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NP and Craig's Interpolation Theorem
LOGIC COLLOQUIUM '82 G. Lolli. G. Long0 and A . Marqa (editors) 0 Ekevier Science Publishers B. V. (North-Holland), I984
345
NP AND C R A I G ' S INTERPOLATION THEOREM
Daniele Mundici
.
L oc Romol a N .76 50060 Donnini
F1 orencc-I t a l y
The t r u t h - v a l u e s of two renowned c o n j e c t u r e s about NP (namely, P f NP and, NP i s n o t c l o s e d under complementation) depend on t h e d i f f i c u l t y i n w r i t i n g down C r a i g ' s i n t e r p o l a n t s i n s e n t e n t i a l l o g i c . The g e n e r a l connection between NP and i n t e r p o l a t i o n i s s t u d i e d by b l e n d i n g i d e a s and t e c h n i q u e s from both model t h e o r y and computation t h e o r y .
0.
Introduction.
W e f i x throughout an a1 habet
c
and r e g a r d Boolean e x p r e s s i o n s a s p a r t i c u l a r words over I f B-+C i s a tautology, then Craig's i n t e r p o l a t i o n theorem y i e l d s an i n t e r p o l a n t I , t h a t i s , a Boolean e x p r e s s i o n I such t h a t B + I and IC are t a u t o l o g i e s , and t h e v a r i a b l e s o c c u r r i n g i n I are e x a c t l y t h o s e which j o i n t l y occur i n B and C. I n Theorem 2 and C o r o l l a r y 6 w e prove t h e f o l l o w i n g r e s u l t (where TAUT&* i s t h e set of t a u t o l o g i e s and is the set of words over
2.
c*
c):
A t l e a s t one of t h e f o l l o w i n g s e n t e n c e s i s t r u e :
(I)
TAUT
i s accepted i n d e t e r m i n i s t i c polynomial t i m e (viz..P=NP);
(11)
TAUT
i s n o t a c c e p t e d i n n o n d e t e r m i n i s t i c polynomial t i m e NP i s n o t c l o s e d under complementation);
(viz., (111)
-
-
I n t e r p o l a t i o n i s polynomially i n t r a c t a b l e , v i z . , f o r every f u n c t i o n Cp if $J i s computable i n d e t e r m i n i s t i c polynomial t i m e , t h e n f o r some t a u t o l o g y B C , $(B,C) f a i l s t o be an i n t e r p o l a n t f o r B+ C.
:c*xc*
c*,
For t h e proof w e use a m i x t u r e of t e c h n i q u e s from computation and model t h e o r y . Assume now t h a t t h e upper bounds f o r computations are r e l a x e d from t h e set 3 of polynomials t o any set 3 2 9 c l o s e d und e r sum and composition. Then t h e above r e s u l t s t i l l h o l d s r e l a t i v e perhaps w i t h a d i f f e r e n t d i s t r i b u t i o n of t o t h e new upper bounds t r u t h - v a l u e s among ( 1 ) - ( 1 1 1 ) . T h i s i s proved i n Theorem 7. Thus i t might be i n t e r e s t i n g t o f i n d two sets 3 and 3' ( i f any) ass i g n i n g d i f f e r e n t t r i p l e t s of t r u t h v a l u e s t o ( I ) - ( I I I ) : as a matter of f a c t , when 3 i s t h e set of a l l f u n c t i o n s , o r even, when '5 c o n t a i n s t h e e x p o n e n t i a l , t h e n t r i v i a l l y ( I ) becomes t r u e and (11)
--
3 46
D.MUNDICI
and (111) f a l s e . A t t h e o p p o s i t e extreme, when '5 is restricted t o t h e set of p o l y n o m i a l s , i t i s w i d e l y c o n j e c t u r e d t h a t ( I ) i s f a l s e , hence, e i t h e r of (11) or (111) i s t h e n t r u e . The above r e s u l t s y i e l d j u s t one more c o n n e c t i o n between t h e modelt h e o r e t i c a l n o t i o n of i n t e r p o l a t i o n , and computation t h e o r y . For a n o t h e r such c o n n e c t i o n , i n c61 i t i s proved t h a t i f t h e r a t e of growth of t h e l e n g t h of i n t e r p o l a n t s f o r any t a u t o l o g y B + C can be kept below some polynomial i n t h e l e n g t h of B and C , t h e n every f u n c t i o n which i s computable i n d e t e r m i n i s t i c polynomial time h a s c i r c u i t d e p t h growing p r o p o r t i o n a l l y t o t h e l o g a r i t h m of t h e input l e n g t h . T h i s as w e l l as o t h e r r e s u l t s c o n c e r n i n g t h e complexity of C r a i g ' s i n t e r p o l a t i o n theorem i n s e n t e n t i a l and i n f i r s t - o r d e r l o g i c a r e surveyed i n t h e f i n a l s e c t i o n of t h i s p a p e r . For t h e g e n e r a l r o l e of C r a i g ' s i n t e r p o l a t i o n theorem i n ( a b s t r a c t ) model t h e o r y , s e e , e . g . , [I1 , [8J and [91.
1.
Preliminaries.
For
A
an a r b i t r a r y s e t ,
d e n o t e s t h e set of words o v e r
A*
t h e set of a l l f i n i t e s t r i n g s of symbols from length
For paper
of
1w1
aCA,
an
A.
For
i.e.,
A,
the
w€A*,
i s t h e number of o c c u r r e n c e s of symbols i n
w
stands f o r
aa...a
(n
times).
Throughout
w.
this
d e n o t e s t h e f o l l o w i n g set of symbols:
c
{ A , v , T p ) .( , X , 0, 1 ) .
=
c,
Boolean e x p r e s s i o n s are u n d e r s t o o d as p a r t i c u l a r words over
accord-
i n g t o t h e f a m i l i a r f o r m a t i o n rules s t u d i e d i n s e n t e n t i a l l o g i c . p o s i t i o n a l v a r i a b l e s a r e words o v e r where t h e s u b s c r i p t
bl...b
i n binary notation.
For
,
var B
E
c*
I var
and
number of e l e m e n t s i n t h i s s e t .
B
I
The v a r i a b l e s i n
t h e o r d e r g i v e n by t h e i r s u b s c r i p t s . Letting
now
XI=
we mean t h a t
B,
B
b = lvar Bl (read:
. . .bn
'
an a r b i t r a r y Boolean e x p r e s s i o n ,
r e s p e c t i v e l y d e n o t e t h e set of v a r i a b l e s o c c u r r i n g i n
bits.
Xbl
i s t h e sequence of d i g i t s of a number
n B
of t h e form
{X,O,I)
Pro-
x
Elements of and
var B {0,1)
x €{O,l}b
satisfies
and
B,
,
inherit
are c a l l e d by
B)
c o n s i d e r e d as a Boolean f u n c t i o n
the
B:{O,l}b-+{O,I),
NP and Craig's Interpolation Theorem
takes value
on i n p u t
1
.. ,xb)).
( =(xl,.
x
341
One can r e c o v e r
f a m i l i a r s e m a n t i c s of s e n t e n t i a l l o g i c upon i d e n t i f y i n g "true"
and
Mod B =
For a r b i t r a r y
tautology
terminology,
( t o the f i r s t
n
Mod B r ( f i r s t n b i t s ) of t h e models of
bits)
Mod B = { O , l } b
f o r any two Boolean e x p r e s s i o n s t a u t o l o g y , an i n t e r p o l a n t var I
Here, as u s u a l ,
logies. In case
var B
n var
C
and
C
a
$
and
either i
B
g:
-
c- p,ll3
c
f o r every
{o,i)",
=(c
1
(resp.,
polynomial t i m e . strings
with regard whether tion
T u r i n g machine i n time
P=NP
one-one
. .. , g ( c n ) ,
A s usual,
P
we s h a l l b r i e f l y
s t a n d s f o r t h e c l a s s of sets
(that is,
f o r any
whether S E A * ,
NP if
say
of
whose c h a r a c t e r i s t i c f u n c t i o n
i n d e t e r m i n i s t i c polynomial t i m e ;
o r even,
bounded
(resp., nondeterministic)
t o n o n d e t e r m i n i s t i c polynomial t i m e .
(that is,
c
i s computable by a determi-
f
nondeterministic)
which are a c c e p t e d
i s computable)
(but otherwise arbi-
,...,CJEC*.
i s computable i n d e t e r m i n i s t i c
f
i s a tautology.
n a t u r a l l y i n d u c e s a one-one map:
g
by a polynomial i n t h e l e n g t h of t h e i n p u t , that
C
r ( c ) = c o n c a t e n a t i o n of g ( c l ) ,
with
I n s t e a d of s a y i n g t h a t nistic
((iB)V(C)).
mapping t h e symbols of
o n t o t h e s e t of t r i p l e t s of b i t s ;
r :C*
a
I such
a r e tauto-
I-C
or
Throughout t h i s p a p e r w e a l s o u s e a f i x e d function
is
B+C
(never occurring i n t h i s paper) Craig's
i n t e r p o l a t i o n theorem s t a t e s t h a t
trary)
) yields,
Boolean e x p r e s s i o n
B-I
a
Mod B f $.
(see [l]
such t h a t
is
B
i s an a b b r e v i a t i o n of
B-C =
and
B
that is
I,
var B f l var C,
=
by
i s t h e set
B.
is satisfiable iff
B
;
C r a i g ' s i n t e r p o l a t i o n theorem i n s e n t e n t i a l l o g i c
that
i s defined
w e a l s o set
n,
iff
Mod B
{ x € [ O p l ] b l x k B}.
I n model-theoretical of r e d u c t s
The set
"false".
with
0
the
with
1
NP
is the
same,
I t i s n o t known
i s c l o s e d under complementaSbNP
then
A*\
S€NP).
