International Journal of Computer and Information Sciences, Vok 3, No. 2, 1974

One,Way Acceptors and Languages Eugene S. Santos 1

Received April 1973; revised December 1973

In the present paper, a parallel presentation of the theories of abstract families of languages (AFL) and abstract families of deterministic languages (AFDL) is given. This is done by introducing two families of languages. One of them is the one-way nondeterministic family of languages (1NFL). A 1NFL is a family of languages closed under special marked substitution and inverse nondeterministic a-gsm mapping. The deterministic counterpart of 1NFL is 1DFL. It is shown that 1NFL and 1DFL are equivalent to AFL and AFDL, respectively. These families of languages are then used to characterize, side by side and with alternate proofs, the families of languages accepted by AFA and AFDA. Moreover, it is also shown that 1NFL and 1DFL can be used to characterize the families of languages accepted by a closed class of 1NBA and 1DBA, respectively.

1. I N T R O D U C T I O N

In the last few years several general fo rmula t ions of one-way acceptors have been p r o p o s e d by var ious authors . A m o n g them are abs t rac t families o f acceptors ( A F A ) ez) and ba l loon au toma ta . ~5) These families of acceptors were in t roduced in o rde r to p rov ide a unified a p p r o a c h to the s tudy o f famil ies o f languages accepted by the var ious types o f acceptors which occurred in the l i terature.

A n A F A is essential ly a family o f nondete rmin is t ic one-way acceptors . By consider ing only the determinis t ic acceptors in an A F A , one obta ins an A F D A . tl) Since A F D A is essential ly the restr ic t ion o f A F A to the deter- minis t ic case, cer ta in re la t ions mus t exist between the families o f languages accepted by an A F A and the cor respond ing A F D A . However , a l though it

This work was supported in part by the University Research Council of Youngstown State University, Youngstown, Ohio.

1 Department of Mathematics, Youngstown State University, Youngstown, Ohio.

t4t �9 1974 Plenura Publishing Corporation, 227 West 17th Street, New York, N.Y. 1001t. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilraing, reeording, or otherwise, without written permission of the publisher.

142 Santos

was shown in Refs. 1 and 2 that the family of languages accepted by AFA and AFDA can be characterized, respectively, by AFL ~2) and AFDL. I1~ the two cases were treated separately. Thus, it is not clear how AFL and AFDL are related to each other.

In the present paper a parallel presentation of the theories of A FL and AFDL are given. This is done by introducing two families of languages. One of them is the one-way nondeterministic family of languages (INFL). A 1NFL is a family of languages closed under special marked substitution and inverse nondeterministic a-gsm mapping. The deterministic counterpart of 1NFL is 1DFL. It is shown that 1NFL and 1DFL are equivalent to AFL and AFDL, respectively. By using INFL and 1DFL, we are able to characterize, side by side and with alternate proofs, the families accepted by AFA and AFDA.

It is known ~51 that the family of languages accepted by a closed class of one-way nondeterministic balloon automata (1NBA) is an AFL, and hence a 1NFL. Moreover, it follows from known results ~4,5~ that the family of languages accepted by a closed class of 1DBA is a 1DFL. In the present paper the converses are also shown to be true. Moreover, as in the case of AFA and AFDA, the two converses are presented in parallel and established side by side using the same proof.

Although taken individually most of the results presented in this paper are either known or minor extensions of known results, nevertheless, since the same proofs are applicable to both deterministic and nondeterministic cases, they put the theories of AFL and AFDL on the same footing, and hence enable us to see how determinism and nondeterminism affect the structures of the families of languages accepted. Moreover, unlike the conventional proofs, the alternate proofs presented in this paper could be modified and carried over to the two-way cases.

2. O N E - W A Y L A N G U A G E S

In this section two families of one-way languages are introduced and their relationships with existing families of languages are discussed.

Notation. Let Z be an arbitrary set. Then Z* denotes the free semigroup with identity e generated by 27, and Z+ = Z* -- {e}. Moreover, if x ~ Z*, then lg(x) denotes the length of x.

