Discrete Applied Mathematics 15 (1986) 117-119 117 North-Holland

N O T E

A PROPERTY OF REAL-TIME TRELLIS A U T O M A T A

Sheng YU Department of Mathematics, University of Turku, Finland

Received 5 February 1986

Trellis a u t o m a t a have been studied in [1-4]. Here , a p roper ty o f real- t ime trellis a u t o m a t a is deduced. This p roper ty can be used to prove that certain languages are not accepted by any real- t ime trellis au toma ta . Two theorems will be established. The second one is a general izat ion o f the first one. The reader is referred to [1-4] for defini t ions and backg round mater ia l .

Theorem 1. Le t L c_ Z* be a real-time trellis language. Then there ex&ts a posi t ive

integer N such that the fo l lowing condit ion holds f o r any x 6Z,* and an infinite se-

quence o f n o n e m p t y strings {Yi}i~N over the one-letter alphabet {a}, where a 6 Z .

I f f o r every i = 1, 2 . . . . .

(1) lyit < lyi+l l, (2) xy i E L, and

(3) there is no y ' ~ {a} * and lyil < ly'l < lyi+ 11 such that xy ' EL , then there exists an infinite subsequence {Yi~}je N such that

(a) lYizl- lYg, I = lYi31- lY~I . . . . . K, and (b) there exist integers to, t I . . . . . tr where r= lx l and 1 <_ti<__N f o r i = 0 , 1 . . . . . r

such that K divides tot 1 ... t r.

Proof. Let T b e a trellis a u t o m a t o n accepting L in real t ime, and let N b e the n u m b e r

o f states o f 7". Let x e Z * and let y ~ {a} ÷, i = 1, 2 . . . . . satisfy the condi t ions (1), (2) and (3). See Fig. 1. Assume that the input is x fol lowed by a sufficiently long string o f a ' s . Consider the diagonal line l 0. Since 10 is only affected by the string o f a ' s , it is easy to prove that l 0 is ul t imately periodical with threshold at most N and the per iod being to, l < t o < N . Consider the d iagonal line /1. A state of this line depends only on the previous states of this line and line l 0. So, ll must be ul t imately periodical with threshold at mos t N 2 and the per iod dividing totl, for some 1 <tl<_N. With the same arguments , we can show that lr is ul t imately periodical with threshold at most N r and its per iod dividing tot 1 . . . t r where l<_ti<_N for i = 1, 2 . . . . . r. By (1) and (2), there exists an i such that [Yi I > N r and xy i E L. Thus, there is at least one final state which appears at line l~ af ter N r steps. By consider- ing the periodical appearances o f this final state and the condi t ion (3), we are guaran teed to find a subsequence {Y6}i~N which satisfies (a) and (b). []

0166-218X/86/$3.50 © 1986, Elsevier Science Publishers B.V. (North-Holland)

1 1 8 S. Yu

computat ion X

step _ ~ _ _ a a . . . a

2 ".

a . . .

l 12 11 lo T '

Fig. 1.

The following examples show applications o f Theorem 1.

Example 1. L l = {anbmn I m, n > 0 } is not a real-time trellis language.

Proo f . Assume L 1 is accepted by a real-time trellis a u t o m a t o n T. Then there exists a constant N satisfying Theorem 1. Choose a prime P > N . Let x = a P and Yi=b iP for i = 1, 2 . . . . . which obviously satisfy (1), (2) and (3). It is clear that any subse-

quence o f {Yi}i~N which satisfies (a) must also satisfy K = c P , for some integer

c > 0 . But cP does not divide toq ""tr, l<-ti<-N for i = 0 , 1 . . . . . r. This is a con- t radict ion to (b). Therefore, L 1 cannot be a real-time trellis language. []

Example 2. L 2 = {a2" ln>0} and L 3 = {aPlp is a prime} are not real-time trellis

languages.

2 i Proo f . For L 2 , choose x = e and v i = a , i = 1, 2 . . . . . It is easy to see that the choice satisfies the condit ions (1), (2) and (3) o f Theorem 1. But there is no subsequence o f y ' s which satisfies (a). So, L 2 cannot be a real-time trellis language. Similar

arguments apply on L 3 as well. []

L 2 has been shown in [3], with a different technique, not to be a real-time trellis language. But the technique o f [3] cannot be applied o n L 1 . The two examples above demonst ra te that Theorem 1 is a more powerful tool.

In Theorem 1, the sequence o f y ' s is restricted to a sequence o f strings over a one- letter alphabet. We can natural ly make the following extension. The p r o o f is

omitted.

Theorem 2. Let L c_ 27* be a real-time trellis language. Then there exists a positive integer N such that the following condition holds for any x e 27* and an infinite se-

A property o f real-time trellis automata 119

quence o f strings {Yi}i~N from {w} +, for some w e - r +. I f for every i= 1, 2 . . . . .

(1) lyil < lyi+~l, (2) x y i E L , a n d

(3) there is no y ' ~ {w} + and [Yi[ < [Y'[ < [Yi+ 1[ such that xy' e L , then there exists an infinite subsequence {Yij}j~ N such that

(a) [yg2l-tyi, l=ly6l-ly~2l . . . . . K, and (b) Kd iv ides toq . . . t r wherer=lx ] and l < t o < N Iwl, l < t i < N , i = 1 , 2 . . . . . r. []

References

[1] C. Choffrut and K. Culik 11, On real-time cellular automata and trellis automata, Acta lnformatica 21 (1984) 393-407.

[2] K. Culik II, J. Gruska and A. Salomaa, Systolic trellis automata, Part I, Intern. J. Computer Math. 15 (1984) 195-212.

[3] K. Culik 11, J. Gruska and A. Salomaa, Systolic trellis automata, Part II, Intern. J. Computer Math. 16 (1984) 3-22.

[4] O.H. lbarra and S.M. Kim, Characterization and computational complexity of systolic trellis automata, Theoret. Comput. Sci. 29 (1984) 123-153.