Graph isomorphism models for non interleaving process
algebra
Citation for published version (APA):
Baeten, J. C. M., & Bergstra, J. A. (1994). Graph isomorphism models for non interleaving process algebra. (Computing science notes; Vol. 9404). Technische Universiteit Eindhoven.
Document status and date: Published: 01/01/1994
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Eindhoven University of Technology
Department of Mathematics and Computing Science
Graph Isomorphism Models for Non Interleaving Process Algebra
by
J.C.M. Baeten and I.A. Bergstra
Computing Science Note 94/04 Eindhoven, January 1994
COMPUTING SCIENCE NOTES
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Mathematics and Computing Science
Eindhoven University of Technology.
Since many of these notes are preliminary
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editors:
prof.dr.M.Rem
Graph Isomorphism Models for
Non Interleaving Process Algebra
J.C.M. Baeten
Department of Computer Science, Eindhoven University of Technology, P.O.Box 513, 5600 MB Eindhoven, The Netherlands
J.A. Bergstra
Programming Research Group, University of Amsterdam, Kruislaan 403, 1098 SJ Amsterdam, The Netherlands
and
Department of Philosophy, Utrecht University, Heidelberglaan 8, 3584 CS Utrecht, The Netherlands
We present a simple and intuitive model for the syntax of ACP based on graph isomorphism. We prove an expressivity result, and use the model to determine the number of states of a process.
1980 Mathematics Subject Classification (1985 revision): 68055,68010,68045. 1987CR Categories: F.1.2, D.3.1, F.3.1, D.1.3.
Key words & Phrases: process algebra, graph isomorphism, non-interleaving, state count. Note: This research was supported in part by ESPRIT basic research action 7166, CONCUR2. The second author is also partially supported by ESPRIT basic research action 6454, CONFER.
1.
INTRODUCTION.1
The purpose of this paper is to provide a very simple model of the syntax of ACP [BEK84]. This model, based on graph isomorphism, provides a clear explanation of the meaning of the primitives of ACP but it satisfies fewer axioms. In particular, it is non-interleaving in the sense of [BAB93].
We feel that the graph isomorphism model is the simplest and most convincing one presented thus far in the literature on ACP. Of course it has various drawbacks: because it is non-interleaving and because it is concrete (in the sense of [BA B88]) it is not very well suited for equational protocol verification. The practical merit of the graph isomorphism model is that it allows a precise state count of systems. We propose to use this model if the numher of states of a process description is referred to. To this end we provide some examples.
We are unaware of a similar model ill the literature. Of course most constructions have been given
"Iready in [BEK85] but that paper failed to identify the graph isomorphism structure as a model for the syntax of ACP in its own right.
ACKNOWLEDGEMENT: The authors thank S. Mauw and P. Rambags (both Eindhoven University of Technology) for useful coillments.
2 J.C.M. Baeten & J.A. Bergstra
2. PROCESS GRAPHS MODULO ISOMORPHISM.
2.1 DEFINITION. We introduce a set of process graphs as follows. Let A be a given set, and let K, A be two infinite cardinal numbers with K 2: A. A process graph 9 of cardinality < K, with out-degree (branching degree) < A, over a set of labels A is a quadruple (S, ->, begin, end) where
• S is a set, the set of states, begin E S, the staJt state • end E S, the end state
----. ~ S x A x S is the transition relation, and we have the following conditions:
1 < lSI < K
begin", end
\is E S I{I E S : 3a E A (s,a,l) E -.}I < A, the out-degree is < A
{s E S : 3a E A (5, a, begin) E ->} = 0, the st,"1 state has no incoming edges {s E S : 3a E A (end, a,s) E ->} = 0, the end state has no outgoing edges.
We call any state different frolll the start state or the end state an interior stale. We write s ~ t for (s,a,l) E ->. We refer to the four components of a process graph 9 by S(g), ->(g), begin(g) and
end(g), respectively. Ij(A, K, A) is the set of all process graphs over a set of labels A of cardinality < K with out-degree < A. Ij(A, 1\0, 1\0) is the set of linite proccss graphs, Ij(A, K, 1\0) contains only finitely branching process graphs.
Note that we require that start and end state are always different. This allows us to give an intuitive
definition for alternative composition (without root unwinding). An alternative to the present definition
is to allow several end states (and several start states). We restrict ourselves to the simplest definition here.
2.2 DEFINITION. Let g,h E Ij(A, K, A). A bijection <p between statcs of 9 and states of h is called an iSOI1101phism if:
I. cp(begin(g)) = begin(h), <p(end(g)) = end (h)
a a
2. 5 -> I ~ <p(s) -> <p(I).
We say g,h are isomorphic, 9 = h, if there is an isomorphism between 9 and h.
Obviously, isomorphism is an equivalence relation on process graphs. We can divide out this
equivalence, and obtain the algebras Ij(A, K, A)/~. Basically, this means that in these algebras the
names of the states do not matter. This allows liS to take disjoint unions of state spaces in the following definitions of operators on process graphs modulo is()ll1orphism.
2.3 DEFtNITtON. We define several operators on process graphs modulo isomorphism, i.e. on the algebras Ij(A, K, A)/~. First, constants.
I. Atomic action. Let a E A. a = ({b, e), {(b, a, e)}, b, e). 2. Deadlock. Ii = ({b, e), 0, b, e)
Abusing notation, we usually write a for a. Next, operators.
Graph isomorphism models for non interleaving process algebra 3
3. Alternative composition. Let g,h E (j(A, K, A) he given. Assume that the set of states of 9 is disjoint from the set of states of h (since we- consider process graphs modulo isomorphism, we can always ensure that this is the case). The set of states of g+h consists of the interior states of g, the interior states of h, and two new states begin, end. The transition relation is given by:
a. all transitions between interior states of 9 or h
h. a transition begin ~ end whenever there is a transition begin(g) 8-. end(g) or begin(h) 8 - . end(h)
c. for interior sin g, a transition begin ~. s if begin(g) ~. s, a transition s "-. end if s "-. end (g) d. for interior sin h, a transition begin ~. s if begin(h) ~ s, a transition S "-. end if s ~ end(h). Note that a + a ~ a, and
9
+ h ~ h + g,9
+ (h + k) ~ (g + h) + k,9
+0 ~
9
for all g,h,k.4. Sequential composition. Let g,h E (j(A, K, A) be given with disjoint state sets. The set of states of
g·h consists of the interior states of g, the interior states of h, the states begin(g), end (h) and a new state link. begin(g·h) = begin(g), end(g·h) = end(h).