Definition. A family of languages is an ordered pair (5r Z), or simply So i f Z is understood, where (1) Z is an infinite set of symbols, (2) for each LE ~LP there exists a finite subset Z 1 of Z such that L C_ZI*, and (3) L 4: (empty set) for some L e ~ .

One-WayAcceptors and Languages 145

In what follows Z will always denote a given infinite set of symbols, and Z with a subscript will denote a finite subset of Z.

Notation. Let L1, L2 C Z*. Lz �9 L,, or LtL,. is the set {xy : x e L1, y E L~,. Moreover, if LI -= {x}, then we shall write xL2 instead of{x} L , , etc.

Definition. A mapping h from ZI* into Z2* is a homomorphism iff h(xy) = h ( x ) h ( y ) for all x, y eZa*. If h(x) = e implies x = e, then h is said to be e free. If L C Z~*, then h(L) --- {h(x): x e L}. If L _C Z2*, then h-~(L) = {x c_-Z2*: h(x) e L}. h -~ is called an inverse homomorphism.

Definition. An abstract family of languages (AFL) (2) is a family of languages closed under the operations of u , . , + , e-free homomorphism, inverse homomorphism, and intersection with regular sets. A full AFL is an AFL dosed under arbitrary homomorphism.

Definition. If L1 _CZI* , L~_C Z2* , and a, b e Z -- (E 1 w Z2) , then aL 1 t3 bL2 is called a marked union, LzaLz a marked product, (Lla) + a marked + , and (Lza)* a marked *.

Definition. A pre-AFL (8~ is a family of languages closed under marked product, marked § inverse homomorphism, intersection with regular sets, and w{e).

Definition. A marked nondeterministic a-gsm is a 7-tuple

where:

1.

2.

3.

4.

5.

M = ( S, Z~ , Z2 , 8, )~, so , F),

S, Z1, and Z2 are finite sets (of states, inputs and outputs, respectively);

3 is a function from S • (Z1 u {r $}) into finite subsets o r s x {0, 1};

is a function from S • (Z1 u {r $}) into finite subsets ofZ~* ;

So e S is the start state; and

F _C S is the set o f accepting states.

If for each s e S and a e Z 1 ~3 {r $}, 3(s, a) and ;~(s, a) contain at most one element, then M is called a marked a-gsm.

Remark. r and $ stand for left and right endmarkers, and are two symbols not in Z.

Definition. Let M be a marked (nondeterministic) a-gsm. I f F = S, then M is called a marked (nondeterministie) gsm.

Notation. N denotes the set of all positive integers.

Notation. Let M = (S , / ' 1 , Z2 ,3 , ~, so, F) be a marked nondeter-

144 Santos

ministic a-gsm. The relation ~-- on S • (2:1 t_) {r $}) • N • Z~* is defined as follows. For w ---- ala2 "'" an, each a~ ~ Z1 w {r $), s, s' e S, 1 ~< i ~< n, and x, y e Z~*, (s, w, i, x) ~-- (s', w, i', xy) if a~ ~ a, (s', d) ~ 3(s, a), y ~ A(s, a), and i' -- i + d ~< n. The symbol ~---* denotes the reflexive, transitive closure of ~--. For each x ~Z~* let

M(x) = {y ~Z2*: (s o , r x $, 1, e) ~---* (s, r x $, lg(x -k 2), y)

for some s E F}

Moreover, let M(L) = ~)~L M(X) and M-I(L) = {x: M(x) n L @- ~}. (Nondeterministic) a-gsm and (nondeterministie) gsm are defined as above

with the endmarkers removed.

Proposition 2.1. Let M0 be a function from 2:1" into 2~. The following statements are equivalent:

1. There exists a marked (nondeterministic) a-gsm M~ such that Ml(x) = Mo(x) for all x E 2:1".

2. There exists a (nondeterministic) a-gsm Mz such that Ms(x) = Mo(x) for all x ~ ZI*.

3. There exists a marked (nondeterministic) gsm M3 such that M3(x) = Mo(x) for all x s 2:1".

Proposition 2.2. Let &o be a family of languages. LP is closed under inverse (nondeterministic) a-gsm mapping, i.e., M - I ( L ) ~ for every (nondeterministic) a-gsm M and L e ~ , if and only if 5r is closed under inverse (nondeterministic) gsm mapping and intersection with regular sets.

