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Journal ArticleDOI

Simulations of asynchronous evolution of discrete systems

15 Jun 1999-Simulation Practice and Theory (Elsevier)-Vol. 7, Iss: 4, pp 309-324

TL;DR: Aerent simulations of asynchronous evolution of discrete systems performed with the research software Discrete System Evolution, lead to the first experimental results predicted by the convergence theorem of asynchronous iterations of discrete dynamic systems partitioned into blocks.
Abstract: Recently, a convergence theorem of asynchronous iterations of discrete dynamic systems partitioned into blocks has been proved [2]. This theorem is verified with several asynchronous block strategies. It also generalizes the chaotic iterations. DiAerent simulations of asynchronous evolution of discrete systems performed with the research software Discrete System Evolution (DSE), lead to the first experimental results predicted by this theorem. ” 1999 Elsevier Science B.V. All rights reserved.

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Simulations of asynchronous evolution of discrete
systems
Jacques M. Bahi
a
, Christian J. Michel
b,
*
a
Laboratoire de Calcul Scienti®que, Antenne de Belfort, UMR CNRS 6623, BP 527, 90016 Belfort, France
b
Laboratoire de Recherche en Informatique, Institut Polytechnique de S
evenans, Rue du Ch
^
ateau
a
S
evenans, 90010 Belfort, France
Received 3 June 1998; received in revised form 1 February 1999
Abstract
Recently, a convergence theorem of asynchronous iterations of discrete dynamic systems
partitioned into blocks has been proved [2]. This theorem is veri®ed with several asynchronous
block strategies. It also generalizes the chaotic iterations. Dierent simulations of asynchro-
nous evolution of discrete systems performed with the research software Discrete System Evo-
lution (DSE), lead to the ®rst experimental results predicted by this theorem. Ó 1999 Elsevier
Science B.V. All rights reserved.
Keywords: Boolean iterations; Asynchronous algorithms; Discrete dynamic systems; Numerical simula-
tions; Research software
1. Introduction
The convergence results in the asynchronous continuous framework are well-
known and are based on a contraction hypothesis with respect to a maximum norm,
see e.g. [4,6,5,8,1,3]. However, this approach cannot be applied to the discrete frame-
work. Indeed, this hypothesis in the continuous framework leads to constant func-
tions in the discrete framework and a new study is necessary for the discrete case.
We have supposed that F is a contraction with respect to a vectorial distance [2].
The contraction with respect to a vectorial distance was ®rst introduced by [7] in or-
der to study a particular case of asynchronous iterations, namely the chaotic ones.
The discrete system considered here has n components. Each component i takes a
®nite number of values x
i
; i 2 1; ...;n
fg
. This system is partitioned into a blocks.
www.elsevier.nl/locate/simpra
Simulation Practice and Theory 7 (1999) 309±324
*
Corresponding author. Fax: 33 3 84 58 30 30; e-mail: christian.michel@utbm.fr
0928-4869/99/$ ± see front matter Ó 1999 Elsevier Science B.V. All rights reserved.
PII: S 0 9 2 8 - 4 8 6 9 ( 9 9 ) 0 0 0 1 0 - 5

Each block i has n
i
components with
P
a
i1
n
i
n. The value of a block i is denoted by
X
i
and the value of the block system, by X X
1
; . . . ; X
a
. The dynamic of the system
is described according to a function f
f x f
1
x
1
; . . . ; x
n
; . . . ; f
n
x
1
; . . . ; x
n
:
This function f is partitioned into a compatible way, i.e.
f x F X F
1
X
1
; . . . ; X
a
; . . . ; F
a
X
1
; . . . ; X
a
:
The state of a system (resp. a block system) at the time t is represented by x
t
(resp.
X
t
), or, more precisely by
x
t
X
t
x
t
1
; . . . ; x
t
n
ÿ
X
t
1
; . . . ; X
t
a
ÿ
:
De®nition 1.1. Consider the Cartesian product E
Q
n
i1
E
i
, where E
i
represents the
®nite set of possible values x
i
of the component i of the discrete system. The vectorial
distance d: E E ! 0; 1
f g
n
is de®ned by for all x; y 2 E E
x; y ! dx; y dx
1
; y
1
; . . . ; dx
n
; y
n
;
where
dx
i
; y
i
1 if x
i
6 y
i
;
0 if x
i
y
i
:
De®nition 1.2. Consider a discrete system whose dynamic is described according to a
function f : E ! E. The boolean matrix Bf associated with f is de®ned by its
general term b
ij
; i; j 2 f1; . . . ; ng, so that
b
ij
1 if the ith component of f depends on x
j
;
0 otherwise:
As Bf is a boolean matrix, its only possible eigenvalues are 0 or 1.
Example 1.1. The following simple example shows that to have the boolean matrix
Bf is equivalent to have the connexion graph of the discrete system. Consider a
discrete system with three components 1, 2 and 3 (cellular automata, processors,
neurons, etc). Assume that the notation 1 ! 2 means that component 1 informs
component 2. If these components are connected as shown in the ®gure below
then, the matrix Bf associated with any mapping f f
1
; f
2
; f
3
describing the
dynamic of the system according to the above graph, is
310 J.M. Bahi, C.J. Michel / Simulation Practice and Theory 7 (1999) 309±324

