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11. Linear automata
11. Linear automata
In this chapter we present results on finite and infinite linear automata. The main goal emphasis will be placed on problems of finite automata realization. A theorem on the realizability of any finite automaton with the help of a linear automaton defined over a field of characteristic 0 will be presented. A sufficient condition for the realizability of a finite automaton with the help of a linear automaton defined over a field of characteristic p, where p is a prime, will be formulated. It will be also shown that there exist finite automata which cannot be realized with the help of a linear automaton defined over any field of characteristic p > 0. The particular case of a linear automaton, i.e. a shift register defined over a field of real numbers which generates cyclic sequences, will be also discussed. Finite automata reproducing finite subsequences generated by these registers will be investigated. The presented results deal with decomposition and cyclic representation. Linear automata have wide applications in digital technique. Linear automata are used for generation of linear codes, detection and correction of errors, construction of testing sequences, and generation of pseudo-random sequences of numbers. They are also used in experiments that require Monte Car10 methods, in the protection of data stored in computer systems, and radiolocation.
I1 . I . Introducing remarks In this subchapter an automaton will be meant as a 5-tuple (S, C, 3, a, y), where S, C and R are nonempty sets of states, input alphabet and output alphabet, respectively, 3: S x C + S is a transition function, and y is an output function. In the case of Mealy automaton the output function is y: S x C + R and in the case of Moore automaton y: S + R . As in Chapter 2 by p we denote the transition function of a Mealy automaton, and by h the transition function of a Moore automaton. The remaining definitions used in this subchapter are the same as in the previous chapters. The definition of a linear automaton introduced here is more general than the definition used previously, namely we do not assume the finiteness of the sets S, C and R. If S, C and R are all finite, then the automaton is finite.
348
11. Linear automata
Let F be a field and n E No.By F , we denote the vector space of column vectors of dimension n over F. A linear automaton is an automaton (S, C, R, a, p) with the following properties: (a) there exists a field F and integers k, m , r E No such that S = F,, C = F,, n= F,, (b) there exist: a k x k matrix A, a k x m matrix B, an r x k matrix C and an r x m matrix D, all over F, and such that for each (s, o)E S x C a(s,
0 )= As
+ Bo
P(s. 6)= Cs + D o
(11.1) (11.2)
Taking into account the fact that the automaton acts in discrete time we will
also use the notation = As,+ BG,
+ Do,
O, = Cs,
(11.3) (11.4)
where S,E S , 6,E C and O, E R for f E No. For the reason that field F and matrices A, B, C and D entirely characterize the linear automaton, instead of the standard notation of the automaton as a 5-tuple (S, C, Q, p) we shall also denote automaton by an 8-tuple (F, A, B, C , D, k, m, r ) or by a 5-tuple (F, A, B, C, D) if no confusion will arise.
a,
Fig.11 . I . Diagram of a linear automaton to Example 11.I Example 11.1.
(a,
Let A, B, C, D) be a linear automaton defined over the field Q2 of integers modulo 2 such that
:],[ :1,
11 . I . Introducing remarks A=
[
B=
349 C = [ O , 1 1 , D=[lI.
The diagram of the automaton is presented on Figure 11.1. The matrix equation (11.3) can be written as k linear equations, and Equation (11.4) as r linear equations. Thus each linear automaton can be presented in the form of a net consisting of adders, multipliers and delay elements with delay elements implementing the memory of an automaton. Symbolic representations of these elements are illustrated on Figure 11.2. The automaton from Example 11.1 is presented in the form of a net on Figure 11.3. Equations (11.3) and (11.4)express the change of the state of an automaton at the moment t + 1 of time and the output signal at moment t as functions of state and input signal at moment t . The following theorem determines the state to be reached by the automaton and the output signal to be generated if at the moment 0 the automaton starts from the state so and the input sequence is finite.
G t
- 7*=Q%
Gt
-
?*=5
Fig.ll.2. Components used to the construction of a linear automaton as a net: (a) adakr, (b) multiplier and its simplified version if a = 1, (c) delay
350
11. Linear automata
w Fig.ll.3. Automaton from Example 11.1 presented in the form of a net Theorem 11.1. Let a linear automaton (F, A, B, C, D) be given. (a) For each t~ M t
s,= A's,
+ C
A'-j-lBoj
j=O
(b)
For each t E No t
u,=CA'S,+
(11.5)
C El-,oj, j=O
where D if t = O CAI-' if t > O Proof. (a) (b)
By induction on t ; omitted. Using (a) we have
o,=CS, + DO,= C(A's,
= CA's,
+
t- 1
+ C A'-J-~BO~) + DO, j=O
I- 1
C CA'-j-'Boj
j=O
+ Do,.
Let A , = ( F , A , , B , , C,, D1) and A,= (F, A,, B,, C,, DJ be linear automata. We say that A , is similar to A, if there exists a nonsingular matrix P such that
I
11.1. Introducing remarks
A, = PA,P-' B,= PB,
c,=C , P ' D2 = D1
35 1
(11.6)
Theorem 11.2. Let linear automata A , = ( F , A,, B,, C,, D1) and A , = ( F , A,, B,, C,, D2) be given. If A, is similar to A,, then A, and A, are isomorphic.
Proof. Let P be a nonsingular matrix such that Equations (11.6) hold. Let 9:Fk+ Fk be such a function that cp(s) = Ps for each s E Fk The inverse P1 exists, hence cp is the bijection. Thus
and furthermore
Q.E.D.
The converse of Theorem 11.2 is not true because there exist isomorphic automata which are not similar. As an illustration let us take the following example. Example 11.2.
If A,=((&, [3], 0, 0, 0) and A,=(Q7, [ 5 ] , 0, 0, 0), then A, and A, are isomorphic but not similar because the matrix [3] is not similar to [ 5 ] over the field Q7 of integers modulo 7. Let a linear automaton A = (F, A, B, C, D) be given. A polynomial p,,(x) = det(xI - A) is said to be a characteristic polynomial of A. On the other hand, for each polynomial k p(x) = 2: a$, i=O
where ak= 1, we can define a companion matrix A such that det(x1- A) = p ( x ) in the following way
11. Linear automata
352
-
0
1
0
...
0
0
0
0
1
... 0
0
... ... ... ... ...
A=
0
0
...
0
-ao - a l - a 2
0
... - Uk-1
... 1 - a,
1I .2. Linear realizations of finite automata
In this chapter we use the most general definition of the realization of an automaton. Let us recall that an automaton A’ = (S’, c’, 8 , Q’, p’) is said to be a realization of an automaton A = (S, C, 3, Q, p) if there exists a triple (a,I, of functions such that (a) a: S -+P(S’), (b) 1: C+Z’, (c) [: R’--+Q, (11.7) c cr(a(s, o))l, (d) (W, E S x WUm, ~(4) (11.8) (el (V(S9 0)E S x C)[S(P’(a(s)), I(@)) = P
c)
a
Theorem 11.3. For any finite automaton A = (S, C, 3, R, p) and any field F of characteristic 0 there exists a linear automaton A’ = ( F , A, B, C, D, 1, 1, 1) such that A’ is a realization of A. For given A , linear automaton A’ can be obtained effectively
.
a’
Proof. Assume that S ’ , and p’ are the set of states, the transition function and the output function of A’, respectively. If K is a subfield of a field F of characteristic 0, then there exists the isomorphism cp: K -+ Q such that 4 = @a) for a certain a E K, 4 E Q and Q being the field of rational numbers. Symbol 4 will denote an element a € K such that
4 = Ha). Assume that S = {s,, s2, ..., s,) and C = (ol,02,..., 0”).Let n = 1 +rnax(u, v). For each i~ No we define three functions, namely fi: &-+No and h: N,,++Mo, in the following way: (a) J ( O )= i(modn), g,(O) = t i / nl, (b) f,(x + 1) = gi(x)(modn) and gi(x + 1) = Lgi(x)/ nl for each x 2 0, (c) h(i) = p L f ( x + 1) = 01. Note that J ( x ) is the x-th right digit when the number i is written out in n-ary notation, and h(i)+ 1 is the number of digits in the n-ary expansion of i.
