ON
THE
DEGREE
Andreas
Weber
Fachbereich
D-6000
OF AMBIGUITY
and H e l m u t
Informati~,
Frankfurt
OF FINITE
AUTOMATA
Seidl,
J. W. G o e t h e - U n i v e r s i t ~ t ,
am Main,
West Germany.
Abstract
We show that the d e g r e e
maton
(NFA)
with
(c I ~ 2.0566).
whether
the authors
mial-time
finite
We p r e s e n t
the degree
valuedness
of a m b i g u i t y
n states,
is not greater
an a l g o r i t h m
which decides
of a m b i g u i t y
obtain
in
of a NFA is finite
[14] a c o r r e s p o n d i n g
of a n o r m a l i z e d
algorithm
of a n o n d e t e r m i n i s t i c
if finite,
which
finite
upper
transducer
decides w h e t h e r
than
finite
+ c1"n
in p o l y n o m i a l
or not.
bound
(NFT),
auto-
2n ' l ° g ~ n
time
Additionally,
for the finite
and also a polyno-
the v a l u e d n e s s
of a NFT is
or not.
Introduction
Let x be an input
string
of a n o n d e t e r m i n i s t i c
and also of a n o r m a l i z e d
guity
of x in M is d e f i n e d
The v a l u e d n e s s
strings
M
finite
of x in M~
on the a c c e p t i n g
(the v a l u e d n e s s
of an input
of M')
string
of M
transducer
by the n u m b e r
is d e f i n e d
paths
finite
(NFT)
M'.
of a c c e p t i n g
by the n u m b e r
for x in M'.
is the m a x i m a l
automaton
paths
degree
for x in M.
output
of a m b i g u i t y
of a m b i g u i t y
depending
M,
of ambi-
of d i f f e r e n t
The d e g r e e
(M') or is infinite,
(NFA)
The degree
of
(valuedness)
on w h e t h e r
or
not a m a x i m u m exists.
The degree
which
of a m b i g u i t y
only r e c e n t l y
lence p r o b l e m
problem
generalized
attention
and NFT's:
sequential
machines
of a m b i g u i t y
time
[12].
that
the e q u i v a l e n c e
not g r e a t e r
or not the d e g r e e
[13],
with
moves)
of NFA's
in p o l y n o m i a l
conjecture
follows
is d e c i d a b l e
in p o l y n o m i a l
of
is un-
the e q u i v a l e n c e
finite v a l u e d n e s s
[12]
the equiva-
the e q u i v a l e n c e
than d can be tested
of a NFA
parameters,
that the e q u i v a l e n c e
~-(input)
d, however,
d, it can be tested
of a m b i g u i t y
and that
of the E h r e n f e u c h t
of NFT's with
integer
structural
in c o n n e c t i o n
(NFT's w i t h o u t
integer
From a g e n e r a l i z a t i o n
any fixed
are
It is w e l l - k n o w n
is P S P A C E - c o m p l e t e
[5]. For any fixed
w i t h degree
Given
received
for NFA's
for NFA's
decidable
and the v a l u e d n e s s
[4].
time w h e t h e r
(the v a l u e d n e s s
of a NFT
621
[6]i
is greater
space
than d. Chan and Ibarra
algorithms
is finite,
teger.
which
and also w h e t h e r
Moreover,
complete.
Using
algorithms
placed
which
In section
Such an e x p l i c i t
in
of a m b i g u i t y
[2].
dependently
criterion,
in
biguity
In section
of us,
which
[14]
(cf.
Ibarra
can be tested
~-moves,
a criterion
of a N F A
(c I
with
the m e t h o d s
for the i n f i n i t e
in p o l y n o m i a l
and R a v i k u m a r
exhibit
time).
over both r e s u l t s
and to the
+ c1"n
be a c h i e v e d
in e x p o n e n t i a l
carry
are d i f f e r e n t
from those
it is s u f f i c i e n t
for e v e r y
time
de-
(In-
an e q u i v a l e n t
By simple
reduc-
to the d e g r e e
size of p r o d u c t s
of am-
of m a t r i c e s
input
paths
in
to c o n s i d e r
[2] and
NFA's
string x, we i n v e s t i g a t e
of x in the NFA,
[10].
First
of a r e s t r i c t e d
a graph which
and we use
"pumping
of all,
type.
describes
arguments"
in
graphs.
