1P = PSPACE:
A.
Simplified
Proof
SHEN
Academy
of Sciences,
Moscow,
Rassia,
CIS
Abstract, Lund et al. [1] have proved that PH is contained
in 1P. Shamir [2] improved this
technique and proved that PSPACE = 1P. In this note, a slightly simphfied version of Shamir’s
proof IS presented, using degree reductions instead of simple QBFs.
Categories
and Subject Descriptors:
F. 1. 2 [Computation by Abstract Devices]: Modes of
and nondetemnmzs??<
probabdlstcc
compatatzon:
F. 1.3 [Computation by
computation—Altemattort
Abstract Devices]: Complexity classes—relation amo~zg corrzplc~i~ classes: F.4. 1 [Mathematical
Logic and Formal Languages]; Mathematical
Logic—proof
theo~
General
Terms: Theory
Additional
Key Word~ and Phrases: Interdctwe
proofs, PSPACE
1. Introduction
It is well
known
to show that
language
that
1P is contained
in PSPACE.
some PSPACE-complete
of true
‘“” Q,lxnB(x,
fiers) and QI
Quantified
Boolean
... x,,), where B(xl
. . . Q,l ● {V, 3}.
So, for equality,
language
has an IP-protocol.
Formulas
(QBF),
“o. x.)
is a Boolean
that
it is enough
We use the
is, formulas
formula
(without
Qlxl
quanti-
Each Boolean
formula
B( xl .0” x,, ) corresponds
to a polynomial
b( x, “”” x.)
where aA~isreplaced
bycx. p, 7a
byl–aandav~
bya*
~=a+~
— a . /3 ( = 1 — (1 – a)(l
– ~ )). Its value coincides
with the value of B on
boolean
arguments
(0 = False,
Let I’(x,.
. . ) be a polynomial.
(AXP)(...
(E&)(...)
(Rxl’)(x,...)
)=
P(O)...
=F’(O,...
is absent.
Note
P and
-x)
all x“
with
RxP coincide
The work of the author was supported
Air Force contract AFOSR 89-0271.
Author’s
Moscow,
P(1),..),
n >1
RxP has the same variables
that
polynomials
). P),,...),
)*
=Pmod(x’
(i.e.,
The polynomial
1 = True).
Define
three
by National
are replaced
as P; in AxP
on Boolean
x).
and E.xP, variable
x
arguments.
Science Foundation
address: Institute
of Problems of Information
103051, u1. Ermolovoi,
19. Russia, CIS.
by
grant CCR 89-12586 and
Transmission,
Academy
of Sciences,
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Jwrndl “f LheA,.cmd,c,n tm C~mput]ng ivl~chm~q. VOI 39 NO J, Oct”bc, 1992, pp b78-880
1P = PSPACE:
Let
know
S(xl
Simplified
~ when
the operations
over
a allowing
1 for any input
less than
879
““. x~ ) be a polynomial
an IP-protocol
probability
Proof
it is false.
(Ax,
some
finite
P to persuade
field
V that
F. Assume
S(ul
we
UI “”” u~, u = F when it is true and with probability
Let U be a polynomial
obtained
from S by one of
Ex, Rx). Let the degree
of S with
respect
to x be less than
some constant
d (known
to V). We construct
persuade
V that U(CI “”” cl) = e with probability
when it is true and with probability
< c + d/#F
a protocol
denotes
a as a procedure
the cardinality
that
. . . UL) = v with
of F.) This
protocol
uses
1 for
when
~
allowing
P to
c1 ““” cl, e
(Here, #F
any input
it is false.
called
on~
once.
2. A Construction
Case A.
P wants
polynomial
U(yl
of IP-protocol
““” y~) =Axs(x,
B
y~ ““” y~).
to persuade
V that U(CI . . . cl) = e. P sends V the coefficients
of a
s(x) = S(x, c1 “o” cl). If degree(s) > d or s(0)s(l)
# e, V rejects.
Otherwise,
V sends P a random
element
must persuade V that S(r, c1 “” o cl) = s(r).
Case E.
U(yl
Replace
s(0)s(l)
Case R.
.“” yl)
= ExS(X,
y,
r c F. Now
(using
protocol
a),
P
“o. yl).
by s(0) * s(l).
U(x, y,
“’” y~) =
RXS(.X,
y~
““” yl).
P wants to persuade
V that ICI(f, c1 “o. c1) = e. P sends V the coefficients
of
s(x) = S(x, c, “.. c,). If de~ee(s)
a polynomial
> d or s(0) + (s(1) – s(0))f #
e, ‘V rejects (note that s(0) ; (s(1) – s(O~f is the value of s(x) mod (Xz – ~) at
f). Otherwise,
must persuade
P can
fool
V sends P a random
element
V that S(r, c1 “o” co = s(r).
V
polynomials
s(x)
not greater
than
either
during
a
r = F. Now
(probability
and S(x, c1 “”” cl) coincide
less
(using
than
at the random
protocol
E) or
point
if
a),
P
different
r (probability
d/#F).
Let + = Qlxl
.-. Q~x~l?(xl
. . . x,,) be a QBF; QI . . . Q,, G {V, 3}. Consider
a polynomial
b(xl
“”” x.) corresponding
to B(xl
“”” x.) and apply (sequentially)
operations
RX1, RX2,...,
RX,*,
qil x,, 7
Rxl,
Rx2,...,
Rxr, _,,
qn-lxn-1>
RX,, RX2,
qlxl>
where q, = A or E if Q, = ‘d or 3, respectively.
After these operations,
we get
a constant
equal to O or 1, depending
on the truth value of ~. P can persuade
V that this constant
is 1 using the reduction
steps described.
Ultimately,
the
equality
b(u ~ .”” u,, ) = v must be checked for some UI . ~. u., v; V can do this
880
alone because
exceed
the
formula
(number
B is known.
of operations
The
probability
A, E, R) . (maximal
of
error
A.
SHEN
does
not
degree)
(#F)
If the length of QBF was 1, then number
of operations
is O(lz) and maximal
degree is 0(1) (degree of t does not exceed 1, R-operations
reduce it to 1 and
later
all degrees
trivial
for
are not
greater
than
2). If #F
is about
14, the probability
to O when 1 + CO.So we can use F = Z/pZ
where p is a prime
length
(p can be chosen by P or V because primality
testing
error tends
logarithmic
numbers
sense of Shamir
of this
size).
It is easy to see that
Verifier
is weak
of
of
is
in the
[2].
ACKNOWLEDGMENTS.
original
proof
version
of
Laborato~
I am grateful
to Prof. Michael
Sipser who explained
the
of IP-PSPACE
to me, and suggested that I write this simplified
it.
I
also
want
for Computer
to
Science
thank
MIT’s
for their
Mathematics
Department
and
hospitality.
REFERENCES
1.
2.
LUND, C., FORTNOW’, L., KARLOFF. H., AND NISAN, N.
Algebraic
proof systems. J. ACM 39, 4 (Oct. 1992), 859-868.
SHAMIR, A.
1P = PSPACE, J. ACM 39, 4 (Oct. 1992), 869-877.
RECEIVED
MAY
1991;
REVISED NOVELIBER 1991; ACCEPTED JULY 1991
Journal oi the Assocmtlon for Computmg Machmerv, Vol 39, No 4, Odoba
1992
methods
for interactive