11.2. Linear realizations of finite automata
353
The functionf, and h have following properties: (i) for each k and 0 Ii < n h(nk + i ) = h(k) + 1,
(11.9)
for each k, x and 0 I i < n
(ii)
fd+i(O)
(11.10) (11.1 1) ( 1 1.12)
=i
fd+i(x + 1) =fAx) if k # 0 then f,+i(h(nx + i)) =f,(h(x)).
Define a coding triple (a,L, [) as follows: (i) for any oic Z ~ ( 0=~i ,)
(ii)
for any S E S
a(si)= ( x b E M. 1 If,(h(x)) I u, 1 Ifx@) 5 v for 0 I k < h(x))
(iii) if x is such an element of F that 1 I f , ( h ( x ) ) I u and 1 I f x ( k ) < v for 0Ik < h(x), then
Otherwise [ ( x ) = c , where c is a fixed element of R. Assume that A = C = [n] and B = D = [l]. Now we prove that the linear automaton (F, [ n ] , [l],[nl, [l]) is a realization of A = (S, C. 3. R, p) such that the coding triple (a,L, [) is defined as above. At first we show that for 1 I i l u, i E a(si). This means that a transforms the set S into the set of nonempty subsets of S ’ . We have that i E No
If(0) = i and g,(O) = 01
3 &(I) = 0
h(i) = pxK.(x + 1) = 01 = 0
* f,(h(i))= i,
i.e. 1 If(h(i)) lu. Hence, no number m E consequently
M
exists such that O I m < h(i), and
11. Linear automata
354 Assume that s E S, oi E C and x E a(s), which implies that
(11.13) Thus
a’@,i(oi))= Ax+ Bi(oi)= nr + i(oi)= nx + i. Let y = nr+ i. To prove that (11.7) holds it suffices to show that a;).Using (11.9) - (11.12) we get
YE
a@($,
and moreover
by (11.13). Now we have to prove that (11.8) holds. Let y = n x +i. 1 Sf,(h(y)) Iu, 1 If,(k) I v for 0 I k < h(y), we have
Since y > O and
Therefore A’ is a realization of a finite automaton A. Note that using the construction presented in the proof, A’ can be obtained effectively. Q.E.D. Next, we will examine the question whether the characteristic 0 of the field is a necessary condition for a linear realizability of a finite automaton. The answer is that not always it depends on the field. In the sequel let p be a prime. The additive group K generated by the unity element of any field of characteristic p is a subfield isomorphic to the field ($ of integers modulo p. If cp(a)=q for some a € K and q E ($, then for convenience, the element 4 will be identified with a, and instead of a we shall write q.
355
11.2. Linear realizations of finite automata
Theorem 11.4. Let A = (S, Z, a, R, p) be a finite automaton and F be a field of characteristic p which contains a transcendental element z over Q. Then there exists a linear automaton A' = (F, A, B, C, D) such that A' is a realization of an automaton A. Furthermore, for a given automaton A, linear automaton A' can be obtained effectively if z is known.
Proof. An element z is transcendental over $.so
where bi, c j € (0. 1, ..., p - 1 ) for O l i l q and O l j l r if and only if q = r and bi = ci for each i such that 0 I i 5 q. Let S = (sl, s2, ..., s), and C = (q,02, ..., 0").Assume also that n = 1 + max(u, v). We define function h: ( 1 , 2, ..., n} + F in such a way that h(i) = zi.
Now the coding triple (a,L, [) can be defined as follows: (i) for any q~ C L(Oi) = h(i). (ii) for any sic S a(si)= ( x I ( 3 k ~No)(3co,cl, ..., C k E F) [c,E
(iii)
( 1 , 2,
..., u ) . C
~ E (1,
2,
..., v)
for O s i e r , and
for any X E F, where F is any field, if x is such an element that
[C,E
<(XI
( 1 , 2, ..., u ) , C = P(s,,
~ E (1,
2,
..., v )
for O l i < k ] then
ack-,ack-2...aco) else b E R, where b is fixed.
Let A = C = [ P ] and B = D = [ l ] . The proof that automaton (F, [z"], 111, [PI, [l]) realizes A is omitted because it is analogous to the corresponding part of the proof of Theorem 11.3. Q.E.D.
356
1 1 . Linear automata
Theorem 11.5. Let the automaton A = (S, C, a, R, j3) be defined by means of Table 11.1. Furthermore, let F be a field of characteristic p in which every element is algebraic over Q. No linear automaton A’ over F is a realization of A.
Proof. By contradiction.
Assume that A’ = (F, A , B, C , D) is a linear automaton which realizes A. Let be a coding triple. We shall show that the sets a(s& and a(sl) are disjoint. Suppose x E a@,) (7 a ( s l ) .
(a, L,
c)
Table 11.1. Automaton for which there does not exist any linear realization over any field of characteristic p
By (11.8) we have that
and thus
Using (11.7), for each function of A’):
XE
A similar reasoning leads to
a(s,) and y E a(s,) we have that
(a’ is the transition
11.2. Linear realizations of finite automata
357
By repeated application of these statements n times, for each X E a(so)we have
a(so) if zi = Bt(1) for 0 Ii < n
n-1
Anx+ C Aizi E i=O
a(sJ if zi E (Bt(O), Bt(1)) for 0 Ii < n,
(11.15)
and zi = Bt(0) for at least one i.
Let A = [ay] and K be the smallest subfield of F which contains the extension of Q, by elements all, a12,..., a,,,,. Since all elements in F are algebraic over Q,, K is a finite extension of $. Therefore K is a finite field. Each element in any matrix A' ( i 2 0 ) is an element of K. Since K is finite, there exist positive or more generally: Ar=Ar+U. Then we have integers r, S E M such that A'=
Let X E a(so),and furthermore let zi =
Bi(0) if i = r + j s for O l j l p - 1 BL(1) otherwise.
Then
=A'+PSx+
r+ps-l
C
i=O P- 1
P-1 A i B ~ ( l ) + C Ar+LB~(0)
=Ar+Px+ C AiBL(l) i=O i # r +j t
i=O
i#r+jt
358
11. Linear automata
=A'+PFX+
r+ps-1
C
A'Bt(l)+
i=O i#r+jt
= A'+pSx +
P-1 2 Ar+&Bt(l) i=O
r +ps- 1 C A'Bt(1) E a(s,). i=O
However the application of (11.15) yields r +ps- 1 A'+~x+ C i=O
a(sl).
Thus a(s&na(s,)f 0, is a contradiction with (1 1.14).
-
0 1 0 ... 0
0
... 0
0
0 0 1 A=
... ... ... ... ...
0 0 0
... 0
... 1
4 a l a2 ... ak-2 ak-l -
-
0
Q.E.D.