The concept of a-transducer was introduced in Ref. 2. It can be shown that for every a-transducer M1 there exists a nondeterministic a-gsm M2 such that M~(x) = M2(x) for all input x, and conversely.

Definition. An abstract family of deterministic languages (AFDL) ~1) is a family of languages closed under marked union, marked *, and inverse marked gsm mapping.

Definition. Let L _CZI*. For each a ~Z~ let L~ _CZ~*. Let r be the function defined by r(e) = e, r(a) = La for each a ~ Z1, and "r(xy) = r(x) r (y) for all x, y ~ Z~*. Then ~- is called a substitution and ~,(L) = U~L r(x). If .r(a) Ca(Z~ -- Z~)* for each a ~ Z~, then ~- is called a marked substitution. If -r(a) C_a(Z~--Z~)*a for each a s Z 1 , then ~- is called a double marked substitution.

Definition. Let ~ be a family of languages. ~ is closed under special marked substitution iff r(Z~*) ~ ~ for every Z~ __C Z and marked substitution

One-Way Acceptor$ and Languages '145

~" such that for each a e / 1 , "r(a)= aL~, where L~ ~ ~ . Closure under special double marked substitution is similarly defined.

Proposition 2.3. Let .L~ be a family of languages closed under inverse a-gsm mapping. Then ~q~ is closed under special marked substitution if and only if ~ is closed under special double marked substitution.

Definition: Let ~,a be a family of languages closed under special marked substitution. If ~ is closed under inverse nondeterministic a-gsm mapping, then .LP is a one-way nondeterministic family of languages (1NFL). If ~f is closed under inverse a-gsm mapping, then ~qo is a one-way deterministic family of languages ( 1D FL).

Theorem 2.1. (a) Every 1NFL is a full AFL, and conversely.

(b) Every 1DFL is an AFDL, and conversely. (c) Every 1NFL is a 1DFL, and every 1DFL is a pre-AFL.

The proof of the above theorem is omitted since the results are either known or follow from other known results.

Theorem 2.2. Let ~ be a 1DFL. Then (i) ~-(L) ~ ~ for every L _C Zi* and L ~ s and for every marked substitution r such that for each a e / 1 , .r(a) = aR~, where R~ is a regular set; and (ii) ,-(R) e Sa for every regular set R _ Zx*, and for every marked substitution -r such that for each a e Z1, 7(a) = aL~, where L~ e 5f.

Proof. Let L _C/1" and for each a e Z i let L~ C Z~*, where Z i C~ Z~ = ~ . Let ~- be a marked substitution such that ~(a) = aL~ for each a e / 1 . Let h be a homomorphism from Z i u Z~ into Z1, where h(a) = a for all a e Z~ and h(a) = e for all a ~ Z i . It can be verified that z(L) = h-i(L) c~ r(Zi* ).

Remark. Theorem 2.5 is also valid if marked substitution is replaced by double marked substitution.

The following notation will be used in the subsequent section.

Notation. Let ~ be a family of languages. Then c~D(~q~) denotes the smallest 1DFL containing ~ , and c~N(~) denotes the smallest 1NFL containing ~q'.

We shall conclude this section by noting the fact that all theorems proven in Ref. 3 are also valid if pre-AFL is replaced by 1DFL.

3. ABSTRACT FAMILIES OF ACCEPTORS

In this section we shall characterize, side by side and with a simpler proof, the families of languages acceptable by AFA and ADFA.

146 Santos

Definition. A nondeterministic abstract device is a 6-tuple

w h e ~ :

1.

2.

3.

4.