In [14]
chieve
[10]
of a m b i g u i t y
2n ' l ° g 2 n
is t e st a b l e
is resequen-
[2]).
all a c c e p t i n g
these
in
the authors
ideas used
we show that
3 we introduce
of a m b i g u i t y
relation).
the degree
than
is P S P A C E -
(a g e n e r a l i z e d
input-output
in-
two e x p o n e n t i a l - s p a c e
machine
cannot
of a NFA
given
the latter p r o b l e m
if the d e g r e e
upper b o u n d
which
than an a r b i t r a r y
that
is not g r e a t e r
of NFA's,
of NFA's with
IN0
Then,
length-preserving
two p o l y n o m i a l -
of a m b i g u i t y
they o b t a i n e d
of a s e q u e n t i a l
if finite,
gree
The
reduction,
2 of this paper we show that
used
tions,
it is g r e a t e r
solve both problems,
with
with n states,
2.0566).
[2] e x h i b i t e d
the d e g r e e
they were able to prove
a simple
by the v a l u e d n e s s
tial m a c h i n e
over
decide w h e t h e r
the authors
extend
an upper b o u n d
a criterion
the a b o v e - m e n t i o n e d
for the finite
for the infinite
in p o l y n o m i a l
In the first
time.
valuedness
The results
in order
of a NFT,
of NFT's,
are p r e s e n t e d
section we summarize
methods
valuedness
which
can be tested
in section
some d e f i n i t i o n s
to a-
and to d e f i n e
4.
and notations.
I. D e f i n i t i o n s
A nondeterministic
w here
Q and
finite
Z denote
Q0,F~ Q d e n o t e
automaton
nonempty,
sets of initial
(NFA)
finite
is a 5-tuple
sets of states
and final
states,
and
M=(Q,Z,6,Q0,F),
and input
symbols,
6 is a subset of
Qx~xQ.
A path of length m for x from p to q in M is a string
Xmq m 6 Q ( Z Q ) *
n = q0xlql ....
so that x = X l . . . x m 6 Z*, p = q0 6 Q, q = q m 6 Q, and for
622
all
i = I .... ,m
path
(qi_1,xi,qi) 6 6. We d e f i n e
for x f r o m p to q in M e x i s t s } .
Note
~
:: { ( p , x , q ) 6 Q w Ze×Q
I a
t h a t 6 is a s u b s e t of ~. We re-
n a m e ~ by 6.
A p a t h ~ f r o m p to q is c a l l e d
a n d q is a final
all x 6
state.
Z* , for w h i c h
form:
da(M))
in M,
i. e. da(M)
The
paths
= sup
states are
of the d e g r e e s
{daM(x)
"<--->"
to the d e g r e e
t h e n M is c a l l e d
is the n u m b e r
of a m b i g u i t y
of a m b i g u i t y
of M
(short
of all x 6 ~*
of a m b i g u i t y
path
in M.
relation
in M.
If no
reduced.
w i t h a state q 6 Q
(short form:
f r o m p to q and f r o m q to p in M exist.
is an e q u i v a l e n c e
L e t Q be d i v i d e d
daM(x))
if it is on no a c c e p t i n g
A s t a t e p 6 Q is said to be c o n n e c t e d
if p a t h s
state
by M, is the set of
I x 6 ~*}.
useless,
irrelevant
s t a t e of M is u s e l e s s ,
p <---> q),
(short form:
for x in M. The d e g r e e
is the s u p r e m u m
if p is an i n i t i a l
accepted
p a t h in M exists.
of x 6 ~* in M
state q 6 Q is c a l l e d
Useless
path,
the l a n g u a g e
an a c c e p t i n g
~he d e g r e e of a m b i g u i t y
of all a c c e p t i n g
accepting
L(M),
N o t e that
on Q.
into the e q u i v a l e n c e
classes
QI' .... Qk w.r.t.
M is s a i d to be of type
Q0 c Q 1
& FCQk
I, if the f o l l o w i n g holds:
k
k-1
U
QixZxQi U U
~ D (QxT~xQ) c
i=I
i =I
"<--e>".
&
M is said to be of type
t h a t the f o l l o w i n g
Q0c
g r a p h GM(X)
(i=I, .... k) e x i s t
such
holds:
{pl } & F c {qk}
L e t M be of type
2, if s t a t e s p i , q i 6 Qi
Qix~xQi+1.
