled transition system T(~R) =~, (V*, E,~R). Later on, we will also consider rooted labelled rewrite systems (R, r), i.e. labelled rewrite systems R together with some axiom r �9 V*. The transition graph of (R, r) for any rewrite relation "*R is then given by T(~*R, r) = , , ({ v �9 V* I r ~ v }, r , ~ R ) .

2 Dually, the result holds also if one considers suffix rewriting.

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The second device we will use in this paper to generate infinite graphs are hypergraph grammars. Let F = Un>l Fn be a graded set of labels such that 57 C_ F2. A hyperarc of arity n is then a word As1 . . . s , labelled by A E Fn joining

the vertices s t , . . . , sn in that order. In particular, an ordinary arc st "~ s~ is the word Asls2. Accordingly, a hypergraph is a set of hyperarcs, while a graph is a set of (binary) arcs. Finally, we denote by [G] the set of all terminal arcs of a graph G, i.e. [G] =af{As l s2 E G [ A E 57}.

Def in i t i on 2.5. A (hyper)graph grammar G is a quadruple (N, 57, R, Go) where

- N _C F \ 57 is the set of graded nonterminals, - 57 is the set of terminals, - R is a finite set of rules of the form Axa . . . xn t> H where A E Fn, H is a

finite hypergraph over 57 U N, and x a , . . . , x , are distinct vertices of H, and - Go is an initial finite hypergraph over 57 U N.

A graph grammar is said to be deterministic, if there is only a single rule for each nonterminal.

Graph grammars can be used to generate infinite graphs by means of graph rewritings. A graph rewriting G --~R G ~ consists of replacing a nonterminal hyperarc X = As1 . . . sn of G (called the redex) by a copy of H where Axl �9 �9 �9 xnt> H is a rule of R such that the vertices si and zi are identified, i.e.

G' = (G \ { X }) U ( Bg( t l ) . . .g(tm) I B t l . . .tm E H }

where g is a matching function mapping xi to si and the other vertices of H injectively to new vertices outside of G. To denote a graph rewriting at redex X we also write G --+R,x G t. Since such a rewriting is context-free, we may define a complete parallel rewriting :=~R as follows.

G =~ G' if G ~ o . . . o ~ G' R R ,X t R,X,~

where { X 1 , . . . , Xn } is the set of all hyperarcs of G. Henceforth, the grammar G will be deterministic. The infinite graph Go, (Go) generated by G starting from Go is then inductively defined by

G~ = Go, Gn(Go):~R Gn+l(Go), and Go,(Go)= U [Gn(a~ n > _ o

Since G is deterministic, G ~ (Go) is unique up to graph isomorphism. Finally, we call a graph G regular if there exists a graph grammar G = (N, 57, R, Go) such that G = Go, (Go). Figure 1 shows an example for a deterministic graph grammar with a single rewriting rule, as well as the generated infinite transition graph. Later on, we will need the basic fact that the restriction of a regular graph G to the vertices accessible from a given vertex r is effectively a rooted regular graph.

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A

a a a a

I C ~ o b ~ (dAt e ~ d b e ~ r "'"

e b A qm ) =

Figure 1. A graph grammar and the generated infinite transition graph.

The classes of infinite-state systems representable by the devices introduced so far can be classified as follows. A transition graph G is said to be a pushdown process (PDP) graph (respectively, a Basic Process Algebra (BPA) graph) if there exists a rooted rewrite system (respectively, a rooted alphabetic rewrite system) R, r such that G is isomorphic to the prefix transition graph of R accessible from r, i.e. T(~-~, r). Furthermore, a transition graph G is called a Basic Parallel

Process (BPP) graph if there exists a rooted alphabetic rewrite system R, r such that G is isomorphic to the transition graph of R accessible from r modulo commutation of nonterminals, i.e. T ( ~ , r ) / - .

Overall, we obtain the taxonomy of classes of infinite-state systems, as depic- ted in Figure 2. It is known that the class of normed BPA graphs, BPA graphs, PDP graphs, and regular graphs form a strict hierarchy [CM90]. Moreover, the class of PDP graphs coincides with the class of rooted regular graphs of finite degree [Cau92]. On the other side the class of BPP graphs and the strictly larger class of Petri Net transition graphs 3 is incomparable with all the other classes [Chr93]. We complete this picture by providing in Figure 3 examples closing the remaining separation gaps. The same examples will also be used in the subsequent sections in order to illustrate the power of the bisimulation collapse.

tl 3 (PDP

t 6 BPP Petri Nets

Figure 2. A taxonomy of classes of infinite-state transition graphs.

s For a formal definition see e.g. [JE96].