-
0 ,B=
.
0
, C = [ l , 0, 0, ..., 01, D=O,
- 1-
then A will be called a linear shift register, or simply a shift register. Figure 11.4 presents the net representation of the shift register. An infinite sequence (a,),t=0, 1, 2, ..., such that its general term is defined by means of (11.5) is called a computation of a shift register. A computation is cyclic if there exists a positive integer T such that = w, for any ? E No.Each T fulfilling this condition is called a cycle length of (a,).If T is the smallest cycle length of the computation, then we say that T is the fundamental cycle length. In this chapter we assume that shift registers are defined over the field of real numbers, i.e. F = El. Furthermore, we assume that the characteristic polynomial of shift register is a function of a complex variable z . By so we denote the initial state of shift register.
359
11.3. Linear sh@ registers
Fig.1 I .4. Linear shvt register
Theorem 11.6. Let @, A, B, C, D, k, 1, 1) be a shift register, and p ( z ) be its characteristic polynomial. Furthermore, let b E R be a fixed element of the input alphabet. If: (a) all roots of p ( z ) are single and lie on the unit circle, i.e. each root zj has the form zj=exp(iqj), where 01qjc27c,j=0, 1, .... k - 1 and i = d - 1 and (b) there exist a positive integer T and positive integers to, t l , ..., tk-l such that qj=27c$/Tand for e a c h j e (0, 1, ..., k - I ) g.c.d.($, T ) = l and
(c)
XO XO Sof
*
, where xo = b (1 -
k- 1
C aj)-l,
j=O
then the computation of the shift register is cyclic and its fundamental cycle length equals T, where T > k and T f ( t j - + ) I q for each j, j * e (0,1, ..., k-11, j f f and q~ Z\(O). Proof. The general term of (or)given in the form of (11.5) is not convenient in our considerations. We make the use of the fact that any computation of shift register can be expressed by a recurrence equation k-1
o,+t- c ujor+j= or j=0
If
(3,
= b, the solution of the above equation is given by
where q, is the solution of the homogeneous equation
11. Linear automata
360
k- 1 @,+&-
c ajw,+j= 0,
j=O
Because all roots of p ( z ) are single and different from 1, then a = 0, k- 1 g(l)=p(l)= 1 - C uj#O, and j=O
k- 1 7\s, = C c,exp(icpjt), j=O
(11.16)
where co, cl, ..., ck-l are constants depending of the initial state of register. Using the assumption (b), it follows from (11.16)that W,+F
k- 1
k-1 C c,exp(icpi(t + r ) ) + b / (1- C a,)-*
j=O k- 1
j=O
k- 1
= C c,exp(i(cpjt + 2ntj)) + b / (1- C aj) = a,. j=O j=O
T is the fundamental cycle length by virtue of (b), because
w , , ~= w, for each
f E
No
implies qjT=2n9 for each j~ (0, 1, ..., k- 1).
To prove that T # ( f j - f j , ) / qwe write p ( z ) in the form
k-1
By virtue of the assumption (a) all roots of the characteristic polynomial are single, hence
exp(i2mjI r ) f exp(i2ntj, I r )
I I .3. Linear shut registers
361
for different j , j * E (0, 1, ..., k - 1 ) , thus tj f ti.. Therefore T # (5 - ti.) I q for each q E Z\(O). Because for each j E (0, 1, ..., k - 1 ) ti and T are relatively prime, 0 < f j / T < 1 and t j # t j . for differentj, j * E (0, 1, ..., k - l ) , then T > k . Finally, note that the assumption (c) means that the constant computation (x,,) does not belong to the set of computations of the shift register. Q.E.D. Let A = @3, A, B, C, D, k, 1, 1) be a shift register fulfilling the assumptions of Theorem 11.6, and let so be its initial state. In the sequel A is denoted by the 6-tuple (S, C. a, R, h, so>. For A we construct a subautomaton A , T(@ such that its state set S, consists of states reachable from so applying words x , XCT, xo2, ..., xoT-1, where x is any word of the input alphabet, which length is not greater then T and Q is a fixed input symbol. The transition function of 4, is the transition function of A restricted to the set S,xZ, and the output function does not depend on input signal. Formally, 4, T(a)=(S,, C,, a,, Q ,, A,, so>, where Z, is a finite subset of M, S,= (a(so, XCT")IXE C,*, I x l I T , Q € C,, n = 0 , 1, ..., 1 - l ) , a,: S,xZ, +S, is a transition function such that (s, q ) Dom(a,)d,(s, ~ q)=a(s, q), A&) = Cs for each s E S,, R, = 3.,(S,J. The automaton A , .T(@ is incomplete. We define a cover xu= (Bo, B , , ..., B,- 1) on its state set in such a way that Bj for j = 1, 2, ..., T - 1 (1) so€ B, and E C)wS E Bj"S' q) 6 Dam@,)* a&, q) E B(j + I k a r l . (2)
w7)
a,€
The cover defined above is not always a partition. In the sequel we shall investigate only the case when X, is the partition of S, Thus, if A , T(o)= (S, C,, a, R,, ha, so), then we can define a total function a': S,xC,+S, such that for each 7 ) E C and S E Bj, where Bj is the block of L, (1) JYs. 7 ) ) = + 1)mOa-p (2) ay
qj
Let 4, ,(@= (S, C,, a',, Q,, h, so>. It is evident that A , T(O) is not determined uniquely. For any fixed 7 ) & ~ the automaton d , generates a cyclic sequence of the fundamental cycle length T and furthermore, it generates all finite subsequences generated by A when A is in the state so and the input sequences are taken from the set (xd Ix E C' and Ixl I Tand CT E C and j E No). For 4, let us consider the automaton &, ,@)=(S, C,, a'J without outputs. A cycle-type partition of the state set S of an automaton A = (S, X, a) is an input-independent partition R such that A / R is strongly connected.
1 1 . Linear automata
362 Theorem 11.7. For T.
there exists a periodic representation with the period
Proof. The partition X, is a nontrivial partition of S, Each block of this partition has the same number T1card(S,) of elements. Let for a fixed q E C,, where v = T-”card(Sd, be defined as follows: a?,, = {Co, C , , ..., C, (*) for a fixed f o E S, Co= (*)
if
(a’(fo,7y) l j = 1, 2, ..., T ) , q-1
SE
S\ u Ci.then for each j=O
c,= (a’&,
7 y ) l j = 1, 2,
q E (1, 2,
..., v )
..., TI.
Each block of a?,, contains T elements and x,-a?,,=O. Hence, by virtue of Gnymala-Busse [6] for AT, there exists a nontrivial periodic representation V with the period T such that V is reducible to a strictly periodic automaton with period 1. Q.E.D.
fl)
The next theorem deals with the serial decomposition of
4,T(@.
Theorem 11.8. There exists a state behavior realization of 4, by a serial connection A, @A2 of an autonomous automaton A, and some automaton A,.
fl)
Proof. For any automaton B there exists a state behavior realization by a serial connection A , @ A , , where A, is an autonomous automaton if and only if there exists a nonmvial periodic representation for B, i.e. a periodic representation with the period T > 1 (cf. Grzymala-Busse [6]).
Q.E.D.
The above considerations are illustrated by example 11.3.