I n t h e f o l l o w i n g theorem t h e s e problems are r e l a t e d t o t h e d e g r e e of
D.MUNDICI
3 48
d i f f i c u l t y i n w r i t i n g down
Craig's interpolants i n sentential logic.
A t l e a s t one of t h e f o l l o w i n g s t a t e m e n t s h o l d s t r u e :
2 . Theorem.
(i)
P=NP;
(ii)
NP
i s n o t c l o s e d under complementation;
( i i i ) ( I n t r a c t a b i l i t y of s e n t e n t i a l i n t e r p o l a t i o n ) :
function
:
c*x c*-
c*,
$ is
if
computable i n d e t e r m i n i s t i c
polynomial t i m e , t h e n t h e r e i s a t a u t o l o g y and
var B A var C
for
B-C.
f
!if, such t h a t
f o r every
with
B-C
B,CEX*
i s n o t an i n t e r p o l a n t
$(B,C)
For t h e proof w e p r e p a r e t h e f o l l o w i n g lemma: 3.Lemma.
For
an a r b i t r a r y nonempty set t h e f o l l o w i n g
S C{O,I}*
are e q u i v a l e n t : (a)
SENP;
(b)
there e x i s t s a function
F:{l}*
-c*
d e t e r m i n i s t i c polynomial t i m e , such t h a t , have, f o r e a c h
Proof of Lemma 3 .
1x1 and
s
Fn;
and (b)
+
(a).
i s , on i n p u t
F~
and
Cook's
as f o l l o w s :
XE{O,I)*,
XI
=yl ,
(b)
.
Fnl ;
n
bits).
f i r s t compute
f i n a l l y guess a
We s h a l l o b t a i n t h e r e q u i r e d NP-complete
(see [2J ) .
,...,h)
lal,a2,...,a k )
Let
A =
and
ak=the b l a n k
v a r i a b l e s of 1
j Gk,
p
Fn
O,
symbol.
S
i n t h e l e n g t h of t h e i n p u t .
b e t h e set of s t a t e s of
T,
with
be t h e a l p h a b e t of Fix
are, f o r each
nB 1 t,i,j,q
t h e following:
We
Theorem 7 . 3 . 9 J .
be a n o n d e t e r m i n i s t i c T u r i n g machine a c c e p t i n g
bounded by a polynomial {O
by
F
well-known argument t o p r o v e t h a t t h e s a t i s f i a b i l i -
s h a l l c l o s e l y f o l l o w t h e n o t a t i o n and t e r m i n o l o g y of [5, T
n =
y E [O,l}r
...,xn=yn.
t y problem f o r Boolean e x p r e s s i o n s i s
So l e t
~ ( i ~w )e ,
=
A f a s t nondeterministic algorithm f o r
r = lvar
(a) modifying
= Mod F n r ( f i r s t
Sn{O,l)n
then let y != Fn
such t h a t
letting
n 2 1:
l v a r Fnlgn,
accepting
which i s computable i n
in t i m e
Let
h
the halting state.
T,
with
and l e t
a =1, a = O 1
m = p(n).
(OGtgm,
2 The
O,
349
NP and Craig's Interpolation Theorem
HEAD(t,i),
SYMB(t,i,j),
(1)
and
STATE(t,q),
respectively saying t h a t a t t i m e square
i,
Consider t h e
t,
a . i s p r i n t e d on t a p e J i , and T i s i n s t a t e q.
has t a p e p o s i t i o n
T
c o n j u n c t i o n of t h e f o l l o w i n g s e n t e n c e s , which uniformly
d e s c r i b e t h e b e h a v i o r of a t each t i m e
T
over any i n p u t of l e n g t h
t , each tape square
p r i n t e d on i t ,
n:
h a s p r e c i s e l y one symbol
i
i s scanning p r e c i s e l y one s q u a r e , and
T
T
i s i n p r e c i s e l y one s t a t e ; a t e a c h time
i
symbol on
if
t,
i s n o t over s q u a r e
T
t',
T
the
halts;
t h e computation s t a r t s i n s t a t e
o v e r t h e l e f t hand
0
end of
w i t h t h e t a p e o n l y c o n t a i n i n g b l a n k symbols ( w i t h
m
t h e p o s s i b l e e x c e p t i o n of t h e s q u a r e s initially
or
then
does n o t change;
a t some t i m e
the input,
i,
each t a p e square
m
through
through m+n-I
m+n-1);
has e i t h e r
1
p r i n t e d on i t ;
0
t h e changes i n t a p e symbols, head p o s i t i o n and s t a t e obey
T's
i n s t r u c tions.
w e assume t h a t t h e i n p u t i s p r i n t e d
N o t i c e t h a t , f o l l o w i n g c5],
rn
t h e tape squares and on
through
m, h, k
and t h e i n s t r u c t i o n s of
among t h o s e d i s p l a y e d i n l y w r i t t e n down i n
our
(A)
(F)
is
m+n-I.
I t i s well-known
can be w r i t t e n down as Boolean e x p r e s s i o n s
(F)
[5
t a k e c a r e of
(D)
and
t h a t (A),(B),(C) o n l y depending
are
and whose v a r i a b l e s
indeed, such expressions a ~ e x p l i c i t -
(1):
, p.236,237, e x p r e s s i o n s ( 1 ) - ( 5 ) , ( 7 ) , ( 8 ) , ( 9 ) ] :
corresponds t o ( 7 ) A (8)A (9)
T,
on
(1)A ( 2 ) A ( 3 ) , therein
.
( B ) i s (41, ( C )
i s (5).
and
The f o l l o w i n g Boolean e x p r e s s i o n s
(E):
m- 1
2m
STATE( 0 , O ) A HEAD(0.m) A ( i& SYMB( 0 , i ,k ) A (i=c+n SYMB(0, i ,k ) ) ;
(D)
mcn-I
i.A = m (sYMB(o,i,l)v S Y M B ( O , ~ , ~ ) ) .
Let
F,
=
(A)A
...A (F);
let
s = I v a r Fn I
-
I.
I t i s no
loss
of
D. MUNDICI
350
g e n e r a l i t y t o assume t h a t t h e first SYm(O,m,l), riable
Xr
x r t {O,l), F,;
...,SYMB(O,m+n-1,l)
of
n
r = 0,.
.. , s ,
are
Fn
i n t h e g i v e n o r d e r . Assume e a c h
i s assigned a b i t
F,
v a r i a b l e s of
va-
( i n t u i t i v e l y , a truth-value)
i n such a way t h a t
xo,
.. . , x
S
satisfies
i n symbols,
(2)
X
o'.
,x
*
t
Then i n t h e l i g h t of
we obtain a collection
(1)
" a t time
of t h e form
-
Fn
t ,
such and such s t a t e , and t a p e s q u a r e p r i n t e d on i t " ,
f o r each
of s t a t e m e n t s
h a s such and such p o s i t i o n , T i s i n
T
OGtLm
i
and
h a s such and such symbol OGiC2m
A
.
Now t h e d e f i n i -
Fn. t o g e t h e r w i t h ( 2 ) e n s u r e t h a t i s not a brute l e c t i o n OP i n c o h e r e n t s t a t e m e n t s , b u t r a t h e r d e s c r i b e s a l e g a l t i o n of
c e p t i n g computation of
m
tape squares
on some i n p u t
T
through
m+n-1
.
y E {O,lIn
colac-
p r i n t e d on t h e
To f i n d t h e symbol
y . originally J (j = 0 , . , n - l ) , observe t h a t i f x . = l p r i n t e d on t a p e s q u a r e m + j J t h e n SYMB(O,m+j,l) i s " t r u e " and, by ( 1 ) w e have t h a t y . = 1 .
..
J
x . = 0 , t h e n SYMB(O,m+j,l) J i s " t r u e " , s o t h a t by ( l ) ,
On t h e o t h e r hand, i f by ( E ) , that 0
SYMB(O,m+l,2)
a =1
1 ,... ,n-1,
and
a = 0).
Mod F~
Conversely ,i f
V
of
T
xo,.. /=
, xs
XO'...'xn-l'xn,... (3)
In definitive,
2
so t h a t
..
on i n p u t
E S
1
T.
xo
i n t i m e a t most
m,
s a f e l y assume t h a t t h e i n p u t i s p r i n t e d on t a p e s q u a r e s m+n-1, Using x
t h e whole t a p e (1)
,...,x n
S
having
Therefore, I n symbols,
t h e n t h e r e i s an a c c e p t i n g computation
... , xn-1
X0'
J
,...,xn- 1 E S. b i t s ) E s n { 0 . 1 ) ~.
n
*Xn-l
y. = x.
y . = 0 (recall J f o r each j =
i s a c c e p t e d by
implies
Fn
r (first
xo..
. ,xn-1
i s " f a l s e " and,
2m+l
many s q u a r e s ,
0.1.
and w e
m
can
through
...,2 m .
w e can unambiguously a s s i g n t r u t h v a l u e s t o all the
v a r i a b l e s of
of i n f o r m a t i o n c o n t a i n e d i n p u t a t i o n w e t h e n have, (4)
X0'.
Therefore,
xo,
.. ,xS
...,xn-1
v.
