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\lecturetitle{Advanced}{12}{Measuring the Complexity of
Proofs (Part 2)}{August}{1}

\noindent
In the previous lecture, we motivated, through a couple of examples,
the idea of a proof system based on axioms and inference rules.  We
then defined the Sequent Calculus and briefly explained why this proof
system is both sound and complete.  We ended by asking whether the
Sequent Calculus has proofs of polynomial size for every tautology.
 
In this lecture, we will finish covering the material that was
originally planned for Lecture 11.  We will define precisely what a
proof system is and show that there is a proof system in which every
tautology has a polynomial-size proof if and only if $\NP$ is closed
under complement.  As will be explained, we can then work towards
a proof that  $\P \neq \NP$ by proving exponential lower bounds for
increasingly more powerful proof systems.  

The plan for the next three lectures is to investigate the power of
some of these systems.  The goal will not be to present the most
definite results, but to highlight the strong and fruitful connections
between proof complexity and computational complexity.  

The tautology that has been used most frequently in establishing lower
bounds for various proof systems is perhaps the pigeonhole principle.
So our first result will be to show that the pigeonhole principle
(that is, the propositional tautology expressing this principle) has
polynomial-size Frege proofs.  This will be established by exploiting
the fact that input bits can be added and compared to a threshold $k$
by $\NC^1$ circuits.

Second, we will consider the Resolution proof system, which can be
viewed as a version of the Sequent Calculus in which all formulas are
variables and the only rule is the cut rule.  We will show that a
certain restriction of Resolution can not prove the pigeonhole
principle by studying, in terms of decision trees, the computational
complexity of a search problem associated with the tautology.

Third, we will present a general method for proving lower bounds on
the complexity of proof systems called the interpolation method.  The
idea is to establish that small proofs of certain tautologies can be
translated into small circuits of a related computational problem.  We
will illustrate how this method has been used to prove lower bounds
for the Cutting Planes proof system.  Lines in a Cutting Planes proof
are linear inequalities of the form $a_1 x_1 + \cdots + a_n x_n \geq
A$.  New lines can be inferred from previous ones by addition and by
multiplication and division by positive integers.  The goal is to
obtain the contradiction $0 \geq 1$, which then refutes the initial
set of inequalities.  Cutting Planes are more powerful than Resolution
and no more powerful than the Sequent Calculus and Frege systems.  

Finally, if time permits, we will briefly survey results concerning
some other proofs systems such as Frege systems in which all formulas
are of constant-depth.  Once again, techniques and ideas from
computational complexity play a key role in establishing lower bounds
for these systems.


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