Stabilization test for conveyors

Copyright @ IFAC Manufacturing, Modeling, Management and Control, Patras, Greece, 2000

STABILIZAnON TEST FOR CONVEYORS

K. V. Sbahbazyan, Y.H.Sboukouryan

Institute for Informatia and Automation Problems of National Academy of&iences of the Republic ofArmenia

Yerevan. Armenia

Abstract: A stabilization problem for conveyors is discussed. A network of functional elements with one-pulse delay is called a conveyor. The main result is the least upper bound of the length of stabilization test. This is valid for the class of isomorphic conveyors. The results were used for the stabilization test of logical nets. Copyright02000 IFAC Keywords: Parallel networks, Pipelines, Stabilizing networks, Tests, Network analyzers, Automata theory.

The object of this paper is conveyor stabilization test, see (CudIjavcev, et al., (985). Examples of conveyors are systolic arrays, see (Kung, (980) and CRAY's conveyor. The conveyors can be considered as a model of electronic schemes.

the automaton 'P; assign a variable zij' The variable zij is called an output variable of the automaton 'P;. .I:

By zij

denote this value on the k-th step of

automaton. Analysis of conveyors puts forward a problem of conveyor stabilization for a fixed input An algorithm of stabilization test for conveyors is proposed. The stabilization test is based on decomposition of processes that run in the conveyors, see (Shahbazyan. 1990, (998). This allows receiving the least upper bound of the test length for an arbitrary class of isomorphic conveyors.

By the previous assumptions, the canonical equations for the automaton ~ have the fonn :

(1) where k=1.2.. ...

A network of functional elements with one-pulse time delay is called a conveyor. Let us detail this defmition.

,At, At'" ,tA,~·

At

where

n=n\u...un.

Here A is a finite alphabet,

J;: A

j"

~

At ·

i=I ..... M.

n

t(q.x)=x and ~(q.x)=j;(q) for all

q, x E

At,

Now let us construct a network K from automata '1'1 ..... 'I'M, i.e., the conveyor K. Let be a set of all input and output variables of automata '1'1.... , 'I'M and let be an arbitrary partition into classes where each class n; (;=1 .....s) contains not more than one input variables of one of automata '1'1•...• 'I'M. Let us identify the variables of each class, i.e., on k-th step of conveyor K the following conditions hold: ~=I ifx.yeIl; (i=1 .....s).

Let '1'1 •...• 'I'M be automata of a conveyor. Suppose ~=( Ai"

qi.l: E

By definition.

each automaton 'P; is a functional element with onepulse delay. To each j-th component (j=1 ,... ,nJ of input symbol of the automaton 'P; assign a variable

A set of automata ('PI ..... 'P,.,) with variable identifying rule, described by the partition of n, is called a conveyor.

xij' This means that variable xij receives the value

of j-th component of input symbol of automaton 'P;. The variable xij is called an input variable of the

Consider a class f1; that contains the output variable

automaton ~. By X~ denote this value on the k-th

Z }I

(recall that it is the only output variable in this

class). To establish the identity of the variables of n, asswne

step of automaton perfonnance. In the same way, to each j-th component (j= 1,.... mJ of output symbol of

149

k k Xis=Zj['

XisEn i ,

u=('P;. 'P)E VG if and only if 8(x jrJ= Zis for some I

k=1,2, ...

ands.

n

Consider now a class i that does not contain the output variable. To establish the identity of the variables of nj, assume

X~s=X~,

xjs,Xr[En i ,

k=1,2, .. .

In the converse case, to each conveyor graph G coresponds a class tp(G) of conveyors such that for any KE tp(G) holds G=G(K). In the sequel, consider along with the conveyor graph G(K) the graph

(3)

-

Let 8 be a map n~n where 8 is given by the following definition. If a class fl; does not contain output variable, then fix a variable XE fl; and settle 8(v)=x for all VE fl;. The variable x is called an input variable of the conveyor K. A set of input variables of the conveyor K is denoted by In(K). If fl; contains an output variable z, then settle 8(v)=z for all VE i. The variable z is called an output van'able of the conveyor K. A set of output variables of the conveyor K is denoted by Out(K).

graph G(K) by elimination of vertices of In(K) with incident edges. By G (K)

is denoted the factor-

graph of G(K) . The automaton stabilization problem is as follows. Let 'P=(A.Q.Z.q>.~qo) be an arbitrary automaton. Suppose that a stable input XE A is given, i.e., suppose that automaton input word is x'. The automaton 'f' is stabilized by the input x if there is no>O such that cp(qo.x·+I)=cp(qo.x·) for n>no.

n

Arranging the variables of In(K), we receive an input alphabet A(K) of conveyor K:A(K)=A/[·(K)/. Similarly, arranging the variables of the set Out(K), we receive an output alphabet Z(K) of conveyor K:Z(K) =A /OuI(K)/.

Further, we use the following notations. Let 'f' be an automaton 'f'=(A.Q.Z.q>. ~. qo). a be any word of input alphabet A, and a any symbol of A . Put by definition cp(q.A)=q. cp(q.aa)=cp(q>(q.a). a),

Combining the canonical equations (I) of automata {IP; } (i=l.· ··.MJ and the identification rules (2),(3), we obtain the canonical equations for conveyor K

~ (q.a)=~(q.aiIJ ~(q. a];) ... ~(q.a). where a]i=a, ... aj if a=al ... a .. n2i. Suppose an automata 'f'=(A.Q.Z. q>. ~qo), is given. A finite rooted oriented tree T is said to be a symple conditional test see (Cudtjavcev, et al.,1985) for the automaton 'P if the following conditions hold: I) a symbol ayE A is assigned to each vertex v of the tree T; 2) a symbol ZpE Z is assigned to each edge p of the tree T the different edges. outgoing from one vertex, have different labels.

