- Email: [email protected]
Some remarks on subclass containment problems for several classes of DPDA'S
26 July 1984
Information Processing Letters 19 (1984) 9-12 North-Holland
SOME REMARKS OF DPDA’S
ON SUBCLASS
CONTAINMENT
PROBLEMS
FOR
SEVERAL
CLASSES
Michio OYAMAGUCHI Faculty oj Engineering, Mie University,
Tsu - shi 514, Japan
Communicated by L. Boasson Received February 1984
Keywords:
Subclass containment, deterministic sary stacking derivation
pushdown
automaton,
nonsingular
dpda’s. proper dpda’s, dpda’s with neces-
Introduction
putation
The subclass containment problems for deterministic pushdown automata (dpda’s) have received 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 language 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).
+ 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 configuration 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 CLE Z*, and live if L(C)#@ If cs += c and there exists no E computation from c, then we denote c by config( We denote CONFIG the set of reachable configurations 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.
1. Notations
2. Undecidability of containment (dpda, NO)
A dpda is a 6-tuple M =I (Q, r, Z, A, c,, F) (states, pushdown symbols, input alphabet, transition 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-
A pair {(p, w), (p’, ww’)} of configurations is strongly reachable if there exists a reachable configuration (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)
from c for input
a E I: U (c}
as (q, VA)
9
INFORMATION
Volume 19, Number 1
such that for any strongly
reachable (resp. reachaof configurations, ble) pair ((p. w), (p, ww’)} (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, containment (dpda, N,) is undecidable, because containment (dpda, WN,) is so [4, Theorem 4.11. Lemma 1.9(WN,)
(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 CONFIG(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 computation 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 implies 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 computations of M: cs -+ b (r, h(w,)B) (r, h(wt)B)
+ & (q, h(w,)A)
-) t$ (P. h(w,)Ah(w’))
= d,
26 July 1984
- 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 ble.
1. Containment
(dpda,
N,)
is undecida-
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) U
{ua”bmh(u)
In 2 1) 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’).
= 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
LETTERS
=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)
PROCESSING
Lemma2.L,~9((Fg)~h(ti)+g(b)foraNuinXC where 5%‘~(P, NM, NQ, NSD}.
INFORMATION
Volume 19. Number 1
PROCESSING
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 P(V). Let M be a dpda accepting v~X*andn,m,m’>,O,wheren>m, c, 4”
L, E
L, such that for u,
(q,, u) -,dv (qz, udv)
+ da” (q3,
udvda” )
+ bm(q4,
udvda”-“)
-,b”-m (q5,
udvd)
+ bm’(q, udvd)
(= c,)
I&)-H(u,v)u{b}+G(u,v), = {b}*G(u,
L(c4)n((c}X*
u {e}Y*)
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 udvda”) = (qJ, udvda”) for n, m > 0. then n (9,. = m holds, because H(u, v) + G(u, v). Hence, in the case where M increases the stack height, M passes through pairwise inequivalent configurations. Thus, M E NM and N E NSD holds. Consider the case where the stack height of M decreases. 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 computation from config(xb!) for input y E {c}X* u any subcomputation c, +a c2 for input OL, {e}y*, 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 the following four types: Type 1:
X* u X*(d}X*,
Type 2 :
(X*{d))2((a)*
Type 3 :
(X*{d})2{a}+{b}+{c}X*.
Type 4:
(X~~d~)~~a~‘~b~‘~e~Y~.
(H(u,
v) = G(u’, v’) A G(u, v) = H(u’, v’))
v (H(u,
(2) = G(u, v),
where G(u, v) = {cu, eg(o)}.
+ u{a)-{b)*)7
We write type(x) = i if x belongs to the type i, Clearly, if type(x) = i and type(y) = j, lgid4. holds, 1 d i, j Q 4. And if type(x) = 1. then i=j then obviously x = y. Consider the case where type(x) = 2. Note that there exist no u, v, u’, v’ in X+ such that
v) = G(u’, v’) A G(u, v) = G(u’, v’)),
h(G) # g(B) for all u in X-.
(1) n # m = config(udvda”)
v),
H(u, v) = {cc, eh(E)},
H(u, v) + G(u.
because have
and
L(c,)
Note that
(= c5)
( = c6)
26 July 1984
LETTERS
Hence,
we
* config(u’dv’da”),
(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”, 11
Volume
19, Number
1
INFORMATION
PROCESSING
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, for WE (P, NM, NQ, NSD}.
W) is undecidable
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’.
LETfERS
26 July 1984
References [l] B. Courcelle. Some negative results concerning dpda’s. Inform. Process. Lett. lS(5) (1984) 285-289. [2] E.P. Friedman. A note on nonsingular deterministic pushdown 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-realtime deterministic pushdown automata, J. Assoc. Comput. Mach. 29 (1982) 1166-1181.
