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Grabmayer, C. A.

Klop, J.W.

Luttik, S. P.

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Grabmayer, C.

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Reflections on a Geometry of Processes

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Reflections on a Geometry of Processes
Clemens Grabmayer∗ Jan Willem Klop† Bas Luttik‡
June 10, 2005
Abstract
In this note we discuss some issues concerning a geometric approach to process alge-
bra. We mainly raise questions and are not yet able to present significant answers.
1 Periodic Processes
Our point of departure is the axiom system BPA in Table 1 together with guarded recursion.
x+ y = y+ x
x+(y+ z) = (x+ y)+ z
x+ x = x
(x+ y) · z = x · z+ y · z
(x · y) · z = x · (y · z)
Table 1: BPA (Basic Process Algebra)
We are in particular interested in non-linear recursion, where products of recursion variables
are allowed, in contrast with linear recursion exemplified by 〈X|X = aY+ b, Y = cX+ dY〉
yielding only regular (finite-state) processes. Non-linear recursion also allows infinite-state
processes, such as the counter 〈C|C = uDC, D = uDD+ d〉 (with actions u, d for “up” and
“down”) or the process Stack that is definable by the infinite set of linear recursion equations
over BPA (cf. the left-hand side of Table 2), and more remarkably, by the finite set of non-
linear recursion equations (cf. the right-hand side of Table 2).
This simple framework is already rich in structure. In [1] this framework was linked with
context-free grammars (CFG’s), in particular with those in (restricted) Greibach normal form.
∗Vrije Universiteit Amsterdam. Postal address: Department of Computer Science, De Boelelaan 1081a, 1081
HV Amsterdam, The Netherlands. E-mail: clemens@cs.vu.nl .
†Vrije Universiteit Amsterdam, Radboud Universiteit, and CWI. Postal address: Department of Computer
Science, De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands. E-mail: jwk@cs.vu.nl .
‡Eindhoven Technical University and CWI. Postal address: P.O. Box 513, 5600 MB Eindhoven, The Nether-
lands. E-mail: s.p.luttik@tue.nl .
118
Sλ = 0·S0 +1·S1
Sdσ = 0·S0dσ +1·S1dσ +d·Sσ
(for d = 0 or d = 1, and any string σ )
S = T·S
T = 0·T0 +1·T1
T0 = 0+T·T0
T1 = 1+T·T1
Table 2: Stack, an infinite linear and a finite non-linear BPA-specification
There the fact was established that while the language equality problem for CFG’s is unsolv-
able, the process equality problem for CFG’s is solvable. A priori this is not implausible,
because a process has much more inner ‘structure’ than a language (the set of its finite ter-
minating traces). The decidability was demonstrated by Baeten, Bergstra, and Klop in [1] as
a corollary of a result concerning the periodical geometry or topology of the corresponding
process graph. In Figure 1 the periodicities of two examples are exhibited: of Stack on the
left-hand side, and of the process 〈X|X = bY+ dZ, Y = b+ bX+ dYY, Z = d+ dX+ dZZ〉
on the right-hand side (this graph repeats three finite graph fragments α , β and γ as is also
illustrated in Figure 2 below).
00 0
0
0
0
1
1
1111
10
0 0
0 00 0
10
1
1111
1
b
d b
b
db
d
b d
b
b d b
b
db
b
d b
d
bd
b
bdb
d b
d b d b
Figure 1: Tree-like periodic processes
The geometric proof in [1] is complicated. For the corollary of the decidability more
stream-lined approaches have subsequently been found by using tableaux methods and other
arguments (cf. Caucal in [7], Hu¨ttel and Stirling in [11], and Groote in [10]). Also, the geo-
metric aspects have been studied, for example by Caucal in [8] and by Burkart, Caucal, and
Steffen in [5]. Actually, the related notion of context-free graph was introduced by Muller and
Schupp [12] already in 1985.
We feel that there is still much to be explained about the geometric aspects of process
graphs. We present a question concerning the fact that periodic graphs in BPA come in two
kinds: ‘linear’ graphs as on the left-hand side, and ‘branching’ graphs as on the right-hand
side in Figure 2.
Question 1 Is it decidable whether a system E of equations (in Greibach normal form) yields
a linear (type I) or a branching (type II) graph?
119
01
2
3
4
5
6  



	


b
a
a
c
c
b
b
d
d
d
b
d
b
c
a
c
a
a
c
a
b g
Figure 2: ‘Linear’ periodic graphs (type I, left), ‘branching’ periodic graphs (type II, right)
Another graph of type II is the ‘butterfly’ process graph in Figure 3 of the recursive BPA-
specification 〈X|X = a+ bY+ fXY, Y = cX+ dZ, Z = gX+ eXZ〉. The relevance of the
                                                                                                                     