Bf
0 1 1
1 0 1
0 1 0
0
@
1
A
:
Proposition 1.1 (detailed in [7]). df x; f y 6 Bf dx; y; for all x; y 2 E E,
where the componentwise order relation 6 is de®ned in f0; 1g
n
by 0 6 0 6 1 6 1.
De®nition 1.3. The function f is a contraction if the spectral radius qB of the
associated matrix Bf is equal to 0. Bf is called the contraction matrix of f.
Proposition 1.2 (detailed in [7]). If the function f is a contraction on E
Q
n
i1
E
i
then
there exists a unique x
2 E so that x
f x
. x
is called the ®xed point of f.
Remark 1.1. The contraction concept is related to the connexion graph of the discrete
system. Indeed, an iteration function f describing the dynamic of the system is a
contraction if and only if its connexion graph has no cycles. Note that there are an
in®nity of iteration functions associated with a given connexion graph.
All iteration functions associated with this connexion graph are contractions.
De®nition 1.4. The block vectorial distance is de®ned as follows
dX ; Y dX
1
; Y
1
; . . . ; dX
a
; Y
a

such that
dX
i
; Y
i
1 if X
i
6 Y
i
;
0 if X
i
Y
i
:
2. Totally asynchronous discrete model
De®nition 2.1. Let the strategy fJtg
t2N
be a sequence of non-empty subsets of
f1; . . . ; ag at the time t. Let fs
i
j
tg
t2N
; i; j 2 f1; . . . ; ag, be a sequence of integers at
the time t satisfying the three following conditions:
(i) s
i
j
t t ÿ r
i
j
t with 0 6 r
i
j
t 6 t, r
i
j
t being the delay of the block j compared
to the block i. These delays may be generated by dierent communications and
computation sizes of the elements of the system. This model takes no synchroni-
zation hypothesis.
(ii) 8i; j 2 f1; . . . ; ag, lim
t!1
s
i
j
t 1, i.e. the delays associated with the block i
are unbounded but follow the iterations of the system.
J.M. Bahi, C.J. Michel / Simulation Practice and Theory 7 (1999) 309±324 311

