Asymptotic behavior of stochastic automata in stationary random media

ASYMPTOTIC BEHAVIOR OF STOCHASTIC AUTOMATA

IN STATIONARY RANDOM MEDIA

E. N. Vavilov, S. D. l~idei~man,

and A~ I. l~zrokhi

UDC 62-507

In this p a p e r we c a r r y out a de ta i l ed a n a l y s i s of the a s y m p t o t i c b e h a v i o r of f ini te (with a number of s t a t e s 2n) a u t o m a t a B2n(k, v 0, vl) in a s t a t i o n a r y r a n d o m m e d i u m C (Pi, P~). Many known au tomata [1-5] be long to this c l a s s .

r We es t ab l i sh that a s equence of au toma ta {Ban(k , v0, vi)}n=t has as its l imi t an infinite (with a countab le

n u m b e r of s t a tes ) a u t o m a t o n B(k, v0, vt) whose p robab i l i s t i e c h a r a c t e r i s t i c s of b e h a v i o r a r e the l imi t (for n --- ~) of the p r o b a b i l i s t i c c h a r a c t e r i s t i c s of the a u t o m a t o n B2n(k, v0, vl). Due to this we can f ind s imp le e x - p r e s s i o n s for the calculation of the important statistical characteristics of the behavior of automata with a

large number of states.

The analysis of the behavior of an infinite automaton [6] allowed a detailed classification of the asymptotic behavior of finite automata to be carried out. In particular, a sequence of finite automata was considered asymptotically optimal if an optimal automaton was its limit. In this paper we show that inthe eases that are most

interesting from a practical viewpoint, where the limit automaton B(k, v0, vl) is optimal, quasioptimal, or a t t r a c t i v e (the c l a s s i f i c a t i o n will be g iven below), t h e r e ex i s t s no s t a t i o n a r y d i s t r ibu t ion of p robab i l i t i e s of s t a t e s of the au tomaton . T h u s , an a n s w e r is found to the p r o b l e m p o s e d .by Va r shavsk i i in [2] (p. 57).

An i m p o r t a n t p a r t of the w o r k is the s tudy of the g e n e r a t i n g funct ions U (n) (z) and Ua(Z) of p robab i l i t i e s of an ac t ion change by au tomata . H e r e we p r e s e n t an a l g e b r a i c m e t h o d fo r inves t iga t ing these funct ions . The p r o b l e m be ing c o n s i d e r e d by us is a p a r t i c u l a r c a s e of the p r o b l e m of o n e - d i m e n s i o n a l r a n d o m walk on a s e m i - ax is . A de ta i led inves t iga t ion of the p r o b l e m of o n e - d i m e n s i o n a l r a n d o m walk , mak ing use of ano the r m a t h e - m a t i c a l tool , is found in the m o n o g r a p h s by ]3orovkov [7], Koro lyuk [8], and T a k a c s [9].

w 1 . D e t e r m i n a t i o n o f A u t o m a t a B a n ( k , Vo, Vl) a n d B ( k , v0 , v l )

We sha l l p r o c e e d f r o m the d e s c r i p t i o n of the b e h a v i o r of an a u t o m a t o n in a s t a t i ona ry r a n d o m m e d i u m C (Pl, P2) found in [1, 2J. The m e d i u m f o r m s the input v a r i a b l e of the a u t o m a t o n (the r e a c t i o n of the medium) , which c a n a s s u m e one of two va l ue s : s = - 1 (penalty) and s = +1 (nonpenalty). The s t a t e s of the a u t o m a t o n a r e d e c o m p o s e d into two r eg ions L 1 and I e. The reg ion L 1 (Ie) conta ins s t a t e s m a r k e d by the ac t ion fl (f2)"

F o r the ac t ion fm the m e d i u m C (Pi, P2) i m p o s e s a penal ty on the au toma ton with a p robab i l i ty Pm; q m = 1 - Prn is the p robab i l i ty of a nonpenal ty , m = 1, 2. F o r the sake of being definite we sha l l a s s u m e that P2 > Pl- Let the s t a t e s of the r e g i o n ~ have n u m b e r s - 1 , - 2 . . . . . - n . . . . . while the s t a t e s of the r eg ion L a have n u m b e r s 0, 1, 2, . . . , n - 1 . . . . . We spec i fy the ru le of change of the in te rna l s t a t e s of au toma ta under the ac t ion of the input v a r i a b l e s. We denote by r 0i(n)(t))the n u m b e r of the s ta te in which the au toma ton B(k, v0, vl) (B2n(k, v0, v~)) is a t the ins tant t , t = 0, 1, 2 . . . . , when the s ignal s(t) is r ece ived . Then

x~(t q- 1 )= xp~(t) (1 + s(t2 q- 1)) + q~= (t) (1 ---s 2 (t'nt" 1)) ,"

~;t(t) = I~ (t) - - sign g (t) with probability %, [~ (t)q-k sign ~;(t) with probability 1 - - %,

~ (t) = I ~ (t) - - sign ~ (t) with probabi!ity 1 - - %, /~P (t) -~ k sign ~p (t) with probability vl.

H e r e we a s s u m e that s i g n 0 = 1. F o r a f ini te au toma ton B2n(k , v0, vl) the ru le is the s a m e , but it is addi t ional ly r e q u i r e d that r = m a x { - n , rain [n - 1, r

T r a n s l a t e d f r o m Kibe rne t ika , No. 5, pp. 3 -11 , S e p t e m b e r - O c t o b e r , 1977. Or ig ina l a r t i c l e submi t t ed Oc tobe r 16, 1975.

0 0 1 1 - 4 2 3 5 / 7 7 / 1 3 0 5 - 06o9 807.50 �9 1978 P lenum Publ i sh ing C o r p o r a t i o n 639

I ~ffH)k-1 ___L~o t

[ t . . . .

~/ Region Li 'Region I.e.

~4)k 1- "

N l-~'a

Fig. 1. Graph of the automaton B(k, Vo, ,1) for s = +1.

