Information Processing Letters 19 (1984) 9-12

North-Holland

26 July 1984

SOME REMARKS ON SUBCLASS CONTAINMENT PROBLEMS FOR SEVERAL CLASSES OF DPDA’S

Michio OYAMAGUCHI

Faculty oj Engineering, Mie University, Tsu - shi 514, Japan

Communicated by L. Boasson Received February 1984

Keywords: Subclass containment, deterministic pushdown automaton, nonsingular dpda’s. proper dpda’s, dpda’s with neces- sary stacking derivation

Introduction

The subclass containment problems for de- terministic pushdown automata (dpda’s) have re- ceived much attention in recent years and several results on the problems have been given (see [3,4]). The containment problem relative to a class hp, written as containment (dpda, U), is the problem of deciding for a dpda M whether there exists a machine in the class 6p accepting the same lan-

guage as M. In this article, we show that containment (dpda,

N,) is undecidable for the class N, of nonsingular dpda’s (Theorem 1). and containment (dpda, U) is

undecidable for any %‘g~ (P, NM, NQ, NSD} where P, NM, NQ and NSD are respectively the classes of proper dpda’s, dpda’s with necessary modes, dpda’s with only necessary quintets and dpda’s with necessary stacking derivation [5,3] (Theorem 2).

1. Notations

A dpda is a 6-tuple M =I (Q, r, Z, A, c,, F) (states, pushdown symbols, input alphabet, transi- tion rules, initial configuration, accepting modes) [4]. A configuration is a member c = (q, u) E Q X r*, and the height of c is ICI= @I. If u = VA, A E r

and (q, A) ha (p, w) E A, then we write the com-

putation from c for input a E I: U (c} as (q, VA)

+ a (P. VW>.

A sequence of computations c, jal c, + . . - da- c, is written as c, +a c, where Q = a,. . .a,,. We write the language accepted from a configura- tion c as L(c). Two configurations c and c’ are equivalent, c = c’, if L(c) = L(c’). The language accepted by M is L(M) = L(c,). A configuration c is reachable if c, + p c for some CL E Z*, and live if L(C)#@ If cs += c and there exists no E computa- tion from c, then we denote c by config( We denote CONFIG the set of reachable configura- tions of M. Henceforth, without loss of generality, we assume that

(1) the initial configuration c, is in Q x I’, (2) (4. A) -,a (p, w) in A implies WI < 2, and (3) all reachable configurations are live.

For a subclass V of dpda’s, we let 9’(U) = {L(M) I M E 59 }. Let D, be the class of dpda’s with empty stack acceptance.

2. Undecidability of containment (dpda, NO)

A pair {(p, w), (p’, ww’)} of configurations is strongly reachable if there exists a reachable con- figuration (q, WA), A E I’, such that (q, A) jB (p, r) and (q, A) +I (p’, w’) for some & y in Z*. A dpda M in D, is weakly (w-) nonsingular (resp. nonsingular) if there exists a positive constant n,

0020-0190/84/S3.00 0 1984, Elsevier Science Publishers B.V. (North-Holland) 9

Volume 19, Number 1 INFORMATION PROCESSING LETTERS 26 July 1984

such that for any strongly reachable (resp. reacha-

ble) pair ((p. w), (p, ww’)} of configurations,

(p, w) = (p, ww’) implies ]w’] d no [2,4]. Let WN,, N, be the classes of w-nonsingular, nonsingular

dpda’s, respectively. We show that 9(WN,)=y(N,). Hence, con-

tainment (dpda, N,) is undecidable, because con- tainment (dpda, WN,) is so [4, Theorem 4.11.

Lemma 1.9(WN,) =9’(N,,).

Proof. 5?(WN,)a9(N0) is obvious. To show p(WN,,) E.V(N,), let M = (Q, P, Z, A, c,, F) in WN,. Without loss of generality, we can assume that M is real-time [4]. Let a homomorphism h: (r x Q x P u r)* -) P* satisfy that h(A) = A, h([A, q, C]) = A for A, C E r, q E Q. Then, we construct a dpda M’ = (Q, I”, Z, A’, c,, F) where r’ = r x Q x P u r and A’ satisfies that for q, p in Q. A, B,, B, in r, w in (P u ( E}),

(i) (q, A) -+a (P, B,B,)E A *(eA’) --ra (~9 [B,, qv Al 4) E A’

for all A’ such that h(A’) = A,

(ii) (q, A) -,a (P, w) E A -(q,A’) --+“(p,w)~A’

for all A’ such that h(A’) = A.