F
n ' Since t h e l a t t e r i s a
by d e f i n i t i o n of !=
E S
X O p . . . l X n-1 ' by j u s t coding t h e p i e c e s
legal
com-
Fn :
Fn
implies that there is
yo,
...,y s I=
Fn
35 1
NP and Craig's Interpolation Theorem
with
yo=xo,.
n
s
(5)
.. , Y , - ~ = X ~ - ~ . C _ Mod Fn
{O,ljn
Now ( 3 ) and ( 5 )
I n symbols,
r (first n
j o i n t l y y i e l d a f i r s t d e s i r e d c o n c l u s i o n about
(a) 3 (b)
To complete t h e proof t h a t
[5,
But t h i s i s well-known
( A ) A ( B ) A (C)A(F) ;
t o hold f o r
portional t o
p4(n),
t h e r e i n i s a t worst pro-
(1)-(9)
i s so simple.
and t h a t ( 1 ) - ( 9 )
The same
c l u s i o n h o l d s i n t h e p r e s e n t c a s e , w i t h t h e same argument. c l u d e s t h e proof t h a t
+ (b)
(a)
Assume s t a t e m e n t s
Mod I n
with
r
the
( i i i ) i n Theorem 2 a r e both
c*
I: {If*+
which i s computable
NP
(6)
s
=
I?
is satisfiable},
{ x ~ j o , ~ ) ~-'(XI l
map d e f i n e d i n s e c t i o n 1 . S i n c e t h e set of s a t i s f i a b l e Boolean e x p r e s s i o n s t h e n s o i s t h e set
( s e e r5, 7.3.51) =
In = I ( 1 " ) . w e
n21:
Proof of L e m m a 4. is in
T h i s con-
I
i n d e t e r m i n i s t i c polynomial t i m e , such t h a t , l e t t i n g have f o r each
con-
and completes t h e proof of lemma 3
( i i ) and
Then t h e r e i s a f u n c t i o n
false.
see
i s claimed t o h o l d ,
m8
where an upper bound of t h e form
p.2381
a f t e r o b s e r v i n g t h a t t h e l e n g t h of
4.Lemma.
F.
w e must show t h a t t h e map
d e s c r i b e d above i s computable i n d e t e r m i n i s t i c polynomial
I n I----, Fn
time.
bits).
r
{ x € {O,l]*]
S
g i v e n by
is satisfiable].
-'(x)
T h i s c l e a r l y f o l l o w s from t h e d e f i n i t i o n of
-S =
assumed t o be f a l s e , t h e n t h e set
r
.
Since
is in
S
{O,l{*\
(ii) i s
NP, t o o .
Withcut l o s s of g e n e r a l i t y t h e r e are n o n d e t e r m i n i s t i c T u r i n g machines T
T'
and
-
accepts
S
and a polynomial
p
i n t i m e bounded by
exist functions
H':
H,
such t h a t p.
~I}*-b~*
(7)
s
n 1o.1)~
s n {o.i)."
accepts
n
Hn = H ( 1 ) ,
n21:
~~r
=
Mod
=
Mod H,:
=+
and
S
TI
(b)) t h e r e
which are computable i n d e t e r -
m i n i s t i c polynomial t i m e such t h a t , l e t t i n g
w e have f o r each
T
B y Lemma 3 ( ( a )
(first
n
bits),
I (first
n
bits).
and
HI
n
= H'(ln),
D. MUNDICI
352
Since
and
T
b o t h a c t i n time bounded by
T'
t i o n of t h e proof of
p , by an e a s y i n s p e c -
w e can s a f e l y s t i p u l a t e t h a t , i n
Lemma 3
ad-
dition, =
v a r H r\ v a r HA
(8)
n
[SYMB(O,m,l)
,...,SYMB(O,m+n-1,l))
w e j u s t u s e t h e same symbols f o r t h e f i r s t
HA,
t h e n rename t h e o t h e r v a r i a b l e s of
condition
is satisfied.
(8)
Hn
n
v a r i a b l e s of
w e have t h a t
xO , . . . , ~ n - l E S
and
so that
Consider now t h e Boolean e x p r e s s i o n
..., xn - l , x n ,...,x
xo,
Hn
( i f necessary)
i f t h e l a t t e r c o n j u n c t i o n were s a t i s f i a b l e
H n h HA:
t h e s i s ) , say
;
n5
(absurdum hypo-
HnA HA , t h e n by ( 7 )
satisfies
, which i s i m p o s s i b l e .
Therefore
we get: (9)
Hn-iHv
i s a tautology, f o r each
n
S i n c e w e a r e assuming t h a t
(iii) i s f a l s e , l e t
that is,
( i i i ) i n Theorem 2 :
example t o
n& 1.
$
$
be a c o u n t e r -
i s computable i n
t e r m i n i s t i c polynomial t i m e , and misses no i n t e r p o l a n t s .
,
= $(Hn
1HA);
then
So l e t
deIn
has t h e following properties:
In
(10)
H n I In
and
(11)
v a r In
{SYMB(O,~,I)
(12)
t h e mapping
In-
a r e t a u t o 1o g i e s ;
i H A
,...,S Y M B ( O , ~ + ~ - ~ , ~ ) ;) .
=
i s computable i n d e t e r m i n i s t i c
In
In
polynomial t i m e . Clause
(12)
i s a consequence of our assumptions about
w i t h t h e f a c t t h a t t h e maps
n
+ Hn
and
computable i n determin s t i c polynomial t i m e . in (13)
( 1 0 ) . and from Mod H~
Hence, by (14)
and
(8
r (first
n
(11)
bits)
C_
w e have Mod I
n
.
Mod In.
S i m i l a r l y , from t h e second t a u t o l o g y i n
(15)
a r e both
From t h e f i r s t t a u t o l o g y
(7) we get
S n{O,lJnC
and from
n w HA
together
(8),(11),(7)
s n {O,l)n
C_
we obtain
ModlIn
.
( l o ) , w r i t t e n as
HI
n
-
71
n'
NP and Craig's Interpolation Theorem From
(14)
and
(15), recalling
(16)
Mod I n = S n j 0 , l ) "
353
we get
(6)
is satisfiable
{ X € { O , I } ~ ~r - ' ( x )
=
w h i c h c o m p l e t e s t h e p r o o f o f our L e m m a .
5 . End of t h e p r o o f of Theorem 2. W e s h a l l prove t h a t if
(iii)
I:{l)**C*
To t h i s p u r p o s e , l e t
M
and
a r e b o t h f a l s e , t h e n P=NP.
(ii)
b e as g i v e n by L e m m a 4 , and l e t
b e a d e t e r m i n i s t i c T u r i n g machine computing e a c h
bounded by a p o l y n o m i a l
q
i n the length
B E E *a,s compute
(D2)
w r i t e down
n =
(D3)
using
w r i t e down
(D4)
check whether
cess
M
Ir(B)l r(B)
3lBI;
=
In; In.
c a n b e c a r r i e d o u t i n d e t e r m i n i s t i c p o l y n o m i a l time
(D4)
q ( n ) , and
r(B)
i n t h e l e n g t h OP
L e t now
B.
\In(
i s bound-
b i t s . The a b o v e p r o -
B
i n t i m e bounded by a p o l y n o m i a l
Therefore we conclude t h a t , under our assumptions,
T h i s c o m p l e t e l y provets
TAUTSZ*
t a u t o l og i e s
n
provides t h e required d e c i s i o n procedure f o r s a t i s -
(Dl)-(D4)
holds.
ensure t h a t
M
i s a sequence of
f i a b i l i t y of any Boolean e x p r e s s i o n
P=NP
on
r(B);
as c l a i m e d , s i n c e t h e p r o p e r t i e s o f e d by
of t h e i n p u t .
follows:
(Dl)
Notice t h a t
i n time
for satisfiability is,
A fast(deterministic)decision p r o c e d u r e
input
n
In
t h e Theorem.
d e n o t e t h e set o f B o o l e a n e x p r e s s i o n s w h i c h
are
. A t l e a s t o n e of t h e f o l l o w i n g s t a t e m e n t s h o l d s t r u e :
6.Corollary. (I)
TAUT
i s accepted i n d e t e r m i n i s t i c polynomial time;
(11)
TAUT
i s n o t accepted i n n o n d e t e r m i n i s t i c polynomial time;
(111)
same a s s t a t e m e n t
P r o o f . I t i s well-known
(iii) that
i n Theorem
TAUT
is i n
2.
P
iff
P=NP
(see [2]).
S i m i l a r l y , TAUT i s i n NP i f f NP i s c l o s e d u n d e r c o m p l e m e n t a t i o n (see, e.g.,
[3.
1.11
).
Now a p p l y Theorem
2.
D.MUNDICI
354
The above C o r o l l a r y i s s t a b l e under r e l a x a t i o n of t h e upper bounds for
computations, a s w e s h a l l
( d e t e r m i n i s t i c and n o n d e t e r m i n i s t i c )
7
s e e i n Theorem functions
below.
f : PI-
.A
El
%
A s usual,
3
set
C
N
5
all
i s c l o s e d under composition
N
5
i f f t h e composition of any two f u n c t i o n s i n in
t h e set of
denotes
i s still a function
c l o s u r e under sum i s s i m i l a r l y d e f i n e d .
;
7. Theorem.
9C%
Let
be an a r b i t r a r y set c o n t a i n i n g t h e polynoThen a t l e a s t one
m i a l s and c l o s e d under composition and sum.
3
( p e r h a p s depending on
)
of t h e f o l l o w i n g s t a t e m e n t s h o l d s t r u e :
TAUT i s a c c e p t e d by some d e t e r m i n i s t i c T u r i n g machine i n t i m e
(Ig)
3
bounded by a f u n c t i o n of (113)
( i n t h e l e n g t h of t h e i n p u t ) ;
i s n o t a c c e p t e d by any n o n d e t e r m i n i s t i c T u r i n g machine
TAUT
- c*
i n t i m e bounded by a f u n c t i o n of For every
(1113)
p
:c * X
c*
3.
;
,
4,
if
a
i s computable by
,
d e t e r m i n i s t i c T u r i n g machine i n time bounded by a f u n c t i o n of then t h e r e i s a tautology such t h a t
C , with
B--,
-
n var
var B
B,CEC*,
i s n o t an i n t e r p o l a n t f o r
H(B.C)
B
f @,
C
C.