The graph of the automaton ]3{k, v 0, vl) for s = +1 is p r e s e n t e d in Fig. 1. By the dashed l ines we have shown the b ranches of the graph of the automaton ]32n{k, v 0, v~), n = / k , t = 1, 2 . . . . . In the ca se s = - 1 the graph has the s ame f o r m , only t rans i t ions f rom the reg ions to the single s ta te t a k e p l a c e with a probabi l i ty 1 - vl, while t rans i t ions into the depth of the regions to k s ta tes take place with the probabi l i ty vi.

In the following, if there is no need to indicate the ac tual va lues of the quant i t ies k, v0, v l ,we shall use the notation ]3(k, v0, v 1) - ]3 and]32n(k, v 0, ,1) -- B2n- The au tomata B2n defined above a r e au tomata of the l inear type [10] and they contain many known c l a s s e s of au tomata . Thus , for example , the au tomata B2n(1, v0, vl) a r e known in the l i t e r a tu r e as au tomata with quas i l i nea r (selective) tact ics [4, 5]. The automaton ]32n(1, 0, 1 / 2 ) is the Krylov automaton; the au tomaton ]32n(n- 1, 0, 0) is the "naive" automaton ofKr insk i i , andthe automaton B2n(1, 0, 0) is the c l a s s i c a l automaton with l inear tac t ics ( introduced by Tse t l i n ) [1 -3 ] . The au tom- ata B2n(k, 0, 0) a re of the type W{k) in the case of an in teger k (in the p r e s e n t work k does not depend on the p a r a m e t e r s of the m e d i u m , as is a s s u m e d in [11]).

w C l a s s i f i c a t i o n o f t h e A s y m p t o t i c B e h a v i o r o f A u t o m a t a ( n ) _

Let Ua, mtPj) (Ua, m(Pj) ) be the probabi l i ty of a change of the act ion fj of the automaton ]32n(B) a f t e r m cycles with a s t a r t f rom the s ta te of the region Lj with the number a; 7(n)i(Ta,j) is the ave rage t ime up to the change of the act ion fj of the au tomaton B2n('B), j = 1, 2. The s t a t e s of the "region f r o m which a change of action a f t e r a s ingle beat is poss ib l e will be ca l l ed initial and wil l be denoted by u~n)~ (Pi) - u~I(Pj ) , Ui, m(Pj) - Um(Pj), T~,~. ) -

(n) ~' '~ J Tj , Ti, j -- Tj for i = 0 (j =2 ) or i = - - 1 (j =1) . L e t p l < P2 for the sake of being definite. We introduce the following definitions.

Definition 1. A sequence of finite au tomata has as its l imi t an infinite automaton if for any m, ~,n~m(Pj)- g ~

Ua,m(Pj ) as n ~ r162 j = 1, 2.

Definition 2. An infinite au tomaton in a s t a t ionary random medium p o s s e s s e s the following p rope r t i e s :

I t is opt imal (antioptimal) if with a probabi l i ty equal to one it l eaves a region with a g r e a t e r ( lesser) p robabi l i ty of a penal ty and i f w i t h a p o s i t i v e probabi l i ty it does not leave a region with a l e s s e r (greater) probabi l i ty of a penalty; it is s t r i c t ly opt imal (antioptimal) if h e r e the a v e r a g e wander ing t i m e until a change of the action f2 (ft) is finite;

it is quas iop t ima l ( an t i -quas iop t ima l ) if with a probabi l i ty equal to one it leaves both r eg ions , but the a v e r a g e wander ing t i m e among the s t a t e s of the reg ion with a l e s s e r (greater) probabi l i ty of a penalty is infinite;

i t is a t t r ac t ive if with a pos i t ive probabi l i ty it does not leave each reg ion and r epu l s ive if with a a probabi l i ty equal to one it leaves both regions and the a v e r a g e t imes up to the change of act ion a r e f ini te , and it does not m a t t e r whether the p robabi l i t i e s of the change of act ion and the a v e r a g e t imes up to the change a r e equal.

Definition 3. A sequence of finite au tomata is sa id to be asympto t ica l ly opt imal , a sympto t ica l ly q u a s i - op t imal , a sympto t i ca l ly a t t rac t ive , o r asympto t ica l ly repu ls ive , if t h e l i m i t infinite automaton p o s s e s s e s the co r respond ing p r o p e r t y .

If the a v e r a g e t imes up to the change of act ion a r e f in i te , then the ma thema t i ca l expecta t ion (MEX) of the penal ty of the automaton by the m ed i um af te r a s~ngle e y c l e o f f u n c t i o n i n g i s c a l c u l a t e d by the t rad i t iona l exp re s s ions [1-5].

In addit ion, it is cons ide red as natura l to de te rmine as follows the a v e r a g e d MEX of the penalty with the a s sumpt ion tha t absorp t ion can take p lace in one of the regions:

640

l

M (B, C) = "-2 2 (ptr!'~ + P~r~'l)" (l) i=O, -- I

Here r is the probabil i ty that absorpt ion in the region Lj o c c u r r e d with the condition that in the initial instant the automaton was in a state with the numbers i, i = 0, - 1 . At the same time we a s sume that the initial s tates a re equally probable.

Definition 4. We shall say (according to [1-3]) that the automaton B has expedient behavior in a statior, a ry random medium C(PI, P2) if the inequality M(B, C) < (Pl + P 2 ) / 2 holds.

w 3. B a s i c R e s u l t s

We introduce the following notation: pj = (1 - ul)pj + u0qj; cij = 1 - l~j; 5j = l~j - kqj;Q = ( [k / (k+l) ] ~0)/(1 -

v 0 -- ul).

