Pelayo, Fernando L.

Valverde, Jose C.

Elsevier - Publisher Connector 

Notes on “Modeling the dynamics of concurrent computing systems” 

Computers and Mathematics with Applications 64 (2012) 661–663
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Computers and Mathematics with Applications
journal homepage: www.elsevier.com/locate/camwa
Notes on ‘‘Modeling the dynamics of concurrent computing systems’’
1. Introduction
In [1], wemodeled concurrent computing systems as discrete dynamical systems, which were then used to extract some
conclusions about the evolution of different states of Petri nets, thanks to the theory of discrete dynamical systems. This
opened up a new perspective for working with concurrent computing systems. In this sense, inspired by the results in [2–4],
we defined a metric, based on the Bayre and Hausdorff ones, which provided a structure of complete metric space for the
set P ({0, 1}n), which represented the domain of the evolution operator of the system.
For these same purposes, in [5], the authors made a partial attempt to get a model. Based on formal power series, they
obtained an algorithm that allows one to check whether a given weighted Petri net corresponds to a continuous polynomial
dynamical system. But, in this context, different initial states of the Petri net (differentmarkings) could correspond to different
dynamical systems.
In this appendix, in order to describe a more accurate model, we give a quasi-pseudometric that allows us to recognize
when the set constituting a state of the system is contained inside another previous state in the evolution of the orbit, and
hence when the variation of the states is under control.
Quasi-metric spaces were first studied by Wilson [4]. We recall some definitions which will be used later on this paper;
see also [3]. Let X be a set, and let
d : X × X → [0,∞)
be a function such that, for all x, y, z ∈ X ,
(i) d(x, x) = 0,
(ii) d(x, y) ≤ d(x, z)+ d(z, y).
Then d is called a quasi-pseudometric on X . If d is a quasi-pseudometric on X , then its conjugate, denoted by d′ on X , is such
that d′(x, y) = d(y, x) for all x, y ∈ X . Obviously, d′ is a quasi-pseudometric. Let d = max{d, d′}. Then d is also quasi-
pseudometric on X .
Now, if d is a quasi-pseudometric such that d(x, y)+d(y, x) > 0 for all x ≠ y, then d is said to separate points in X . A quasi-
pseudometric that separates points is said to be a separating quasi-pseudometric. When d is a separating quasi-pseudometric,
the topology τd on X induced by d is Hausdorff.
If a quasi-pseudometric d on X satisfies d(x, y) = d(y, x) for all x, y ∈ X in addition to (i) and (ii), then d is called
pseudometric. A pseudometric d that satisfies d(x, y) = 0 if and only if x = y is a metric.
On the other hand, if a quasi-pseudometric d on X satisfies d(x, y) = 0 if and only if x = y in addition to (i) and (ii), then
d is called quasi-metric. A symmetric quasi-metric d is a metric.
This addendum is organized as follows. In the next section, we set out themodel, defining the corresponding phase space
which is endowed with a pioneer quasi-pseudometric. As a result, it is demonstrated that the state space with the topology
induced by this metric is a compact space. Finally, some conclusions about the dynamics of this kind of dynamical system
and an outlook of intended further research directions are given.
2. Petri nets modeled as discrete dynamical systems
The discrete dynamical systemwhich encodes amarked Petri ordinary netN = (P, T , F ,M) is the triple (X,N,Φ), where
• X = P ({0, 1}n) is the set of all subsets of {0, 1}n, n being the number of places of the marked Petri ordinary net. This is
a finite set of 2(2
n) elements.
0898-1221/$ – see front matter© 2012 Elsevier Ltd. All rights reserved.