For t h e proof w e modify L e m m a s 3 and 4 as f o l l o w s :
8. Lemma.
@
Assume
f
Then t h e r e e x i s t s
F:
i s accepted by a
non-
i n t i m e bounded by a f u n c t i o n
f e z
S E{0,1}*,
d e t e r m i n i s t i c T u r i n g machine (1)"-
T
and
z*
S
by a d e t e r -
which i s computable
3 ,
m i n i s t i c T u r i n g machine i n t i m e bounded by a f u n c t i o n of letting lvar Proof.
w e have f o r each
Fn = F ( I n )
F
~
n, I
and ~
s
n {o,i)"
=
Write down
Fn
T
i s now g i v e n by
and n o t e t h a t
of t h e p r e s e n t Lemma.
Fn
f
such t h a t
n>l: Mod
~~r
Argue e x a c t l y as i n t h e proof of Lemma 3
upper bound f o r
.
(first
n
bits).
( ( a ) =3 ( b ) ) ;
( i n s t e a d of
p
the
therein).
s a t i s f i e s t h e second r e q u i r e m e n t
T o see t h a t t h e mapping
n W F n
i s computable
by a d e t e r m i n i s t i c T u r i n g machine i n t i m e bounded by a f u n c t i o n of by analogy w i t h t h e f i n a l o b s e r v a t i o n i n t h e proof of Lemma 3,
'5 ,
first
note that lFnl i s a t most p r o p o r t i o n a l t o m4 , t h a t i s , p r o p o r t i o n a l 4 can be w r i t t e n down i n time n o t much g r e a t e r t o f ( n ) . Again, than
IFn)
,
F,
say f o r definiteness lFn12
.
But t h e f u n c t i o n
f8(n)
355
NP and Craig's Interpolation Theorem
2 !FA2 still is in 9.Lemma.
(119) and
).
r
{0,1)
€
*\ z
Z
be defined by
c*\TAUT
. 3Clearly,
1°C
=
p ( x ) is
{O,l>*I
r
,
is accept-
2
in time bounded by
Y
is in
.
w
we can safely assume
c*
HI : {I}*-
w'.
=
By Lemma 8
3
)
zn{o,I)n
=
Mod
zn(O,l}n
=
Mod
in
Hl;r
Since
(1113)
sum,
deterministic Turing
such a way
5
(using the
that
(first n
bits),
(first n
bits).
and
-
Arguing now as in the proof of Lemma 4 one shows that is a tautology.
W' in
there are functions
which are computable by
3
-
Z=
is closed under
machines in time bounded by the same function U E closure properties of
a
NP, hence the set
is accepted by a nondeterministic Turing machine
time bounded by a function ~ ' € 3 Since
H,
such that letting
is a tauto~ogy).
By assumption, and by definition of
ed by a nondeterministic Turing machine €unction ~
3 ,
n21
(x E {O,l}n
z c_ {o,I)*
Let
a taut 01ogy
a
which is computable by a deterministic
we have for each Mod In =
Proof.
(1119) are both false. Then there is
machine in time bounded by a function of
In = I(ln)
.
3
by the assumed closure properties of
- x*
Assume
function I: {I}* Turing
3,
Hn is assumed to be false, let
- I Hn '
$
be
computable by a deterministic Turing machine in time bounded by some function
bE
4
tautology, $(Hn
3
, 1 HA);
, with the property that whenever B -C is a (B,c) is an interpolant for B + C. Let In
the mapping
n !-+
In
Turing machine in time bounded by some function d € g tained as a suitable composition of the functions u,b some polynomial).
The mappingnI,n
(d
can be ob-
together with
is now proved to satisfy all
our requirements by the same argument as in the end of the proof Lemma
4.
This completes the proof o€ Lemma
3
Turing machine
accepting
Thus Theorem 7
is proved.
of
9.
Arguing now as in section 5 , using Lemmas 8 and 9 closure properties of
=
is computable by a deterministic
and the
, one easily produces a deterministic TAUT
in time bounded by a function of
5.
D. MUNDICI
356
10.
F u r t h e r Topics.
I n t h i s f i n a l s e c t i o n w e survey
what i s known on t h e complexity
of
e s h a l l s t a t e a number of r e s u l t s C r a i g ' s i n t e r p o l a t i o n theorem. W c o n c e r n i n g t h e r a t e of growth of i n t e r p o l a n t s , b o t h i n s e n t e n t i a l Boolean e x p r e s s i o n s for s e n t e n t i a l l o g i c
and i n f i r s t - o r d e r l o g i c .
are p a r t i c u l a r words over a l p h a b e t
c
as d e f i n e d i n s e c t i o n 1 .
Sentences of f i r s t - o r d e r l o g i c are u n d e r s t o o d as p a r t i c u l a r over some s u i t a b l e a l p h a b e t ).
( s ee [I]
rules
c' ,
words
according t o t h e f a m i l i a r formation
I n s e n t e n t i a l l o g i c t h e precise determination
of t h e r a t e of growth
of i n t e r p o l a n t s i s an open
(and i m p o r t a n t )
The Pollowing Theorem s t a t e s t h a t if s e n t e n t i a l i n t e r p o l a n t s
problem.
t u r n out t o grow polynomially, i n d e t e r m i n i s t i c polynomial
t h e n e v e r y f u n c t i o n which i s computable
(Turing) t i m e ,
has c i r c u i t depth
p o r t i o n a l t o t h e l o g a r i t h m of t h e i n p u t l e n g t h .
pro-
T h i s would p r o v i d e
a p o s i t i v e s o l u t i o n t o a c e n t r a l open problem of computation t h e o r y (see [I I] )
.
Assume t h e r e e x i s t s a polynomial
10.1 Theorem. whenever
an i n t e r p o l a n t
with
I
111
t h e r e s t r i c t i o n of
f
d e p t h Fn See [6
Recall function
,
< p ( 1BI +
F1 , F 2 ,
to
,
2.3.23)
f : {O,l}n-
I
tO.1)
,
Then f o r e v e r y f u n c t i o n
c i r c u i t s , with
d e l a y complexity of
computing
n
c>O,
.
n = 1,2,...
. that the delay
complexity
of a Boolean
i s t h e d e p t h oP t h e smallest d e p t h
(over our f i x e d b a s i s
Boolean e x p r e s s i o n
F
such t h a t , f o r some
f o r each
)
{A, V
. As u s u a l ,
any
i s r e g a r d e d as a Boolean f u n c t i o n o v e r i t s own
B
variables, via the identification
propagate t o
ICl).
... of
[O,lJn,
< c - l o g2 n
Theorem 2 . g
(from b 0
circuit for
such t h a t
which i s computable i n d e t e r m i n i s t i c polynomial
(0'1)
t i m e t h e r e i s a sequence
ProoP.
p
i s a t a u t o l o g y i n s e n t e n t i a l l o g i c , one can f i n d
B-C
f : {O,l]*-
for t h e n e c e s s a r y background:
See f l 0 , 2.21
B
1 = 'ltrue'l
i s , roughly,
and
0 = "false"; the
t h e t i m e needed f o r i n p u t s t o
t h e o u t p u t , i n t h e Pastest c i r c u i t computing
B.
The
f o l l o w i n g Theorem t h e n s t a t e s t h a t t h e t i m e needed by t h e f a s t e s t
NP and Craig's Interpolation Theorem
c i r c u i t t o compute terpolant
( t h e Boolean f u n c t i o n c o r r e s p o n d i n g t o ) any
needed t o compute e i t h e r of 10.2 Theorem. d<620) and I
C
or
B
C:
F o r i n f i n i t e l y many
t h e r e i s a tautology
d€N
(and s t a r t i n g w i t h
some
i n s e n t e n t i a l l o g i c , with
B+C
B
having t h e i r d e l a y complexity S d , such t h a t e v e r y i n t e r p o l a n t
h a s a d e l a y complexity
Proof.
in-
B - + C may happen t o be g r e a t e r t h a n t h e time
for
I
351
17,
See
d
I
>d +
(1/3)log2(d/2).
Theorem 2.51
As remarked above, i n s e n t e n i a l l o g i c t h e r e i s a t p r e s e n t no deEini-
t i v e e s t i m a t e oP t h e r a t e oP growth of ICl
, where
Theorem
a s a f u n c t i o n of IBI and
111
i s a smallest l e n g t h i n t e r p o l a n t € o r
I
(See [6,
B-C.
1.93 f o r an upper bound, and t r y t o improve i t ) .
By c o n t r a s t
i n f i r s t - o r d e r l o g i c w e have: 10.3 Theorem. ( i ) In t h e arithmetical hierarchy t h e r e i s a
TI1 - f u n c t i o n
g i v i n g an upper bound f o r t h e l e n g t h oP P i r s t - o r d e r (*)
whenever
B+C
interpolant ( i i ) No
c
1
I
-function
with
B+C
as i n
(*)
[ 6 , Theorem 3.13
Theorem
b
ICl). can g i v e an
.
Proof.
,
b(lBI+
111
( i . e . , no r e c u r s i v e f u n c t i o n )
111
[4
i n t e r p o l a n t s , i.e..
i s v a l i d i n f i r s t - o r d e r l o g i c , t h e r e i s an for
upper bound f o r ( i ) See
b: N+pl
.
(11)
T h i s c a n be e x t r a c t e d from
11.
Due t o i t s a s y m p t o t i c c h a r a c t e r , t h e above Theorem
10.3 ( i i ) g i v e s
no i n f o r m a t i o n on t h e p o s s i b l e l e n g t h s of i n t e r p o l a n t s € o r s h o r t plications.
The 'following i s a non-asymptotic r e s u l t :
We c a n w r i t e down a v a l i d i m p l i c a t i o n i n f i r s t - o r d e r
10.4 Theorem.
logic,
with
B---+C
l B ( .lC(<1145
pol a n t w e have : .*2
III> 2'
Proof.
See
im-
&,
Theorem 3 . 4 .
1
such t h a t whenever
seven t w o ' s
.
I
i s an i n t e r -
D.MUNDICI
358
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.
. .
Mundici D . , Compactness, i n t e r p o l a t i o n and F r i e d m a n ' s t h i r d problem, Annals of Mathematical Logic 22 ( 1 982) 197-21 1
.