By the definition of M’, note that (q, W)E CONFIG implies that (q, h(w)) E CON-

FIG(M), (cl, W) 3 (9, h(w)) and w E (I” - r)* (r u

{E}). So, L(M) = L(M’) holds. Further, note that if (q. wA’w’) E CONFIG and A’ = [A, p, C] E r x Q x l?, then (p, h(w)C) E CONFIG and for some input j3, (p, C) + L (q, h(A’w’)) is a compu- tation of M.

To show M’ E N,, consider any two reachable configurations c = (q, w) and c’ = (p, ww’) of M’ such that c = c’ and ]w’] > 0. Let w = wi A’ and A’ = [A, r, B] E r x Q x P. Note that ]w’] > 0 im-

plies lw] > 0 (since M is real-time, M E D,, and c = c’) and the existence of such A’. Then, by the above arguments, we have the following computa- tions of M:

cs -+ b (r, h(w,)B) + & (q, h(w,)A) = d,

(r, h(wt)B) -) t$ (P. h(w,)Ah(w’)) = d’,

for some inputs OL, I$ y. Note that d = d’ by c = c’. Let d + L e = (q’, h(w,)) for some q’ and let d’

10

- L e’ = (p’, h(w,)u), where 6 is a minimal input such that (q, A) - 6 (q’, E). It is well known that ]6] is bounded by a fixed constant, say k,. So, if ]Ah(w’)] > k,, certainly d’ + L (p’, h(w,)u) for some p’, LA. In this case, the pair (e, e’} is strongly reachable and e = e’. so that ]u] is bounded by a fixed constant no, because M E WN,. It follows that ]w’] = ]h(w’)] < no + k,. Hence, M’ E N,.

Thus, 9(WN,)c9(N,) holds. •I

Theorem 1. Containment (dpda, N,) is undecida-

ble.

3. Other containment problems

Recently, several non-real-time subclasses (P, NM, NQ, NSD) were introduced in [3,5] to investigate the equivalence problems and to check whether the classes have the same generation capacity as that of general dpda’s. Courcelle [l] gave a negative solution to the latter problems by showing that

L, = {ua”b”g(u) In 2 1)

U {ua”bmh(u) In, m 2 1, n + m}

for some homomorphisms h, g is not in 4P(NSD), but inp(D,,). Here, ii is the mirror image of u. We

define a similar language to L,. For any two homomorphisms h, g : X* + Y *

where h(x) f c, g(x) Z E for any x in X, let

L, = { udvda” b*cir, udvda” b”eh( ti) In 2 1,

u,vEX+}

u { udvda” bmcb, udvda” b”eg( t) In, m > 1,

n#m,u,vEX+},

where

XnY=@ and (XuY)n{a,b,c,d,e}=~.

Clearly, L, E 9’(D,). For VE {P, NM, NQ, NSD} we show that L, EZ(V) iff h(S) f g(Q) for all u in X + . Thus, Post’s correspondence problem reduces to containment (dpda, 59’).

Lemma2.L,~9((Fg)~h(ti)+g(b)foraNuinXC where 5%‘~ (P, NM, NQ, NSD}.

Volume 19. Number 1 INFORMATION PROCESSING LETTERS 26 July 1984

Proof. (=): If h(B) = g(B) for some u in X+, then

L, E9(V). Let a dpda M accept L, and let u satisfy that

h(a) = g(k). Clearly while M reads ui, i > 0, and then reads

du’, k > 0, the stack height of M must increase. And while M reads a”, n > 0, after reading uidukd, the stack height must increase. Note that config (u’du’da”) = config(uidukdam+d), where m’ > 0, iff i = k, but the finite state control of M cannot determine whether i = k. Similarly, while M reads b”, m > 0, after reading uidukda”, the stack height must decrease in order to check whether n = m. So, in the case where k = i, the computation from

config(u’du’d) increases the stack for input a” and then decreases it for input b”. Since we have config(u’du’da”) = config(u’du’da”‘) for any n, n’ > 0, it holds that MB NSD and ME NM. Under the computation from config(u’du’da”) for input b”, the number of the stack symbols popped by non-c moves (i.e., the number of non-removable elements) is unbounded for n = 1, 2, . . . . Thus, M E P holds. Further, by

config(u’du’da”b”) = config(u’du’da”b”‘)

for any m, m > 0

we have M Q NQ. (e): If h(n) # g(u) for all u E X+ , then L, E

P(V). Let M be a dpda accepting L, such that for u,

v~X*andn,m,m’>,O,wheren>m,

c, 4” (q,, u) -,dv (qz, udv)

+ da” (q3, udvda” )

+ bm (q4, udvda”-“) (= c,)

-,b”-m (q5, udvd) (= c5)

+ bm’ (q, udvd) ( = c6)

and

I&)-H(u,v)u{b}+G(u,v),

L(c,) = {b}*G(u, v),

L(c4)n((c}X* u {e}Y*) = G(u, v),

where

H(u, v) = {cc, eh(E)}, G(u, v) = {cu, eg(o)}.