&
~ n (Q×~XQ) c
2 such t h a t L(M)
= (V,E)
Qix~.xQi u
k-1
u
i=I
{ q i } x Z x { P i + 1 }.
~ ~, and let x = X l . . . x m 6 L(M).
of the a c c e p t i n g
V := {(q,i) 6 Qx{0, .... m}
k
u
i=I
paths
The
for x in M is d e f i n e d by
I (Pl,Xl---xi,q)
6 6 & ( q , x i + 1 . . . X m , q k)
6 6} a n d
E := {((p,i-1), (q,i)) 6 V 2 [ i 6 {I ..... m}
Note
t h a t the n u m b e r
the d e g r e e
Let M=(Q,E,6,Q0,F)
cluded
F'cF.
in M
of p a t h s
of a m b i g u i t y
from
(Pl,0)
to
(qk,m)
in GM(X)
equals
of x in M.
!
a n d M ' : ( Q ' , Z ' , 6 ' , Q 0 ,F')
(short form:
& (P,xi,q) 6 6}.
M' c M ) ,
if Q' c Q ,
be two NFA's.
E' c E, 6' c 6 ,
M'
is in-
Q0 'c Q0 and
623
2. An U p p e r
In this
Bound
section
MI,...,~cM
we
leads
Theorem
the
degree
where
c I := I +
finite
lower
bound
In the
(2/3).iog23
lemma
of is
any NFA M with
N
da(M)
~,
$ ~
i=I
t h e n we
n states
and
bound
of N F A ' s
2 n" (I+I/64)
we c o n s i d e r
the c a s e
Let
with
q0 6 Q0'
implies
q 6 Q.
qF 6 F,
for any
following
lemmas
Lemma
of a m b i g u i t y .
2n ' l ° g 2 n
+ c~-n,
I for
n states
the m a x i m a l
is o p t i m a l .
see
with
The best
[14]).
one e q u i v a l e n c e
Then,
NFA
such
that
for any x 6 E*,
da(M) < ~.
there
Let
is at m o s t
for x f r o m p to q in M.
(q,v,p) 6 6. We a s s u m e
two
degree
"<--->".
p 6 Q be c o n n e c t e d
In the
< 2n ' l ° g 2 n .
than
of N F A ' s
I: L e t M = ( Q , Z , 6 , Q 0 , F ) be a r e d u c e d
This
finite
greater
(for d e t a i l s
Lemma
Proof:
da(M)
of t h e o r e m
with
w.r.t.
path
NFA's
~ 2.0566.
class
one
n states
d a ( M i) a n d N < 2 n - 3 2 n / 3 .
show:
of M is n o t
the u p p e r
of a m b i g u i t y
of A m b i g u i t y
result:
of a m b i g u i t y
we k n o w
first
for
da(M) <
following
know whether
degree
Degree
that
M be a N F A w i t h
Then,
We do n o t
2 such
2 with
to the
1: L e t
the F i n i t e
construct
of type
If M is of type
This
for
there
we consider
2: L e t M be a
6 Z*
are
such
in
a NFA
I such
~ da(M)
<
~ 2 i+I.
M=(Q,Z,6,Q0,F).
(q0,u,p),
paths
Thus,
(q,w,qF) ,
for x f r o m p
da(M)
Proofs
to q.
= ~. ~
|
of the n e x t
[14].
NFA with
of type
that
two d i f f e r e n t
i ~ 0 : daM(ux(vx)iw)
can be f o u n d
M I,...,MNcM
and u , v , w
that
n states.
We c a n c o n s t r u c t
that N < 2n and
the
following
NFA's
assertions
are
true :
N
(I)
da(M)<~
(2)
da(M) = ~ ~ B i 6
Lemma
3: L e t M be a N F A
M 1,...,~cM
are
Z d a ( M i) < ~.
i=I
{I .... ,N}: d a ( M i) = ~.
2 such
~ da(M)
<
true :
da(M)<~
(2)
da(M) = ~ ~ 3 i 6
4: L e t
I with
that N<
n states.
32n/3" a n d
We c a n c o n s t r u c t
the
following
NFA's
assertions
N
(I)
Lemma
of t y p e
of type
~ d a ( M i) < ~.
i=I
{I ..... N}: d a ( M i) = ~.
M be a N F A
of type
2 with
n states
such
that
0 <da(M)<
~.
624
Then,
da(M)
Proof:
bered
~ 2 n'lOg2n.