The transition relation is given by:
a. all transitions hetween states of gar h that arc still in the set of states of g·h b. it transition s ~ link whenever there is a transition s~. end(g)
c. a transition link ~. s whenever there is a transition begin(h) "-. s.
Examples: fig. la shows a process graph (modulo isomorphism) for 8·a. The start slale is denoted by a small unlabeled incoming arrow, the end state by a small unlabeled outgoing arrow. Note that this graph is not isomorphic to the graph of
o.
Fig. I b shows a process graph for a·a + a·a. Note that this graph is not isomorphic to the graph of a·a, so9
+9 ~ 9
does not hold for all process graphs. Similarly, (a + b)·a is not isomorphic to a·a + b·a (if b '" a), so (g + h)·k ~ g·k + h·k does not hold in general. We do have the identity (g·h)·k ~ g·(h·k) for all graphs.--0
0 - - - - 0 - -
a
Fig. I a.o·a.
-~a
a
a
a
Fig. Ih. a·a + a·a.
5. Parallel composition. Let g,h E (j(A, K, A)bc given. Let a partial, commutative and associative function y: A X A -7 A be given, the communication fUllction. The set of states of 9
II
h is the cartesian product of the states of 9 and the states of h. begin(gII
h)=
(begin(g), begin(h), end(gII
h)=
(end(g), end (h).4 J.C.M. Baeten & J.A. Bergstra
a. for each state 5 in g, and each transition I ". l' in h, there is a transition (5,1) ". (5,1') b. for each state I in h, and each transition S ". 5' in h, there is a transition (5,1) ". (5',1)
c. for each pair of transitions
(S,I)~'
(5',1),(S,I)~'
(5,1') such that y(a,b) is defined, say y(a,b) = C, there is a transition (5,1) "-. (5',1').Note that a
II
b z a·b + b·a (if y(a,b) is undefined). However, a·aII
b is not isomorphic to a·(a·b + b,a) + b,a,a (the fonner graph has 6 states, the latter 7).6. Left merge. The graph of 9
IL
h has the same states as the graph of 9II
h, and the same transitions except that the transitions (begin(g), begin(h) ~. (5, I) with I '" begin(h) are omitted.7. Communication merge. The graph ofg
I
h has the same states as the graph ofgllh, and the same transitions except that the transitions (begin(g), begin(h)~' (5, I) with I=
begin(h) or 5=
begin(g) are omitted.8. Encapsulation. Let 9 E (j(A, K, A) be given, and let H <;; A. The graph of dH(g) has the same states as the graph of g, and the same transitions except that all transitions s ~~ s' with a E H are omitted. 9. Renaming. Let 9 E (j(A, K, A) be given, and let f: A -4 A be a given function. The graph of Pf(g) has the same states as the graph of g, the same begin and end, and each transition 5 ". 5' is replaced by a transition 5
f(~.
5'.10. Conditional operator. Let 9 E (j(A, K, A) be given. The graph of true :---7 9 is the same as the graph
of g, and the graph of false :-4 9 is obtained from the graph of 9 by removing all edges starting in the begin state. The ternary if .. fhen ... eise ... operator is defined by (b is a boolean):
9 <I b I> h = (b :-4 g)
+
(~g :-4 h).It is more involved to define a conditional operator over a general boolean algebra (other than {Irue, false», as we did in [BAB92]. We sketch part of this in section 6.
11. Finite state operator. Let g E (j(A, K, A) be given, let 51 be a finite set, let act: A x 51 -4 A be a pmtial function, let eff: A x 51 -4 51 be a total function and let T E 51. The set of states of AT(g) is the
the cartesian product (5(g) - {end(g))) x 51 together with the singleton {end(g)). begin(AT(9)) = (begin(g), T), end(AT(g)) = end(g).
The transition relation is given by:
a. Suppose 5".5' is a transition in g, I E 51 and aCI(a, I) is undefined. In this case, we do not have a transition.
b. Suppose 5 ~. 5' is a transition in g, 5' '" end(g), I E 51 and aCI(a, I) is defined. In this case, we
have a transition (5,1) "-. (s',u), where b
=
aCI(a,l) and u=
ell(a,I).c. Suppose 5 ". end(g) is a transition in g, I E 51 and aCI(a, I) is defined. In this case, we have a
Graph isomorphism models for non interleaving process algebra
5
12. Priority operator. Let 9 E (j(A, K, A) be given, and let < be a given partial ordering on A. The set of states of 8dg) is the set of states of g, and the set of transitions is a subset of the set of transitions of g, given by:
S "-. s' is a transition in Bdg) if for all b> a we do not have a transition s
~
s" in g.13. Single exit iteration operator (sec [BEBP93a]). We start by defining a ternary operator sei. Let g,h,k E (j(A, K, A) be given with disjoint state sets. The set of states of sei(g,h,k) is the set of states of g·k together with the set of the interior states of h. The begin state is begin(g), the end state is end(k). The transitions arc those of g·k, as given in 4, the transitions between interior states of h, and moreover:
b. a transition S
a...
link whenever there is a transition S "-. end(h) in h c. a transition link "-. s whenever there is a transition begin(h) "-. s in h. Note that g·h=
sei(g, 8, h). MOI'cover, we dcfine:gEilh = sei(g, g, h) (see [FOZ93]) g*h = h
+
gEilh.2.4 DEFtNtTtON. Our model makes it possible to define a cardinality function on process graphs
modulo isomorphism. Further, we can define the reverse of a process. Finally, we define an extra operator~, that limits a process to its set of reachable stales.