doi:10.1016/j.camwa.2011.12.079
662 Correspondence / Computers and Mathematics with Applications 64 (2012) 661–663
• Φ : τ × X → X is the evolution operatorΦ verifying
1. Φ(0, A) = A ∀A ∈ X , i.e.,Φ0 = idX ,
2. Φ(1, A) = B A, B ∈ X , where
– A = {x1, . . . , xk}, where xi ∈ {0, 1, . . . , k}n encodes markings of the marked Petri ordinary net N ,
– B = ∪ki=1 Bi,
– Bi = ∪tj=1{yji}, i.e. the union of all (t) possible reachable markings from xi, defined by xi[Ri⟩yji, Ri being the set of
transitions of the net enabled at marking xi,
3. Φ(t,Φ(s, A)) = Φ(t + s, A) ∀t, s ∈ τ , ∀A ∈ X .
At this point, we define an original metric on {0, 1}n, inspired by the Bayre metric (see [2]), given by
d(x, y) = 1
2l(x∩y)
− 1
2r
, x, y ∈ {0, 1}n, (1)
where l(x ∩ y) is the length of the longest common initial part of the binary vectors x and y.
Theorem 1. The function d defined in (1) is a metric.
Proof. Effectively, from the definition, it is obvious that d is symmetric and that
d(x, y) = 0⇔ x = y.
On the other hand, for all x, y, z ∈ {0, 1}n, it is true that
d(x, y) ≤ d(x, z)+ d(z, y).
To check this, it is sufficient to observe that for all x, y ∈ {0, 1}n the function d(x, y) shows the coincidence grade of the
initial part of x and y. So, if by reduction to the absurd, we suppose that for some x, y, z ∈ {0, 1}n it holds that
d(x, y) > d(x, z)+ d(z, y),
and we call k, l,m the length of the longest common initial part of the pairs of vectors (x, y), (x, z), and (z, y), then k < l,m.
Note that if k ≥ l (or k ≥ m) then
1
2k
− 1
2n
≤ 1
2l
− 1
2n
≤ 1
2l
− 1
2n
+ 1
2m
− 1
2n
,
which is inconsistent with the supposition made before.
But, if k < l,m, the coincidence grade of the initial part of x and z and also of z and y is greater than k. Thus, x, y, z have
a initial coincident part whose length is the minimum of l and m, which is greater than k, which contradicts the statement
that k is the longest common initial part of the pair of vectors x and y. 
At this point, taking into account this metric d, we can define the distance between a vector x ∈ {0, 1}n and a subset B of
vectors of {0, 1}n, i.e., an element in P ({0, 1}n), in this manner:
d(x, B) = min{d(x, y) : y ∈ B}.
Consequently, we can measure the distance between two elements A, B in P ({0, 1}n) as
D(A, B) = max{d(x, B) : x ∈ A}. (2)
Now, in view of the reasoning above, it is easy to check the following result.
Theorem 2. The application D defined in (2) determines a quasi-pseudometric on P ({0, 1}n), which is not a quasi-metric.
Proof. By the definition of D, it is obvious that, for every A ∈ P ({0, 1}n), D(A, A) = 0. On the other hand, D inherits the
triangle inequality property from the metric d.
However, we do not have the property of symmetry. For instance, if we consider a Petri netwith three nodes and consider
the subsets A, B ∈ P ({0, 1}3), given by A = {(0, 0, 1)}, B = {(0, 0, 0), (0, 0, 1)}, then it is easy to check that D(A, B) = 0,
while D(B, A) = 1
22
− 1
23
≠ 0. This also proves that the condition D(A, B) = 0 does not always imply that A = B. 
Remark 1. As in [3], we have a quasi-pseudometric verifying D(A, B) = 0 if an only if A ⊆ B.
Therefore, we will denote it d⊆ from now on.
In a similar way, another quasi-pseudometric d⊇ could be defined.
Coming back to the context of Petri nets, observe that, if A ⊆ B, then the reachable set of markings from A is included in
the reachable set of markings from B.
Thus, if we consider A, B ∈ X with A ⊆ B and k ∈ N such that Φ(k, B) = A, then d⊆(A, B) = 0, and we can ensure that,
as the system evolves, the reachability of B is reduced into the reachability of A, and so we have the evolution of the system
under control. This infers the consideration of quasi-pseudoperiodic orbits as a tool to control the evolution of the system.
