6.841 Advanced Complexity Theory
Apr 27, 2009
Lecture 21
Lecturer: Jakob Nordstr
¨
o
m
Scribe: Rishi Gupta
Today and Thursday we will give an (extremely) selective overview of
Proof Complexity
. We will cover
some major concepts, give an overview of the area, prove a classic result, and develop some nice tools along
the way.
Introduction to Proof Systems
Proof Complexity is about figuring out how hard it is to prove things. But what is a proof? I claim 25957
is a product of two primes. What would we accept as proof of this fact? Some possibilities:
•
“Left to the reader”. In a sense, every statement is its own proof.
•
Proof by exhaustion:
compute 25957 mod 2 and every odd number less than 25957, and see that
exactly 2 of them are 0.
•
25957 = 101
·
257, check that 101 and 257 are prime.
Observation by Cook:
A proof should be
efficiently
verifiable.
In particular, a
proof system
P
for a
language
L
is a deterministic function
P
(
x, π
)
∈ {
0
,
1
}
, polynomialtime in
x
and
π
, where
x
∈
L
⇒
there exists
π
such that
P
(
x, π
) = 1 (a proof exists)
x
6∈
L
⇒
for all
π,
P
(
x, π
) = 0 (there are no fake proofs)
Note that there is no restriction on the length of
π
.
Propositional Proof Complexity
We’re interested in proving
tautologies
, namely (boolean) formulas that are true no matter how you assign
values to the variables. For instance, (
x
∨
¯
x
)
∈
TAUT. Why are they interesting?
•
For a while, people thought this would help with the P vs NP question.
•
It provides a framework for quantifying/understanding limits of different kinds of mathematical rea
soning. Proof systems have varying strengths, like the complexity classes we’re familiar with.
•
Commercial SATsolvers are absurdly quick, and routinely solve formulas that have millions of variables.
In fact, good SATsolvers are now roughly Θ(
n
). This subfield is called automated theorem proving.
A proof system has complexity
g
if every
x
∈ P
has a proof of size
g
(

x

). If
g
is polynomial we say
P
is
a polynomiallybounded [propositional] proof system, or a pbpps for short.
Observation by Cook and Reckhow (1979): NP = coNP if and only if there exists a polynomiallybounded
propositional proof system for TAUT (easy exercise).
Corollary: If there is no pbpps for TAUT, P
6
= NP, since P is closed under complementation.
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 Spring '09
 MadhuSudan
 Logic, Computational complexity theory, Automated theorem proving, Interactive proof system, proof systems

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