1. The cardinality of a process graph, Igl, is the cardinality of its set of states. We can compute: a. Ig· hi = Igl + Ihl -1
b. Ig
+
hi = Igl+
Ihl -2 c. IgII
hi = Igl x Ihl.2. Reverse operator. If 9 E (j(A, K, A), then g-1 has the same set of states as g, begin(g-1) = end(g), end(g-1)
=
begin(g) and S ~t
is a transition in g-1 whenevert "--.
s is a transition in g. Note that this operator commutes with all opcrators defined so far. It does not, however, commute with the following operator.3. Reachability operator. Let a process graph 9 E (j(A, K, A) be given. We define its set of reachable states, reach(g) <;; 8(g) inductively:
a. begin(g) E reach(g)
b. ifsE reach(g), and s "-. s' is a transition in g, then s' E reach(g).
Now we define ~(g) as follows. Thc set of states of ~(g) is reach(g) u (end(g)), with same begin state and end state, and only the transitions between rcachable states. With the help of this reachability operator, wc can formulate new versions of well-known identities, for instance we have ~(ll·g) = 8 for all process graphs g.
2.5 THEOREM. The models (j(A, K, A) arc non-intcrleaving (in the sense of [BAB93]).
PROOF: Consider the process a·a
II
b (a ct b, and y(a,b) is undefined). If we have an interleaving model, then this process should equal a·(a·b + b·a) + b·a·a. However, we found in 2.3.5 that this is6 J.C.M. Baeten & J.A. Bergstra
We conclude that the expansion theorem docs not hold in the models (j(A, K, A), and thus they are
non-interleaving models.
3. BISIMULATION.
We look at the familiar notion of bisill1ulatioTl in the present setting. To this end, consider the following
definition.
3.1 DEFINITION. We detine the t,ulliliar notion of bisimulation on process graphs. Let g,h E (j(A, K, A).
A relation R between states of 9 and states of h is called a hi.lill1l1/aliol1 if:
I. R(begin(g), begin(h)), R(end(g), end(h)) and a begin or end state is not related to another state;
2. if R(s, t) and S ~. s', then there is a t' such thai t ". t' and R(s', 1'); 3. if R(s, t) and t ". t', then there is a s' such that S ~. s' and R(s', t').
We say g,h are hi.limi/ar, 9 co h, if there is an bisimulation between 9 and h.
It is well-known that bisilllulation is an equivalence relation 011 process graphs. We can divide out this equivalence. and obtain the algebras (j(A, K, A)/t::!. Since hisimulation is also a congruence for all operators defined in section 2.3, we call define these operators on these algebras. We cannot, however,
define the cardinality operator or the reverse operator any more. For the reachability operator, we have ~(g) !=! g, so this operator becomcs the identity on process graphs modulo bisimulation.
3.2 DEFINITION. Let a process graph 9 E (j(A, K, A) be given. We say states s,t of 9 are bisimulation equivalent, S i::Z t, iff there is an bisimulatioll R hetween 9 and 9 such that R(s, t).
It is obviolls that this defines an equivalence relation on states of g. We can divide out this
equivalence relation, and obtain the reduced graph of g.
3.3 DEFINITION. Let a process graph 9 E (j(A, K, A) be given. The reduced graph of g, g/", has as states the set of equivalence classes of bisilllulatinn equivalent states of g, begin(g/!=l) = {begin(g)), end(g/!=!) = {end(g)) and s/!=! '1_. s'l co ill S~. s' .
3.4 REMARK. The graph isomorphism model is non-interleaving. The bisilllulation model can be obtained from this model as a homomorphic image, and is interleaving. The bisimulation model, however, is not the least identifying Illodel that is interleaving; in other words, there are models in
between the graph isomorphism model and the bisimulation model that are still interleaving. Whether
sllch model are useful, we do not know. Therefore, we choose not to present such a model here.
4. EXPRESSIVITY.
In this section, we prove an expressivity resull for the algebras G(A, ~o, ~o)/=='. We show that every finite process graph modulo isomorphism can be obtained from a single graph by lIsing alternative, sequential, parallel composition and iteration, renllllling and encapsulation operators. We do need an
Graph isomorphism models for non interleaving process algebra 7
infinite set of atomic actions and an appropriate choice of the communication function in order to obtain this result.
4.1 ATOMIC ACTIONS.
Suppose we have a countable set of atomic actions A. Divide A into a countable set B and a disjoint countable set C. Suppose we have a bijection i from the set of finite subsets of B to C. Define a communication function yas follows:
1. y(a,b) = i({a,b)) ifa,b E B
2. y(a, its)) = y(i(s), a) = ilIa} u s) if a E B, s a tinite subset of B 3. y(i(s), its')) = its us') if s,s' arc finite subsets of B.
Notc that this definition makes ycotlllllutative and associative. Notice that this definition amounts to a
free communication function on the set B. This is similar to the approach we used in [BAB93].
4.2 THE SEED PROCESS.
Let a,b,c,d,e,f,g,h,k be distinct atoills from B. The seed process P is given in fig. 2. This process has 4 states, and is maximally connected: every state except end has an olltgoing edge to every state except begin and every state except begin has an incoming edge from every state except end.
If Q is the same as process P except that the k-edgc is omitted, then we have P ~ Q
+
k. Note that a further decomposition of P using the operators of section 2 is not possible.a
9
d f }---II--,---~/'
C}
bFIGURIO 2. Seed process.
4.3 THEOREM. Let FE (j(A, ~o, ~o). Then there is a graph G ~ F and G can be constructed using only:
I. the seed process P
8 J.C.M. Baeten & J.A. Bergstra 3. additional atoms from B.
We will construct the graph G in several stages.
4.4 OBTAINING ENOUGH NODES.
First of all, we construct a graph G1 that has at least as many nodes as F, and is maximally connected. Moreover, all edges of G1 have distinct labels. Take a number N ~ 1 such that 2N ~ IFI - 2 (the number of internal nodes of F). Choose a set of distinct atoms {ak,j : 1 S k
s
9, 1 S j S N} ~ B. Define for each j the renaming Ij by:Ij(a)=a1,j, Ij(b)=a2,j, Ij(c)=a3,j, Ij(d)=a4,j, Ij(e)=a5,j, 1j(f)=a6,j, Ij(g)=a7,j, Ij(h)=aa,j, Ij(k)=ag,j Define Pj= Pfj(P). This gives N copies of P, with all edge-labels distinct.