5.

s = (K, Z', /-,, /, jr g),

K and 27 are infinite abstract sets, and P and I are nonempty abstract sets;

f i s a function f r o m / ' * x / in to / - '* u { ~};

g is a function from F* into the finite subsets of F* such that g(e) = {e}, and e ~ g(7') implies 7' = e;

for each 7' ~ g ( F * ) = U~r,g(7) there exists an element I~ e l satisfying f(7 ' , I) = 7" for all 7" such that ~ ~ g(7"); and

for each u ~ I there exists a finite set/ '~, _C _P with the following property: i f / '1 _C/', 7' ~/~*, and f(7, u) ~ ;~, then

f@, u) ~ (/'1 u/ '~)*.

If, in addition, for each 7' ~ T*, g(y) contains at most one element, then s is called an abstract device.

Definition. Let s = (K, Z, F, I , f , g) be a nondeterministic abstract device. A nondeterministie acceptor ofs is a 5-tuple D = (/(1, Z1 ,3 , q0, F), where:

1. /s and Z1 are finite subsets of K and Z1, respectively;

2. FC_K;

3. qo E K1 ; and

4. 3 is a function from /s x (Z1 u {e, r $}) x g(P*) into the finite subsets ofK~ x Isuch that the set GD = {7': 3(q, a, 7') ~ ~ for some q and a} is finite.

If, in addition, for each q ~/(1, 3(q, e, e ) - ~ , and for each q ~/(1 and 7' E g(P*) either (i) 3(q, a, 7') contains at most one element for all a ~ Z 1 u {r $} and 3(q, e, 7 ' )= ~ , or (ii) 3(q, e, 7') contains at most one element and 3(q, a, 7') = ~ f o r all a ~ Z1 u {r $}, then D is called an acceptor ofs

Notation. Let s = (K, Z, 1", L f, g) be a nondeterministic abstract device and D = (/s Z1 ,3 , q0, F) a nondeterministic acceptor of s The relation ~-- on Ks x (Z1 w {r $})* x P* is defined as follows: For a ~ 271 u {e, r $}, (p, aw, y) ~-- (p', w, y'), if there exist 7"" and u such that 7'" ~ g(7'), (p ' , u) ~ 8(p, a, 7'"), and f(7', u) = 7". Moreover, ~---* denotes the reflexive, transitive closure of ~-. The language accepted by D is L(D)----{w~Zl*: (q0, r e) ~--* (q, e, e) for some q ~F}. Furthermore, let cg~(s = {L(D): D

One-Way Acceptors and Languages ~47

is an acceptor of g2} and %v(g2) = {L(D): D is a nondeterministic acceptor of t2}.

Knowledgeable readers will observe that TD(O), where g2 is an abstract device, is the family of languages accepted by an AFDA, m while ~N(,Q) is the family of languages accepted by an AFA t"~ with endmarkers. It is easily shown that the addition of endmarkers does not enlarge the family of languages accepted by an AFA.

In what follows if the range of a function contains ~ , then the value of the function is ~ except where otherwise stated.

Theorem. 3.1. Let ~ = (K, S, F, L f , g) be a nondeterministic abstract device. Then CdD(O ) (resp. ~u(s is a 1DFL (resp. 1NFL).

Proof. Let 2:1,272_C27, where 21c~12---- ~ . For each a ~ I 1 let "r(a) = aLaa, where L, = L(D~), O~ = (K~, ! 2 , 3 ~ , q~,Fa) is a (non- deterministic) acceptor of g2, and K~ n Kb = ~ for all a v ~ b. Let D -- (Ko ,11 w 12 ,3 , q0 ,F ) be the (nondeterministic) acceptor of O where K0 = K1 W {77: q e/s U {q0}, /s = U~z~ h a , each 77 a new symbol in K, F = U~z~ F~, and for all ~, e g(F*):

1. 3(po, r e) = {Po,I~};

2. 3(po, a, e) = {(,~, u): (p, u) e 3o(q~, r e)} for all a e 21 ;

3. 3(?/, b, 7) = {(P, u): (p, u) e 3~(q, b, V)} for all a e 11, q e Ka, and b e I . u {e};

4. ~(77, a, 7) = 3,(q, $, ~) for all a e 22~ and q e K~ ;

5. 3(q, e, 7) = 3~(q, e, V) for all a e !~ and q e K~ ;

6. 3(q, a, e) = {(~, u): (p, u) e 3~(q, r e)} for all a eZ' 1 and q eF ; and

7. $(q, $, e) = {(q, I~)} for all q e F.