252

(]) a a a

b b b

(2) '~ a a I [BPA

J d O d O d O

d d d d (4) "~ c""

bill d d

... [BPA

Root: Z [ Z~AZ[ Z - ~ D [ A - ~ AA[ ~-~[ A " ~ D [ V---~ D [

Root: Z z --~ xz x-~xx x-~s

BPP Root- A I A .-~ AB B--~e

(3)

a a a

~ e d ~ ~ " " (5)

a c a a a B O p

d d d d

r vp i q0A --~ q0AA q0A q2 q2A q0A --~ ql qlA ql q3A ~ A

(6)

'BPP R~t.: A , ~ . l ~

I A--~ B I c

d

�9 -. (7) a a a ...

d d ' d ,c d

q0A ~ q0AA qlA qlAA q 0 i --~ q0 ql A ql | q0A --~ ql ql ql

a a a

"'" qoz --~ q0AZ @z qlZ q0A --~ q0AA q0A qIA q0A --~ q0 q lA --~ ql

BPP Roo~ A

A - ~ B - - ~

F i g u r e 3. Examples of infinite-state transition graphs.

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3 F a c t o r i z a t i o n o f B P P g r a p h s

When interpreting transition graphs as the operational semantics of processes one often considers an equivalence on the set of states which captures when two processes are said to possess the same behaviour. Notably, the bisimulation equivalence has attracted a lot of investigation. In this section we consider the effect of the bisimulation collapse on BPP graphs. It will be shown that the class of normed BPP graphs is closed under this operation, whereas the full class of BPP graphs is not.

De f in i t i on 3.1. A binary relation R between states is a bisimulation if whenever (p, q) E R then for each a E ,U:

1. p -~ p' implies 3 q'. q -~ q' A (p', q') E R, and 2. q -~ q' implies 3 p'. p -~ p' A (p', q') E R.

Two states p and q are said to be bisimulation equivalent or bisimilar, written p ~ q, if (p, q) E R for some bisimulation R.

To be of practical interest decidability of bisimulation equivalence is an im- portant aspect. For finite state systems decidability follows obviously from the fact that all finitely many binary relations on the set of states can be enumer- ated. Hence it suffices to check whether a relation is a bisimulation. Surprisingly, decidability can also be extended to some classes of infinite state systems, in par- ticular to the class of BPA graphs [CHS92] which will be studied in the next section, as well as the class of BPP graphs [CHM93a], for which we will consider the factorization problem in this section.

We start by considering bisimulation bases for BPP graphs which characterise bisimulation equivalence as the least congruence of a finite relation. To ease notation we fix in the remainder of this section a rooted alphabetic rewrite system (n , r) with 7~ = (V, ,U, R), and the associated transition graph T( R---+, r ) / - - .

De f in i t i on 3.2. The norm of a vertex s o f a transition graph T, written as Ilsll, is the length of the shortest transition sequence within T from s to a terminating state. A vertex is said to be normed if its norm is finite, while an alphabetic rewrite system is called normed it every X E V is normed in the associated transition graph.

De f in i t i on 3.3. Any binary relation B C V • V* induces an unlabelled rewrite system

= , , { ( w u w ' , w w ' ) I (u,v) e B , w , w ' e V * } . B

The reflexive, symmetric, transitive closure ~ is then always a congruence relation, i.e. it is an equivalence relation which additionally satisfies u ++~ v implies xuy e+* B xvy, for all z, y E V*.

De f in i t i on 3.4. A binary relation B C_ V x V* is called fundamental if it sat- isfies the following three conditions.

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- B is functional: if (X, a) E B and (X,/3) E B then a =/3. - S is factorizing: im(S) C (VN \ dora(B))* - B is norm-preserving: if (X,c~) e B then IlXll = I1~11.

Since any fundamental relation B is confluent and noetherian each c~ E V* posesses a unique normal form ~ $ B.

Def in i t i on 3.5. Let B C V x V* be a fundamental relation of bisimilar vertices which is maximal wrt. inclusion. Then a variable X E V is called prime wrt. bisimilarity if X ~ dora(B).