Now put G1 = OH(P1
II
P2II ... II
PN), with H = Ho U H1 U H2Ho = {ak,j : 1 S k S 9, 1 S j S N} H1 = {i(s) E C: lsi < N}
H2={i(s)E C:3kE {7,8,9},jE {1, ... ,N}ak,jE S 1\ 3kE {1, ... ,6),jE {1, ... ,N}ak,jE s}.
Ho and H1 together ensure that the only steps possihle in G1 are communicalions involving a step from each of the components, H2 ensures that if one component does a termination step (a step to end) then all components do so simultaneously.
Now put G2 = ~(G1). Note that G2 has exactly 2N internal nodes and • exactly one transition from begin to each internal node and to end
• exactly one transition from each internal node to each internal node (including itself) • exactly one transition from each internal node to end.
Moreover, all transitions have distincl lahels.
4.5 EXACT NUMBER OF NODES.
The second step is 10 rcduce G2 so that we obtain exactly the right number of nodes. Let .p be an injection from S(F) into S(G2) that respects begin and end. Such an injection exists by choice of N. Let H3 contain all labels of edges of h2 that start from or end in a node outside the range of.p.
Put G3 = ~oOH3(G2). Then G3 has exactly IFI nodes and still has the fUl1her properties ofG2 above.
4.6 MULTtPLE STEPS.
Let M be the maxilllulll number of distinct edges between any pair of nodes in F. We will modify G3. so that each transition is replaced hy M distinct transitions. To this end, let b1, ... , bM, C1, ... , CM be fresh atoms in B (distinct from all ai,j and pairwise distinct). Put G4 = b1 + ... + bM, G5 = C1 + ... +
CM. Next,
G6 = OH'(G3
II
G4*G5), with H' = H4 U H5 u H6 U H7 H4 = Ibm, Cm : 1 S m S M}Hs = {i(s) E C : lsi = N}
H6 = {i(s U {cm)) E C : i(s) labels a non-terminating step in G3 and 1 ~ m S M} H7 = {i(s U (b m)} E C : i(s) lahels a terminating step in G3 and 1 S m S M}.
Graph isomorphism models for non interleaving process algebra 9
The encapsulation here ensures that the only stcps possible in G3 are communications between a step of G3 and a step of the other component, and moreover that both components only terminate together.
Now put G7 = ~(G6). G7 has exactly IFlnodes and
• exactly M transitions from begin to each internal node and to end
• exactly M transitions from each internal node to each intel.'nal node (including itself) • exactly M transitions from each internal node to end,
Moreover, all transitions have distinct labels.
We have now constructed a graph into which F can be emhedded, after a suitahle relabeling of edges. What rcmains now is to define this renaming. and trim away all superfluous edges.
4.7 NUMBER OF EDGES, LABELS
or
EDGES.Now we define a renaming function f and an cncapsulation set H" as follows:
Take a pair of nodes (n,m) in G7. If either n = end(G7) or m = begin(G7), do nothing. Otherwise, there are M edges from n to m. say with labels d1, ... , dM.
Since <jl is a bijection between S(F) and S(G7). n
=
<jl(s), m=
<jl(I) for certain nodes s,1 in F. Supposc there are K edges between S and I in F, with labels e1, ... , eK, D S; K S; M. Put f(d1) = e1, ... , f(dK)=
eK, ancl put dK+1, ... , dM into H".Do the same for every nOde-pair in G7. Since all labels in G7 are distinct, f is well-defined, and dom(f) and H" are disjoin!. Define
G = dH"0pf(G4). By construction we have G ~ F.
S.
ApPLICATION: COUNTING STATES.The graph isomorphism model allows us to determine the number of states of a process. As an example, we consider the familiar Alternating Bit Protocol. We take the description of [BAW9Dj, and recast this in order to lise iteratioll operators instead of recursive equations. We remark that [BEBP93b] contains a description ancl verification of a simplified ABP (cluc to PARROW [PAR85]), also using iteration instead of recursion.
We assume that we have a data set D. with IDI = n, that are to be transmitted from sender S to receiver
R
using unreliable channels K,L. B = {D,1}. The communication links are as shown in fig. 3. We use the standard communication function given by y(rk(x), Sk(X)) = Ck(X) (see [BA W90]). We have the following specifications.~_K
6 L 5
10 5.1 CHANNELS. K = (
L
r2(db)-(i'S3(db)+
i'S3{-1)) ) * 15 dED,bEB L =(L
r5(b)·(i·ss(b) + i·ss(1·)) ) * 15 bEBJ.C.M. Baeten & J.A. Bergstra
Following the definitions in section 2, we find that K has 12n + 3 states, and L has 15 states. The graph of L is shown in fig. 4. The labels in the second part are the same as those in the first part, and are omitted. The graph of K is similar, except that the branching in the begin state and the middle state is of size 2n.
FtGURE 4. Acknowledgement .channel L.
FtCiURE 5. Reduced graph of L.
We can divide out bisilllulatiotl equivalence, and then the states in the second part of the graph are
cancelled. We show the reduced graph of L in fig. 5. We find in this case that K has 6n + 3 states, and L 9. If we want to define these reduced processes in ollr syntax, it is not sufficient to lise the operators of section 2, but have to follow in essence the construction of section 4.
To sketch this in case of L, we take three copies of the seed process P, with sets of atoms a1 , ... ,ag resp. b1 , ... ,bg resp. C1 , ... ,Cg, take the parallel composition, encapsulate all actions except the following
12 ternary communications, apply the following renaming and lastly apply the reachability operator. rename the communication of a1, b2 and C2 and of as, b4 and C5 into r5(1)
Graph isomorphism models for non interleaving process algebra 11
• rename the communication of a3, b6 and C5, of a3, b5 and C6, of a5, b6 and C3 and of a6, b6
and C3 into i
rename the communication of a4, b3 and C5 into s6(1)
• rename the communication of a4, b3 and C4 into S6(0)
• rename the communication of a4, b6 and C4 and of a5, b3 and C4 into S6(~).