Example 113. Let A = (JB, A, B, C, 0, 2, 1, 1) be a shift register such that A=[
-: ]
and so=
[ ]
Register A generates cyclic computations of the fundamental cycle length T = 4 . Let ‘c = 2 and (3 = 0. For C = (0, 1), the diagram of A,, 4(o) is shown on Figure 11.5. For A2, 4(o) we compute the cover xo= (Bo, B,, B,, B 3 ) , where
1 1 3 . Linear shiji registers
363
0
Fig.115. Diagram of A,, 4(0) to Example 11.3
Thus ~0 is the partition of the state set of A2, 4(0). We construct A,, 4(o) with diagram presented on Figure 11.6. Now we calculate the partition "el = {Co, C , , C,, C3) of the state set of A2,J0) which gives
1 1 . Linear automata
364
Fig.ll.6.Diagram of &, 4(0) to Example 11.3 Let z= (oo,or)be an alphabet, and let cl: be coding function such that
xo+r, c2:
-+r, c3: X 0 + r
a1
I I .3. Linear shvt registers
365
Fig.Il.7. Autonomous automaton A,
Fig.1 I .8.Serial &composition of
A,,4(o)
Table 11.2. Automaton A, to Example 11.3
366
11. Linear awomata
Then the state behavior of A,, 4(o) can be realized by the serial connection of the autonomous automaton A, given by means of the diagram shown on Figure 11.7 and the automaton A, = (S,, q,dJ defined with the help of Table 11.2. In Table , is not explicitly presented 11.2 the state set S,= (ooa0.Q ~ Q , , Q , Q ~ , Q ~ Q , ) and in a current moment of time. A picture of the connection is shown on Figure 11.8. Exercises
11.1. Let A = (S, C, 3, R, A) be a Moore automaton. If there exists n E M such that for each x E CnC' and for each sl, s, E S the condition d(s,, x ) = &s,, x ) holds, then such an automaton is called feedback-free. Show that (F, A, B, C,D) is a feedback-free automaton if and only if A" is nilpotent, i.e. A" = 0 and (n=1 v ( n 2 2 A A"-'#O)). 11.2. Prove that if A, and A, are similar linear automata, and pAi(x) is the characteristic polynomial of Ai (i = 1, 2), then pA1(x)= pA2(x). 11.3. Let A = (S, C, a) be an automaton such that for a fixed n E E.9 the state set S = Sn)= ( Q ~ Q , . . . Q ~loi E C) and for each s E S, O E C we have that a($, Q) = dW1)o,where if s = op,. ..on,then dwl)= Q ~ Q , ..q,, . (a) Show that for A there exists a shift register which is a state behavior realization of A. (b) Prove that the group of automorphisms of A is trivial, i.e. card(G(A))= 1. 11.4. Let 4, gi and h be the functions defined as in the proof of Theorem 11.3. Show that they are recursive. 11.5. Let A = (S, C, 3) be a strongly connected permutation automaton such that I is its characteristic group, and H is a subgroup of i. For A there exists a linear state behavior realization over Q if and only if (i) there exists a commutative normal subgroup N of such that for each [XI E N the condition [x]p= [el holds, (ii) C c N [ x ] for a certain Ex] E i, (iii) H n N = [el (Hartmanis, Walter [I]). Show that for the automaton defined by means of Table 11.3 there exists a linear state behavior realization.
Exercises
367
Table 11 3. Automaton to Exercise I I5
a Table 11.4. Automaton to Exercise 11.6
11.6. Let an automaton A = ( S , C, a) be given, and let there exist injections a: S + F , and I: C j F , , , , where F =Q, such that a(d(s. a))=Aa(s)+ Bt(s) + a . Then we say that for A there exists an extended linear realization. Let us suppose that B is an automaton without outputs. If V,* is a fmed analog of an extension V, associated with isomorphisms and V,* is strongly connected, then for V,* there exists an extended linear realization over Q if and only if for B there exists a linear state behavior realization over (Q (Mikolajczak [4]). Prove that if B is an automaton defined by means of Table 11.4, then there exist no extended linear state behavior realization for the fixed analog V,* of any extension. 11.7. If for an automaton A = ( S , C, a) such that a(s,, 6)= a(s2, 6) for some 6 E C = (0, 1 ) and sl, s2 E S there exists a linear state behavior realization over Q with the identity function coding inputs, then a(s,, a)=a(s,, 6) for each o E C. Furthermore, if for some S, E S a(s,. 0) = a(s. 0) = a(s, 1) for all S E S (Hartmanis, Davis [ll). (a) Prove that for the automaton A presented in Table 11.5 there exists a linear state behavior realization over Q. (b) Prove that there exists a quotient automaton A l x , where x is an SP partition, for which there does not exist a linear state behavior realization.
11. Linear automata
368 Table 11 5.Automaton to exercise 11.7
s -
a 0
1
S1
s5
s4
s,
s3
s7
%
'8
'2
'6
%
s3
s,
s1
'6
SS
s4
ss
11.8. A linear automaton A = (F, A, B, 0, 0) without outputs, where F is a finite field, is strongly connected if and only if E(A) = (f: F , +F , IAx) = x + v, where Av = v] (Mason. Smith [l]). Let I be the k x k identity matrix. Prove that (a) if 1 is not the eigenvalue of A, then A is strongly connected if and only if E(A) = G(A)= (I] (b) if A is a nilpotent matrix, then A is strongly connected if and only if G ( A ) = (I]. 11.9. Let A = @ , A, B, C, 0) be a linear shift register, and let (03 be a nonfinite sequence of its input symbols. (a) Show that if lim o , = b , then the computation of the register cowern-w
ges to the limit g = b(1
-
k- 1
C a,)-1.
r=O
Exercises (b)
369
Rove that if (i) (a,,)= (b), where b f 0, (ii)
k- 1 C a,=l,
r=O (iii) Oca,cl for r = 0 , 1, ..., k - 1 , then the computation of A is divergent. (c) Show that if (a,,)= (0) and la,l+ lull + ... + I U ~ -c~1,I then the computation of A is convergent to 0 for any initial state of A.
Bibliographic note
The state of knowledge concerning linear automata up to the middle of sixties is presented in the monographs by Gill [3] and Harrison [ l ] and is not discussed here. Results dealing with shift registers, not only linear, are summarized in the monograph by Golomb [ll. The theory of linear automata is developed in a few directions. One of them deals with problems of linear realizations of finite automata. These group of problems was studied by Herman [l], Ecker [l-31, Zelcstein [l], Hartmanis and Walter [l], Mikolajczak [4], Reusch [l, 21, Stucky and Walter [l], Zeigler [l], Ury [l]. Second direction is concerned with the operation-preserving functions and is connected with the works of Hartmanis and Davis [l], Gallaire [l], Fuzimoto [l], Ecker [3], Mason and Smith [l, 21. Time-varying linear automata were investigated for the first time by Deuel [l, 21, and infinite linear automata by Gallaire et al. [l] Gossel and Poschel [l] have introduced the notion of L-linear automaton which is in general a time-varying automaton. Another generalization of a linear automaton, was introduced by Garg and Sing [l]. A model equivalent to shift register, called k-machine, was introduced in 1970 by Grodzki [l]. Linear k-machines were investigated by Zakowski [l], Chmiel and Zakowski [l], Krzesniak and Zakowski [l], and also by Stoklosa [1-8] and Stoklosa and Zakowski [l]. The results presented in this chapter dealing with realizations were obtained by Herman [l]. The results dealing with shift registers are taken from Stoklosa [3-51.