T h e o r e m 3.6 ( U n i q u e D e c o m p o s i t i o n ) . Any normed vertex ~ E V* can be expressed uniquely as a product of primes [CHM93b].

This unique decomposition property of normed BPP processes implies that bisimulation equivalence for normed BPP graphs can be characterized as the least congruence of a finite fundamental relation, as stated in the following co- rollary.

C o r o l l a r y 3.7. I f Tg is normed, then there exists a finite fundamental relation B C V x V* such that two vertices a,/3 E V* are bisimilar i f f~ +-~/3. Moreover, B can effectively be computed from Tr

Such a fundamental bisimulation base can now be used to reduce the rewrite system under consideration in order to obtain a finite representation for the factor graph of the original system.

T h e o r e m 3.8. The class of normed BPP graphs is effectively closed under fac- torization wrt. bisimulation equivalence.

Proof. Let 7g = (V,~U,R) be a labelled rewrite system with root r E V*, and B be a bisimulation base for Tg. The factor graph of T ( ~ , r)/ = is then

isomorphic to the rooted transition graph modulo commutation of (TU, r ~) where n ' = (V \ dom(B),,U,,R'),r' = r J~ B, and R' = { (A -~ a $ B) [ A dom(B) and ( A 4 a ) e R } . []

We close this section by giving an example that this closure property no longer holds for the general case.

P r o p o s i t i o n 3.9. The class of BPP graphs is not closed under fuctorization wrt. bisimulation equivalence.

Proof. Consider the unnormed BPP graph (6) of Figure 3 whose factor graph is (7). We will show that the transition graph (7) is not a BPP graph.

Let so, s l , . . , be the vertices in the upper row, and t a , t 2 , . . , be the vertices in the lower row of the graph (7). Since so is terminating it is clearly labelled with e, written as e(s0), mad the root sl must be labelled with a variable. So let

255

wlog. Al(sl) and 7D(tl) r A1. Thus we have Al(sl) -~ e(so),Al(Sl) -~ 7D(tl),

and 7D (tl) -~ 7D (tl). Now assume A1 occurs again in a word labelling some si, i > 1, i.e. A1 a i - 1 (si).

Hence, Ala,_l(s~) 2+ a,_l(s,_l), and Ala,-l(Si) -~ 7Da,-l(t~). Moreover, the

transition sequence ai- l(S,-1) b~ 2 Al(s l ) impl ies 7Dai-l( t i ) b~ 2 7DAI(t~). However, this yields a contradiction, since the transition 7pAl(t2) -~ 72 is not possible in the graph (7).

Thus we may conclude wlog. A2(s2). Applying the same arguments as above we see that also As cannot occur a second time in a labelling of some s~, i > 2. Proceeding with this construction, we obtain an infinite sequence of different alphabetic labels for sl, s2,. . . , which is obviously not possible for BPP graphs.

[]

Although the class of BPP graphs is not closed under factorization wrt. bisimulation, it was recently shown that finiteness up to bisimulation is decidable for the strictly more expressible class of Petri net transition graphs [JE96].

4 F a c t o r i z a t i o n o f B P A g r a p h s

Another, practically probably much more important class, the BPA graphs (or context-free processes) has attracted a lot of investigation concerning decidab- ility of bisimulation equivalence [BBK87, Cau90, Gro91, HS91, CHS92, HM94, HJM94, BCS95]. However, only in [Cau90] the structure of the factor graphs wrt. bisimulation has been considered. He has shown that the class of normed BPA graphs is effectively closed under factorization by bisimulation. This result does not longer hold for the general unnormed case, as illustrated by the trans- ition graph (2) of Figure 3. Its factorization is the transition graph (3) which in turn is not a BPA graph due to the existence of a vertex with unbounded in-degree. Moreover, it is known that the factorization of PDP graphs already yields graphs which are no longer regular [CM90]. As an example consider the transition graph (4) which has the non-regular graph (5) as its factorization graph.

The main theorem of this paper states that the bisimulation collapse of each (possibly unnormed) BPA graph is effectively a regular graph of finite out-degree. Since the finiteness of regular graphs is decidable, simply by checking that there are no cyclic dependencies in the (hyper)graph grammar, our construction yields, as a corollary, a decision procedure for the finiteness problem of context-free processes wrt. bisimulation [MM94]. It should be noted, however, that our quotient construction is not limited to the case where the resulting quotient is finite.