It can be verified that T(II* ) = L(D). Thus cdD(~ ) and ~N(~) are closed under special double marked substitution. Next, let

M = ( S, Ia ,12 , 3, y, so , F)

be a (nondeterministic) a-gsm and Do = (/s qo, Fo) be a (non- deterministic) acceptor of ~ . Let k be an integer such that lg(2(s, a)) ~< k for all s e S and a E Z' 1 . Let D = (1(1, S1 ,3 ' , Po, Fx) be the (nondeterministic) acceptor of g2, where K 1 ----- [Ko • S • (Z' a u {0}) • {0, 1, 2,..., k}] u {Po}, F l = { ( q , s , O , O ) : q ~ F o , s s F } , and for every q e K o , s e S , a ~ S ~ , and 7 ~ g(P*):

1. 3'(po, r e) = {((q, so, 0, 0), u): (q, u) ~ 3o(qo , r e)}; 2. 8'((q, s, a, n - -1) , e, 7) = {((q', s, a, n), u): (q', u) e 3o(q, n(h(s, a)), ~,)}

if lg(2(s, a)) > n :> 0;

t48 Sanr

3. 3'((q, s, a, n), e, y) = {((q', s, 0, 0), u): (q', u) e 30( q, n(,'~(s, a)), y} if lg0~(s, a)) = n > 0;

4. 3'((q, s, a, n), e, 7) = {((q', s, a, n), u): (q', u) e 3o( q, e, 7)} if n > 0; 5. 3'((q, s, a, 0), e, y) = {((q, s, 0, 0), I,)} if Ig(A(s, a)) ~- 0; 6. 3'((q, s, 0, 0), a, 7) = {((q, s', 0, a), I,): s' e 3(s, a)}; 7. 3'((q, s, 0, 0), $, 7) = {(q', s, 0, 0), u): (q', u) e 80(q, $, ~)}; and 8. 3'((q, s, 0, 0), e, 7) = {((q', s, 0, 0), u): (q', u) ~ 8o( q. e, 7)};

where n(x) stands for the nth symbol in x. It can be verified that L(D) = M-I(L(Do)). Thus ~D($2) is closed under inverse marked a-gsm mapping, and ~N(f2) is closed under inverse marked nondeterministic a-gsm mapping. Hence ~D($2) is a 1DFL and fiN(f2) is a 1NFL.

Theorem 3.2. For every family ~ of languages there exists an abstract device $-2 such that CgD(/2) = cgD(~) and ~fN(f2) = CgN(Sa).

PrOof. For each L ~ s let E be a new symbol in Z. Let f2 = (K, Z, F, I, f , g) be the abstract device where K is an infinite set, F = 2J, I = ~q" u I W {e},

{YeU if u e 2 t ~ f (y, u) = if u s s 7 ~ ~u

otherwise

and g(e) = {e} and g(y) = {ao} for all e v a Y ~ 27% where ao is a fixed element of 27. We shall show that aQ has the desired properties. Let L ~ 2 ' and L _C 271". Let D = ({Po}, 271,8, Po, {Po}) be the acceptor of $2, where for every y e {e, ao}, we have 3(po, r y) = {(Po, L)}, 3(po, $, y) = {(Po, L)}, and 3(po, a, ~,) = {(Po, a)} for all a s 221 . Clearly, L = L(D). Thus, by Theorem 3.1, cgD(~a ) __ cgD(~2 ) and ~gN(~) _C cgN(/? ). Conversely, suppose

D = ( KI , I 1 , 3, qo , F)

is a (nondeterministic) acceptor of /2 . Let M = (S, 2:1, Z 2 , 3 1 , •, so, F1) be the marked (nondeterministic) a-gsm where S = / ( 1 • {e, ao}, so = (qo, e), F1 = {(q, e): q eF} , for every q e / ( 1 , 7 e {e, ao}, and a eZ71:

