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\lecturetitle{Advanced}{15}{The Interpolation Method}{August}{4}

\noindent
In this lecture, we introduce the {\em Cutting Planes} proof system
and prove an exponential lower bound using the {\em interpolation
method}.  Like Resolution, Cutting Planes is a refutation 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 derive the inequality $0 \geq 1$.
This establishes that the initial set of inequalities is
unsatisfiable.  We will show that Cutting Planes are more powerful
than Resolution and no more powerful than the Sequent Calculus and
Frege systems.

The interpolation method can be seen as an extension of the decision
tree method we used in the previous lecture to prove a lower bound for
Tree Resolution.  There, we converted a refutation of an unsatisfiable
set of clauses into a decision tree of the same size for the following
problem: given as assignment to the variables, output a clause that is
falsified by that assignment.  We then showed that the decision tree
has to be large, which implies that the refutation is large.  

In this lecture, we convert a Cutting Planes refutation of a set of
inequalities of a certain form into a circuit of roughly the same size
that computes a related function.  By choosing a particular set of
inequalities, we will ensure that this function can be computed only
by large circuits, which implies that the refutation has to be large.

The interpolation method has been used to obtain lower bounds for
several other proof systems, including Resolution.  This is our fourth
and last example the strong and fruitful connections between
computational complexity and proof complexity.  To summarize, here are
the four examples we saw:
\begin{enumerate} 

\item We showed that a polynomially bounded proof system exists if
and only if $\NP$ is closed under complement.

\item We designed a polynomial-size Sequent Calculus proof of the
pigeonhole principle using an $\NC^1$ iterated addition circuit.

\item We proved a lower bound on decision trees to prove a lower bound
for Tree Resolution.

\item We used a circuit complexity lower bound to prove a lower bound
for Cutting Planes.

\end{enumerate}

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