4.1 T h e M a i n L e m m a

Key towards the proof of the main result is the observation that the factorization of BPA graphs can be characterised by three rewriting relations. 1) a prefix

256

rewriting relation, capturing the context-free part, 2) a suffix rewriting relation taking care of bisimilar unnormed parts, and 3) a prefix reduction relation, which realizes the required right cancellation.

In order to formalise this idea, let us fix an alphabet V and a label set Z, as well as finite relations P, Q, R in V* x Z x V*. Moreover, let

GP, Q,R,r -~af (V*,E,,~-~ U ~ U ~::::~,r) Q R

the graph rooted at r E V* which is generated by the prefix transitions of P, the suffix transitions of Q, and the prefix reduction transitions of R. Then we can effectively construct a graph grammar generating a transition graph isomorphic to Gp, Q,R,r, as stated in the following lemma.

Main L e m m a 4.1. For finite relations P , Q , R C_ V* x E x V*, and a root r E V*, Gp, Q,n,r is effectively a regular graph (of finite out=degree).

Proof. We will construct a deterministic graph grammar ~ generating Gp, Q,R,r (henceforth only G for short) according to the length of the words denoting the vertices. The key idea is to partition G into a finite graph Go plus finitely many isomorphic connected components, each corresponding to a hyperarc in the graph grammar to be constructed. To start with, note that the set Va of vertices of G is the following rational language:

Va = Dorn( P ) . N * U l m ( P) .N*UN*.Dorn( Q )ON* . Im( Q)UDom( R ) . N * U I m ( R).

Now let rnp (resp. mQ) be the following length of words in P (respectively in Q):

rnp --d~ max{ min{ ]u], I v] } ] 3a (u _5~ v) E P }, m q =dr max{ min{ luI, IvI } I 3a (u -st v) E Q },

and let m~ (resp. rn~) be the greatest length of the left hand sides (resp. right hand sides) of R:

m~ --at max{ luI I Sa Sv (u 4 v) E R}, m~ =dr max{ IvI I Sa Su (u 4 v) E R}.

Taking rnpre =dr max(me, rntR), re,u! =dr mQ and U =,f max(mpre+m,uf, m~), we restrict the vertex set to VM =af { u E V a I M <_ lul }. Furthermore, any word u E VM is decomposed as u = p~euqu where IpuI = rnpr~ and IquI "- m,ul. Hence every vertex u E VM has now the property that prefix and suffix redices do not overlap. Let us ignore for the moment the prefix reductions of R by considering

H u

the subgraph of G generated by the prefix transitions of P and the suffix trans- itions of Q. Let now H~, for any vertex u E VM, be the restriction Hl{v I lul_<l.I}

257

of H to the vertices of length at least the length of u, and Cu be the connected component of Hu containing u. We then distinguish the frontier Iu of Cu

I ~ = , , { v e V c . I I v l - l u l } ,

and the border Ju of Cu

J u = d f { v E V v . ] 3 a 3 w , ( w - ~ ' v ) e H U H -1 A I w l < N l < l v l } ,

as depicted in the following Figure. NI

C l a i m : For any u E VM, the vertex sets Iu and Ju can effectively be determ- ined.

Proof. We denote by P0 (resp. Q0) the unlabelled transitions of P U p - 1 (resp. Q u Q - l ) , i.e.

Po =d~{(u,v) I 3 a , u -st v E P U P - 1 } , and

Qo=df{ (u , v ) [ 3 a , u -~ v E Q U Q - 1 } ,

which allow to express the connectivity wrt. P (resp. Q). Now we complete P0 and Q0 to ~0, respectively Q---o, in order to guarantee that the obtained rewrite relations are only applicable to words of length at least M, and, moreover, that the words obtained by rewriting have again length of at least M. Thus we define

P~0 =dr { (UW, VW) I U Po v and Iwl = mpre - min(lul, Ivl) }, and

Q---~ =d, { (WU, WV) I u Qo v and Iwl = m,~s - min(lul, Ivl) }

Using ~00 by prefix and Q---'o by suffix, we associate now with each u E VM the following two finite relations:

(I,,) =d, { (P, q) lP,, ~ P ^ q= __'~ q A lul--Ipc,,ql }, Po Qo

(su) =~, { (p, q) I p~ *_~ p ^ q. ~ q ^ lul < Ipc~ql ^ Po Qo

(p ~ p' ^ Ip'c,,ql < lul) v (q ~ q' A Ipc,,q'l < lul) } Po Qo

Observe that both relations can effectively be constructed since { p ] p,, , *__.~ p }, Po

as well as { q I qu ~ q } are regular sets c.f. Lemma 2.4. Oo

So overall we have {pcuq I (p, q) E (Iu) } = In and {pcuq I (P, q) E (Ju) } = Ju. []

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The next step in the construction consists of a completion of C,, to the following graph:

G~=~tC~U{vw-%z6G l v-%z6R A vw6Vc.} by adding the R-transitions from the vertices of Cu, as illustrated in the following Figure.