THEOREM 1. The infinite automaton B(k, v0, vl) is the l imit of the sequence of finite automata {Bzn(k, Uo, uI)}~=I- H e r e ; ( 2 ~ < +~; for 5 i > 0 , l i m - c , ~ ! = T . , , < - t - ~ ; for 6 , 40 ,1 imT(~)=T~ . /= -{ - co .

n . ~ a , l a,1

For the p roof of Theorem, 1 we note at to show that for n -- % Uatn)(z) -- U a (z), I zl <

t ] ~ ~' (z) =

once that by vir tue of the continuity theorem [12] it is necessary 1, where u(n)(z) and Ua(z ) a r e generat ing functions of the form

Z rn

m----O rn=0

THEOREM 2. A sequence of automata {B2n(k , Vo, vi)}n~=l in C(Pl, P2) posses ses the followit~g behavior ; Fo r v 0 + vl < 1 it is expedient, for u 0 + vl = 1 it is unimportant, and for % + vi > 1 it is n0nexpedie~.

4 If ~0 + vl < 1 v 0 < k / ( k + 1), then the sequence of automata LB2n}n=l in C (Pl, P2) for Pl < Q - P 2 is a s y m p - totically optimal , for pl < Q < P2 it is asymptot ical ly s t r ic t ty optimal, for pl = Q < Pz it is asymptot ical ly quas iopt imal , for Pl < P2 < Q it is asymptot ical ly a~tractive, and for Q < Pl < P2 it is asymptot ical ly r e - pulsive.

If v 0 + vl > 1, u 0 > k / ( k + 1), then the sequence of automata {B2a}n= 1 in C(Pl, P2)for Pl - Q < 02 is asymptot ical ly antiopt~mal, for p~ < Q = [92 it is asymptot ical ly anti-quasioptimal, for Q < p~ < �89 i r i s asymto- t ica l ly at t ract ive, and for Pi < 132 < Q it is asym0tot ical ly repulsive.

eo If u 0+ vl < 1, v 0 ~ k / ( k + l ) ( v 0 + vt > l , v 0_< k / ( k + l ) ) , then the sequence of automata{Bzn}n= i in

C @l, Pz) is asymptot ica l ly repuls ive (asym0tot ical ty at tract ive) .

The resul ts just p resen ted allow us to formulate the conditions of asymptot ic optimality for known c lasses of automata.

Automata with select ive tactics [4,5] form an asymptot ical ly s t r ic t ly optimal sequence in C (Pi, P2) when the conditions

1 1 2 Vo

re-I- v t < 1, r e < T ' P i< "1 ~ ( v o + vi) <p~

a r e satisfied. Hence, in pa r t i cu la r , it follows that a sequence of Krylov automata does not possess asymptot ic optimality in C (Pl, P2) (t31, P2 ~ 1). Automata B2n(k, 0, 0) form an asymptot ical ly optimal sequence if Pl < k / ( k + 1) _< P2. We note that automata of the type WC~:) possess asymptot ical ly s t r i c t optimali ty, since in [ t l ] in the role of k the value of In (q2/qI) / In (Pl/P2) was taken.

Automata with l inear tactics B2n(1, 0, 0) form an asymptot ical ly s t r ic t ly optimal sequence in C (!ol, P2) if Pl < 1 / 2 < P2; they form an asymptot ical ly at tract ive sequence for PI, P2 < ! / 2 , an asymptot ical ly repulsive sequence f o r P2, t~ > i / 2 , and an asymptot ical ly quasiopt imal sequence for Pt = 1 / 2 < P2-

w D i s c u s s i o n o f t h e B a s i c R e s u l t s

Our approach to the study of the asymptotic behavior of a sequence of finite automata differs f rom the approach expounded in the well-known works [1-5, 10-15]. We not only study the behavior of the probabil is t ic cha rac t e r i s t i c s when the memory capaci ty is increased, but also establish the existence of a l imiting infinite automaton. The c lass i f ica t ion of such automata in fact provides the descript ion of the possible asymptotic

641

behav io r of a sequence of finite au tomata . H e r e s ide by side with the probabi l i ty of the change of act ion we f indthe a v e r a g e w a n d e r i n g t i m e s _(n) and the i r l imi ts for n ~ oo ' a , j

All this al lows us to give a comple te p ic tu re of the asympto t ic behav ior of au tomata B2no In p a r t i c u l a r , the conditions of asympto t ic opt imal i ty obtained by us a r e s t r i c t e r than in [1-5], where it was a s s u m e d that the au tomata f o r m an asympto t i ca l ly opt imal sequence in C (Pl, P2) for Pl < Q, i .e . , for Pl < (I/2 - v0)/[1 - (v0 + vl)], v0 + vl < 1, v 0 < 1 / 2 for au tomata with se lec t ive tac t ics and for Pl < 1 / 2 for au tomata with l inear t ac t i cs . The sequences of au tomata of Kry lov , Kr insk i i , and Robbins w e r e c o n s i d e r e d asympto t ica l ly opt imal in a l l med ia C{Pl, P2). However , the condit ion Pl < Q guaran tees only a t t rac t ion of the automaton into an opt i - m a l reg ion , and it is n e c e s s a r y to s t ipulate addit ionally that the a t t rac t ion condit ion is not fulfi l led in a "bad" reg ion (P2 -> Q). We note that in [1t] fo r au tomata W(k) the re a r e p r e s e n t e d p r e c i s e l y such two-s ided opt i - mal i ty condit ions.

In [1-5] no dis t inct ion was made be tween sequences of asympto t ica l ly a t t rac t ive and asympto t ica l ly opt i - m a l au tomata , s ince the condition

lira -- 0, Pt<Pz, (2)

was fulf i l led for them. But for the a sympto t i ca l ly opt imal au tomata B2n(k, v 0, v~), lira v~ " )= 67 ~ (see Sec. 6), r t - ~ oo

r(n) - - n while for a sympto t i ca l ly a t t r ac t ive au toma ta i n c r e a s e s as P2 , i .e . , with an i n c r e a s e in n it rapidly goes to infinity. F o r a l imi t ing a u t o m a t a i n t h i s c a s e t h e w a n d e r i n g over a "bad" region may not end at all and the m a t h e m a t i c a l expecta t ion of the penalty is M(B, C) ~ Pl.