Savage J.E.,
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345
NP AND C R A I G ' S INTERPOLATION THEOREM
Daniele Mundici
.
L oc Romol a N .76 50060 Donnini
F1 orencc-I t a l y
The t r u t h - v a l u e s of two renowned c o n j e c t u r e s about NP (namely, P f NP and, NP i s n o t c l o s e d under complementation) depend on t h e d i f f i c u l t y i n w r i t i n g down C r a i g ' s i n t e r p o l a n t s i n s e n t e n t i a l l o g i c . The g e n e r a l connection between NP and i n t e r p o l a t i o n i s s t u d i e d by b l e n d i n g i d e a s and t e c h n i q u e s from both model t h e o r y and computation t h e o r y .
0.
Introduction.
W e f i x throughout an a1 habet
c
and r e g a r d Boolean e x p r e s s i o n s a s p a r t i c u l a r words over I f B-+C i s a tautology, then Craig's i n t e r p o l a t i o n theorem y i e l d s an i n t e r p o l a n t I , t h a t i s , a Boolean e x p r e s s i o n I such t h a t B + I and IC are t a u t o l o g i e s , and t h e v a r i a b l e s o c c u r r i n g i n I are e x a c t l y t h o s e which j o i n t l y occur i n B and C. I n Theorem 2 and C o r o l l a r y 6 w e prove t h e f o l l o w i n g r e s u l t (where TAUT&* i s t h e set of t a u t o l o g i e s and is the set of words over
2.
c*
c):
A t l e a s t one of t h e f o l l o w i n g s e n t e n c e s i s t r u e :
(I)
TAUT
i s accepted i n d e t e r m i n i s t i c polynomial t i m e (viz..P=NP);
(11)
TAUT
i s n o t a c c e p t e d i n n o n d e t e r m i n i s t i c polynomial t i m e NP i s n o t c l o s e d under complementation);
(viz., (111)
-
-
I n t e r p o l a t i o n i s polynomially i n t r a c t a b l e , v i z . , f o r every f u n c t i o n Cp if $J i s computable i n d e t e r m i n i s t i c polynomial t i m e , t h e n f o r some t a u t o l o g y B C , $(B,C) f a i l s t o be an i n t e r p o l a n t f o r B+ C.
:c*xc*
c*,
For t h e proof w e use a m i x t u r e of t e c h n i q u e s from computation and model t h e o r y . Assume now t h a t t h e upper bounds f o r computations are r e l a x e d from t h e set 3 of polynomials t o any set 3 2 9 c l o s e d und e r sum and composition. Then t h e above r e s u l t s t i l l h o l d s r e l a t i v e perhaps w i t h a d i f f e r e n t d i s t r i b u t i o n of t o t h e new upper bounds t r u t h - v a l u e s among ( 1 ) - ( 1 1 1 ) . T h i s i s proved i n Theorem 7. Thus i t might be i n t e r e s t i n g t o f i n d two sets 3 and 3' ( i f any) ass i g n i n g d i f f e r e n t t r i p l e t s of t r u t h v a l u e s t o ( I ) - ( I I I ) : as a matter of f a c t , when 3 i s t h e set of a l l f u n c t i o n s , o r even, when '5 c o n t a i n s t h e e x p o n e n t i a l , t h e n t r i v i a l l y ( I ) becomes t r u e and (11)
--
3 46
D.MUNDICI
and (111) f a l s e . A t t h e o p p o s i t e extreme, when '5 is restricted t o t h e set of p o l y n o m i a l s , i t i s w i d e l y c o n j e c t u r e d t h a t ( I ) i s f a l s e , hence, e i t h e r of (11) or (111) i s t h e n t r u e . The above r e s u l t s y i e l d j u s t one more c o n n e c t i o n between t h e modelt h e o r e t i c a l n o t i o n of i n t e r p o l a t i o n , and computation t h e o r y . For a n o t h e r such c o n n e c t i o n , i n c61 i t i s proved t h a t i f t h e r a t e of growth of t h e l e n g t h of i n t e r p o l a n t s f o r any t a u t o l o g y B + C can be kept below some polynomial i n t h e l e n g t h of B and C , t h e n every f u n c t i o n which i s computable i n d e t e r m i n i s t i c polynomial time h a s c i r c u i t d e p t h growing p r o p o r t i o n a l l y t o t h e l o g a r i t h m of t h e input l e n g t h . T h i s as w e l l as o t h e r r e s u l t s c o n c e r n i n g t h e complexity of C r a i g ' s i n t e r p o l a t i o n theorem i n s e n t e n t i a l and i n f i r s t - o r d e r l o g i c a r e surveyed i n t h e f i n a l s e c t i o n of t h i s p a p e r . For t h e g e n e r a l r o l e of C r a i g ' s i n t e r p o l a t i o n theorem i n ( a b s t r a c t ) model t h e o r y , s e e , e . g . , [I1 , [8J and [91.
1.
Preliminaries.
For
A
an a r b i t r a r y s e t ,
d e n o t e s t h e set of words o v e r
A*
t h e set of a l l f i n i t e s t r i n g s of symbols from length
For paper
of
1w1
aCA,
an
A.
For
i.e.,
A,
the
w€A*,
i s t h e number of o c c u r r e n c e s of symbols i n
w
stands f o r
aa...a
(n
times).
Throughout
w.
this
d e n o t e s t h e f o l l o w i n g set of symbols:
c
{ A , v , T p ) .( , X , 0, 1 ) .
=
c,
Boolean e x p r e s s i o n s are u n d e r s t o o d as p a r t i c u l a r words over
accord-
i n g t o t h e f a m i l i a r f o r m a t i o n rules s t u d i e d i n s e n t e n t i a l l o g i c . p o s i t i o n a l v a r i a b l e s a r e words o v e r where t h e s u b s c r i p t
bl...b
i n binary notation.
For
,
var B
E
c*
I var
and
number of e l e m e n t s i n t h i s s e t .
B
I
The v a r i a b l e s i n
t h e o r d e r g i v e n by t h e i r s u b s c r i p t s . Letting
now
XI=
we mean t h a t
B,
B
b = lvar Bl (read:
. . .bn
'
an a r b i t r a r y Boolean e x p r e s s i o n ,
r e s p e c t i v e l y d e n o t e t h e set of v a r i a b l e s o c c u r r i n g i n
bits.
Xbl
i s t h e sequence of d i g i t s of a number
n B
of t h e form
{X,O,I)
Pro-
x
Elements of and
var B {0,1)
x €{O,l}b
satisfies
and
B,
,
inherit
are c a l l e d by
B)
c o n s i d e r e d as a Boolean f u n c t i o n
the
B:{O,l}b-+{O,I),
NP and Craig's Interpolation Theorem
takes value
on i n p u t
1
.. ,xb)).
( =(xl,.
x
341
One can r e c o v e r
f a m i l i a r s e m a n t i c s of s e n t e n t i a l l o g i c upon i d e n t i f y i n g "true"
and
Mod B =
For a r b i t r a r y
tautology
terminology,
( t o the f i r s t
n
Mod B r ( f i r s t n b i t s ) of t h e models of
bits)
Mod B = { O , l } b
f o r any two Boolean e x p r e s s i o n s t a u t o l o g y , an i n t e r p o l a n t var I
Here, as u s u a l ,
logies. In case
var B
n var
C
and
C
a
$
and
either i
B
g:
-
c- p,ll3
c
f o r every
{o,i)",
=(c
1
(resp.,
polynomial t i m e . strings
with regard whether tion
T u r i n g machine i n time
P=NP
one-one
. .. , g ( c n ) ,
A s usual,
P
we s h a l l b r i e f l y
s t a n d s f o r t h e c l a s s of sets
(that is,
f o r any
whether S E A * ,
NP if
say
of
whose c h a r a c t e r i s t i c f u n c t i o n
i n d e t e r m i n i s t i c polynomial t i m e ;
o r even,
bounded
(resp., nondeterministic)
t o n o n d e t e r m i n i s t i c polynomial t i m e .
(that is,
c
i s computable by a determi-
f
nondeterministic)
which are a c c e p t e d
i s computable)
(but otherwise arbi-
,...,CJEC*.
i s computable i n d e t e r m i n i s t i c
f
i s a tautology.
n a t u r a l l y i n d u c e s a one-one map:
g
by a polynomial i n t h e l e n g t h of t h e i n p u t , that
C
r ( c ) = c o n c a t e n a t i o n of g ( c l ) ,
with
I n s t e a d of s a y i n g t h a t nistic
((iB)V(C)).
mapping t h e symbols of
o n t o t h e s e t of t r i p l e t s of b i t s ;
r :C*
a
I such
a r e tauto-
I-C
or
Throughout t h i s p a p e r w e a l s o u s e a f i x e d function
is
B+C
(never occurring i n t h i s paper) Craig's
i n t e r p o l a t i o n theorem s t a t e s t h a t
trary)
) yields,
Boolean e x p r e s s i o n
B-I
a
Mod B f $.
(see [l]
such t h a t
is
B
i s an a b b r e v i a t i o n of
B-C =
and
B
that is
I,
var B f l var C,
=
by
i s t h e set
B.
is satisfiable iff
B
;
C r a i g ' s i n t e r p o l a t i o n theorem i n s e n t e n t i a l l o g i c
that
i s defined
w e a l s o set
n,
iff
Mod B
{ x € [ O p l ] b l x k B}.
I n model-theoretical of r e d u c t s
The set
"false".
with
0
the
with
1
NP
is the
same,
I t i s n o t known
i s c l o s e d under complementaSbNP
then
A*\
S€NP).
I n t h e f o l l o w i n g theorem t h e s e problems are r e l a t e d t o t h e d e g r e e of
D.MUNDICI
3 48
d i f f i c u l t y i n w r i t i n g down
Craig's interpolants i n sentential logic.