Note that

H(u, v) + G(u. v) for all u, v E X- 1

because h(b) + g(E) for all u in X’. We observe that if (q,. u) = (q,, u’) for u, u’ E

X*, then u = u’ holds, and if (q2, udv) ‘i: (q2. udv’) for v, v’E X*, then v = v’ holds. Further, if

(9,. udvda”) = (qJ, udvda”) for n, m > 0. then n = m holds, because H(u, v) + G(u, v). Hence, in the case where M increases the stack height, M passes through pairwise inequivalent configura- tions. Thus, M E NM and N E NSD holds. Con- sider the case where the stack height of M de- creases. Lex x = udvda”b” for n > m 2 0. By

H(u, v) + G(u, v), we have config * config(xb’) for any d> 0. Further, while M makes the compu- tation from config(xb!) for input y E {c}X* u

{e}y*, any subcomputation c, +a c2 for input OL, where a # c, satisfies ci * c2. Thus, hi E NQ holds. It remains to show that M E P.

Let config = config for input strings x, y where L(config(x)) # 8. Then, x belongs to exactly one set of

Type 1:

Type 2 :

Type 3 :

Type 4:

We write lgid4. then i=j

the following four types:

X* u X*(d}X*,

(X*{d))2((a)* + u{a)-{b)*)7

(X*{d})2{a}+{b}+{c}X*.

(X~~d~)~~a~‘~b~‘~e~Y~.

type(x) = i if x belongs to the type i,

Clearly, if type(x) = i and type(y) = j, holds, 1 d i, j Q 4. And if type(x) = 1.

then obviously x = y. Consider the case where type(x) = 2. Note that there exist no u, v, u’, v’ in

X+ such that

(H(u, v) = G(u’, v’) A G(u, v) = H(u’, v’))

v (H(u, v) = G(u’, v’) A G(u, v) = G(u’, v’)),

because h(G) # g(B) for all u in X-. Hence, we have

(1) n # m = config(udvda”) * config(u’dv’da”),

(2) (n 2 m) A (n - m + n’ - m’)

* config( udva” b”’ ) = config( u’dv’da” b&) .

Using (1) and (2), we can easily show that if x = udvda”, n > 0, then x = y, and if x = udvda”b”,

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Volume 19, Number 1 INFORMATION PROCESSING LETfERS 26 July 1984

n 2 m ) 0, then config = config( Further, if x = udvda”bm and n c m, then y = u’dv’da*‘b”’ where n’ c m’, it J ii’ and g(9) = g(V). In this case,

since {v’ Ig(O) = g(V)} is a finite set, the weight is bounded in the case of configurations equivalent to config( Similarly, this holds in the case where type(x) E { 3,4}. Thus, M E P holds. CI

Theorem 2. Containment (dpda, W) is undecidable

for WE (P, NM, NQ, NSD}.

The machine containment problem for VE (P, NM, NQ, NSD}, i.e., whether or not a dpda is in V, can easily be shown to be undecidable, since the machine M defined in the proof of Lemma 2 is in Viff h(5) # g(b) for all u in X’.

References

[l] B. Courcelle. Some negative results concerning dpda’s. In-

form. Process. Lett. lS(5) (1984) 285-289.

[2] E.P. Friedman. A note on nonsingular deterministic push-

down automata. Theoret. Comput. Sci. 7 (1978) 333-339.

131 Y. ftzhaik and A. Yehudai. New families of non real-time

dpda’s and their decidability results, Theoret. Comput. Sci.,

to appear.

[4] M. Oyamaguchi, Some results on subclass containment

problems for special classes of dpda’s related to nonsingular

machines. Theoret. Comput. Sci. 31(3) (1984) 317-335.

15) E. Ukkonen, The equivalence problem for some non-real-

time deterministic pushdown automata, J. Assoc. Comput.

Mach. 29 (1982) 1166-1181.