In analyzing the e igenvalues of the Markov chain assoc ia ted with an automaton in C (Pl, P~), T s e r t s v a d z e [14, 15] (see a lso [2]) e s t ab l i shed that for Krylov au tomata and au tomata with l inear tac t ics (for Pl < P2 < 1 / 2 ) the re ex is t s an e igenvalue which exponential ly with r e s p e c t to n tends to 1 (there is s t i l l one eigenvalue equal to 1, while the r e s t a r e bounded by a constant a , a < 1). These e s t i m a t e s " . . . enable us to have doubts about the ergodic i ty of the co r re spond ing cha ins , i .e . , the independence of the final dis t r ibut ion of the initial s t a te , when n - - ~. The quest ion of uniqueness of a stationary, d is t r ibut ion of p robabi l i t i e s of s ta tes for n ~ ~ r e m a i n s open" [2] {p. 57). The ana lys i s c a r r i e d out in the p r e s e n t work shows, in p a r t i c u l a r , that the Krylov automaton and the automata with l inear tac t ics ffor Pl < P2 < 1 / 2 ) f o r m asympto t i ca l ly a t t r ac t ive sequences , while for the l imi t ing automaton a s t a t iona ry d i s t r ibu t ionof s ta te p robabi ! i t i es does not exis t at a l l , and the behavior is subs tant ia l ly de t e rmined by the s ta r t ing s ta te .

The a igebra ic methods p re sen ted below allow us to study [16] the asympto t ic behav io r in fact of a b r o a d e r c l a s s of au tomata B2n(k, m , v0, vl), for which t rans i t ions into the depth of the region I~ (L 2) to k s ta tes and f r o m the region ~ (L 2) to m s t a t e s a r e poss ib le , during one cyc le .

The p ropos i t ions of T h e o r e m s 1 and 2 will be va l id if k is r ep l aced by k / m , 6j = ml~j - kqj. Analogous r e su l t s a r e va l id a l so for au tomata for which t rans i t ions into the depth of the reg ion to any number of s ta tes up to k and f r o m the reg ion to may num ber of s t a t e s up to m a re poss ib le .

w 5. C o n s t r u c t i o n a n d A n a l y s i s o f t h e G e n e r a t i n g F u n c t i o n s o f

P r o b a b i l i t i e s o f C h a n g e o f A c t i o n . P r o o f o f T h e o r e m 1

We inves t iga te the wander ing of the au tomaton B o v e r the s t a t e s of the reg ion L 2 up to the change of act ion f2- F r o m the ru le of behav ior of an infinite automaton it follows that

Ua.m+= (Pz) ~- P~U,~-l.rn (P=) "4" q~.ua+~,,.n (p,,), (3) r n = 1 ,2 . . . . ,

u_~ ,0=l , u~,0=O, a----O,l ,2 . . . . . (4)

Multiplying (3) by z m+l and summing over all m , we obtain the di f ference equation for the genera t ing function,

U,, (z)="p~zU,,_~ (z) + ~zU,,+,,(z), a = O, 1, 2 . . . . . (5)

and the boundary condition

U - I (z) = 1. (6)

The solut ion of Eq. (5) wil l be sought in the f o r m U a (z) = X a+l (z). Then re la t ive to X(z) we obtain the equation

~(z) = ~ z + ~z~ TM (z). (7)

642

T h e probabil iLv or2, of i n t e r e s t to us , tha t the a u t o m a t o n changes tlle ac t ion f2, s t a r t i n g f r o m the initial s t a te of the reg ion L2, equals ~2 = U(1). F o r z = 1 Eq. (7) a s s u m e s the f o r m ( X - 1)iX k + xk+l + . . . + X - (P2/q2)} =0. One r o o t of this equat ion is X = 1. We sha l l d e t e r m i n e w h e t h e r the equa t ion

q2

ha s r e a l r oo t s on the s e g m e n t [0 ,1 ] . Since on this s e g m e n t ~)'(X) > 0, ~I,{0) < 0, al l is d e t e r m i n e d by the sigr~ of ~(-s Ifl~ 2 < k / ( k + 3.), then Eq. (8) has a s ing le roo t ~ ~ (0, 1). Unity and cr cons t i tu t e the p o s s i b l e va lues of the gene ra t i ng funct ion Mz) for z = 1. We sha l l s e e k the so lu t ion of Eq. (7) in the f o r m

(z) = zF (w), w = z ~+~ . (9)

Subst i tu t ing (9) into (7), we a r r i v e at the fol lowing a l g e b r a i c equat ion fo r F(w):

F (w) = p~ + q~w(F (w)) , (!0)

We have to f ind a so lu t ion of (10) which is a Mac l au r in s e r i e s with nonnegat ive coe f f i c i en t s tha t eowcerges fo r w = 1.

We denote x = F , r = 132 + c~2wx k+l. T h u s , we m u s t so lve the equat ion x = ~(x); i .e . , we have to f ind the f i x e d p o i n t s of the mapp ing y = if(x). The fol lowing p r o p o s i t i o n [6] is val id .

LEMMA 1. y = ~(x) f o r Iwl <_ 1 is a c o m p r e s s i v e mapp ing of the s e g m e n t ~ , p2((k + 1 ) / k ) ] if P2 < k / i k + 1) and of the s e g m e n t [t~, 1] i f t32 _> k / ( k + 1).

F r o m the l e m m a it fol lows that t h e r e ex i s t s a unique so lu t ion of Eq. (10) which can be obta ined by the i t e ra t ion method . Hav ing t a k e n ~ i n t h e ro l e of F 0 and d e t e r m i n e d a s equence which tends to the r e q u i r e d Soiutlon F n = ~ + ~2W(Fn_l) k+l, we find tha t the so lu t ion d e t e r m i n e d by the i t e r a t i on me thod is a M a c i a u r i n s e r i e s with nonnega t ive coe f f i c i en t s . Subst i tu t ing F(w) into the e x p r e s s i o n (9), we obta in the r e q u i r e d so lu t ion Mz) of Eq . (7).