                                                          
                                                          
0
1
2
3
4
5
6
7
ae
b cf
a
d
g
Figure 3: A ‘butterfly’ process graph.
distinction between type I and type II graphs is made clear below, in order to show that certain
graphs are not of type I or type II.
In the study of BPA-definable graphs an important property is that of being “normed”.
A graph is normed if from every node in it there is a path to a terminating node. (In term
rewriting terminology this is called the weak normalization property WN.) The norm of a
node is then the minimum number of steps to termination. Originally, the decidability of
context-free processes (BPA-definable processes) was established in [1] only for the normed
case. Subsequently this was generalized by Christensen, Hu¨ttel, and Stirling in [9] to all
BPA-definable processes.
Note that the norm of a node in a process graph is preserved under bisimulation: if norms
are pictorially represented by drawing the process graph with horizontal ‘level’ lines, arrang-
ing points with the same norm on the same level (see the graph left in Figure 2 and the graph
120
in Figure 3), then bisimulations relate only points on horizontal lines. Collapsing a normed
graph to its canonical form is a compression in horizontal direction.
An important question is whether BPA-definable processes are closed under minimization
(i.e. under compressing a graph such that it is minimal under bisimulation; the resulting graph
is also called the “canonical” graph). The question whether such a statement does in fact
hold was left open in [1]. Making a graph canonical can alter its geometry considerably.
For instance, consider the counter C mentioned above. The process graph g of C is a linear
sequence of nodes C,DC,DDC, . . . connected by u-steps to the right and d-steps to the left.
The merge C‖C in the process algebra PA has a grid-like graph similar to that of the process
Bag on the left side in Figure 6 below. But if we collapse this graph g for C‖C to its canonical
form by identifying the bisimilar nodes on diagonal lines, we obtain again the graph g for C.
So a grid may collapse to a linear graph.
Normedness plays a part when graphs are compressed to their canonical form. In [5]
Burkart, Caucal, and Steffen give the following example of a BPA-graph that after compres-
sion to canonical form no longer is a BPA-graph: For the process with recursive definition
〈Z|Z = aAZ+ cD, A = aAA+ cD+ b, D = dD〉 in BPA, the graph on the left in Figure 4 is
its associated BPA-process graph, while the graph on the right is the respective minimization,
 
  
	 

Z AZ AAZ AAAZ
DAAZDAZDZD
a a a
bbb
c c c c
d d d d
 
  
	 