G~: ~

g u

The added vertices are denoted by Ku =dr VG. -- Vcu. We define an equivalence - on the set V as follows:

u = v iff (Iu) = (I~) and (Ju) = (J~)

Note that if u = v then Gu is isomorphic to G~. Furthermore, - is of finite index and a set U of representatives is constructible from P, Q, and R. In order to construct the intended graph grammar, we take now a graded alphabet F disjoint of E, and to each u 6 U, we associate a hyperarc (Ju) labelled in F such that

(ju) = fsl...sn with { s l , . . . , s , } = (Iu) U (Jr,) U gu and si # sj if i # j and (ju)(1) # (jr)(1) if u # v 6 U.

Any vertex v 6 VM has a unique representative u 6 U with u - v, and we consider the hyperarc

j~ =dff(Sl ~ C ~ ) . . . ( S , +--C,) where (J~/ = f s t . . . s n ,

and, for every p, q, c 6 V,

(p,q)+--c =at pcq and p+--c =dr P.

For any u E U, we construct the terminal transitions T~, of the right-hand side associated to j= :

T,, =dr {v-5, w 6G,, l lvl =lul or lwl=lul}

which is the subgraph of Gu consisting of arcs linked to Iu, and we construct the following set Nu of the nonterminal hyperarcs associated to ju:

Nu =dr { j~ [ C~ is a connected component of Cu - Tu }

Finally, letting HM =dr { U -5, V 6 g [ [u[ > M and Iv[ > M } the restriction of H to the vertices of length at least M, we construct the initial finite hypergraph a s

Go =df G<_M O { j~ [ C~ is a connected component of HM }

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Overall, we have constructed the following deterministic graph grammar G:

=,, ({ {ju >Tu } eu,G0)

with the property that, for any u E U, Gu belongs to ~~ as well as that ~ ( G 0 ) contains G. Thus we have proved that G is effectively a regular graph (of finite out-degree). []

4.2 Application to BPA graphs

In [CHS92] it is shown that bisimulation equivalence on BPA graphs (or context- free processes) can be characterised as the least congruence of a finite relation B, i.e. for any two vertices u, v we have u ,,, v iff u ~-~ v. Recently, in [BCS95] this result was improved by presenting an algorithm which allows to effectively compute such a bisimulation base. The idea is to exploit a bound on the number of transitions needed to distinguish two nonbisimilar normed vertices for the collection of a finite set of pairs of vertices which is subsequently refined until some fixpoint is reached. In this section we will use these results in conjunction with our Main Lemma 4.1 in order to prove that every BPA graph factorized by bisimulation is effectively a regular graph.

T h e o r e m 4.2. The factorization of any BPA graph wrt. bisimulation equival- ence is effectively a regular graph.

Proof. We fix a rooted alphabetic rewrite system 7~ = (V, S, A, r) with A C_ V • ,U • V* where the set V of variables is partitioned into the set of normed variables VN =dr {X E V I B w E ,U*. X -% e}, and the set of unnormed variables Vv =dr V \ VN, respectively. Furthermore, let B C V* • V* be a bisimulation base of T~, i.e. for all a,/3 E V* we have a --,/3 iff a +-r~/3. Such a bisimulation base can effectively be computed from 7~ by the algorithm given in [BCS95].

As shown in [CHS92, BCS95] the bisimulation base may be split into two relations B1 and B2 such that B = B1UB2, B1 C_ VN x V + , and B2 C V~rVu • V~rVv. Moreover, the first relation B1 is fundamental. Our goal is now to construct from T~ and B a graph grammar G and an initial graph Go with root r' such that ~~ r') is isomorphic to T ( ~ , r ) / , , , .

We start by reducing the rewrite rules A according to B1. Since the relation B1 is functional we obtain T~' = (V - dom(B1), S , A', r .~ 81) where

(A ~ a $ B1) E A' iff (A -~r a) E A and A ~ dora(B1).