Let the e q u i v a l e n c e
and the states
ence w i t h
(*)
The
da[M)
lemma
Proof
Base
Qi
QI' .... Qk w.r.t.
(for i=1,...,k)
of type
M is reduced.
"<--->"
be g i v e n
on Q be num-
in c o r r e s p o n d -
2. Since da(M) > 0, we know:
We show by i n d u c t i o n
L(M) ~ 0,
on k:
~ 2n'[l°gzk]-k+1
follows
of
from
(*)
(for f u r t h e r
details
see
[14]).
(*) :
of induction:
follows:
pi,qi6
the d e f i n i t i o n
Q0={Pl }, F={qk},
classes
da(M)
k=1.
Pl is c o n n e c t e d
w i t h q1" A c c o r d i n g
to lemma
I
= 1.
I n d u c t i o n step: Let k ~ 2. D e f i n e 1 := [k/2]. Divide M into NFA's
1
k
M I = ( U Qi ' Z , 61, {pl } , {ql }) and M 2 = ( U
Qi ' Z ' 62 ' {PI+I } '
i=I
i=I+I
1
1-I
{qk}), w h e r e 61 := 6 D ( U Q i x ~ × Q i U U
{qi} x Z x {Pi+1}) and
i=I
i=I
k
k-1
62 := 6 N ( U
Qi×ZxQi U
U
{qi } × Z × {Pi+1 })"
i=i+I
i=i+I
1
k
M I , M 2 are N F A ' s of type 2 w i t h n I := Z #Qi and n 2 :=
~
#Qi states
i=I
i=i+I
such that
0 < d a ( M I) < ~ and 0 < d a ( M 2) < ~.
Let X = X l . . . x m 6 L(M).
edges
Consider
"from Q1 to QI+I"'
Define
i.e.
J := {j 6 {I ..... m}
daM(x)
I
in the g r a p h GM(X)=(V,E)
((ql,J-1), (Pl+1,j)) 6 E}.
daM(x)
~
hypothesis
follows:
Z
2nl'FiOgz[k/2]]-[k/2]+1-2
j6 J
#j . 2 n'[IOgz (k/2) I-k*2
Note that [log2[k/2]]
show: #D = #J ~ 2 n-1.
Since
that
that
# J > 2 n-1
Define
~ 6 {0,...,t}
= [logz(k/2)].
#J > 2 n-1.
and,
•
Jl < J2 and A31
Let t 6 ~.
We observe:
= j 6Z J daM1(Xl.. "xj-1 ).daM2 (xj+1"''Xm)"
F r o m the i n d B c t i o n
We a s s u m e
the set D of all
D = {((ql,J-1), (Pl+1,j)) 6 E ] j 6 {I ..... m}}.
Yt
Consider
for all
=
A.
32
= #j . 2 n ' [ I O g 2 k ] - n - k * 2
Therefore,
for all
j 6 J , PI+I
=:
n2"[l°g2[k/2]]-[k/2]+1
j6 J
it is s u f f i c i e n t
A.
:= [ q 6 Q
to
I (q,J) 6 V}.
6 Aj c Q , 3 jl,J2 6 J exist
such
A.
:= x 1 " ' ' x j
(xj
.xj2)t xj
...x m. For all
I
I +I""
2 +I
and all q 6 A we observe:
625
( Pl ' x1"''xj
1
(xj1+1"''xj2)T
( q ' (xj1+1"''xj2)T
Moreover,
....
x32+I
' q ) 6 d &
Xm ' qk ) £ 6.
•
there is a path for x31+i
.
1
.
from some state in A n i=I
U Qi
..x32
to PI+I in M (via (ql,xj ,Pl+1 ) £ 5). From this follows:
Thus, da(M) = ~. ~
2
GM(X):
x.31+I"" .x.32
x1"''Xjl
0 ..........
daM(Y t) ~ t.
xj2+1...x m
Jl ..........
J2 ..........
Aj I
Aj 2
m
Q1
QI+IF Pl+I
Qk~
qk
|
This completes the proof of theorem I.
3. A Criterion for the Infinite Degree of Ambiguity,
Which is Decidable
in Polynomial Time
Let M=(Q,Z,6,Q0,F)
(DA)
be a NFA. Consider the following criterion for M:
I H q0 6 Q0
3 p,q, 6 Q B q F 6 F
& (p,v,p),(p,v,q),(q,v,q) 6 6
v
v
H u , v , w 6 E*
:
& (q,w,q F) 6 6
(q0,u,p) 6 6
& p ; q.