In this chapter we present results on finite and infinite linear automata. The main goal emphasis will be placed on problems of finite automata realization. A theorem on the realizability of any finite automaton with the help of a linear automaton defined over a field of characteristic 0 will be presented. A sufficient condition for the realizability of a finite automaton with the help of a linear automaton defined over a field of characteristic p, where p is a prime, will be formulated. It will be also shown that there exist finite automata which cannot be realized with the help of a linear automaton defined over any field of characteristic p > 0. The particular case of a linear automaton, i.e. a shift register defined over a field of real numbers which generates cyclic sequences, will be also discussed. Finite automata reproducing finite subsequences generated by these registers will be investigated. The presented results deal with decomposition and cyclic representation. Linear automata have wide applications in digital technique. Linear automata are used for generation of linear codes, detection and correction of errors, construction of testing sequences, and generation of pseudo-random sequences of numbers. They are also used in experiments that require Monte Car10 methods, in the protection of data stored in computer systems, and radiolocation.
I1 . I . Introducing remarks In this subchapter an automaton will be meant as a 5-tuple (S, C, 3, a, y), where S, C and R are nonempty sets of states, input alphabet and output alphabet, respectively, 3: S x C + S is a transition function, and y is an output function. In the case of Mealy automaton the output function is y: S x C + R and in the case of Moore automaton y: S + R . As in Chapter 2 by p we denote the transition function of a Mealy automaton, and by h the transition function of a Moore automaton. The remaining definitions used in this subchapter are the same as in the previous chapters. The definition of a linear automaton introduced here is more general than the definition used previously, namely we do not assume the finiteness of the sets S, C and R. If S, C and R are all finite, then the automaton is finite.
348
11. Linear automata
Let F be a field and n E No.By F , we denote the vector space of column vectors of dimension n over F. A linear automaton is an automaton (S, C, R, a, p) with the following properties: (a) there exists a field F and integers k, m , r E No such that S = F,, C = F,, n= F,, (b) there exist: a k x k matrix A, a k x m matrix B, an r x k matrix C and an r x m matrix D, all over F, and such that for each (s, o)E S x C a(s,
0 )= As
+ Bo
P(s. 6)= Cs + D o
(11.1) (11.2)
Taking into account the fact that the automaton acts in discrete time we will
also use the notation = As,+ BG,
+ Do,
O, = Cs,
(11.3) (11.4)
where S,E S , 6,E C and O, E R for f E No. For the reason that field F and matrices A, B, C and D entirely characterize the linear automaton, instead of the standard notation of the automaton as a 5-tuple (S, C, Q, p) we shall also denote automaton by an 8-tuple (F, A, B, C , D, k, m, r ) or by a 5-tuple (F, A, B, C, D) if no confusion will arise.
a,
Fig.11 . I . Diagram of a linear automaton to Example 11.I Example 11.1.
(a,
Let A, B, C, D) be a linear automaton defined over the field Q2 of integers modulo 2 such that
:],[ :1,
11 . I . Introducing remarks A=
[
B=
349 C = [ O , 1 1 , D=[lI.
The diagram of the automaton is presented on Figure 11.1. The matrix equation (11.3) can be written as k linear equations, and Equation (11.4) as r linear equations. Thus each linear automaton can be presented in the form of a net consisting of adders, multipliers and delay elements with delay elements implementing the memory of an automaton. Symbolic representations of these elements are illustrated on Figure 11.2. The automaton from Example 11.1 is presented in the form of a net on Figure 11.3. Equations (11.3) and (11.4)express the change of the state of an automaton at the moment t + 1 of time and the output signal at moment t as functions of state and input signal at moment t . The following theorem determines the state to be reached by the automaton and the output signal to be generated if at the moment 0 the automaton starts from the state so and the input sequence is finite.
G t
- 7*=Q%
Gt
-
?*=5
Fig.ll.2. Components used to the construction of a linear automaton as a net: (a) adakr, (b) multiplier and its simplified version if a = 1, (c) delay
350
11. Linear automata
w Fig.ll.3. Automaton from Example 11.1 presented in the form of a net Theorem 11.1. Let a linear automaton (F, A, B, C, D) be given. (a) For each t~ M t
s,= A's,
+ C
A'-j-lBoj
j=O
(b)
For each t E No t
u,=CA'S,+
(11.5)
C El-,oj, j=O
where D if t = O CAI-' if t > O Proof. (a) (b)
By induction on t ; omitted. Using (a) we have
o,=CS, + DO,= C(A's,
= CA's,
+
t- 1
+ C A'-J-~BO~) + DO, j=O
I- 1
C CA'-j-'Boj
j=O
+ Do,.
Let A , = ( F , A , , B , , C,, D1) and A,= (F, A,, B,, C,, DJ be linear automata. We say that A , is similar to A, if there exists a nonsingular matrix P such that
I
11.1. Introducing remarks
A, = PA,P-' B,= PB,
c,=C , P ' D2 = D1
35 1
(11.6)
Theorem 11.2. Let linear automata A , = ( F , A,, B,, C,, D1) and A , = ( F , A,, B,, C,, D2) be given. If A, is similar to A,, then A, and A, are isomorphic.
Proof. Let P be a nonsingular matrix such that Equations (11.6) hold. Let 9:Fk+ Fk be such a function that cp(s) = Ps for each s E Fk The inverse P1 exists, hence cp is the bijection. Thus
and furthermore
Q.E.D.
The converse of Theorem 11.2 is not true because there exist isomorphic automata which are not similar. As an illustration let us take the following example. Example 11.2.
If A,=((&, [3], 0, 0, 0) and A,=(Q7, [ 5 ] , 0, 0, 0), then A, and A, are isomorphic but not similar because the matrix [3] is not similar to [ 5 ] over the field Q7 of integers modulo 7. Let a linear automaton A = (F, A, B, C, D) be given. A polynomial p,,(x) = det(xI - A) is said to be a characteristic polynomial of A. On the other hand, for each polynomial k p(x) = 2: a$, i=O
where ak= 1, we can define a companion matrix A such that det(x1- A) = p ( x ) in the following way
11. Linear automata
352
-
0
1
0
...
0
0
0
0
1
... 0
0
... ... ... ... ...
A=
0
0
...
0
-ao - a l - a 2
0
... - Uk-1
... 1 - a,
1I .2. Linear realizations of finite automata
In this chapter we use the most general definition of the realization of an automaton. Let us recall that an automaton A’ = (S’, c’, 8 , Q’, p’) is said to be a realization of an automaton A = (S, C, 3, Q, p) if there exists a triple (a,I, of functions such that (a) a: S -+P(S’), (b) 1: C+Z’, (c) [: R’--+Q, (11.7) c cr(a(s, o))l, (d) (W, E S x WUm, ~(4) (11.8) (el (V(S9 0)E S x C)[S(P’(a(s)), I(@)) = P
c)
a
Theorem 11.3. For any finite automaton A = (S, C, 3, R, p) and any field F of characteristic 0 there exists a linear automaton A’ = ( F , A, B, C, D, 1, 1, 1) such that A’ is a realization of A. For given A , linear automaton A’ can be obtained effectively
.
a’
Proof. Assume that S ’ , and p’ are the set of states, the transition function and the output function of A’, respectively. If K is a subfield of a field F of characteristic 0, then there exists the isomorphism cp: K -+ Q such that 4 = @a) for a certain a E K, 4 E Q and Q being the field of rational numbers. Symbol 4 will denote an element a € K such that
4 = Ha). Assume that S = {s,, s2, ..., s,) and C = (ol,02,..., 0”).Let n = 1 +rnax(u, v). For each i~ No we define three functions, namely fi: &-+No and h: N,,++Mo, in the following way: (a) J ( O )= i(modn), g,(O) = t i / nl, (b) f,(x + 1) = gi(x)(modn) and gi(x + 1) = Lgi(x)/ nl for each x 2 0, (c) h(i) = p L f ( x + 1) = 01. Note that J ( x ) is the x-th right digit when the number i is written out in n-ary notation, and h(i)+ 1 is the number of digits in the n-ary expansion of i.