This transformation takes care of all bisimulation equivalences obtained by the least congruence of B1. The second step deals with all pairs of bisimilar vertices contained in the least congruence of the remaining relation B2 ~. B1. Let

P = { ( A , a , a ) I (A-Sta) E A ' and a E V ~ } ,

Q = { ( a , $ , f l ) I (a,/3) E B 2 $ B 1 } , and

R = { (A,a,a) l (A--~ a) e zY and a E V~rVtr }.

260

By construction, the prefix transitions of P and R generate the prefix transition graph of A ~. However, using R as a prefix reduction has the benefit that the right-cancellation law for unnormed context-free processes is taken into account, i.e. for any (A -~ c~) �9 A' where a �9 V~rVv, we have

Aj3 ~ a instead of A~ a~Tr a]~ ~ a. zP

Moreover, using Q with suffix rewriting ensures that bisimilar vertices are con- nected by means of $ transitions. Lemma 4.1 states now that the transition graph G rooted at r and generated by prefix rewriting of P , suffix rewriting of Q, and prefix reduction of R is regular, i.e. representable by a graph grammar ~ ' = (NG,,.,V,,Rv,,,G'o) such that G = ~ ~'(G~).

By identifying all vertices in the graph grammar which are connected by $ transitions one obtains, finally, a graph grammar G which generates the factor- ization of T(~, r) with respect to bisimulation. []

Since the finiteness of regular graphs is decidable, simply by checking that there are no cyclic dependencies in the (hyper)graph grammar, we obtain the following corollary, which solves the finiteness problem up to bisimulation for arbitrary context-free processes [MM94, BG96].

Coro l la ry 4.3. It is decidable whether the factorization of a BPA graph with respect to bisimulation equivalence is finite.

It should be noted, however, that our effective quotient construction is not lim- ited to the case where the resulting quotient is finite.

We close this section by illustrating the construction for the prefix transition graph generated from the root X by the following alphabetic rewrite system [MM94].

Y 4 y c Ae~ = Y -~ e

z -~ z c 4 e

For Zl,, we have VN = { Y, C }, Vtr = { X, Z }, and a bisimulation base B, , = { (Z, CZ) }. According to the proof of Theorem 4.2 we thus obtain

P = { (Y, b, YC), (Y, d, e), (C, c, e) }, Q = { (Z, $, CZ) }, _~ = { (X, a, YZ), (Z, c, Z) }.

261

The application of Lemma 4.1 then yields mp = 1, mq = 1, m~ = 1, m~ = 2, mpre = 1, msul = 1, and thus M = 2. For this example, we hence have only a single frontier I = { YZ, C Z } with an empty border J = 0. Overall, Figure 4 summarises the prefix transition graph generated from X by Aex (1), the associated factor graph (2), the regular graph Gp, Q,R,x (3), and the graph grammar generating the factor graph (4).

"~ a b '~, a t"~ b (1) X ~ YZ > YCZ ... (2)

Z Z c.-z~ -~ cz -~ dz -..

�9 C

(3) ' ~ a $ "~ a

oo

~ ~ ~ C3Z .-- . -Z c( J $ $ - c

A

F i g u r e 4. The transition graph, the factor graph, and the construction for Aez.

5 C o n c l u s i o n s

In this paper we have considered the factorization problem wrt. bisimulation equivalence for certain classes of infinite-state processes. Our main result states that factor graphs of BPA processes are effectively regular graphs, i.e. finitely representable by a deterministic graph grammar. Since finiteness is trivially decidable for regular graphs, this result yields, as a corollary, a decision procedure for the finiteness up to bisimulation of arbitrary context-free processes thereby improving on already known algorithms for subclasses of BPA [MM94, BG96].

For the class of BPP processes, it turns out that the subclass of normed processes is closed under factorization, which does not longer hold for the general case. Nevertheless, it is possible even for the strictly larger class of Petri net transition graphs to decide equivalence with a given finite-state system, and finiteness up to bisimulation [JM95, JE96]. However, it remains the question whether we can find some reasonable extensions of BPP such that the factor graphs of BPP processes are expressible in this larger class.

262

References

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Chr93.

CHS92.

CM90.

Gro91.

HJM94.

HM94.

HS91.

JE96.

JM95.

MM94.

Sti96.

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