(4)

where

-

G (K) . The graph G (K) is obtained from the

q; EA", i=l.2.... and d(x;) is the value of

the variable 8(x;) on the k-th step of conveyor K . Crearly, the inner states alphabet Q(K) concides with Z(K).

Consider a path Notice that the automaton K, i.e., the conveyor K, is well defmed: K has the input alphabet A (K), the alphabet of inner states Q(K), the output alphabet Z(K), and canonical equations (4).

where VI is the root of T, the vertex v. has no successors, P;=(Vi. Vj+JJ is the edge of T. The pair of

Now we introduce the following concept. A labelled graph G=(V. U) is called a conveyor graph if the following conditions hold: (i) each vertex VE Vhas an integer label m(v); (ii) m(vj$out(v) , where out(v) is the number of edges outgoing from v.

words

(a VI aV2 ...ay_I , zp I , zp 2 ,..., zp ,....... ) is called the

result of the test T applyed to the automaton 'f' if

zp I ,zp 2 , ... ,zp.-1

=~(qo,ayay ...ay ). I

2

,,-I

As a complexity mesure of the test T we cosider the length of the test T applied to 'I', i.e .• the length of the

It is clear that to each conveyor corresponds a conveyor graph. By G(K)=(Vc;,Vc;) denote the conveyor graph corresponding to the conveyor K. The vertices of G(K) are the input variables of the conveyor K and the automata 'P, ..... 'PM, i.e., VG=ln(KJJ{'P,•.... 'PMJ . The vertice If'; (i= l ... .. M) has the label mi, the vertice XE In(K) has the label I . Thus u=(x. 'PJE VG if and only if 8(x;)=x for some j. Also,

sequence

a ay2 ...ay.-1 . This length we denote by VI

LT(¥'). Now we introduce the concept of stabilization test for conveyors. Let K=(A.Q.Z. q>.~qoJ be a conveyor and XEA .

150

Consider a conveyor K and the class of conveyor

Let an upper bound of LT(K) be known, i.e., LT(K)~C. Recall that LT(K)~A!OwI(K)/. Thus it is easy to define the stabi/ization test for K as a finite oriented tree T labelled as follows: I) a. =x for each vertex of the tree T; 2) each vertex of T either has /Z(K)/ outgoing edges or has no outgoing edges; 3) a vertex v has no outgoing edges in two cases: a) if the labels of (v ".v') and (v·.v) coincide, b) if the path from the root of the tree to v has the length C.

p(K) such that K 'e p(K) implies G(K')

The conveyor K may be considered as an acyclic network of automata . The atuomata of this network are the automata corresponding to the bycomponents

B of graph G (K) . We denote by B the subconveyor corresponding to the bycomponent B. Let B be a vertice on the graph G*. Assign to each vertex B on G* a weight /(B) . If B is empty bycomponent, then /(B)=I . If B is nonempty

It is obvious that the conveyor K can be stabilized if and only if the edges Po-2, Pn-I in (5) have the same labels. Notice that the length of the test essentially depends on the accuracy of the bound LT(K). Our aim is to receive the least upper bound for the class of conveyors that have isomorphic conveyor graphs.

Iv.1

~>. .,..

bycomponent, then I(B)

REFERENCES

Statement I. Let K be an acyclic conveyor, consisting of automata '1'[, .... 'I'M. The length of stabilization test for K is not greater than the length of maximal path

CudIjavcev, V.B., S.V. Aljeshin and A.S. Podcolsin (1985). Introduction into Automata Theory, 300p. Nauka,Gl.red.phis-math.lit., (in Russian), Moscow. Kung H.T. (1980). The structure of Parallel Algorithms. Advances in Computers, 19, pp.65112. Shahbazyan K.V. (1990). On the Representation of Processes in Graphs. Sovjet.Math. Docl. 10, 2. Shahbazyan K.V. (1998). On the Process Algebra in Graphs. Cibernetica, W6, pp.38-42 (in Russian).

in the graph G(K) . Now we consider the decomposition of a conveyor K which has strongly connected graph G(K) . We first examine the strongly connected graph B=(Vs,Es) and its evolute graph R(B)=(VxN, U) where U={(v.t,v·.t+I) / v.v'e Vs, (v,v')eEs, teN) . Consider a partition Vs=Vo UVJ U ... UVk_J such as if (v. v ')e Es and ve JoI, then v 'e JoI+J(fOtOdk)' The maximal number of classes in this partition we denote by k(B). We call this number a splitting index of B. Evidently the partition Vs into k(B) classes is unique (to within the enumeration of classes) . Let us consider the class of conveyors p(B) such that Ke p(B) implies G(K) =B=(Vs,Es). We suppose B is a strongly connected graph with splitting index k(B) . Statement 2. Let K be a conveyor with strongly

connected graph G(K) =B. The following least

IE,I L(K):5 k(B) ·IAI.I:(B). the

length

= k(B) IAI .1:(8) •

Statement 3. The least upper bound of the length of stabilization test for generic conveyor Ke p(G) is equal to maximal length of the path in the weighted graph G*.

We consider an arbitrary network S of automata '1'[, ... , 'I'M. Suppose that there is a stabilization test for each automaton 'l'; (i=I, ... ,M) and the bound LT('I'J is known. Assign to each vertex 'l'; of S the bound LT('I'J as its weight.

upper bound for experiment is valid:

= G(K) .

of stabilization

151