F r o m L e m m a 1 it a l so fol lows that fo r P2 < k / ( k + 1),(r 2 _ [t~2(k + 1 ) ] / k < 1; h o w e v e r , in the c a s e t~ 2 _> k / (k + 1), a 2 = 1, s ince Eq . (10) fo r w = 1 has the unique so lu t ion F(1) = 1, which is ob ta inable by the i t e ra t ion method . T h u s , Ua(z) = ~a+l(z) is a g e n e r a t i n g funct ion which s a t i s f i e s Eq. (5) and the boundary condi t ion (6).

Below we s tudy the s ingle ana ly t ica l so lu t ion ~l(z) of Eq. (7). The fol lowing p r o p o s i t i o n is va l i d r e l a - t ive to al l so lu t ions of this equat ion.

LEMMA 2. F o r {z[ < l , ] X l ( z ) [ < 1, [Ai(z)l > X, i = 2 . . . . . k + t . I f [ z l = i , t h e n f o r t ~ j > k / ( k + t ) , L l =

1, lXil > 1, i = 2 . . . . . k + 1; f o r l~j = k / ( k + l ) , k 1 =X 2 = 1 , IXil > 1, i = 3 . . . . , k + l ; fo r l~j < k / ( k + 1), ~i < 1, A2 = 1 , I~i] > 1, i = 3 . . . . , k + i .

T h e p r o o f is~based on_ the Rouche t h e o r e m . Le t [ z I < 1. We denote ~1 (X) = q2 ~k+t, ~o2~X )~ = ~ - l~2(z), I ~o2 (X) [IM=I -> 1 - P21 z I > q2 = I cpl (X) i iM= t. T h e r e f o r e , the funct ions ~0 2 (X) - ~1 (~) and r (X) have in the c i r c l e [M < 1 the s a m e n u m b e r of z e r o s , i .e . , up to a s ing le z e r o .

F o r z = 1 we r e a s o n ana logous ly . If l~j > k / ( k + 1), then we take the c i r c l e ! X I < 1 + e, w h e r e the sma l l e Ca > 0) is then c h o s e n in an a p p r o p r i a t e m a n n e r . Ifl~j < k / ( k + 1), then the ana lys i s is c a r r i e d out in the c i r c l e tX{ < 1 - e . F i n a l l y , for l3j = k / ( k + l ) in the c i r c l e ~Xl < 1 - e we have to c o n s i d e r the funct ions ~2(X)=

k and %(M = X X ' . g ~ l

T h e w a n d e r i n g of the f ini te a u t o m a t o n Bzn a m o n g the s t a t es of the r e g i o n L 2 is d e s c r i b e d by the s a m e d i f fe rence equat ion , and a l so by the fo l lowing cond i t ions :

u~ .0 (p~)= l , u(~) t -~=O, a = O , 1, n - - l ; a , 0 k P ' $ l " " "~

U in) I n . ] - - tn ,,_~ . . . . . . - u , , ~ _ ~ (p~), i = 1~ � 9 k , m = 1, 2 , . . . .

H e n c e , it fol lows that the g e n e r a t i n g funct ion Ua(n)(z) is the so lu t ion of the p r o b l e m

( n ) _ _ ~ ( n ) U,~ (z) -- p~zUa_l (z) + q~zU~_,, (z), (11)

V ~ (z) = 1, Utn) U ('~ n-~ ( z )= ~+i_~(z), i = 1 . . . . ,k. (12)

643

We seek the solution of the di f ference equation 0-1) in the form ,~a+t (z). Relat ive to ;~(z) we obtain Eq. (7). F o r I zl < 1 all roots of this equation a r e s imple. Indeed, jointly solving the equations P(A) -c~2zi~k+l (z) + l~2z - Mz) = 0 and P,(A) - (k + 1)4zz)~k(z) - 1 = 0, we a r r i v e at the conclusion that they have a genera l solution only for z t f o r which

za+~= (k'-~-]-) ( i k + l ) .

But on the right we have a quantity which is g r e a t e r than 1 for t~2 ~ k / ( k + 1) and equal to 1 for 1~2 = k / ( k + 1).

k + l

Thus , the genera l solution of Eq. (11) gives the express ion U~")(z) = ~,~ A~(z)~+~(z). We satisfy the condi- t=1

t ions 0-2). Then re la t ive to Ai(z ) we obtain a sys tem of a lgebra ic equations having for Izl < 1 a nonzero de t e r - minant of the sys tem

]~ A,(z) = i, ~., A, (z)ZT(z)(~(z)-- 1)= 0, l = 1 , . . . , k t = l i=1

We find A i(z) and subst i tute it into the express ion for u(n)(z). We obtain

U~ "~ (z) A,~+~_,, (z) = a . . _ . ( z ) '

where

(13)

Ll (z) - - 1 . . . . . . . . ~k+l(Z)--I

A, (z) = ~(z) - - 1 . . . . . . . . ~ + l ( z ) - I

F o r z = l , i~ 2 ~ k / ( k + l ) one of the solutions of Eq. (7) equals 1 ; t h e r e f o r e , U ( n ) ( 1 ) = l f o r a n y a ~ 0 . I f z = l , t~ 2 = k / ( k + 1), then Eq. (7) has a mult iple root a n d w e cannot use the express ion 0-3) immediate ly . But by s imple arguments it is poss ib le to es tabl ish that in this case Ua(n)0- ) = 1. Thus , the probabil i ty of a change of act ion by a finite automaton B2n is 1 for any initial s tate.

We shall now prove that in accordance with our definition the automaton B is the l imit of the sequence of automata {B2n~n= 1. We go in Eq. (13) to the l imit for n ~ o o , assuming that Izt < 1. Then, using Lemma 2, we obta in

lira U(~ n) (z) = ~.~+~ (z) = U~ (z), I zl < 1.