It is obvious that this definition of L clocs not add to our understanding of the process. We will omit
slIch descriptions with a minimalnumbcr of states in the sequcl.
We conclude that the number of states of a process depends very much on the specification of the
process in the syntax, and that the simplest ancimost intuitivc notation usually does not have a minimal number of states. 5.2 SENDER. 5 = (50·51) * 8 5b =
L
r1 (d) . sei(s2(db), (r6(1-b)+r6CL))'S2(db), r6(b)) dED for b = 0,1.We find that each 5b has 3n + 2 states, and 5 has 12n + 5 states. The sender will have even more
states if we do not lise the ternary sei operator, but inste"lci the binary iteration operators. The graph of
5
is shown in tig. 6, in case D = {a, 1}. Again, the labels in the second part are the same as in the first part, and are omitted. Dividing out bisimulation equivalence, we get that each 5b has 2n + 2 states,and S has 4n + 4. We show the reduced graph of S in fig. 7.
12
r1 (1)
'1(0)
FIGURE 7. Reduced graph of S, for D = {a, 1}.
5.3 RECEIVER. R = (R1·RO)· Ii
Rb = ( ((
L
f3(db) + f3(1-) dEDJ.C.M. Baeten & J.A. Bergstra
for b=O, 1
We find that each Rb has 2n + 6 states, and R has Bn + 20 states. Dividing out bisimulation equivalence, we get that each Rb has n + 5 statcs, and R has 2n + 9. We show the graph of R in fig. 8, in case 0 ::: {D, 1}, with the same conventions as above, and the reduced graph in fig. 9.
Graph isomorphism models for non interleaving process algebra 13
FIGURE 9. Reduced graph of R, for D = {O, 1}.
5.4 PROTOCOL.
The protocol is now given by ABP = dH(S II K II L II R), with H = {rk(X), Sk(X) : k E {2,3,5,6), X E (D x B) u B u {~}}.
By section 2, the process ABP has (12n + 3)(12n + 5)(8n + 20)15 = 17280n3 + 54720n2 + 30600n + 4500 states. Of course, most of these slates are not reachable. It is much more interesting to determine the number of states of ~(ABP), i.c. thc number of reachable states of the protocol.
Although the expansion theorem does not hold in our graph isomorphism model, a restricted form of it does hold, and we can lise this to calculate the numher of reachable states. This calculation is based on the following two identities:
• ~(ax IL y) = a·~(x II y)
• ~«ax I by) IL z) = c·~(x II y II z) if y(a, b) = c.
Thus, if a parallel composition of processes yields a process that is really sequential (i.e. in all states
there is only one possibility to proceed, either an autonomous step of one component, or a synchronisation between two components), we can use these identities to calculate the state graph. In
the following, we present the results for the Alternating Bil Protocol.
5.5 NUMBER OF REACHABLE STATES.
The ABP starts with an initialisation phase, where the components have not all reached the iteration parts of their behaviour. We show this initialisation phase in fig. 10. Open circles denote n states, one for every clement of 0, closed circles denote just one state. The initialisation phase ends in one of the two closed circles on the top lefl. These two are states the protocol can return to later. We show the iteration behaviour in fig. II. We counl 67n + 3 states in the initialisation phase, plus 1 end state.
The rest of the reachable states of the AS P are shown in fig. I I. The two closed circles on the top
left are the same as those shown in fig. 10. We count S8n + 4 states in this part. Thus, in total we have 125n + 8 states. Dividing out bisil11ulation equivalence, this reduces to 34n + 4 states. Without
14 J.C.M. Baeten & J.A. Bergstra C6(1 ) C6(1 ) C5(1 )
7
C6(O) C6(O) C6(O)FtGURE 10. Initialisation phase of ABP.
6. CONDITIONS.
We can redo the theory of the previous sections in case
we
have conditions over a general booleanalgebra. We use the theory developed in [B;\]392], and just mention a few key items, glossing over many details.
Graph isomorphism models for non interleaving process algebra
15
C6(1 )
C5( 1) __ -o..----s-'-4('-e-'.-)- u
C5(1) s4(e):7
o
S4(d)..C5(O)
C6(O)
n--S-4('"""d),.---r_
rC5(O)C6(O)
FIGURE II. Iteration phase of ABP.
6.1 BOOLEAN ALGEBRA.
Let Ja be a boolean algebra, with constants true, false and operators v, 1\, ~ We lise letters
<p,
\If torange over IE. A valuation is a homomorphism from:ra into {true, false}. For each process x, there is a process <t> :~ X (if
<»,
then x). We rederine the model or process graphs, in order to take conditions on edges into account.16 J.C.M. Baeten & J.A. Bergstra 6.2 PROCESS GRAPHS OVER A BOOLEAN ALGEBRA.
Before, we had that a process graph is a quadruple (8, -', begin, end) with --> (;; 8 x A x 8, or, equivalently, the transition relation is a mapping from 8 x A x 8 into {true, false}. Over a general boolean algebra l8l, the transition relation is a mapping from 8 x A x 8 into l8l, and thus (s ~ t) E l8l. The conditions on the transition relation are reformulated as follows:
• for all valuations V, states s I{t E 8: 3a E A vIs ". t) = true}l < 1-.., • \1s E 8 \1a E A (s ~ begin(g)) = false,
a
• \1s E 8 \1a E A (end(g) -. s) = false.
Isomorphism between two graphs g,h is now defined as expected. A bijection F between states of 9 and states of h is called an isomorphism if:
I. F(begin(g)) = begin(h), F(end(g)) = end(h) 2. for all valuations v, v(s~. t) = v(F(s) ~. F(t)).
Again, g,h are isotnOrl)hic, 9 ::::: h, if there is an isolllorphism between 9 and h.
6.3 OPERATORS.
Definition of the operators in 2.3 is fairly straightforward. Writing down four interesting cases,
condition b. in the definition of g+h becomes:
b'. begin "-. end = (begin(g)
a--.
end(g)) v (begin(h) "-. end(h)).In case of parallel composition, we calculate (s,t) ". (s',t') as the disjunction of all (s
~
s') 1\ (t"-. t')for pairs b,c with y(b,c) = a, plus (s ~ s') in case t=l', plus (t ~ t') in case s=s'. In case of the conditional operator, (begin ~. s) in <I>:-->g eqllals <I> 1\ (begin ~. s) in g.