A t l e a s t one of t h e f o l l o w i n g s t a t e m e n t s h o l d s t r u e :
2 . Theorem.
(i)
P=NP;
(ii)
NP
i s n o t c l o s e d under complementation;
( i i i ) ( I n t r a c t a b i l i t y of s e n t e n t i a l i n t e r p o l a t i o n ) :
function
:
c*x c*-
c*,
$ is
if
computable i n d e t e r m i n i s t i c
polynomial t i m e , t h e n t h e r e i s a t a u t o l o g y and
var B A var C
for
B-C.
f
!if, such t h a t
f o r every
with
B-C
B,CEX*
i s n o t an i n t e r p o l a n t
$(B,C)
For t h e proof w e p r e p a r e t h e f o l l o w i n g lemma: 3.Lemma.
For
an a r b i t r a r y nonempty set t h e f o l l o w i n g
S C{O,I}*
are e q u i v a l e n t : (a)
SENP;
(b)
there e x i s t s a function
F:{l}*
-c*
d e t e r m i n i s t i c polynomial t i m e , such t h a t , have, f o r e a c h
Proof of Lemma 3 .
1x1 and
s
Fn;
and (b)
+
(a).
i s , on i n p u t
F~
and
Cook's
as f o l l o w s :
XE{O,I)*,
XI
=yl ,
(b)
.
Fnl ;
n
bits).
f i r s t compute
f i n a l l y guess a
We s h a l l o b t a i n t h e r e q u i r e d NP-complete
(see [2J ) .
,...,h)
lal,a2,...,a k )
Let
A =
and
ak=the b l a n k
v a r i a b l e s of 1
j Gk,
p
Fn
O,
symbol.
S
i n t h e l e n g t h of t h e i n p u t .
b e t h e set of s t a t e s of
T,
with
be t h e a l p h a b e t of Fix
are, f o r each
nB 1 t,i,j,q
t h e following:
We
Theorem 7 . 3 . 9 J .
be a n o n d e t e r m i n i s t i c T u r i n g machine a c c e p t i n g
bounded by a polynomial {O
by
F
well-known argument t o p r o v e t h a t t h e s a t i s f i a b i l i -
s h a l l c l o s e l y f o l l o w t h e n o t a t i o n and t e r m i n o l o g y of [5, T
n =
y E [O,l}r
...,xn=yn.
t y problem f o r Boolean e x p r e s s i o n s i s
So l e t
~ ( i ~w )e ,
=
A f a s t nondeterministic algorithm f o r
r = lvar
(a) modifying
= Mod F n r ( f i r s t
Sn{O,l)n
then let y != Fn
such t h a t
letting
n 2 1:
l v a r Fnlgn,
accepting
which i s computable i n
in t i m e
Let
h
the halting state.
T,
with
and l e t
a =1, a = O 1
m = p(n).
(OGtgm,
2 The
O,
349
NP and Craig's Interpolation Theorem
HEAD(t,i),
SYMB(t,i,j),
(1)
and
STATE(t,q),
respectively saying t h a t a t t i m e square
i,
Consider t h e
t,
a . i s p r i n t e d on t a p e J i , and T i s i n s t a t e q.
has t a p e p o s i t i o n
T
c o n j u n c t i o n of t h e f o l l o w i n g s e n t e n c e s , which uniformly
d e s c r i b e t h e b e h a v i o r of a t each t i m e
T
over any i n p u t of l e n g t h
t , each tape square
p r i n t e d on i t ,
n:
h a s p r e c i s e l y one symbol
i
i s scanning p r e c i s e l y one s q u a r e , and
T
T
i s i n p r e c i s e l y one s t a t e ; a t e a c h time
i
symbol on
if
t,
i s n o t over s q u a r e
T
t',
T
the
halts;
t h e computation s t a r t s i n s t a t e
o v e r t h e l e f t hand
0
end of
w i t h t h e t a p e o n l y c o n t a i n i n g b l a n k symbols ( w i t h
m
t h e p o s s i b l e e x c e p t i o n of t h e s q u a r e s initially
or
then
does n o t change;
a t some t i m e
the input,
i,
each t a p e square
m
through
through m+n-I
m+n-1);
has e i t h e r
1
p r i n t e d on i t ;
0
t h e changes i n t a p e symbols, head p o s i t i o n and s t a t e obey
T's
i n s t r u c tions.
w e assume t h a t t h e i n p u t i s p r i n t e d
N o t i c e t h a t , f o l l o w i n g c5],
rn
t h e tape squares and on
through
m, h, k
and t h e i n s t r u c t i o n s of
among t h o s e d i s p l a y e d i n l y w r i t t e n down i n
our
(A)
(F)
is
m+n-I.
I t i s well-known
can be w r i t t e n down as Boolean e x p r e s s i o n s
(F)
[5
t a k e c a r e of
(D)
and
t h a t (A),(B),(C) o n l y depending
are
and whose v a r i a b l e s
indeed, such expressions a ~ e x p l i c i t -
(1):
, p.236,237, e x p r e s s i o n s ( 1 ) - ( 5 ) , ( 7 ) , ( 8 ) , ( 9 ) ] :
corresponds t o ( 7 ) A (8)A (9)
T,
on
(1)A ( 2 ) A ( 3 ) , therein
.
( B ) i s (41, ( C )
i s (5).
and
The f o l l o w i n g Boolean e x p r e s s i o n s
(E):
m- 1
2m
STATE( 0 , O ) A HEAD(0.m) A ( i& SYMB( 0 , i ,k ) A (i=c+n SYMB(0, i ,k ) ) ;
(D)
mcn-I
i.A = m (sYMB(o,i,l)v S Y M B ( O , ~ , ~ ) ) .
Let
F,
=
(A)A
...A (F);
let
s = I v a r Fn I
-
I.
I t i s no
loss
of
D. MUNDICI
350
g e n e r a l i t y t o assume t h a t t h e first SYm(O,m,l), riable
Xr
x r t {O,l), F,;
...,SYMB(O,m+n-1,l)
of
n
r = 0,.
.. , s ,
are
Fn
i n t h e g i v e n o r d e r . Assume e a c h
i s assigned a b i t
F,
v a r i a b l e s of
va-
( i n t u i t i v e l y , a truth-value)
i n such a way t h a t
xo,
.. . , x
S
satisfies
i n symbols,
(2)
X
o'.
,x
*
t
Then i n t h e l i g h t of
we obtain a collection
(1)
" a t time
of t h e form
-
Fn
t ,
such and such s t a t e , and t a p e s q u a r e p r i n t e d on i t " ,
f o r each
of s t a t e m e n t s
h a s such and such p o s i t i o n , T i s i n
T
OGtLm
i
and
h a s such and such symbol OGiC2m
A
.
Now t h e d e f i n i -
Fn. t o g e t h e r w i t h ( 2 ) e n s u r e t h a t i s not a brute l e c t i o n OP i n c o h e r e n t s t a t e m e n t s , b u t r a t h e r d e s c r i b e s a l e g a l t i o n of
c e p t i n g computation of
m
tape squares
on some i n p u t
T
through
m+n-1
.
y E {O,lIn
colac-
p r i n t e d on t h e
To f i n d t h e symbol
y . originally J (j = 0 , . , n - l ) , observe t h a t i f x . = l p r i n t e d on t a p e s q u a r e m + j J t h e n SYMB(O,m+j,l) i s " t r u e " and, by ( 1 ) w e have t h a t y . = 1 .
..
J
x . = 0 , t h e n SYMB(O,m+j,l) J i s " t r u e " , s o t h a t by ( l ) ,
On t h e o t h e r hand, i f by ( E ) , that 0
SYMB(O,m+l,2)
a =1
1 ,... ,n-1,
and
a = 0).
Mod F~
Conversely ,i f
V
of
T
xo,.. /=
, xs
XO'...'xn-l'xn,... (3)
In definitive,
2
so t h a t
..
on i n p u t
E S
1
T.
xo
i n t i m e a t most
m,
s a f e l y assume t h a t t h e i n p u t i s p r i n t e d on t a p e s q u a r e s m+n-1, Using x
t h e whole t a p e (1)
,...,x n
S
having
Therefore, I n symbols,
t h e n t h e r e i s an a c c e p t i n g computation
... , xn-1
X0'
J
,...,xn- 1 E S. b i t s ) E s n { 0 . 1 ) ~.
n
*Xn-l
y. = x.
y . = 0 (recall J f o r each j =
i s a c c e p t e d by
implies
Fn
r (first
xo..
. ,xn-1
i s " f a l s e " and,
2m+l
many s q u a r e s ,
0.1.
and w e
m
can
through
...,2 m .
w e can unambiguously a s s i g n t r u t h v a l u e s t o all the
v a r i a b l e s of
of i n f o r m a t i o n c o n t a i n e d i n p u t a t i o n w e t h e n have, (4)
X0'.
Therefore,
xo,
.. ,xS
...,xn-1
v.