626
If M c o m p l i e s
i.e.
da(M)
there
with
= ~.
(DA),
is a N F A M' c M
section
(DA),
show
that
M'
too.
This
leads
only
We can
decide
to the
if M c o m p l i e s
The
graph
G3 =
(Pi,a,qi) 6 6 }.
H p,q6 Q
degree
Thus,
with well-known
(on a R A M u s i n g
with
(DA)
Theorem
(see
3: L e t
decidable
With
O(n6.#Z
is i n f i n i t e .
and hence
i,
2 and
3,
In this
M complies
with
of M is i n f i n i t e ,
or n o t a N F A c o m p l i e s
NFA with
I H a
cost
6
V
(DA)
we
i 6
with
Consider
the
{1,2,3}
:
as follows:
(p,p,q)
can
criterion)
theorem
n states.
~
from
algorithms
M be a N F A w i t h
in time
daM(uvlw)
to l e m m a
E 3 :=
in G 3 a p a t h
the u n i f o r m
i 6~
result:
us to r e w r i t e
graph
[I]).
da(M')
of a m b i g u i t y
be a r e d u c e d
&
all
(DA).
(Q3,E 3) , w h e r e
p ~ q
for
according
(DA),
time w h e t h e r
G 3 allows
:
that
with
((PI'P2'P3)' (ql , q 2 ' q 3 )) 6 Q3 x Q3
= {
that
then,
following
with
in p o l y n o m i a l
(DA) : L e t M = ( Q , ~ , ~ , Q 0 , F )
directed
2 such
complies
2: L e t M be a NFA.
if a n d
we o b s e r v e
is i n f i n i t e ,
of type
we
Theorem
then
If da(M)
to
test
(p,q,q)
in time
whether
or n o t
exists.
O(n6-#Z
+ n 8)
M complies
2 follows:
n states
and
+ n 8) w h e t h e r
an i n p u t
or n o t
alphabet
the d e g r e e
Z.
It is
of a m b i g u i t y
of M is i n f i n i t e .
We r e m a r k
(see
~14]).
degree
The
that
Proof:
Case
and
This
2 of
Q0={PI },
implies:
and with
the p r o o f
of type
2 such
Let
and
improved
theorem
of t h e o r e m
that
da(M)
to O(n6-#Z)
I the e x a c t
in p o l y n o m i a l
the e q u i v a l e n c e
the
states
pi,qi6
with
the d e f i n i t i o n
F={qk} ,
M is r e d u c e d .
are p',q' 6 Q and y C
two d i f f e r e n t
3 c a n be
(finite)
space.
2.
is infinite.
Then,
(DA).
on Q be n u m b e r e d
I: T h e r e
[2]
completes
in c o r r e s p o n d e n c e
know:
in t h e o r e m
of a N F A can be c o m p u t e d
Let M=(Q,Z,6,Q0,F).
"<--->"
given
lemma
with
bound
theorem
5: L e t M be a N F A
M complies
we
With
time
of a m b i g u i t y
following
Lemma
the
paths
B p,q 6 Q
E* such
for y f r o m p'
of
:
( p ' , y l , p ) , (p',yl,q), (p,y2,q'), (q,y2,q') 6 6
QI,...,Qk
w.r.t-
(for i=I ..... k)
type
t h a t p'
to q'
3 y l , Y 2 , y 3 6 ~*
classes
Qi
2. S i n c e
be
da(M)
is c o n n e c t e d
= ~,
with
in M exist.
p ~ q
&
&
y = ylY2
&
(q',y3,p') 6 6. D e f i n e
q',
627
v := y2Y3Yl , then:
with
{p,q}×{v} × {p,q} c 6. Since M is reduced,
M complies
(DA)..
Case 2: For all i 6 {1,...,k},
all p',q' 6 Q i, and all y 6 Z* there
is at
most one path for~ y from p' to q' in M.
Let X = X l . . . X m 6 L ( M )
graph GM(X)=(V,E)
and 1 6 {I,...,k-I}
the set Dl(X)
(note:
of all edges
k > 2). Consider
in the
"from Q1 to QI+I"'
i.e.
Dl(X ) = { ((qi,3-I), (Pi+i,3))6 E i 3 6 {I ..... m}}. Define n := #Q. We
are able to choose x 6 L ( M ) and 1 6 {I,...,k-I} so that #Dl(X) > 2 n-1.