Information Processing Letters 19 (1984) 9-12 North-Holland
SOME REMARKS OF DPDA’S
ON SUBCLASS
CONTAINMENT
PROBLEMS
FOR
SEVERAL
CLASSES
Michio OYAMAGUCHI Faculty oj Engineering, Mie University,
Tsu - shi 514, Japan
Communicated by L. Boasson Received February 1984
Keywords:
Subclass containment, deterministic sary stacking derivation
pushdown
automaton,
nonsingular
dpda’s. proper dpda’s, dpda’s with neces-
Introduction
putation
The subclass containment problems for deterministic pushdown automata (dpda’s) have received 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 language 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).
+ 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 configuration 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 CLE Z*, and live if L(C)#@ If cs += c and there exists no E computation from c, then we denote c by config( We denote CONFIG the set of reachable configurations 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.
1. Notations
2. Undecidability of containment (dpda, NO)
A dpda is a 6-tuple M =I (Q, r, Z, A, c,, F) (states, pushdown symbols, input alphabet, transition 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-
A pair {(p, w), (p’, ww’)} of configurations is strongly reachable if there exists a reachable configuration (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)
from c for input
a E I: U (c}
as (q, VA)
9
INFORMATION
Volume 19, Number 1
such that for any strongly
reachable (resp. reachaof configurations, ble) pair ((p. w), (p, ww’)} (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, containment (dpda, N,) is undecidable, because containment (dpda, WN,) is so [4, Theorem 4.11. Lemma 1.9(WN,)
(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 CONFIG(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 computation 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 implies 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 computations of M: cs -+ b (r, h(w,)B) (r, h(wt)B)
+ & (q, h(w,)A)
-) t$ (P. h(w,)Ah(w’))
= d,
26 July 1984
- 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 ble.
1. Containment
(dpda,
N,)
is undecida-
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) U
{ua”bmh(u)
In 2 1) 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’).
= 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
LETTERS
=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)
PROCESSING
Lemma2.L,~9((Fg)~h(ti)+g(b)foraNuinXC where 5%‘~(P, NM, NQ, NSD}.
INFORMATION
Volume 19. Number 1
PROCESSING
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 P(V). Let M be a dpda accepting v~X*andn,m,m’>,O,wheren>m, c, 4”
L, E
L, such that for u,
(q,, u) -,dv (qz, udv)
+ da” (q3,
udvda” )
+ bm(q4,
udvda”-“)
-,b”-m (q5,
udvd)
+ bm’(q, udvd)
(= c,)
I&)-H(u,v)u{b}+G(u,v), = {b}*G(u,
L(c4)n((c}X*
u {e}Y*)
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 udvda”) = (qJ, udvda”) for n, m > 0. then n (9,. = m holds, because H(u, v) + G(u, v). Hence, in the case where M increases the stack height, M passes through pairwise inequivalent configurations. Thus, M E NM and N E NSD holds. Consider the case where the stack height of M decreases. 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 computation from config(xb!) for input y E {c}X* u any subcomputation c, +a c2 for input OL, {e}y*, 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 the following four types: Type 1:
X* u X*(d}X*,
Type 2 :
(X*{d))2((a)*
Type 3 :
(X*{d})2{a}+{b}+{c}X*.
Type 4:
(X~~d~)~~a~‘~b~‘~e~Y~.
(H(u,
v) = G(u’, v’) A G(u, v) = H(u’, v’))
v (H(u,
(2) = G(u, v),
where G(u, v) = {cu, eg(o)}.
+ u{a)-{b)*)7
We write type(x) = i if x belongs to the type i, Clearly, if type(x) = i and type(y) = j, lgid4. holds, 1 d i, j Q 4. And if type(x) = 1. then i=j then obviously x = y. Consider the case where type(x) = 2. Note that there exist no u, v, u’, v’ in X+ such that
v) = G(u’, v’) A G(u, v) = G(u’, v’)),
h(G) # g(B) for all u in X-.
(1) n # m = config(udvda”)
v),
H(u, v) = {cc, eh(E)},
H(u, v) + G(u.
because have
and
L(c,)
Note that
(= c5)
( = c6)
26 July 1984
LETTERS
Hence,
we
* config(u’dv’da”),
(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”, 11
Volume
19, Number
1
INFORMATION
PROCESSING
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, for WE (P, NM, NQ, NSD}.
W) is undecidable
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’.
LETfERS
26 July 1984
References [l] B. Courcelle. Some negative results concerning dpda’s. Inform. Process. Lett. lS(5) (1984) 285-289. [2] E.P. Friedman. A note on nonsingular deterministic pushdown 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-realtime deterministic pushdown automata, J. Assoc. Comput. Mach. 29 (1982) 1166-1181.