F
n ' Since t h e l a t t e r i s a
by d e f i n i t i o n of !=
E S
X O p . . . l X n-1 ' by j u s t coding t h e p i e c e s
legal
com-
Fn :
Fn
implies that there is
yo,
...,y s I=
Fn
35 1
NP and Craig's Interpolation Theorem
with
yo=xo,.
n
s
(5)
.. , Y , - ~ = X ~ - ~ . C _ Mod Fn
{O,ljn
Now ( 3 ) and ( 5 )
I n symbols,
r (first n
j o i n t l y y i e l d a f i r s t d e s i r e d c o n c l u s i o n about
(a) 3 (b)
To complete t h e proof t h a t
[5,
But t h i s i s well-known
( A ) A ( B ) A (C)A(F) ;
t o hold f o r
portional t o
p4(n),
t h e r e i n i s a t worst pro-
(1)-(9)
i s so simple.
and t h a t ( 1 ) - ( 9 )
The same
c l u s i o n h o l d s i n t h e p r e s e n t c a s e , w i t h t h e same argument. c l u d e s t h e proof t h a t
+ (b)
(a)
Assume s t a t e m e n t s
Mod I n
with
r
the
( i i i ) i n Theorem 2 a r e both
c*
I: {If*+
which i s computable
NP
(6)
s
=
I?
is satisfiable},
{ x ~ j o , ~ ) ~-'(XI l
map d e f i n e d i n s e c t i o n 1 . S i n c e t h e set of s a t i s f i a b l e Boolean e x p r e s s i o n s t h e n s o i s t h e set
( s e e r5, 7.3.51) =
In = I ( 1 " ) . w e
n21:
Proof of L e m m a 4. is in
T h i s con-
I
i n d e t e r m i n i s t i c polynomial t i m e , such t h a t , l e t t i n g have f o r each
con-
and completes t h e proof of lemma 3
( i i ) and
Then t h e r e i s a f u n c t i o n
false.
see
i s claimed t o h o l d ,
m8
where an upper bound of t h e form
p.2381
a f t e r o b s e r v i n g t h a t t h e l e n g t h of
4.Lemma.
F.
w e must show t h a t t h e map
d e s c r i b e d above i s computable i n d e t e r m i n i s t i c polynomial
I n I----, Fn
time.
bits).
r
{ x € {O,l]*]
S
g i v e n by
is satisfiable].
-'(x)
T h i s c l e a r l y f o l l o w s from t h e d e f i n i t i o n of
-S =
assumed t o be f a l s e , t h e n t h e set
r
.
Since
is in
S
{O,l{*\
(ii) i s
NP, t o o .
Withcut l o s s of g e n e r a l i t y t h e r e are n o n d e t e r m i n i s t i c T u r i n g machines T
T'
and
-
accepts
S
and a polynomial
p
i n t i m e bounded by
exist functions
H':
H,
such t h a t p.
~I}*-b~*
(7)
s
n 1o.1)~
s n {o.i)."
accepts
n
Hn = H ( 1 ) ,
n21:
~~r
=
Mod
=
Mod H,:
=+
and
S
TI
(b)) t h e r e
which are computable i n d e t e r -
m i n i s t i c polynomial t i m e such t h a t , l e t t i n g
w e have f o r each
T
B y Lemma 3 ( ( a )
(first
n
bits),
I (first
n
bits).
and
HI
n
= H'(ln),
D. MUNDICI
352
Since
and
T
b o t h a c t i n time bounded by
T'
t i o n of t h e proof of
p , by an e a s y i n s p e c -
w e can s a f e l y s t i p u l a t e t h a t , i n
Lemma 3
ad-
dition, =
v a r H r\ v a r HA
(8)
n
[SYMB(O,m,l)
,...,SYMB(O,m+n-1,l))
w e j u s t u s e t h e same symbols f o r t h e f i r s t
HA,
t h e n rename t h e o t h e r v a r i a b l e s of
condition
is satisfied.
(8)
Hn
n
v a r i a b l e s of
w e have t h a t
xO , . . . , ~ n - l E S
and
so that
Consider now t h e Boolean e x p r e s s i o n
..., xn - l , x n ,...,x
xo,
Hn
( i f necessary)
i f t h e l a t t e r c o n j u n c t i o n were s a t i s f i a b l e
H n h HA:
t h e s i s ) , say
;
n5
(absurdum hypo-
HnA HA , t h e n by ( 7 )
satisfies
, which i s i m p o s s i b l e .
Therefore
we get: (9)
Hn-iHv
i s a tautology, f o r each
n
S i n c e w e a r e assuming t h a t
(iii) i s f a l s e , l e t
that is,
( i i i ) i n Theorem 2 :
example t o
n& 1.
$
$
be a c o u n t e r -
i s computable i n
t e r m i n i s t i c polynomial t i m e , and misses no i n t e r p o l a n t s .
,
= $(Hn
1HA);
then
So l e t
deIn
has t h e following properties:
In
(10)
H n I In
and
(11)
v a r In
{SYMB(O,~,I)
(12)
t h e mapping
In-
a r e t a u t o 1o g i e s ;
i H A
,...,S Y M B ( O , ~ + ~ - ~ , ~ ) ;) .
=
i s computable i n d e t e r m i n i s t i c
In
In
polynomial t i m e . Clause
(12)
i s a consequence of our assumptions about
w i t h t h e f a c t t h a t t h e maps
n
+ Hn
and
computable i n determin s t i c polynomial t i m e . in (13)
( 1 0 ) . and from Mod H~
Hence, by (14)
and
(8
r (first
n
(11)
bits)
C_
w e have Mod I
n
.
Mod In.
S i m i l a r l y , from t h e second t a u t o l o g y i n
(15)
a r e both
From t h e f i r s t t a u t o l o g y
(7) we get
S n{O,lJnC
and from
n w HA
together
(8),(11),(7)
s n {O,l)n
C_
we obtain
ModlIn
.
( l o ) , w r i t t e n as
HI
n
-
71
n'
NP and Craig's Interpolation Theorem From
(14)
and
(15), recalling
(16)
Mod I n = S n j 0 , l ) "
353
we get
(6)
is satisfiable
{ X € { O , I } ~ ~r - ' ( x )
=
w h i c h c o m p l e t e s t h e p r o o f o f our L e m m a .
5 . End of t h e p r o o f of Theorem 2. W e s h a l l prove t h a t if
(iii)
I:{l)**C*
To t h i s p u r p o s e , l e t
M
and
a r e b o t h f a l s e , t h e n P=NP.
(ii)
b e as g i v e n by L e m m a 4 , and l e t
b e a d e t e r m i n i s t i c T u r i n g machine computing e a c h
bounded by a p o l y n o m i a l
q
i n the length
B E E *a,s compute
(D2)
w r i t e down
n =
(D3)
using
w r i t e down
(D4)
check whether
cess
M
Ir(B)l r(B)
3lBI;
=
In; In.
c a n b e c a r r i e d o u t i n d e t e r m i n i s t i c p o l y n o m i a l time
(D4)
q ( n ) , and
r(B)
i n t h e l e n g t h OP
L e t now
B.
\In(
i s bound-
b i t s . The a b o v e p r o -
B
i n t i m e bounded by a p o l y n o m i a l
Therefore we conclude t h a t , under our assumptions,
T h i s c o m p l e t e l y provets
TAUTSZ*
t a u t o l og i e s
n
provides t h e required d e c i s i o n procedure f o r s a t i s -
(Dl)-(D4)
holds.
ensure t h a t
M
i s a sequence of
f i a b i l i t y of any Boolean e x p r e s s i o n
P=NP
on
r(B);
as c l a i m e d , s i n c e t h e p r o p e r t i e s o f e d by
of t h e i n p u t .
follows:
(Dl)
Notice t h a t
i n time
for satisfiability is,
A fast(deterministic)decision p r o c e d u r e
input
n
In
t h e Theorem.
d e n o t e t h e set o f B o o l e a n e x p r e s s i o n s w h i c h
are
. A t l e a s t o n e of t h e f o l l o w i n g s t a t e m e n t s h o l d s t r u e :
6.Corollary. (I)
TAUT
i s accepted i n d e t e r m i n i s t i c polynomial time;
(11)
TAUT
i s n o t accepted i n n o n d e t e r m i n i s t i c polynomial time;
(111)
same a s s t a t e m e n t
P r o o f . I t i s well-known
(iii) that
i n Theorem
TAUT
is i n
2.
P
iff
P=NP
(see [2]).
S i m i l a r l y , TAUT i s i n NP i f f NP i s c l o s e d u n d e r c o m p l e m e n t a t i o n (see, e.g.,
[3.
1.11
).
Now a p p l y Theorem
2.
D.MUNDICI
354
The above C o r o l l a r y i s s t a b l e under r e l a x a t i o n of t h e upper bounds for
computations, a s w e s h a l l
( d e t e r m i n i s t i c and n o n d e t e r m i n i s t i c )
7
s e e i n Theorem functions
below.
f : PI-
.A
El
%
A s usual,
3
set
C
N
5
all
i s c l o s e d under composition
N
5
i f f t h e composition of any two f u n c t i o n s i n in
t h e set of
denotes
i s still a function
c l o s u r e under sum i s s i m i l a r l y d e f i n e d .
;
7. Theorem.
9C%
Let
be an a r b i t r a r y set c o n t a i n i n g t h e polynoThen a t l e a s t one
m i a l s and c l o s e d under composition and sum.
3
( p e r h a p s depending on
)
of t h e f o l l o w i n g s t a t e m e n t s h o l d s t r u e :
TAUT i s a c c e p t e d by some d e t e r m i n i s t i c T u r i n g machine i n t i m e
(Ig)
3
bounded by a f u n c t i o n of (113)
( i n t h e l e n g t h of t h e i n p u t ) ;
i s n o t a c c e p t e d by any n o n d e t e r m i n i s t i c T u r i n g machine
TAUT
- c*
i n t i m e bounded by a f u n c t i o n of For every
(1113)
p
:c * X
c*
3.
;
,
4,
if
a
i s computable by
,
d e t e r m i n i s t i c T u r i n g machine i n time bounded by a f u n c t i o n of then t h e r e i s a tautology such t h a t
C , with
B--,
-
n var
var B
B,CEC*,
i s n o t an i n t e r p o l a n t f o r
H(B.C)
B
f @,
C
C.
For t h e proof w e modify L e m m a s 3 and 4 as f o l l o w s :
8. Lemma.
@
Assume
f
Then t h e r e e x i s t s
F:
i s accepted by a
non-
i n t i m e bounded by a f u n c t i o n
f e z
S E{0,1}*,
d e t e r m i n i s t i c T u r i n g machine (1)"-
T
and
z*
S
by a d e t e r -
which i s computable
3 ,
m i n i s t i c T u r i n g machine i n t i m e bounded by a f u n c t i o n of letting lvar Proof.
w e have f o r each
Fn = F ( I n )
F
~
n, I
and ~
s
n {o,i)"
=
Write down
Fn
T
i s now g i v e n by
and n o t e t h a t
of t h e p r e s e n t Lemma.