1. 3~((q, 7), a) = {((p, 7), 1): (p, e) ~ 3(q, a, ),)} w {((p, ao), 1): (p, u) ~ 3(q, a, 7) for some u~Z~} u {((p, e), 1): (p, u) e 3(q, a, 7) for some u e 2~a} w {((p, 7), 0): (p, e) ~ 3(q, e, 7)} w {((p, ao), 0): (p, u) e 3(q, e, 7) for some u e2J1} w {((p, e), 0): (p, u) e 3(q, e, 7) for some u ~ ~ } ; and

2. )~((q, 7 ) , a ) = { u : u e I l w { e } and ( p , u ) s 3 ( q , a , y ) for some p e Ki} u {~: u ~ s and (p, u) e ~(q, a, 7) for some p ~/(1}.

One-Way Acceptors and Languages t4~

It can be verified that L ( D ) = M-l(r(S0*)), where S o = {ii: u ~ ~o and (p, u) E 3(q, a, y) for some p, q, a, and y}, and for each E ~ S 0 , T(L') = E L L Thus, ~v(O)_C ~,'~(s162 and ~6~N((2)_C ~f,N('f). Therefore s has the desired properties.

It is worthwhile to observe that we essentially used the same proof for both the nondeterministic and deterministic cases to establish Theorems 3.1 and 3.2. This enables us to see how nondeterminism and determinism affect the structure of the family of languages accepted. Moreover, it should be observed that in Theorem 3.2 the same (2 works for both the nondeterministic and deterministic cases.

4. B A L L O O N A U T O M A T A

It is known t~,a~ that the family of languages accepted by a closed class of one-way deterministic balloon automata is closed under inverse gsm mapping, intersection with regular sets, and special marked substitution, and is therefore a 1DFL. Moreover, it is known (5~ that the family of languages accepted by a closed class of one-way nondeterministic balloon automata is a full AFL, and is therefore a 1NFL. In this section we shall show side by side that the two converses are also true.

Definition. A one-way nondeterministic balloon automaton (1NBA) is an 8-tuple B = (/(1, Z'x, N1 , f l , g l , h , , go, F0, where:

1. K1 and s are finite sets (of states and inputs, respectively);

2. N~ is a finite subset of N;

3. F1 _C/s (the set o f accepting states);

4. qo ~ K1 (start state); and

5. f l , gz, and hz are functions from K~ • N into N w {;~}, /(1 • (271 w {r $}) • N1 into 2Klx~~ and N into No, respectively.

If for every q ~ K, a ~ ~ t3 {r $}, and ^/~ N~, g~(g, a, ~) contains at most one element, then B is called a one-way deterministic balloon automaton (1DBA).

Notation. Let B = (Kz, X1, Nz , f l , g l , h l , go, F~) be a 1NBA. For w = aza2 "'" an, each ai~X1 w {r $}, q, q' ~ / (1 ,1 ~< i ~< n, and ~,~,' ~N, let (q, w, i, 7) ~-- (q', w, i', y') if there exist a, d, and m such that ai = a, hz(7) = m, (q', d) ~ gz(q, a, m), v' = f~(q, y), and i ' = i - l - d ~ n . The symbol ~---* is the reflexive, transitive closure of ~--. The set o f words accepted by B is L(B) = {w ~ Z'x*: (qo, r 1, 1) ~---* (p, r lg(w + 2), V) for some p ~/'1 and V ~ N}.

t50 Santos

Definition. A closed system over X is a triple q0 := (X', o*-, ,*') satisfying

the following conditions:

1. ~ is a countably infinite set;

2. o~- is a family of functions from N into N w { ~ } containing cxo and a , where ao is the function that maps N into one, and a is the identity

function on N; and

3. ~ is a nonempty family of functions from N into N such that (a) h(N) is finite for all h ~ ~ , and (b) hz, hz ~ ~ imply that ha e for each function from N into N where (i) ha(N) is finite, and (ii) h3(x) = h3(y) if hz(x) = hz(y) and h2(x) = hz(y).