Otherwise
we would have,
V x6L(M)
i.e.
da(M)
because of case 2:
k-1
~ #Dl(X) < 2 (n-l)" (k-l) ,
1=I
: daM(x) <
<~
Just like in the proof of lemma
with the following
u,y~ E ,
6
E
,
A
((ql, lUl-1),(Pl+l,lUl)) 6 E , ((ql, luyl-1), (Pl+1,1uyi))
:= {q6 Q i (q, lul) 6 V} = {q6 Q I (q, luyl)6 V}.
states r i6 A
Choose
such that
r I 6A
(i > I) as follows:
si 6 A
states
such that it holds:
~ si=sj)
u
QI r
i ~'
(i=1,2,...).
Y
I I
r4 ~
r4
r3 ~
r3
w
~
~
.~
~
%V
~
rl
ql
rl
1+I
pl+1:s0
So
s I
s I
s2
s2=s4=s6=:q
s3
s3=s 5
A
s o := PI+I 6 A.
(V
j 6 {I,...,i}:
iN such
y ;z
Q1
A
Define
(ri,Y,ri_ I)
= ri1+i 2 =: p.
There are i 3 6 IN0 and i 4 6
r2
°kl
such that
(si_1,Y,S i) 6 6 &
p : =r 5 =r 2
L
ri6A
6 ~ such that ril
(i => 0) as follows:
We construct
Choose
si_1=sj_1
Let a I 6 7~ so that y = Ylal.
(rl,Yl,q I) 6 6. Choose
(i=2,3 .... ). There are il,i 2
si6A
x = uyw
properties:
We construct
6 ~
4 we can find a decomposition
Y'~
Y
;I-1 Yl
"
y'3
628
that
si3 = si3+i 4 =: q and i i + i 3 = 0 m o d i 2.
In c o n c l u s i o n ,
(p , yi2
(PI+I
w e know:
, p) £ 6 ,
, yl3
, q) 6 6 , (q , yl4
p a n d q are d i f f e r e n t ,
PI+I"
Choose
v := yi2-jl.
(p,v,p)
Moreover,
Hence,
in
6 6
,
(p,v,q)
6 6 ,
we k n o w t h a t Pl 6 Q0'
(DA).
[10]
be c o n n e c t e d
with
i 4, and d e f i n e
(q,v,q) 6 6.
(Pl 'u'p)
6 6, qk 6 F and
2 contains
for the e x p o n e n t i a l
(for d e t a i l s
(q,w,qk)
£ 6.
see
degree
the c o r e of a b r i e f p r o o f
of a m b i g u i t y ,
which was
intro-
[14]).
on V a l u e d n e s s
We are able to m o d i f y
the t h e o r e m s
of a N F A is r e p l a c e d
in this
section,
I and 3 so t h a t the d e g r e e
by the v a l u e d n e s s
the p r o o f s
finite
where
and F h a v e the same m e a n i n g
Q'Z'Q0
transducer
are g i v e n
A normalized
finite
ql w o u l d
•
t h a t the p r o o f of l e m m a
4. R e s u l t s
stated
otherwise
Then, w e have:
of a c r i t e r i o n
guity
because
' Pl+1 ) £ 6 ,
, q) 6 6.
Jl ~ ( i 1 + i 3 ) / i 2 so t h a t 91 = 0 m o d
M complies with
We r e m a r k
duced
(p , y i l - l y I , ql ) 6 6 , (ql ' al
output alphabet,
in
is
[14].
is a 6 - t u p l e
M = (Q,Z,A,6,Q0,F) ,
as in a NFA,
a
6 is a f i n i t e
of ambi-
The o u t c o m e
A is a n o n e m p t y ,
( Z U { E } ) x A * x Q.
m
A p a t h w i t h i n p u t x a n d o u t p u t z in M is a s t r i n g ~ = q0
H xiziqi
i=I
6 Q ((ZU{e})A*Q)*
such t h a t X = X l . . . x m 6 Z*, Z = Z l . . . z m 6 4*, a n d for all
i=1,...,m
and
(NFT)
of a NFT.
(qi_1,xi,zi,qi) £ 6. A c c e p t i n g
s u b s e t of Q x
paths
are d e f i n e d
just like in
NFA.