On the basis of the continuity theorem this s ignif ies , as was a l ready noted, that lira u( ~)~ = u~,~ for any m.

w 6. C a l c u l a t i o n o f T a , j , ~.(n) T h e A s y m p t o t i c B e h a v i o r o f ~-(n) �9 . a . , j " a , ~

F o r the analysis of the behavior of automata it is necessa ry to calculate the mathemat ica l expectat ion of the t ime the automaton spends in the region Lj up to the change of action. The quantity ra, j is given by the ex -

p r e s s i o n %.i = ~ mua.m(p~) _--U~(1). F r o m Eqs. (9) and (10) it follows that h~(z) = F(w) + (k + l )wF'(w), F'(w)(1 -

w(k + 1)qjFk(w)) = qjFk+l (w). Now tel l~j > k / ( k + 1). In this c a se F0-) = 1, F'(1) = qjSj "1. T h e r e f o r e ,

xa,i = --a67 l, ~=.2 = (a ~- 1) 8~-1 (14)

If l~j = k / ( k + 1), then -ra, j = ~; i .e . , the number of cycles o f func t ion i rgof the automaton B preceding the change of the act ion fj has an infinite MEX. In the case pj < k / ( k + 1) the infinite automaton B with a posi t ive p r o b - abili ty does not change the act ion fj a t a l l

(n) We p r o c e e d to f ind Ta, j . We shall cons ider in m o re detail the finite Markov chain O:ff~ which descr ibes the behavior of the automaton B2n in C (Pi, P2). F r o m a s ta te with the number i, with a posi t ive probabi l i ty , we can reach any other s ta te with the number j a f ter a finite number of s teps. Consequentiy, the Markov chain

644

(both f in i te and infinite) is not t r a n s f e r a b l e . A f ini te M a r k o v cb~ain ~/~ h a s no p e r i o d i c or a b s o r b i n g s t a t e s and , c o n s e q u e n t l y , c o n s i s t s of e rgod ie s t a t e s .

F o r such a e h a i n t h e r e ex i s t s [121the u n i q u e s t a t i o n a r y d i s t r i bu t i on x n = {Xn, i}, i = - n . . . . . 0 . . . . . n - 1 of the p r o b a b i l i t i e s of s t a t e s of the au toma ton . At the s a m e t i m e xn , i > 0. F r o m this and f r o m the w e l l - k n o w n

_(n) t h e o r e m s on Markov cha ins [12] it fo l lows tha t ~a,j a r e a lways f in i te quan t i t i e s . To d e t e r m i n e t h e m we f ind the

d i f f e r e n c e equa t ion fo r ~'~).=~.~ u~")~,m (pj).m. F r o m (4) we have the equa t ion (m + )an ,m+ i = mp2ua , i , rn + mqz x r n = 0

(n) ~ u(n) ~ (n) Hence us ing the fac t tha t fo r f in i te M a r k o v cha ins ~ u(~ = t we a r r i v e a t Ua+k,m + P2 a - l , m + q2Ua+k,m �9 , ~.~ ,

2 a+k,2

H e r e the bounda ry condi t ions f o r the p r o b a b i l i t i e s U(a,n) l e ad to the fo l lowing condi t ions fo r r(n)'a,2"

~1,~ = 0; ~22~,~ = ~2-r ~ = 1, 2 , . . . , k. 0 -a)

F o r the so lu t ion of the p r o b l e m 0-5), (16) we use the f ac t tha t (this c a n be p r o v e d i m m e d i a t e l y ) Eq. (15) has the

p a r t i c u l a r so lu t ion (a + 1)52-1; t h e r e f o r e , f o r I~ 2 ~ k / ( k + 1), ~")~,~ = (a + 1)57 ~ + ~ B}")~ +~. is d e t e r m i n e d

f r o m the s y s t e m of a l g e b r a i c equa t ions

i=1 g= l

F r o m the s y s t e m (17) we f ind that B} n) = D: n)x~n/A_n(1), w h e r e D~ n) is ob ta ined f r o m A-t 0.) by r e p l a c e m e n t of the i - th c o l u m n by the c o l u m n of the f r e e t e r m s of the s y s t e m 0.7). Since a m o n g X i t h e r e is a lways 1, A_n(1 ) does not depend on n, whi le a m o n g the c oeffic ients D{ n) only the coe f f i c i en t wi th the n u m b e r p o s s e s s e d by the roo t of Eq. (8) equa l to 1 depends on n. Le t 1~2 > k / ( k ~(1). Then by L e m m a 2 X l = 1, l Xil > 1, i = 2 . . . . . k + ~. T h e r e f o r e , limB~=)=0, i = 2 . . . . . k + 1., wh i l e l im& ) = 0 , s i n c e in the d e t e r m i n a n t D{ n) a l l e l e m e n t s of the

f i r s t row Lend to z e r o f o r n ~ ~r T h u s , lira ~(~.~) = (a + 1) 5~ -t , which co inc ides with (14L

Le t now/~2 < k / ( k + l ) . Then by L e m m a 2 Xl < 1, X 2 = 1 , fa i l > 1, i = 3 , . . . , k + l . T h e r e f o r e , fo r i : 3 . . . . . k + l , l imB~" '=0. We now s tudy the b e h a v i o r of the s e q u e n c e O n = B/n) .~ +* + B (n) fo r n - - ~. An

r/-)'oo

e l e m e n t a r y e a ! c u l a t i o n shows tha t | = AXl"n[ 1 - X f +* + en], w h e r e A is a p o s i t i v e c o n s t a n t , whi le lira e= = 0.

F r o m what has b e e n j u s t s a i d it fo l lows tha t r(,nl fo r n - ~ tends to ~ exponent ia l ly r e l a t i v e to n. In this c a s e , as we e s t a b l i s h e d f o r the l imi t ing au toma ton , the w a n d e r i n g o v e r the r eg ion Lj with a pos i t i ve p r o b a b i l i t y does not end a t a l l .