a a a
d
c c c
b b b
c
Figure 4: Counterexample against the preservation of BPA-graphs under minimization.
which does not have the periodical structure of a BPA-graph. Note that neither of these graphs
is normed.
Question 2 How can those BPA-graphs be characterized whose canonical graphs are again
BPA-graphs?
We note that Question 2 has already received quite some attention in Caucal’s work. Con-
trasting with the counterexample for the unnormed case given above, in [7] he has shown the
following theorem.
Theorem 1 (Caucal, 1990) The class of normed BPA-graphs is closed under minimization.
The (obvious) link between CFG’s and BPA-definable processes was first mentioned in
[1]. An example is the graph on the right in Figure 1 and in Figure 2 above: it determines as
context-free language (CFL) the language of words having equal numbers of letter b and d.
An intriguing question is the following.
Question 3 How does the classical pumping lemma for CFL’s relate to the periodicity present
in BPA-definable processes?
121
a a a a
bbbb
b b b
c
b b
c
c
b
c
Figure 5: The language L.
Another interesting observation, due to
H.P. Barendregt, is the following. It is well-
known that the language L = {anbncn|n ≥ 0}
is not a CFL. This language can be obtained
as the set of finite traces of the triangular, in-
finite, minimal graph in Figure 5. Intuitively
it is obvious that this graph is not tree-like pe-
riodic. This leads to the next question.
Question 4 Can the fact that the graph in Figure 5
is not a BPA-graph (when established rigorously)
be used to conclude that L is not a CFL, applying
the correspondence between CFL’s and definability
in BPA as well as the ensuing tree-like periodicity?
2 Non-definability of Bag in BPA
The expressiveness of the operations defined by the axioms of BPA is limited; basically only
sequential processes can be defined. The axiom system PA is an extension of BPA with
axioms for the merge ‖ (interleaving) and the auxiliary operator ‖ (left merge). In PA we
00
0 0 0
000
1 1 1 1
1111
1 1 1 1
0
0 0 0
0
0 0 0
1 1 1 1
000
0 0 0
00
1111
1 1 1 1
000
0 0 0
000
0 0 0
1 1 1 1
1111
1 1 1 1
0 0 0
000
0 0 0
000
1 1 1 1
1111
1 1 1 1
t
Figure 6: The minimal process graphs of the process Bag (on the left-hand side), and of a
terminating variant Bagt of Bag (on the right-hand side).
have a succinct recursive definition for the process Bag (over data {0,1}) as follows:
B = 0(0‖B)+1(1‖B).
It has been proved by Bergstra and Klop in [3] that the process Bag cannot be defined by
means of a finite recursive specification over BPA. Considering the minimal process graph
for it in Figure 6, this does not come as a surprise: it is not tree-like, but “grid-like”. Below
we give an alternative proof of this fact.
122
Theorem 2 (Bergstra, Klop, 1984) Bag is not BPA-definable.
Proof (Sketch). Suppose that the process Bag is BPA-definable. Then there exists a recursive
specification E in BPA such that Bag is bisimilar with a tree-like periodic graph g(E) as
defined by Baeten, Bergstra, and Klop in [1]. Then g(E) is a “BPA-graph” according to the
terminology used in [5].1
In [5] Burkart, Caucal, and Steffen have shown that, for every BPA-graph G, the canonical
graph of G is a “pattern graph”, which means that it can be generated from a finite (hy-
per)graph by a reduction sequence of length ω according to a deterministic (hypergraph)
grammar.2 Since Bag is itself a canonical graph and since therefore Bag is the canonical
graph of the BPA-graph g(E), it follows that Bag is a pattern graph.
A theorem due to Caucal in [8] states that all (rooted) pattern graphs of finite degree are
“context-free” according to the definition of Muller and Schupp in [12].3 It follows that Bag is
context-free. However, it is not difficult to verify that Bag is actually not a context-free graph
according to the definition in [12].
In this way we have arrived at a contradiction with our assumption that Bag is definable in
BPA. 
By using Caucal’s theorem, Theorem 1, it is also possible to establish quickly the non-
definability in BPA of many normed graphs. For example, for the terminating version Bagt of
Bag (where Bagt is normed) with the process graph on the right in Figure 6, it can be reasoned
as follows. This graph is canonical, so if it were BPA-definable, then it would be a graph of
type I or type II. However, for a type I graph it holds that the number of nodes in a sphere
B(s,ρ), where s is the center and ρ is the radius, depends linearly on ρ; for a type II graph this
dependance is of exponential form. But for the graph under consideration the number of nodes
in a ball B(s,ρ) only depends quadratically on ρ . Hence this graph is not BPA-definable.
Where do we need the preservation of BPA-definability under minimization? The process
graph of Bagt is clearly not one obtainable by a BPA-definition, as it is not of type I or type II.
But equality of processes is considered here modulo bisimulation—so it is not inconceivable
that there is a BPA-definition E of Bagt such that g(E) after compression to canonical form
can(g(E)) were just the process graph graph(Bagt) for Bagt on the right in Figure 6. So
can(g(E)) = graph(Bagt) holds. But with the preservation property, Theorem 1, we have
can(g(E)) = g(E ′) for some BPA-specification E ′, hence g(E ′), and therefore graph(Bagt),
are of type I or type II, quod non.
1In earlier papers of Caucal (e.g. in [6] and [8]) BPA-graphs were known under the name “alphabetic graphs”.
2
“Pattern graphs” according to this definition used by Caucal and Montfort in [6] are called “regular graphs”