(iii) 8i 2 1; . . . ; a
f g
; Cardft; i 2 Jtg 1, i.e. no block is de®nitively lost.
Then, the asynchronous iterations with delays fr
i
j
tg according to the strategy
fJtg are described by the algorithm
Given X
0
X
0
1
; . . . ; X
0
a
t 0; 1; . . .
i 1; . . . ; a
X
t1
i
F
i
X
s
i
1
t
1
; . . . ; X
s
i
a
t
a
if i 2 J t
X
t
i
if i 62 Jt:
(
1
Remark 2.1.
1. The block i at the time t is either updated (iterated) by using some blocks j with
states X
s
i
j
t
j
available at the previous time s
i
j
t t ÿ r
i
j
t 6 t, i.e.
X
t1
i
F
i
X
s
i
1
t
1
; . . . ; X
s
i
a
t
a
, or not updated, i.e. X
t1
i
X
t
i
.
2. These two alternatives are described by the iteration strategy fJtg
t2N
,
Jt f1; . . . ; ag8t 2 N: if the block i belongs to the strategy J(t) then its state
X
t1
i
is updated by F
i
otherwise the block i is not updated. The iterations consid-
ered are asynchronous, i.e. chaotic iterations with delays.
3. The chaotic iterations are particular cases of Alg. (1) with s
i
j
t t.
Remark 2.2. The set of all possible strategies is not countable. Furthermore, if the
delays are also considered, the evolution complexity of asynchronous iterations is
hard to imagine.
Example 2.1. This simple example gives an evolution of a discrete model at three
components with chaotic and asynchronous iterations. It will show in particular that
(i) a successive substitution function F with a ®xed point does not automatically
imply the convergence of chaotic and asynchronous iterations;
(ii) the chaotic iterations are particular cases of asynchronous iterations;
(iii) the asynchronous iterations do not represent a classical process of successive
substitutions;
(iv) asynchronous iterations with dierent strategies and delays can reach a state
which is the ®xed point in the successive substitutions, but without convergence.
Let the function F given by the following Table 1.
Then, the graph of successive substitutions is given in Fig. 1.
The graph of F is very simple with a ®xed point 3 (no cycles) (Fig. 1). For simplic-
ity reasons, the chaotic and asynchronous iterations considered below are compo-
nentwise iterations, i.e. at each iteration, only one component is updated. The
chaotic function F
i
X is given by the following Table 2.
Then, the graph of chaotic iterations is given in Fig. 2.
Although there is a ®xed point in the successive substitutions, the chaotic itera-
tions do not converge. For example, if the initial state is 4 (1,0,0), then the chaotic
iterations are in a cycle containing the states 4,6,7 (Fig. 2).
312 J.M. Bahi, C.J. Michel / Simulation Practice and Theory 7 (1999) 309±324

Let X
s
be a previous value of X. F
i
X
s
is a value among all possible values of
F
i
X , i.e. 0 or 1 in this example. The asynchronous function F
i
X
s
is given by the
following Table 3.
Then, the graph of asynchronous iterations is given in Fig. 3.
The chaotic iterations are obviously particular cases of asynchronous iterations
(Figs. 2 and 3). Therefore, the asynchronous iterations do also not converge. Other-
Fig. 1. Graph of successive substitutions associated with Table 1.
Table 1
Example of successive substitutions
State X F(X)
0 0,0,0 0,0,1
1 0,0,1 1,0,1
2 0,1,0 0,0,1
3 0,1,1 0,1,1
4 1,0,0 1,1,0
5 1,0,1 0,1,1
6 1,1,0 1,0,1
7 1,1,1 1,1,0
Table 2
Example of chaotic iterations
State X F X F
1
X ; X
2
; X
3
X
1
; F
2
X ; X
3
X
1
; X
2
; F
3
X
0 0,0,0 0,0,1 0,0,0 0,0,0 0,0,1
1 0,0,1 1,0,1 1,0,1 0,0,1 0,0,1
2 0,1,0 0,0,1 0,1,0 0,0,0 0,1,1
3 0,1,1 0,1,1 0,1,1 0,1,1 0,1,1
4 1,0,0 1,1,0 1,0,0 1,1,0 1,0,0
5 1,0,1 0,1,1 0,0,1 1,1,1 1,0,1
6 1,1,0 1,0,1 1,1,0 1,0,0 1,1,1
7 1,1,1 1,1,0 1,1,1 1,1,1 1,1,0
J.M. Bahi, C.J. Michel / Simulation Practice and Theory 7 (1999) 309±324 313

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Gérard M. Baudet1Institutions (1)
TL;DR: A class of asynchronous iterative methods is presented for solving a system of equations corresponding to a parallel implementation on a multiprocessor system with no synchronization between cooperating processes to show clearly the advantage of purely asynchronous Iterative methods.
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Mouhamed Nabih El Tarazi1Institutions (1)
Abstract: Nous presentons dans cet article des resultats de convergence des algorithmes asynchrones bases essentiellement sur la notion classique de contraction. Nous generalisons, en particulier, tous les resultats de convergence de ces algorithmes qui font l'hypothese de contraction en norme vectorielle qui recemment a ete tres souvant utilisee. Par ailleurs, l'hypothese de contraction en norme vectorielle peut se trouver difficile, voire impossible a verifier pour certains problemes qui peuvent etre cependant abordes dans le cadre de la contraction classique que nous adoptons. In this paper we present convergence results for the asynchronous algorithms based essentially on the notion of classical contraction. We generalize, in particular, all convergence results for those algorithms which are based on the vectorial norm hypothesis, in wide spread use recently. Certain problems, for which the vectorial norm hypothesis can be difficult or even impossible to verify, can nontheless be tackled within the scope of the classical contraction that we adopte.

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