Finally, in case of the priority operator, the value of a transition s '!.... s' in 8«g) becomes the
conjunction of its value in 9 and all
~
(s~.
s") for b > a.6.4 REACHABILITY.
Let a process graph 9 E (j(A, K, 1-..) be given. As in 2.4.3, we define its set of reachable states, reach(g) (;; 8(g) inductively:
a. begin(g) E reach(g)
b. if s E reach(g), and there is a vaillation v and an action a such that vIS
~
S') = true, then s' E reach(g).Again, ~(g) is obtained from 9 by reslriction to the set of reachable states.
6.5 BISIMUloATION.
Conditions 2 and 3 of the definition of bisimulation in 3.1 must be reformulated as follows (cf. [BAB92], section 8.2):
2'. if R(s, t) and V is a valuation such that vIs ~ s') = true, then there is a l' such that v(t~. t') = true and R(s', t');
3'. if R(s, t) and v is a valuation sllch thai v(t ~. t') = true, then there is a s' such that vIs ~. s') = true and R(s', t').
Graph isomorphism models for non interleaving process algebra 17 6.6 EXPRESSIVITY.
We can still obtain an analogue of the expressivilY result of section 4 in the present setting. Basically.
the steps in 4.4, 4.5 remain the same, giving the required number of nodes, with exactly one transition
between each pair (with first componenl nol end, second component not begin). Next. we do not proceed as in 4.6. but instead. handle each node pair separately. Let (n.m) be a node pair. Since <I> is a bijection. we can take certain nodes 5.1 in F with n = <1>(5), m = <1>(1). In case all transitions between s and I in F have condition false. do nothing. Otherwise there are K ~ 1 edges between sand t in g. with conditions <1>1 ... <l>K different from false. Take fresh atomic actions bo, b1, ... , bK in B.
We now consider two different cases. First, the case where n = s = begin or m = I = end. In this
case, we know that a transition under consideration can be executed at most once. In this case, we can
put Gs = SOdH(G3 II (<1>1 :~b1 + ... + <l>k:~bk)). where the encapsulation enforces a communication between the edge between nand m in G3 and onc of the bj. This replaces the step in 4.6. Next. we
proceed by a renaming as in 4.7.
Otherwise, the transition under consideration can be executed several times, and we have to use an iteration construct. We also need a termination clause. We lake in this case
Gs = sodH(G3 II (h~b1
+ ... +
<l>k:~bkrbo).where bo will synchronize with each termination step (again enforced by encapsulation).
7.
CONCLUSION,We have presented a simple and intuitive model for the syntax of ACP. This model is non-interleaving.
We can use this model to calculate the number of states of a process. We found that this number of states depends very much on the representation of a process in the syntax. The most intuitive representation usually does not yield the minimal number of states. We have presented an expressivity result, that shows that every finite state graph in the model can be expressed in our syntax, starting from one seed process.
REFERENCES.
[BAB88] l.C.M. BAETEN & l.A. BERGSTRA, GI"bal renaming ,,/,erotors in concrete process algebra,
Inf. & Camp. 78. PI'. 20S-24S.
[BAB92] .I.C.M. BAETEN & .I.A. BERGSTRA, Process algebra with sigl/als and condilions. in
Programming and mathematical method, Proc. Summer School, Marktoberdorf 1990, Germany (M.
Broy. ed.). NATO ASI Series F 88. Springer Verlag 1992. pp. 273-323.
[BAB93] J.C.M. BAETEN & .I.A. BERCiSTRA. Non il/terleaving process algebra, in Proc. CONCUR'93. Hildesheim. Germany (E. Best, ed.), Springer LNCS 71 S, 1993, pp. 308-323.
[BAW90] l.C.M. BAETEN & W.P. WEI.ILAND. Prucess algebra, Cambridge Tracts in Theoretical
Computer Science J 8, Cambridge University Press 1990.
[BEBP93a] l.A. BERGSTRA, I. BETHKE & A. PONSE. Process algehra with iteration, report P9314. Programming Research Group. University of Amsterdam 1993.
18 J.C.M. Baeten & J.A. Bergstra [BEBP93b] J.A. BERGSTRA, I. BI3THKE & A. PONSE, Process "/~e"ra wilh comhinators, report P9319, Programming Research Group, Un i vcrsity of Amsterdam 1993.
[BEK84] J.A. BERGSTRA & J.W. KLOI', Process {l1~e"ro for synchronous comll1unication, lnf. &
Control 60, Pl'. 109-137.
[BEK85] l.A. BERGSTRA & J.W. KLOI', Aigehro of' cOlllll1ltnicatlllg processes with abstractioll,
Theor. Camp. Sci. 37, Pl'. 77-121.
[FOZ93] W.J. FOKKINK & H. ZANTIOMA, Basic I,roass algebra IVltil ileralloll: compleleness of its equatlooal {lXIOmS, report CS-R9368, CWI Amsterdam 1993.
[PAR85] J. PARROW, Fairness I)J-()I}(~rt;es;,/ eroct'ss a/ge/Jra ~ lVith (lppliCaliol1s in communication
prolocol veri/icatlon, Ph.D. Thesis, DoCS X5!03. Dept. of Computer Systems, Uppsala University 1985.
Computing Science Notes
In this series appeared:
91/01 D. Alstein 91/02 R.P. Nederpelt H.C.M. de Swart 91/03 J.P. Katoen L.A.M. Schoenmakers 91/04 E. v.d. Sluis A.F. v.d. Stappen 91/05 D. de Reus 91/06 K.M. van Hee 91/07 E.Poll 91/08 H. Schepers 91/09 W.M.P.v.d.Aalst 91/10 R.C.Backhouse PJ. de Bruin P. Hoogendijk G. Malcolm E. Voennans J. v.d. Woude 91/11 R. C. B ackhouse P.J. de Bruin G.Malcolm E.Voennans J. van der Woude 91/12 E. van der Sluis
91/13 F. Rietman 91/14 P. Lemmens
91/15 A.T.M. Aerts K.M. van Hee 91/16 A.J.J .M. Marcelis
Department of Mathematics and Computing Science Eindhoven University of Technology
Dynamic Reconfiguration in Distributed Hard Real-Time Systems, p. 14.