Fn
f
such t h a t
n>l: Mod
~~r
Argue e x a c t l y as i n t h e proof of Lemma 3
upper bound f o r
.
(first
n
bits).
( ( a ) =3 ( b ) ) ;
( i n s t e a d of
p
the
therein).
s a t i s f i e s t h e second r e q u i r e m e n t
T o see t h a t t h e mapping
n W F n
i s computable
by a d e t e r m i n i s t i c T u r i n g machine i n t i m e bounded by a f u n c t i o n of by analogy w i t h t h e f i n a l o b s e r v a t i o n i n t h e proof of Lemma 3,
'5 ,
first
note that lFnl i s a t most p r o p o r t i o n a l t o m4 , t h a t i s , p r o p o r t i o n a l 4 can be w r i t t e n down i n time n o t much g r e a t e r t o f ( n ) . Again, than
IFn)
,
F,
say f o r definiteness lFn12
.
But t h e f u n c t i o n
f8(n)
355
NP and Craig's Interpolation Theorem
2 !FA2 still is in 9.Lemma.
(119) and
).
r
{0,1)
€
*\ z
Z
be defined by
c*\TAUT
. 3Clearly,
1°C
=
p ( x ) is
{O,l>*I
r
,
is accept-
2
in time bounded by
Y
is in
.
w
we can safely assume
c*
HI : {I}*-
w'.
=
By Lemma 8
3
)
zn{o,I)n
=
Mod
zn(O,l}n
=
Mod
in
Hl;r
Since
(1113)
sum,
deterministic Turing
such a way
5
(using the
that
(first n
bits),
(first n
bits).
and
-
Arguing now as in the proof of Lemma 4 one shows that is a tautology.
W' in
there are functions
which are computable by
3
-
Z=
is closed under
machines in time bounded by the same function U E closure properties of
a
NP, hence the set
is accepted by a nondeterministic Turing machine
time bounded by a function ~ ' € 3 Since
H,
such that letting
is a tauto~ogy).
By assumption, and by definition of
ed by a nondeterministic Turing machine €unction ~
3 ,
n21
(x E {O,l}n
z c_ {o,I)*
Let
a taut 01ogy
a
which is computable by a deterministic
we have for each Mod In =
Proof.
(1119) are both false. Then there is
machine in time bounded by a function of
In = I(ln)
.
3
by the assumed closure properties of
- x*
Assume
function I: {I}* Turing
3,
Hn is assumed to be false, let
- I Hn '
$
be
computable by a deterministic Turing machine in time bounded by some function
bE
4
tautology, $(Hn
3
, 1 HA);
, with the property that whenever B -C is a (B,c) is an interpolant for B + C. Let In
the mapping
n !-+
In
Turing machine in time bounded by some function d € g tained as a suitable composition of the functions u,b some polynomial).
The mappingnI,n
(d
can be ob-
together with
is now proved to satisfy all
our requirements by the same argument as in the end of the proof Lemma
4.
This completes the proof o€ Lemma
3
Turing machine
accepting
Thus Theorem 7
is proved.
of
9.
Arguing now as in section 5 , using Lemmas 8 and 9 closure properties of
=
is computable by a deterministic
and the
, one easily produces a deterministic TAUT
in time bounded by a function of
5.
D. MUNDICI
356
10.
F u r t h e r Topics.
I n t h i s f i n a l s e c t i o n w e survey
what i s known on t h e complexity
of
e s h a l l s t a t e a number of r e s u l t s C r a i g ' s i n t e r p o l a t i o n theorem. W c o n c e r n i n g t h e r a t e of growth of i n t e r p o l a n t s , b o t h i n s e n t e n t i a l Boolean e x p r e s s i o n s for s e n t e n t i a l l o g i c
and i n f i r s t - o r d e r l o g i c .
are p a r t i c u l a r words over a l p h a b e t
c
as d e f i n e d i n s e c t i o n 1 .
Sentences of f i r s t - o r d e r l o g i c are u n d e r s t o o d as p a r t i c u l a r over some s u i t a b l e a l p h a b e t ).
( s ee [I]
rules
c' ,
words
according t o t h e f a m i l i a r formation
I n s e n t e n t i a l l o g i c t h e precise determination
of t h e r a t e of growth
of i n t e r p o l a n t s i s an open
(and i m p o r t a n t )
The Pollowing Theorem s t a t e s t h a t if s e n t e n t i a l i n t e r p o l a n t s
problem.
t u r n out t o grow polynomially, i n d e t e r m i n i s t i c polynomial
t h e n e v e r y f u n c t i o n which i s computable
(Turing) t i m e ,
has c i r c u i t depth
p o r t i o n a l t o t h e l o g a r i t h m of t h e i n p u t l e n g t h .
pro-
T h i s would p r o v i d e
a p o s i t i v e s o l u t i o n t o a c e n t r a l open problem of computation t h e o r y (see [I I] )
.
Assume t h e r e e x i s t s a polynomial
10.1 Theorem. whenever
an i n t e r p o l a n t
with
I
111
t h e r e s t r i c t i o n of
f
d e p t h Fn See [6
Recall function
,
< p ( 1BI +
F1 , F 2 ,
to
,
2.3.23)
f : {O,l}n-
I
tO.1)
,
Then f o r e v e r y f u n c t i o n
c i r c u i t s , with
d e l a y complexity of
computing
n
c>O,
.
n = 1,2,...
. that the delay
complexity
of a Boolean
i s t h e d e p t h oP t h e smallest d e p t h
(over our f i x e d b a s i s
Boolean e x p r e s s i o n
F
such t h a t , f o r some
f o r each
)
{A, V
. As u s u a l ,
any
i s r e g a r d e d as a Boolean f u n c t i o n o v e r i t s own
B
variables, via the identification
propagate t o
ICl).
... of
[O,lJn,
< c - l o g2 n
Theorem 2 . g
(from b 0
circuit for
such t h a t
which i s computable i n d e t e r m i n i s t i c polynomial
(0'1)
t i m e t h e r e i s a sequence
ProoP.
p
i s a t a u t o l o g y i n s e n t e n t i a l l o g i c , one can f i n d
B-C
f : {O,l]*-
for t h e n e c e s s a r y background:
See f l 0 , 2.21
B
1 = 'ltrue'l
i s , roughly,
and
0 = "false"; the
t h e t i m e needed f o r i n p u t s t o
t h e o u t p u t , i n t h e Pastest c i r c u i t computing
B.
The
f o l l o w i n g Theorem t h e n s t a t e s t h a t t h e t i m e needed by t h e f a s t e s t
NP and Craig's Interpolation Theorem
c i r c u i t t o compute terpolant
( t h e Boolean f u n c t i o n c o r r e s p o n d i n g t o ) any
needed t o compute e i t h e r of 10.2 Theorem. d<620) and I
C
or
B
C:
F o r i n f i n i t e l y many
t h e r e i s a tautology
d€N
(and s t a r t i n g w i t h
some
i n s e n t e n t i a l l o g i c , with
B+C
B
having t h e i r d e l a y complexity S d , such t h a t e v e r y i n t e r p o l a n t
h a s a d e l a y complexity
Proof.
in-
B - + C may happen t o be g r e a t e r t h a n t h e time
for
I
351
17,
See
d
I
>d +
(1/3)log2(d/2).
Theorem 2.51
As remarked above, i n s e n t e n i a l l o g i c t h e r e i s a t p r e s e n t no deEini-
t i v e e s t i m a t e oP t h e r a t e oP growth of ICl
, where
Theorem
a s a f u n c t i o n of IBI and
111
i s a smallest l e n g t h i n t e r p o l a n t € o r
I
(See [6,
B-C.
1.93 f o r an upper bound, and t r y t o improve i t ) .
By c o n t r a s t
i n f i r s t - o r d e r l o g i c w e have: 10.3 Theorem. ( i ) In t h e arithmetical hierarchy t h e r e i s a
TI1 - f u n c t i o n
g i v i n g an upper bound f o r t h e l e n g t h oP P i r s t - o r d e r (*)
whenever
B+C
interpolant ( i i ) No
c
1
I
-function
with
B+C
as i n
(*)
[ 6 , Theorem 3.13
Theorem
b
ICl). can g i v e an
.
Proof.
,
b(lBI+
111
( i . e . , no r e c u r s i v e f u n c t i o n )
111
[4
i n t e r p o l a n t s , i.e..
i s v a l i d i n f i r s t - o r d e r l o g i c , t h e r e i s an for
upper bound f o r ( i ) See
b: N+pl
.
(11)
T h i s c a n be e x t r a c t e d from
11.
Due t o i t s a s y m p t o t i c c h a r a c t e r , t h e above Theorem
10.3 ( i i ) g i v e s
no i n f o r m a t i o n on t h e p o s s i b l e l e n g t h s of i n t e r p o l a n t s € o r s h o r t plications.
The 'following i s a non-asymptotic r e s u l t :
We c a n w r i t e down a v a l i d i m p l i c a t i o n i n f i r s t - o r d e r
10.4 Theorem.
logic,
with
B---+C
l B ( .lC(<1145
pol a n t w e have : .*2
III> 2'
Proof.
See
im-
&,
Theorem 3 . 4 .
1
such t h a t whenever
seven t w o ' s
.
I
i s an i n t e r -
D.MUNDICI
358
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.
. .
Mundici D . , Compactness, i n t e r p o l a t i o n and F r i e d m a n ' s t h i r d problem, Annals of Mathematical Logic 22 ( 1 982) 197-21 1
.
Savage J.E.,
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