Definition. Let ~ = ( : ( , ~-, ~ ) be a closed system over Z. A 1NBA o f is a 1NBA B = (Kx,Z~, N~, fx , gz, h~, q0, F~) such that K~ C_ jCr,

2:1 _C Z, hi ~ ~ , and for each q ~ Ki , f~q~ ~ -~, where f~) i s the function from N into N w { ~} defined byf t~ (y ) = fl(q, Y) for all y ~ N.

Notation. Let # be a closed system over Z. r = {L(B): B is a 1DBA of ~} and c~N(q5) = {L(B): B is a 1NBA of ~b}.

Observe that cgo(q~) and cg~(~) are families of languages accepted by closed classes of 1DBA and 1NBA, respectively.

Theorem 4.1. For every family Ar of languages there exists a closed system qb over Z such that cgo(~) = f f n ( ~ ) and cg~(~) = ~N(Se).

Proof. For every L ~ s let E be a new symbol in Z. Let fi be a one-to- one correspondence from Z* into N. Let �9 = (~,Y', ~ , ~rr be the closed system over 27 where:

1. Y = s 2. o~ = {f~: u ~ s W Z w {e}} ~ {a0}, where for every 7 e N,

fu(Y) --= if u ~ ~q~ and fl-x(y) ~ ~u otherwise

3. ~(' = {ho}, where ho(y )= 1 for all y ~ h r.

We shall show that cr = CYD(~) and cr = cgN(s Let L e s and L _C_C 271". Let B = (KI, 21 , {1}.fl, gz, ho ,Po, Fa}, where

Ki = {Po, L, L} U Zx ; F, = {L};f(u, ~) = fu(Y)

for all u ~ Zz w {E, L} and y e N; and g~(Po, r 1) = {(E, 1)}, gz(p, $, 1) = {(L, 1)}, and g~(p, a, 1) = {(a, 1)} for all p ~ Z1 w {E, L} and a ~ Z1. It can be verified that L(B) = L. Thus c6'/~(La) _C c~D(~ ) and ffN(Sr C c~X(q) ). Con-

One-WayAcceptors and Languages 15t

versely, suppose B = (K!, X1, {1},fl , gz, ho ,Po, FI) is a 1DBA (1NBA) of q~. Let M =- (K~ , Sx , Z2 , 3z , Aa , Po , F1) be the marked (nondeterminisfic) a-gsm, where 272 = 271 u 27o, So = {u ~ 27: u ~ -.~ and for some q ~ K1, f~(y) ---- f (q , y) for all y ~ N}, and for every p ~ Kz and a E 271 w {r $}:

1. ~x(p, a) = gl(P, a, 1); and

2. ;~I(P, a) = {w ~ 272*: w ~ 27z w {e} and (w, d) ~ ga(P, a, 1) for some d~{0,1}; or w = ~ for some u ~ where (u, d) ~ gl(p, a, 1) for some d e {13, 1}}.

It can be ver i f ied that L ( B ) = M-m(~'(~-~0*)), where for each E~27o, �9 (E) = E L L Thus ~D(~) C ~ v ( ~ ) and ~N(~) C ~N(~) . Hence ~D(~b) ----- ~D(oW) and ~N(~) = ~N(.W).

It is worthwhile to observe that the same ~0 works for both the deter- ministic and the nondeterministic cases.

REFERENCES

I. W. J. Chandler, "Abstract families of deterministic languages," Proc. ACM Symp. on Theory of Computing (Marina del Rey, California, 1969), 21-30.

2. S. Ginsburg and S. Greibach, "Abstract families of languages," Memoirs Am. Math~ Soc. 1969(87):1-32.

3. S. Ginsburg, S. Greibach, and J. Hopcroft, "Pre-AFL," Memoirs Am. Math. Soc. 1969(87):41-51.

4. S. Ginsburg and J. Hopcroft, "Two-way balloon automata and AFL," J. ACM 17:3-13 (1970).

5. J. Hopcroft and J. Ullman, "An approach to a unified theory of automata," Bell Syst. Tech. J. 46:1793-1829 (1967).

6. G. F. Rose, "Abstract families of processors," J. Computer Syst. Sci. 4(3):193-204 (1970).