Time and Space in Proof Complexity

Abstract

In this thesis we explore the limitations of efficient computation by way of the complexity of proofs.

One of the goals of theoretical computer science is to understand algorithms broadly --

to identify meta-algorithms that may be useful in a variety of cases, understand how they work,

when they work well, and when they don't.

For several important groups of algorithms, there is a relatively simple mathematical

proof that the algorithm is correct for every input. Moreover, when the algorithm is run

on any particular input, this proof specializes to produce a very simple combinatorial proof

of the correct answer for that input. A simple example is a linear program, which may find

an optimal point and give a dual solution proving the optimality, or a graph search algorithm

which may fail to find a path between two points and yield a cut separating them instead.

Often times, self-contained proofs that the answer to some query is yes or no are just as

important as the answer itself. When the algorithm is meant to identify problems in some

system, or opportunities, the proof may actually represent actionable information in some domain.

A good example is when a SAT solver is used to validate a system that is being designed. When the

validation fails, the proof indicates where the problem is with the current design.

Proof complexity attempts to place limits on how efficient such algorithms can be by showing that

these proofs must sometimes be very complex, growing rapidly as the problem scales up. More broadly,

proof complexity attempts to understand the proving power of traditional formal systems for logic, and

to corroborate hypotheses like P != NP.

We develop novel combinatorial techniques for studying proof complexity in several well-studied proof

systems. We give substantial quantitative improvements on existing results, and also, develop a new

method that allows us to study the time and space needed for a proof simultaneously and prove ``tradeoff''

lower bounds. We show that small reductions in space can sometimes lead to large increases in time, even

in situations where the algorithm has large amounts of space to work with. Previously no such results

were known in that situation.