We f inal ly c o n s i d e r the m o s t e o m p l e x c a s e : i~ 2 = k / ( k + 1). In th is c a s e Eq. (8) has a mu l t i p l e r o o t X, = = 1 , t),il > 1 i = 3 . . . . k + l whi le Eq. (15) has the p a r t i c u l a r so lu t ion r(n) = (a + l ) 2 / k a a d t h e g e n e r a l

' ' a,2

so lu t i on of the f o r m .(~,.~.~ = (a +k 1)' FNi,,+N(n)(a4 -" 1)+2Nin'~.~+L w h e r e N~ n) a r e d e t e r m i n e d by the s y s t e m of

i ~ 3

algebraic equa t ions

k+1 k+~

= - - V - - . ~ , ~ = I , .,k. (18)

T h e d e t e r m i n a n t of the s y s t e m (18) C (n) is ob t a ined f r o m A_a(1) by r e p l a c e m e n t of the f i r s t c o l u m n with

the c o l u m n 0 , and the s e c o n d w i t h the c o l u m n 1 , N n) = 2n)~n[C n ) / c i = 3, k, w h e r e C~ n)

0 k

has a f in i t e l imi t fo r n ~ ~. T h u s , lira N~ n) = 0 , i = 3 . . . . , k. But then f r o m the f i r s t equat ion of the s y s t e m

(18) we f ind tha t lirnNl~)_0. F i n a l l y , N (n) = 2n[CJn) /C (n)] and Ca (n) has a f in i te p o s i t i v e Limit fo r n - - ~ T h u s , n->o o *

845

T a ( , n • --* ~ for n ~ ~ , f o r n . Th i s resu l t c o r r e s p o n d s to the fact that a v e r a g e wander ing t ime of the l imi t ing the automaton B ove r the reg ion Lj up to a change is infinite.

T h u s , T h e o r e m 1 is comple te ly proved.

In conclusion, we cons ider the ca se k = 1. Then for I~ 2 ~ 1/2, ~a,2(n) = (a + 1)6~ 1 + ([(~+l _ 1)6~.l~.n]/ ( 1 - k t ) ) . I f v 0= vi = 0 , a = 0 , then~-2 (n)= (1-X~'n)5~ "i. F r o m (8) f o r k = l w e f i n d h = p 2 / q 2 . Thus , T2(n)= [(q2/P2) n - 1]/(q2 - P2) coincides with the known re su l t (see, for example , [10]). Fo r P2 = 1 / 2 , v 0 = vl = 0, T(,n~ = ( 2 n - - a ) ( a + 1).

w 7. A n a l y s i s o f t h e B e h a v i o r o f t h e I n f i n i t e A u t o m a t o n

B ( k , v 0, v l ) in a S t a t i o n a r y R a n d o m M e d i u m

This ana lys i s is ba sed on the r e su l t s p r e s e n t e d in Sec. 6, L e m m a l , and the wel l -known t h e o r e m s on infinite Markov chains [12, 17]. We shal l cons ider the following case .

Str ict ly Optimal Automata B(k, v 0, vi). F o r s t r i c t opt imal i ty of an automaton it is n e c e s s a r y , by def ini- t ion, to have ~t < 1 and ~2 = 1, ~2 < +~ . This c o r r e s p o n d s to the inequality l~l < k / ( k + 1) < P2 or

Pi[l~(%~%)l~k+l %' (19)

p~ [1 - - (v 0 -~ vt)] ~ k-~ ' ] - --%" (20)

Since pl < P2, for v0+ vl -> 1 , ~ m 1~2. Thus , w e h a v e to a s s u m e that v0+ vi < 1. In this c a s e , if v 0 _> k / ( k + 1), then the inequality (19) is not s a t i s f i ed for any values of Pi. Consequent ly , for s t r i c t opth~na[ity the foUowing inequali t ies m u s t be fulfil led: v 0 + vl < 1, v 0 < k /{k + 1), Pl < Q < P2, Q = [ (k / (k + 1)) - v0] / [1 - (v 0 + vl)]. We p r o c e e d with the descr ip t ion of the p r o p e r t i e s of the Markov chain cor responding to this case . The p r o b - abil i ty of r e tu rn ing to the s ta te with the number 0 equals 1~2~1 + c~2 ak = c~ 2 + l~2al < 1. Thus , this s ta te is non- r e c u r r e n t . (In [17], p. 390, such a s t a te is cons ide red as r e c u r r e n t and is s a i d t o be t r a n s i t i v e . E v e r y w h e r e in the following we confine ou r se lves to the l ess de ta i led but m o r e widely used c l a s s i f i ca t ion p r e s e n t e d in [12].) But s ince the Markov chain descr ib ing the behav io r of the au tomaton is i r reduc ib le , all i ts s ta tes a re nonrecu r ren t and t h e r e is no s ta t ionary probabi l i ty d is t r ibut ion [12].

On the bas i s of Eq. (1) the MEX of the penalty of the s t r i c t ly opt imal au tomaton is M(B, C) = Pi < (Pl + P2) /2 ; i .e . , the s t r i c t ly opt imal au tomaton p o s s e s s e s expedient behavior .

The r e s t of the poss ib i l i t i e s a r e ana lyzed analogously [6].

The authors e xp re s s the i r gra t i tude to V. S. Koro[yuk for his ex t r eme ly useful and fr iendly d iscuss ion of this paper .

LITERATURE CITED

I. M.L. Tset|in, Investigations in the Theory of Automata and Modeling of Biological Systems [in Russian], Nauka, Moscow (1969).

2. V.I. Varshavskii, Collective Behavior of Automata [in Russian], Nauka, Moscow (1973). 3. V.G. Sragovieh, The Theory of Adaptive Systems [in Russian], Nauka, Moscow (1976). 4. N.P. Kandeiaki and G. N. Tsertsvadze, "On the behavior of certain classes of stochastic automata in

random media," Avtom. Teiemekh., No. 6 (1966). 5. V. Ya. Vaiakh, "On the behavior of an automaton with selective tactics in stationary random media,"

Kibernetika, No. 4 (1968). 6. S.D. E~idel'man andA. I. I~zrokhi, ~Adaptive properties of discrete automata functioning in stationary

random media,', Avtom. VychisI. Tekh., No. 5 (1976). 7. A.A. Borovkov, Stochastic Processes irltheTheory ofQueuingSystems [inRussian], Nauk~, Moscow (1972). 8. V.S. Korolyuk, Boundary-Value Problems for Complex Poisson Processes [in Russian], Naukova Dumka,

Kiev (1975). 9. L. T a k a c s , Combina to r i a l Methods in the Theory of Stochast ic P r o c e s s e s , Wi l ey - In t e r s c i ence (1967).