11.2. Linear realizations of finite automata
353
The functionf, and h have following properties: (i) for each k and 0 Ii < n h(nk + i ) = h(k) + 1,
(11.9)
for each k, x and 0 I i < n
(ii)
fd+i(O)
(11.10) (11.1 1) ( 1 1.12)
=i
fd+i(x + 1) =fAx) if k # 0 then f,+i(h(nx + i)) =f,(h(x)).
Define a coding triple (a,L, [) as follows: (i) for any oic Z ~ ( 0=~i ,)
(ii)
for any S E S
a(si)= ( x b E M. 1 If,(h(x)) I u, 1 Ifx@) 5 v for 0 I k < h(x))
(iii) if x is such an element of F that 1 I f , ( h ( x ) ) I u and 1 I f x ( k ) < v for 0Ik < h(x), then
Otherwise [ ( x ) = c , where c is a fixed element of R. Assume that A = C = [n] and B = D = [l]. Now we prove that the linear automaton (F, [ n ] , [l],[nl, [l]) is a realization of A = (S, C. 3. R, p) such that the coding triple (a,L, [) is defined as above. At first we show that for 1 I i l u, i E a(si). This means that a transforms the set S into the set of nonempty subsets of S ’ . We have that i E No
If(0) = i and g,(O) = 01
3 &(I) = 0
h(i) = pxK.(x + 1) = 01 = 0
* f,(h(i))= i,
i.e. 1 If(h(i)) lu. Hence, no number m E consequently
M
exists such that O I m < h(i), and
11. Linear automata
354 Assume that s E S, oi E C and x E a(s), which implies that
(11.13) Thus
a’@,i(oi))= Ax+ Bi(oi)= nr + i(oi)= nx + i. Let y = nr+ i. To prove that (11.7) holds it suffices to show that a;).Using (11.9) - (11.12) we get
YE
a@($,
and moreover
by (11.13). Now we have to prove that (11.8) holds. Let y = n x +i. 1 Sf,(h(y)) Iu, 1 If,(k) I v for 0 I k < h(y), we have
Since y > O and
Therefore A’ is a realization of a finite automaton A. Note that using the construction presented in the proof, A’ can be obtained effectively. Q.E.D. Next, we will examine the question whether the characteristic 0 of the field is a necessary condition for a linear realizability of a finite automaton. The answer is that not always it depends on the field. In the sequel let p be a prime. The additive group K generated by the unity element of any field of characteristic p is a subfield isomorphic to the field ($ of integers modulo p. If cp(a)=q for some a € K and q E ($, then for convenience, the element 4 will be identified with a, and instead of a we shall write q.
355
11.2. Linear realizations of finite automata
Theorem 11.4. Let A = (S, Z, a, R, p) be a finite automaton and F be a field of characteristic p which contains a transcendental element z over Q. Then there exists a linear automaton A' = (F, A, B, C, D) such that A' is a realization of an automaton A. Furthermore, for a given automaton A, linear automaton A' can be obtained effectively if z is known.
Proof. An element z is transcendental over $.so
where bi, c j € (0. 1, ..., p - 1 ) for O l i l q and O l j l r if and only if q = r and bi = ci for each i such that 0 I i 5 q. Let S = (sl, s2, ..., s), and C = (q,02, ..., 0").Assume also that n = 1 + max(u, v). We define function h: ( 1 , 2, ..., n} + F in such a way that h(i) = zi.
Now the coding triple (a,L, [) can be defined as follows: (i) for any q~ C L(Oi) = h(i). (ii) for any sic S a(si)= ( x I ( 3 k ~No)(3co,cl, ..., C k E F) [c,E
(iii)
( 1 , 2,
..., u ) . C
~ E (1,
2,
..., v)
for O s i e r , and
for any X E F, where F is any field, if x is such an element that
[C,E
<(XI
( 1 , 2, ..., u ) , C = P(s,,
~ E (1,
2,
..., v )
for O l i < k ] then
ack-,ack-2...aco) else b E R, where b is fixed.
Let A = C = [ P ] and B = D = [ l ] . The proof that automaton (F, [z"], 111, [PI, [l]) realizes A is omitted because it is analogous to the corresponding part of the proof of Theorem 11.3. Q.E.D.
356
1 1 . Linear automata
Theorem 11.5. Let the automaton A = (S, C, a, R, j3) be defined by means of Table 11.1. Furthermore, let F be a field of characteristic p in which every element is algebraic over Q. No linear automaton A’ over F is a realization of A.
Proof. By contradiction.
Assume that A’ = (F, A , B, C , D) is a linear automaton which realizes A. Let be a coding triple. We shall show that the sets a(s& and a(sl) are disjoint. Suppose x E a@,) (7 a ( s l ) .
(a, L,
c)
Table 11.1. Automaton for which there does not exist any linear realization over any field of characteristic p
By (11.8) we have that
and thus
Using (11.7), for each function of A’):
XE
A similar reasoning leads to
a(s,) and y E a(s,) we have that
(a’ is the transition
11.2. Linear realizations of finite automata
357
By repeated application of these statements n times, for each X E a(so)we have
a(so) if zi = Bt(1) for 0 Ii < n
n-1
Anx+ C Aizi E i=O
a(sJ if zi E (Bt(O), Bt(1)) for 0 Ii < n,
(11.15)
and zi = Bt(0) for at least one i.
Let A = [ay] and K be the smallest subfield of F which contains the extension of Q, by elements all, a12,..., a,,,,. Since all elements in F are algebraic over Q,, K is a finite extension of $. Therefore K is a finite field. Each element in any matrix A' ( i 2 0 ) is an element of K. Since K is finite, there exist positive or more generally: Ar=Ar+U. Then we have integers r, S E M such that A'=
Let X E a(so),and furthermore let zi =
Bi(0) if i = r + j s for O l j l p - 1 BL(1) otherwise.
Then
=A'+PSx+
r+ps-l
C
i=O P- 1
P-1 A i B ~ ( l ) + C Ar+LB~(0)
=Ar+Px+ C AiBL(l) i=O i # r +j t
i=O
i#r+jt
358
11. Linear automata
=A'+PFX+
r+ps-1
C
A'Bt(l)+
i=O i#r+jt
= A'+pSx +
P-1 2 Ar+&Bt(l) i=O
r +ps- 1 C A'Bt(1) E a(s,). i=O
However the application of (11.15) yields r +ps- 1 A'+~x+ C i=O
a(sl).
Thus a(s&na(s,)f 0, is a contradiction with (1 1.14).
-
0 1 0 ... 0
0
... 0
0
0 0 1 A=
... ... ... ... ...
0 0 0
... 0
... 1
4 a l a2 ... ak-2 ak-l -
-
0
Q.E.D.