The v a l u e d n e s s
strings
of x 6 Z* in M
(short form:
z 6 4" so t h a t an a c c e p t i n g
in M. The v a l u e d n e s s
of M
path with
(short form:
set { V a l M ( X ) I x 6 ~*}. M o r e o v e r ,
ValM(X))
i n p u t x and o u t p u t
val(M))
we d e f i n e
:= m a x ( { 1 } U{# 6 n ({p} × {a} x A * x {q})
diff(6)
:= m a x ( { 0 } U { I Iz11-1z211
Theorem
6 6} U {Izl
4: L e t M = (Q,Z,A,6,Q0,F)
z exists
is the s u p r e m u m of the
val(6)
(Pi,a,zi,qi)
is the n u m b e r of all
I
H a £ Z
I p,q6 Q , a6~U:{E}
V i=1,2
B pi,q i6 Q :
I B p , q 6 Q : (p,£,z,q) 6 6}).
be a N F T w i t h n s t a t e s
}),
and finite
629
valuedness. Then, the v a l u e d n e s s of M is not g r e a t e r than
2 ( n - 5 ) - l o g z n + c3-(n-I) , val(~)n-1
(1+diff(~))n-1 . #An4-diff(~)
w h e r e c 3 := 21 + (2/3)-iog23 ~ 22.0566.
T h e o r e m 5: Let M be a NFT.
It is d e c i d a b l e in p o l y n o m i a l time w h e t h e r
or not the v a l u e d n e s s of M is infinite.
References
[I]
A. Aho, J. Hopcroft,
C o m p u t e r Algorithms.
J. Ullman (1974): The Design and A n a l y s i s of
A d d i s o n - W e s l e y , Reading, Mass..
[2]
T.-h. Chan, O. Ibarra (1983): On the F i n i t e - V a l u e d n e s s
Sequential Machines. TCS 23, pp. 95-101.
[3]
K. Culik II, J. K a r h u m ~ k i (1985): The E q u i v a l e n c e P r o b l e m for Sing l e - V a l u e d 2-Way T r a n s d u c e r s is Decidable. R e s e a r c h Report, CS 85-24,
U n i v e r s i t y of Waterloo.
[4]
K. Culik II, J. K a r h u m ~ k i (1985): The E q u i v a l e n c e of F i n i t e V a l u e d
T r a n s d u c e r s (on HDTOL Languages) is Decidable. T e c h n i c a l Report,
U n i v e r s £ t y of Waterloo.
[5]
T. Griffiths (1968): The U n s o l v a b i l i t y of the E q u i v a l e n c e P r o b l e m
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[6]
E. Gurari, O. Ibarra (1983): A N o t e on F i n i t e - V a l u e d and F i n i t e l y
A m b i g u o u s Transducers. Math. Systems Theory 16, pp. 61-66.
[7]
M. H a r r i s o n (1978): I n t r o d u c t i o n to Formal L a n g u a g e Theory.
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[8]
K. H a s h i g u c h i (1982): L i m i t e d n e s s T h e o r e m on F i n i t e A u t o m a t a w i t h
Distance Functions. JCSS 24, pp. 233-244.
[9]
J. Hopcroft, J. Ullman (1979): I n t r o d u c t i o n to A u t o m a t a Theory,
L a n g u a g e s and Computation. A d d i s o n - W e s l e y , Reading, Mass..
P r o b l e m for
Ad-
[10] O. Ibarra, B. R a v i k u m a r (1986): On Sparseness, A m b i g u i t y and other
D e c i s i o n Problems for A c c e p t o r s and Transducers. STACS 1986, pp.
171-179.
[11] Ch. Kintala, D. W o t s c h k e
nite Automata. A c t a Inf.
(1980): A m o u n t s of N o n d e t e r m i n i s m in Fi13, pp. 199-204.
[12] R. Stearns, H. Hunt III (1985): On the E q u i v a l e n c e and C o n t a i n m e n t Problems for U n a m b i g u o u s R e g u l a r Expressions, R e g u l a r Grammars, and Finite Automata. S I A M J. Comp. 14, pp. 598-611.
[13] L. Stockmeyer, A. Meyer (1973): Word P r o b l e m s R e q u i r i n g E x p o n e n tial Time (Preliminary Report). Proc. 7th A C M Symp. on P r i n c i p l e s
of P r o g r a m m i n g Languages, pp. I-9.
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von e n d l i c h e n A u t o m a t e n und Ubersetzern. G o e t h e - U n i v e r s i t ~ t , Ffm..