Implication. A survey of the different logical analyses "if...,then ... ", p. 26.
Parallel Programs for the Recognition of P-invariant Segments, p. 16.
Perfonnance Analysis of VLSI Programs, p. 31.
An Implementation Model for GOOD, p. 18. SPECIFICATIEMETHODEN, een overzicht, p. 20.
CPO-models for second order lambda calculus with recursive types and subtyping, p. 49.
Tcnninology and Paradigms for Fault Tolerance, p. 25. Interval Timed Petri Nets and their analysis, p.53. POLYNOMIAL RELATORS, p. 52.
Relational Catamorphism, p. 31.
A parallel local search algorithm for the travelling salesman problem, p. 12.
A notc on Extensionality, p. 21.
The PDB Hypenncdia Package. Why and how it was built, p. 63.
Eldorado: Architecture of a Functional Database Management System, p. 19.
An example of proving attribute grammars correct: the representation of arithmetical expressions by DAGs, p. 25.
91/17 A.T.M. Aerts P.M.E. de Bra K.M. van Hee 91/18 Rik van Geldrop 91/19 Erik Poll 91/20 A.E. Eiben RV. Schuwer 91/21 J. Coenen W.-P. de Roever J.Zwiers 91/22 G. Wolf 91/23 K.M. van Hee L.J. Somers M. Voorhoeve 91/24 A.T.M. Aerts D. de Reus 91/25 P. Zhou 1. Hooman R Kuiper 91/26 P. de Bra G.J. Houben J. Paredaens 91/27 F. de Boer C. Palamidessi 91/28 F. de Boer 91/29 H. Ten Eikelder R van Geldrop 91/30 J.C.M. Baeten F.W. Vaandrager 91/31 H. ten Eikelder 91/32 P. Struik 91/33 W. v.d. Aalst 91/34 J. Coenen
Transfonning Functional Database Schemes to Relational Representations, p. 21.
Transfonnational Query Solving, p. 35.
Some categorical properties for a model for second order lambda calculus with subtyping, p. 21.
Knowledge Base Systems, a Fonnal Model, p. 21.
Assertional Data Reification Proofs: Survey and Perspective, p. 18.
Schedule Management: an Object Oriented Approach, p. 26.
Z and high level Petri nets, p. 16.
Fonnal semantics for BRM with examples, p. 25.
A compositional proof system for real-time systems based on explicit clock temporal logic: soundness and complete ness, p. 52.
The GOOD based hypertext reference model, p. 12.
Embedding as a tool for language comparison: On the CSP hierarchy, p. 17.
A compositional proof system for dynamic proces creation, p. 24.
Correctness of Acceptor Schemes for Regular Languages,
p. 31.
An Algebra for Process Creation, p. 29.
Some algorithms to decide the equivalence of recursive types, p. 26.
Techniques for designing efficient parallel programs, p. 14.
The modelling and analysis of queueing systems with QNM-ExSpect, p. 23.
Specifying fault tolerant programs in deontic logic,
91/35 F.S. de Boer J.W. Klop C. Palamidessi 92/01 J. Coenen J. Zwiers W.-P. de Roever 92/02 J. Coenen J. Hooman 92/03 J.C.M. Baeten J.A. Bergstra 92/04 J .P.B. W.v .d.Eijnde 92/05 J.P.H.W.v.d.Eijnde 92/06 J.C.M. Baeten J.A. Bergstra 92/07 R.P. Nederpelt 92/08 R.P. NederpeJt F. Kamareddine 92/09 R.C. Backhouse 92/10 P.M.P. Rambags 92/11 R. C. B ackhouse J.S.C.P.v.d.Woude 92/12 F. Kamareddine 92/13 F. Kamareddine 92/14 J.C.M. Baeten 92/15 F. Kamareddine 92/16 R.R. Seljee
92/17 W.M.P. van der Aalst
92/18 R.Nederpelt F. Kamareddine 92/19 J.C.M.Baeten J.A.Bergstra S.A.Smolka 92/20 F. Kamareddine
Asynchronous communication in process algebra. p. 20.
A note on compositional refinement. p. 27.
A compositional semantics for fault tolerant real-time systems. p. 18.
Real space process algebra. p. 42.
Program derivation in acyclic graphs and related problems. p. 90.
Conservative fixpoint functions on a graph. p. 25. Discrete time process algebra. pA5.
The fine-structure of lambda calculus. p. 110. On stepwise explicit substitution. p. 30.
Calculating the Warshall/Floyd path algorithm. p. 14. Composition and decomposition in a CPN model. p. 55. Demonic operators and monotype factors. p. 29.
Set theory and nominalisation. Part I. p.26. Set theory and nominalisation. Part II. p.22. The total order assumption. p. 10.
A system at the cross-roads of functional and logic programming. p.36.
Integrity checking in deductive databases; an exposition. p.32.
Interval timed coloured Petri nets and their analysis. p. 20.
A unified approach to Type Theory through a refined lambda-calculus. p. 30.
Axiomatizing Probabilistic Processes: ACP with Generative Probabilities. p. 36.
92/21 F.Kamareddine 92/22 R. Nederpclt F.Kamareddine 92/23 F .Kamarcddine E.Klein 92/24 M.Codish D.Dams Eyal Yardeni 92/25 E.Poll 92/26 T.H.W.Beclen W.J.J.Stut P.A.C.Verkoulen 92/27 B. Watson G. Zwaan 93/01 R. van Geldrop 93/02 T. Verhoeff 93/03 T. Verhoeff 93/04 E.H.L. Aarts J.H.M. Korst P.J. Zwietering 93/05 J.C.M. Baeten C. Verhoef 93/06 J.P. Veltkamp 93/07 P.D. Moerland 93/08 J. Verhoosel 93/09 K.M. van Hee 93/10 K.M. van Hee 93/11 K.M. van Hee 93/12 K.M. van Hee 93/13 K.M. van Hee
Non well-foundedness and type freeness can unify the interpretation of functional application, p. 16.