-
0 ,B=
.
0
, C = [ l , 0, 0, ..., 01, D=O,
- 1-
then A will be called a linear shift register, or simply a shift register. Figure 11.4 presents the net representation of the shift register. An infinite sequence (a,),t=0, 1, 2, ..., such that its general term is defined by means of (11.5) is called a computation of a shift register. A computation is cyclic if there exists a positive integer T such that = w, for any ? E No.Each T fulfilling this condition is called a cycle length of (a,).If T is the smallest cycle length of the computation, then we say that T is the fundamental cycle length. In this chapter we assume that shift registers are defined over the field of real numbers, i.e. F = El. Furthermore, we assume that the characteristic polynomial of shift register is a function of a complex variable z . By so we denote the initial state of shift register.
359
11.3. Linear sh@ registers
Fig.1 I .4. Linear shvt register
Theorem 11.6. Let @, A, B, C, D, k, 1, 1) be a shift register, and p ( z ) be its characteristic polynomial. Furthermore, let b E R be a fixed element of the input alphabet. If: (a) all roots of p ( z ) are single and lie on the unit circle, i.e. each root zj has the form zj=exp(iqj), where 01qjc27c,j=0, 1, .... k - 1 and i = d - 1 and (b) there exist a positive integer T and positive integers to, t l , ..., tk-l such that qj=27c$/Tand for e a c h j e (0, 1, ..., k - I ) g.c.d.($, T ) = l and
(c)
XO XO Sof
*
, where xo = b (1 -
k- 1
C aj)-l,
j=O
then the computation of the shift register is cyclic and its fundamental cycle length equals T, where T > k and T f ( t j - + ) I q for each j, j * e (0,1, ..., k-11, j f f and q~ Z\(O). Proof. The general term of (or)given in the form of (11.5) is not convenient in our considerations. We make the use of the fact that any computation of shift register can be expressed by a recurrence equation k-1
o,+t- c ujor+j= or j=0
If
(3,
= b, the solution of the above equation is given by
where q, is the solution of the homogeneous equation
11. Linear automata
360
k- 1 @,+&-
c ajw,+j= 0,
j=O
Because all roots of p ( z ) are single and different from 1, then a = 0, k- 1 g(l)=p(l)= 1 - C uj#O, and j=O
k- 1 7\s, = C c,exp(icpjt), j=O
(11.16)
where co, cl, ..., ck-l are constants depending of the initial state of register. Using the assumption (b), it follows from (11.16)that W,+F
k- 1
k-1 C c,exp(icpi(t + r ) ) + b / (1- C a,)-*
j=O k- 1
j=O
k- 1
= C c,exp(i(cpjt + 2ntj)) + b / (1- C aj) = a,. j=O j=O
T is the fundamental cycle length by virtue of (b), because
w , , ~= w, for each
f E
No
implies qjT=2n9 for each j~ (0, 1, ..., k- 1).
To prove that T # ( f j - f j , ) / qwe write p ( z ) in the form
k-1
By virtue of the assumption (a) all roots of the characteristic polynomial are single, hence
exp(i2mjI r ) f exp(i2ntj, I r )
I I .3. Linear shut registers
361
for different j , j * E (0, 1, ..., k - 1 ) , thus tj f ti.. Therefore T # (5 - ti.) I q for each q E Z\(O). Because for each j E (0, 1, ..., k - 1 ) ti and T are relatively prime, 0 < f j / T < 1 and t j # t j . for differentj, j * E (0, 1, ..., k - l ) , then T > k . Finally, note that the assumption (c) means that the constant computation (x,,) does not belong to the set of computations of the shift register. Q.E.D. Let A = @3, A, B, C, D, k, 1, 1) be a shift register fulfilling the assumptions of Theorem 11.6, and let so be its initial state. In the sequel A is denoted by the 6-tuple (S, C. a, R, h, so>. For A we construct a subautomaton A , T(@ such that its state set S, consists of states reachable from so applying words x , XCT, xo2, ..., xoT-1, where x is any word of the input alphabet, which length is not greater then T and Q is a fixed input symbol. The transition function of 4, is the transition function of A restricted to the set S,xZ, and the output function does not depend on input signal. Formally, 4, T(a)=(S,, C,, a,, Q ,, A,, so>, where Z, is a finite subset of M, S,= (a(so, XCT")IXE C,*, I x l I T , Q € C,, n = 0 , 1, ..., 1 - l ) , a,: S,xZ, +S, is a transition function such that (s, q ) Dom(a,)d,(s, ~ q)=a(s, q), A&) = Cs for each s E S,, R, = 3.,(S,J. The automaton A , .T(@ is incomplete. We define a cover xu= (Bo, B , , ..., B,- 1) on its state set in such a way that Bj for j = 1, 2, ..., T - 1 (1) so€ B, and E C)wS E Bj"S' q) 6 Dam@,)* a&, q) E B(j + I k a r l . (2)
w7)
a,€
The cover defined above is not always a partition. In the sequel we shall investigate only the case when X, is the partition of S, Thus, if A , T(o)= (S, C,, a, R,, ha, so), then we can define a total function a': S,xC,+S, such that for each 7 ) E C and S E Bj, where Bj is the block of L, (1) JYs. 7 ) ) = + 1)mOa-p (2) ay
qj
Let 4, ,(@= (S, C,, a',, Q,, h, so>. It is evident that A , T(O) is not determined uniquely. For any fixed 7 ) & ~ the automaton d , generates a cyclic sequence of the fundamental cycle length T and furthermore, it generates all finite subsequences generated by A when A is in the state so and the input sequences are taken from the set (xd Ix E C' and Ixl I Tand CT E C and j E No). For 4, let us consider the automaton &, ,@)=(S, C,, a'J without outputs. A cycle-type partition of the state set S of an automaton A = (S, X, a) is an input-independent partition R such that A / R is strongly connected.
1 1 . Linear automata
362 Theorem 11.7. For T.
there exists a periodic representation with the period
Proof. The partition X, is a nontrivial partition of S, Each block of this partition has the same number T1card(S,) of elements. Let for a fixed q E C,, where v = T-”card(Sd, be defined as follows: a?,, = {Co, C , , ..., C, (*) for a fixed f o E S, Co= (*)
if
(a’(fo,7y) l j = 1, 2, ..., T ) , q-1
SE
S\ u Ci.then for each j=O
c,= (a’&,
7 y ) l j = 1, 2,
q E (1, 2,
..., v )
..., TI.
Each block of a?,, contains T elements and x,-a?,,=O. Hence, by virtue of Gnymala-Busse [6] for AT, there exists a nontrivial periodic representation V with the period T such that V is reducible to a strictly periodic automaton with period 1. Q.E.D.
fl)
The next theorem deals with the serial decomposition of
4,T(@.
Theorem 11.8. There exists a state behavior realization of 4, by a serial connection A, @A2 of an autonomous automaton A, and some automaton A,.
fl)
Proof. For any automaton B there exists a state behavior realization by a serial connection A , @ A , , where A, is an autonomous automaton if and only if there exists a nonmvial periodic representation for B, i.e. a periodic representation with the period T > 1 (cf. Grzymala-Busse [6]).
Q.E.D.
The above considerations are illustrated by example 11.3.