in the later paper [5] by Burkart, Caucal, and Steffen. Because the use of the attribute “regular” for process
graphs could lead to wrong associations, we avoid this terminology from (hyper)graph rewriting here.
3Note that the class of “context-free” graphs in Muller and Schupp’s definition does not coincide with the
graphs associated with “context-free” processes (the class of BPA-graphs), but that it forms a strictly richer class
of graphs corresponding to the class of transition graphs of push-down automata.
123
3 The strange geometry of Queue
After the paradigm processes Stack and Bag, we now turn to the third paradigm process
Queue (the first-in-first-out version with unbounded capacity). Table 3 gives the infinite BPA-
specification.
Q = Qλ = ∑d∈D r1(d) ·Qd
Qσd = s2(d) ·Qσ +∑e∈D r1(e) ·Qeσd
(for d ∈ D, and σ ∈ D∗)
Table 3: Queue, infinite BPA-specification
As before, the endeavour is to specify Queue in a finite way. It was proved by Bergstra
and Tiuryn [4] that the system BPA is not sufficient for that; in fact, they showed that Queue
cannot even be defined in ACP with handshaking communication (see [2] for a complete
treatment of the axiom system ACP). But Queue has a finite recursive specification in ACP
with renaming operators (see Table 4, the specification is originally due to Hoare using the
‘chaining’-operation).
Q = ∑d∈D r1(d)(ρc3→s2 ◦∂H)(ρs2→s3(Q)‖ s2(d) ·Z)
Z = ∑d∈D r3(d) ·Z
Table 4: Queue, finite ACP-specification with renaming
An ambitious question is the following.
Question 5 Is there a geometric (topological) property of processes definable by handshaking
communication?
Finally, we turn to geometric properties of the process Queue. Surprisingly, it is unex-
pectedly problematic to draw the process graph of Queue in a ‘neat’ way (cf. also Figure 7),
similar to Stack and Bag. We would like to uncover the ‘deep’ reason for this difficulty.
Question 6 Is it possible to fit g(Queue) in the binary tree space?
References
[1] J. C. M. Baeten, J. A. Bergstra, and J. W. Klop. Decidability of bisimulation equivalence
for process generating context-free languages. Journal of the ACM, 40(3):653–682,
1993.
124
3 The strange geometry of Queue
After the paradigm processes Stack and Bag, we now turn to the third paradigm process
Queue (the first-in-first-out version with unbounded capacity). Table 3 gives the infinite BPA-
specification.
Q = Qλ = ∑d∈D r1(d) ·Qd
Qσd = s2(d) ·Qσ +∑e∈D r1(e) ·Qeσd
(for d ∈ D, and σ ∈ D∗)
Table 3: Queue, infinite BPA-specification
As before, the endeavour is to specify Queue in a finite way. It was proved by Bergstra
and Tiuryn [4] that the system BPA is not sufficient for that; in fact, they showed that Queue
cannot even be defined in ACP with handshaking communication (see [2] for a complete
treatment of the axiom system ACP). But Queue has a finite recursive specification in ACP
with renaming operators (see Table 4, the specification is originally due to Hoare using the
‘chaining’-operation).
Q = ∑d∈D r1(d)(ρc3→s2 ◦∂H)(ρs2→s3(Q)‖ s2(d) ·Z)
Z = ∑d∈D r3(d) ·Z
Table 4: Queue, finite ACP-specification with renaming
An ambitious question is the following.
Question 5 Is there a geometric (topological) property of processes definable by handshaking
communication?
Finally, we turn to geometric properties of the process Queue. Surprisingly, it is unex-
pectedly problematic to draw the process graph of Queue in a ‘neat’ way (cf. also Figure 7),
similar to Stack and Bag. We would like to uncover the ‘deep’ reason for this difficulty.
Question 6 Is it possible to fit g(Queue) in the binary tree space?
References
[1] J. C. M. Baeten, J. A. Bergstra, and J. W. Klop. Decidability of bisimulation equivalence
for process generating context-free languages. Journal of the ACM, 40(3):653–682,
1993.
124
010 011
0100 0101 0110
100
1000 1001 1010
110 111
0
0
0
1
1
1
0
0
1
0
α
δ
β
γ
1
1100 1101 1110 11111011
101
0111
000 001
0000 0001 0010 0011
λ
0
00 01 10 11
0 1
Figure 7: Attempt at drawing Queue in ‘tree space’.
[2] J. C. M. Baeten and W. Peter Weijland. Process Algebra. Number 18 in Cambridge
Tracts in Theoretical Computer Science. Cambridge University Press, 1990.
[3] J. A. Bergstra and J. W. Klop. The algebra of recursively defined processes and the
algebra of regular processes. In J. Paredaens, editor, Proceedings of ICALP’84, volume
172 of LNCS, pages 82–95. Springer, 1984.
[4] J. A. Bergstra and J. Tiuryn. Process algebra semantics for queues. Fundamenta Infor-
maticae, X:213–224, 1987.
[5] O. Burkart, D. Caucal, and B. Steffen. Bisimulation collapse and the process taxonomy.
In Proceedings of CONCUR’96, 1996.
[6] D. Caucal and R. Montfort. On the transition graphs of automata and grammars. In
Proceedings of WG 90, volume 484 of LNCS, pages 61–86. Springer, 1990.
[7] D. Caucal. Graphes canoniques de graphes alge´briques. Theoret. Inform. and Appl.,
24(4):339–352, 1990.
[8] D. Caucal. On the regular structure of prefix rewriting. Theoretical Computer Science,
1992.
[9] S. Christensen, H. Hu¨ttel, and C. Stirling. Bisimulation equivalence is decidable for all
context-free processes. Information and Computation, 121:143–148, 1995.
[10] J. F. Groote. A short proof of the decidability of bisimulation for normed bpa-processes.
Information Processing Letters, 42:167–171, 1992.
[11] H. Hu¨ttel and C. Stirling. Actions speak louder than words: Proving bisimilarity for
context-free processes. In Proceedings of LICS’91, pages 376–386, 1991.
[12] D.E. Muller and P.E. Schupp. The theory of ends, pushdown automata, and second-order
logic. Theoretical Computer Science, 1985.
125


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