A useful lambda notation, p. 17.
Nominalization, Predication and Type Containment, p. 40.
Bonum-up Abstract Interpretation of Logic Programs,
p.
33.A Programming Logic for Fro, p. 15.
A modelling method using MOVIE and SimCon/ExSpect,
p. 15.
A taxonomy of keyword pattern matching algoritbms,
p. 50.
Deriving the Aho-Corasick algoritbms: a case study into the synergy of programming methods, p. 36.
A continuous version of the Prisoner's Dilemma, p. 17 Quicksort for linked lists, p. 8.
Deterministic and randomized local search, p. 78.
A congruence theorem for structured operational semantics with predicates, p. 18.
On the unavoidability of metastable behaviour, p. 29 Exercises in Multiprogramming, p. 97
A Formal Deterministic Scheduling Model for Hard Real-Time Executions in DEDOS, p. 32.
Systems Engineering: a Formal Approach Part I: System Concepts, p. 72.
Systems Engineering: a Formal Approach Part II: Frameworks, p. 44.
Systems Engineering: a Formal Approach Part III: Modeling Methods, p. 101. Systems Engineering: a Formal Approach Part IV: Analysis Methods, p. 63.
Systems Engineering: a Formal Approach Part V: Specification Language, p. 89.
93/14 J.C.M. Baeten J.A. Bergstra 93/15 J.C.M. Baeten J.A. Bergstra R.N. Bo1 93/16
H.
Schepers J. Hooman 93/17 D. A1steinP. van der Stok 93/18 C. Verhoef 93/19 G-J. Houben 93/20 F.S. de Boer 93/21 M. Codish D. Dams G. File M. Bruynooghe 93/22 E. Poll 93/23 E. de Kogel
93/24 E. Poll and Paula Severi 93/25
H.
Schepers and R. Gerth93/26 W.M.P. van der Aalst 93/27 T. Kloks and D. Kratsch 93/28 F. Kamareddine and
R. Nederpelt
93/29 R. Post and P. De Bra 93/30 J. Deogun
T. Kloks D. Kratsch
H.
Mill1er 93/31 W. Korver93/32 H. ten Eike1der and H. van Ge1drop
On Sequential Composition, Action Prefixes and Process Prefix, p. 21.
A Real-Time Process Logic, p. 31.
A Trace-Based Compositional Proof Theory for Fault Tolerant Distributed Systems, p. 27
Hard Real-Time Reliable Multicast in the DEDOS system, p. 19.
A congruence theorem for structured operational semantics with predicates and negative premises, p. 22. The Design of an Online Help Facility for ExSpect, p.21. A Process Algebra of Concurrent Constraint Program-ming, p. 15.
Freeness Analysis for Logic Programs - And Correct-ness?, p. 24.
A Typechecker for Bijective Pure Type Systems, p. 28. Relational Algebra and Equational Proofs, p. 23. Pure Type Systems with Definitions, p. 38.
A Compositional Proof Theory for Fault Tolerant Real-Time Distributed Systems, p. 31.
Multi-dimensional Petri nets, p. 25.
Finding all minimal separators of a graph, p. 11.
A Semantics for a fine A-calculus with de Bruijn indices, p.49.
GOLD, a Graph Oriented Language for Databases, p. 42. On Vertex Ranking for Permutation and Other Graphs,
p.
11.Derivation of delay insensitive and speed independent CMOS circuits, using directed commands and
production rule sets, p. 40.
On the Correctness of some Algorithms to generate Finite Automata for Regular Expressions, p. 17.
93/33 L. Loyens and I. Moonen 93/34 93/35 93/36 93/37 93/38 93/39 93/40 93/41
I.C.M. Baeten and lA. Bergstra W. Ferrer and P. Severi J.C.M. Baeten and J.A. Bergstra J. Brunekreef J-P. Katoen R. Koymans S. Mauw C. Verhoef W.P.M. Nuijten E.H.L. Aarts
D.A.A. van Erp Taalman Kip KM. van Hee
P.D.V. van der Stok M.M.M.PJ. Claessen D. Alstein A. Bijlsma 93/42 P.M.P. Rambags 93/43 B.W. Watson 93/44 B.W. Watson 93/45 E.J. Luit I.M.M. Martin 93/46 T. Kloks D. Kratsch J. Spinrad 93/47 W. v.d. Aalst P. De Bra G.I. Houben Y. Komatzky 93/48 R. Gerth
ILIAS. a sequential language for parallel matrix computations. p. 20.
Real Time Process Algebra with Infinitesimals, p.39.
Abstract Reduction and Topology, p. 28.
Non Interleaving Process Algebra, p. 17.
Design and Analysis of
Dynamic Leader Election Protocols in Broadcast Networks, p. 73.
A general conservative extension tbeorem in process algebra, p. 17.
Job Shop Scheduling by Constraint Satisfaction, p. 22.
A Hierarchical Membership Protocol for Synchronous Distributed Systems, p. 43.
Temporal operators viewed as predicate transformers,
p. 11.
Automatic Verification of Regular Protocols in
PIT
Nets,p. 23.
A taxomomy of finite automata construction algorithms, p. 87.
A taxonomy of finite automata minimization algoritbms,
p. 23.
A precise clock synchronization protocol,p.
Treewidth and Patwidth of Cocomparability graphs of Bounded Dimension, p. 14.
Browsing Semantics in tbe "Tower" Model, p. 19.
Verifying Sequentially Consistent Memory using Interface Refinement, p. 20.
94/01 P. America
M. van der Kammen R.P. Nederpelt O.S. van Roosmalen H.C.M. de Swart 94/02 F. Kamareddine
R.P. Nederpelt 94/03 L.B. Hartman
K.M. van Hee
The object-oriented paradigm, p. 28.
Canonical typing and n-conversion, p. 51.
Application of Marcov Decision Processe to Search Problems, p. 21.