On the other hand, as d⊆(A, B)+ d⊆(B, A) > 0 for every A, B ∈ P ({0, 1}n), d⊆ separates points. Hence, the topology τd⊆
on X induced by d⊆ is Hausdorff. In addition, we can prove the following theorem.
Correspondence / Computers and Mathematics with Applications 64 (2012) 661–663 663
Theorem 3. (P ({0, 1}n),D) is a compact (topological) space with the topology τd⊆ induced by the quasi-pseudometric d⊆.
Proof. Note that, if two subsets A, B are different, then the minimum distance between them could be
1
2n−1
− 1
2n
= 2− 1
2n
= 1
2n
.
Thus, every element A ∈ P ({0, 1}n) could be considered an open set in the topology τd⊆ induced by the quasi-pseudometric
d⊆, because it coincides with the open ball B(A, 12n+1 ), with center A and radius
1
2n+1 .
Therefore, using the topological notion of compact space introduced by Alessandrov and taking into account that
P ({0, 1}n) is a finite set, for every cover of P ({0, 1}n) constituted by a family of open sets, we can find a finite number
of open sets of this family which contain all the space P ({0, 1}n). Hence, P ({0, 1}n) is a compact (topological) space. 
3. Conclusions and further research directions
The quasi-pseudometric d⊆ expresses the difference between each ordered pair of elements belonging to P ({0, 1}n).
Actually, it is equal to 0 when the first set is a subset of the second one.
So, we could define kinds of quasi-pseudoperiodic orbit or quasi-pseudofixed point, which are in fact periodic or fixed
orbits considering the quasi-pseudometric d⊆. This infers a new possibility in the study of the dynamics, because when the
evolution of the initial set arrives at a subset of it, we have the future evolution under control.
Nevertheless, describing the orbit structure of a given Petri net seems difficult. Obviously, one could count all the diverse
orbits, but, for a state space big enough, this could be very hard.
Those questions which can be studied by means of the differentiability of the evolution operator, as an attraction of
certain orbits, are also now very difficult to state.
Another open problem is to analyze the perturbations, or even what a perturbation could be here, of these kinds of
discrete dynamical system. Often, this problem is formalizedmathematically by adding a parameter in the expression of the
evolution operator. But, in our case, the evolution operator is not given by a formula, and even to formalize a perturbation
of a system is a problem.
Acknowledgments
This work has been partially supported by projects TIN2009-14312, PAC08-0173-4838, PEII09-0184-7802, MTM2008-
03679, and CGL07-66440-C04-03.
References
[1] J.L. Guirao, F.L. Pelayo, J.C. Valverde, Modeling dynamics of concurrent computing systems, Comput. Math. Appl. 61 (2011) 1402–1406.
[2] J.W. Baker, E.P. Vink, A metric approach to control flow semantics, Ann. New York Acad. Sci. 806 (1996) 11–27.
[3] J. Rodriguez-Lopez, S. Romaguera, O. Valero, Denotational semantics for programming languages, balanced quasi-metrics and fixed points, Int. J.
Comput. Math. 85 (3) (2008) 623–630.
[4] W.A. Wilson, On quasi-metric spaces, Amer. J. Math. 53 (1931) 675–684.
[5] M.V. Foursov, C. Hespel,Weighted Petri nets and polynomial dynamical systems, in: Proceedings of the 17th International Symposium onMathematical
Theory of Networks and Systems, pp. 1539–1546.
Fernando L. Pelayo
Department of Computing Systems,
University of Castilla - La Mancha, Spain
E-mail address: FernandoL.Pelayo@uclm.es.
Jose C. Valverde∗
Department Mathematics,
University of Castilla - La Mancha, Spain
E-mail address: Jose.Valverde@uclm.es.
22 October 2011
∗ Corresponding author. Tel.: +34 967599200; fax: +34 967599224.


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