Example 113. Let A = (JB, A, B, C, 0, 2, 1, 1) be a shift register such that A=[
-: ]
and so=
[ ]
Register A generates cyclic computations of the fundamental cycle length T = 4 . Let ‘c = 2 and (3 = 0. For C = (0, 1), the diagram of A,, 4(o) is shown on Figure 11.5. For A2, 4(o) we compute the cover xo= (Bo, B,, B,, B 3 ) , where
1 1 3 . Linear shiji registers
363
0
Fig.115. Diagram of A,, 4(0) to Example 11.3
Thus ~0 is the partition of the state set of A2, 4(0). We construct A,, 4(o) with diagram presented on Figure 11.6. Now we calculate the partition "el = {Co, C , , C,, C3) of the state set of A2,J0) which gives
1 1 . Linear automata
364
Fig.ll.6.Diagram of &, 4(0) to Example 11.3 Let z= (oo,or)be an alphabet, and let cl: be coding function such that
xo+r, c2:
-+r, c3: X 0 + r
a1
I I .3. Linear shvt registers
365
Fig.Il.7. Autonomous automaton A,
Fig.1 I .8.Serial &composition of
A,,4(o)
Table 11.2. Automaton A, to Example 11.3
366
11. Linear awomata
Then the state behavior of A,, 4(o) can be realized by the serial connection of the autonomous automaton A, given by means of the diagram shown on Figure 11.7 and the automaton A, = (S,, q,dJ defined with the help of Table 11.2. In Table , is not explicitly presented 11.2 the state set S,= (ooa0.Q ~ Q , , Q , Q ~ , Q ~ Q , ) and in a current moment of time. A picture of the connection is shown on Figure 11.8. Exercises
11.1. Let A = (S, C, 3, R, A) be a Moore automaton. If there exists n E M such that for each x E CnC' and for each sl, s, E S the condition d(s,, x ) = &s,, x ) holds, then such an automaton is called feedback-free. Show that (F, A, B, C,D) is a feedback-free automaton if and only if A" is nilpotent, i.e. A" = 0 and (n=1 v ( n 2 2 A A"-'#O)). 11.2. Prove that if A, and A, are similar linear automata, and pAi(x) is the characteristic polynomial of Ai (i = 1, 2), then pA1(x)= pA2(x). 11.3. Let A = (S, C, a) be an automaton such that for a fixed n E E.9 the state set S = Sn)= ( Q ~ Q , . . . Q ~loi E C) and for each s E S, O E C we have that a($, Q) = dW1)o,where if s = op,. ..on,then dwl)= Q ~ Q , ..q,, . (a) Show that for A there exists a shift register which is a state behavior realization of A. (b) Prove that the group of automorphisms of A is trivial, i.e. card(G(A))= 1. 11.4. Let 4, gi and h be the functions defined as in the proof of Theorem 11.3. Show that they are recursive. 11.5. Let A = (S, C, 3) be a strongly connected permutation automaton such that I is its characteristic group, and H is a subgroup of i. For A there exists a linear state behavior realization over Q if and only if (i) there exists a commutative normal subgroup N of such that for each [XI E N the condition [x]p= [el holds, (ii) C c N [ x ] for a certain Ex] E i, (iii) H n N = [el (Hartmanis, Walter [I]). Show that for the automaton defined by means of Table 11.3 there exists a linear state behavior realization.
Exercises
367
Table 11 3. Automaton to Exercise I I5
a Table 11.4. Automaton to Exercise 11.6
11.6. Let an automaton A = ( S , C, a) be given, and let there exist injections a: S + F , and I: C j F , , , , where F =Q, such that a(d(s. a))=Aa(s)+ Bt(s) + a . Then we say that for A there exists an extended linear realization. Let us suppose that B is an automaton without outputs. If V,* is a fmed analog of an extension V, associated with isomorphisms and V,* is strongly connected, then for V,* there exists an extended linear realization over Q if and only if for B there exists a linear state behavior realization over (Q (Mikolajczak [4]). Prove that if B is an automaton defined by means of Table 11.4, then there exist no extended linear state behavior realization for the fixed analog V,* of any extension. 11.7. If for an automaton A = ( S , C, a) such that a(s,, 6)= a(s2, 6) for some 6 E C = (0, 1 ) and sl, s2 E S there exists a linear state behavior realization over Q with the identity function coding inputs, then a(s,, a)=a(s,, 6) for each o E C. Furthermore, if for some S, E S a(s,. 0) = a(s. 0) = a(s, 1) for all S E S (Hartmanis, Davis [ll). (a) Prove that for the automaton A presented in Table 11.5 there exists a linear state behavior realization over Q. (b) Prove that there exists a quotient automaton A l x , where x is an SP partition, for which there does not exist a linear state behavior realization.
11. Linear automata
368 Table 11 5.Automaton to exercise 11.7
s -
a 0
1
S1
s5
s4
s,
s3
s7
%
'8
'2
'6
%
s3
s,
s1
'6
SS
s4
ss
11.8. A linear automaton A = (F, A, B, 0, 0) without outputs, where F is a finite field, is strongly connected if and only if E(A) = (f: F , +F , IAx) = x + v, where Av = v] (Mason. Smith [l]). Let I be the k x k identity matrix. Prove that (a) if 1 is not the eigenvalue of A, then A is strongly connected if and only if E(A) = G(A)= (I] (b) if A is a nilpotent matrix, then A is strongly connected if and only if G ( A ) = (I]. 11.9. Let A = @ , A, B, C, 0) be a linear shift register, and let (03 be a nonfinite sequence of its input symbols. (a) Show that if lim o , = b , then the computation of the register cowern-w
ges to the limit g = b(1
-
k- 1
C a,)-1.
r=O
Exercises (b)
369
Rove that if (i) (a,,)= (b), where b f 0, (ii)
k- 1 C a,=l,
r=O (iii) Oca,cl for r = 0 , 1, ..., k - 1 , then the computation of A is divergent. (c) Show that if (a,,)= (0) and la,l+ lull + ... + I U ~ -c~1,I then the computation of A is convergent to 0 for any initial state of A.
Bibliographic note
The state of knowledge concerning linear automata up to the middle of sixties is presented in the monographs by Gill [3] and Harrison [ l ] and is not discussed here. Results dealing with shift registers, not only linear, are summarized in the monograph by Golomb [ll. The theory of linear automata is developed in a few directions. One of them deals with problems of linear realizations of finite automata. These group of problems was studied by Herman [l], Ecker [l-31, Zelcstein [l], Hartmanis and Walter [l], Mikolajczak [4], Reusch [l, 21, Stucky and Walter [l], Zeigler [l], Ury [l]. Second direction is concerned with the operation-preserving functions and is connected with the works of Hartmanis and Davis [l], Gallaire [l], Fuzimoto [l], Ecker [3], Mason and Smith [l, 21. Time-varying linear automata were investigated for the first time by Deuel [l, 21, and infinite linear automata by Gallaire et al. [l] Gossel and Poschel [l] have introduced the notion of L-linear automaton which is in general a time-varying automaton. Another generalization of a linear automaton, was introduced by Garg and Sing [l]. A model equivalent to shift register, called k-machine, was introduced in 1970 by Grodzki [l]. Linear k-machines were investigated by Zakowski [l], Chmiel and Zakowski [l], Krzesniak and Zakowski [l], and also by Stoklosa [1-8] and Stoklosa and Zakowski [l]. The results presented in this chapter dealing with realizations were obtained by Herman [l]. The results dealing with shift registers are taken from Stoklosa [3-51.