10. ~ . M. S i l ' v e s t r o v a , Markov Fini te Automata of the L inear Type and Adaptive Queuing Sys tems [in R u s - s ian] , P r e p r i n t of Ins t i tu te of Cybe rne t i c s , Academy of Sciences of the Ukrainian SSR, Kiev (1974).

11. V . A . Andryushchenko, E. N. VaviLov, and L. P. Lobanov, ,,Synthesis of au tomata asympto t ica l ly op t i - m a l in s ta t ionary random medka,,, Kiberne t ika , No. 1 (1972).

646

12. W. F e l l e r , A n Introduct ion to Probab i l i ty Theory and Its Appl ica t ions , Vo[. 1, 3 rd ed. , W i i e y q a t e r - sc ience (1968).

13. V . A . Voikonski i , "Asymptot ic p r o p e r t i e s of the behav io r of the s i m p l e s t au tomata in a g a m e , '~ Prob[ . P e r e d a c h i Inf . , 1., No. 2 (1965).

14. N . P . Kandelaki and G. N. T s e r t s v a d z e , "On the r a t e of convergenc e of asympto t ica l ty opt imal au to - maton sequences , " in: Automata , Hybr id and Control l ing Machines [in Russ ian] , Nauka, Moscow (1972).

15. G . N . T s e r t s v a d z e , "On the a sympto t i c p r o p e r t i e s of opt imal au tomata in statio~nary random m e d i a , , Avtom. Telemek~h., No. 8 (1968).

16. E . N . ~ . M) Vavilov, S. D. l~idel 'man, and A. L l~zrokhi, "On the pecu l ia r i t i es of the asympto t ic behav io r of s tochas t ic au toma ta , " DokL Akad. Nauk UkrSSR, No. 8 0-977).

17. P. L. Henneken and A. T o r t r a , P robab i l i ty Theory and Some of Its Applicat ions ~ u s s i a n t rans la t ion] , Nauka, Moscow (1974).

PROPERTIES OF A CLASS OF a-GROUPS

L. B. Smikun UDC 51:62-50

The p rob l em of the equivalence of au tomata re la t ive to semigroups has been cons ide red in [1, 2] in con- nection with the theory of d i sc re te c o n v e r t e r s [3, 4]. In [5] a t heo rem on the reducibi l i ty of the equivalence p r o b l e m for s emig roups with reduct ion to the p rob l em of equivalence re la t ive to the max imum subgroups is p roved . A c l a s s of c~-groups (groups with ra t io a of the quas io rde r s ) is defined there and a theorem is p ro v ed on the solvabi l i ty of the equivalence p rob l em re la t ive to a - g r o u p s with a solvable p rob lem of the equality of words .

The c l a s s of a--groups is inves t iga ted in th is paper . Some fea tu res of giving the quas io rde r ra t io a in the group a r e clarif ied�9 Group - theo re t i c a l p r o p e r t i e s of the c l a s s of a - g r o u p s a r e cons idered . Special a t - tention is pa id to a p roof of the c [osedness of a c l a s s of re la t ive ly different g roup- theore t i ca l operat ions . Namely , these p r o p e r t i e s a f ford the poss ib i l i ty of cons t ruc t ing s i m p l e r a - g r o u p s (which a r e f ree and finite groups) f rom m o r e complex ones. On the o ther hand, r e su l t s of a "negat ive" nature a f fo rd the poss ib i l i ty of finding groups on which giving the q u a s i o r d e r ra t io a is imposs ib le in pr inc ip le .

A c lass of f i -groups with a s o l v g o l e p r o b l e m o f e q u i v a l e n c e of au tomata [61, which is at leas t not a l ready the c l a s s of P - g r o u p s , is known ~c this t ime . However , it is comple te ly p robab le that some a s se r t i ons can be p r o v e d only fo r the s i m p l e s t groups with a so lvable equivalence p r o b l e m , which a r e unquestionably a - g r o u p s ~ and not for al l g roups of this c l a s s .

The study of the p r o p e r t i e s of the c l a s s of a - g r o u p s is due to this c i r c u m s t a n c e .

Let us p r e s e n t the fundamental r e su l t s f rom [5J�9

We call a group G with an ex t r ac t ed finite se t Y genera t ing G a Y-group. We cal l a sequence of e iements gl . . . . . gn, �9 - in > 1) a Y - t r a j e c t o r y if for any i, --1 �9 - gi gi+l E Y. Let us cons ider the Y-group G on which the q u a s i o r d e r ra t io a is given (reflexive and t rans i t ive) . Now gl -< g2(a) means that the e lements gl and g2 a r e in a r a t io a . If gi -<g2(~), then g2 -> gl (a). The ra t io gl -< g2 Ca) & g2 -< gl (a) is an equivalence ra t io which we ca l l a - e q u i v a l e n c e and denote by gl ~ g2 (a) :

g, < g 2 (~) "~g, <: gz(~) ag~ 7 '~ g,. (~),

6 (g, ~) = {,~ E 6; g < ,~ (~)}.

F o r any se t H ~ G, G (H, a ) = { g E G; Yh E H, h_< g (a)}. We ca l l the e iements gt and g2 s im i l a r , g~ - g2 (~), if for any h ~ G the following conditions a r e sa t is f ied:

1) glh ~ gl (a) r g2 h ~ g2(a);

2) glh ~ G(gl, a) r g2h 6G(g2, a).

T r a n s l a t e d f r o m Kiberne t ika , No. 5, pp. 12-19, S e p t e m b e r - O c t o b e r , 1977. Original a r t i c l e submi t ted F e b r u a r y 27, 1976.

0011-4235/77/130520647807.50 �9 1978 Plenttrn Publishing Corpora t ion 647