Space in Proof Complexity

Propositional proof complexity is the study of the resources that are needed to prove formulas in propositional logic. In this thesis we are concerned with the size and space of proofs, and in particular with the latter. There are different approaches to reasoning that are captured by corresponding proof systems, each with its own strengths and weaknesses and need for resources. The simplest and most well studied proof system is resolution, and we try to get our understanding of other proof systems closer to that of resolution. In particular we look at polynomial calculus, which captures reasoning with polynomials, and cutting planes, which captures reasoning with linear inequalities. In resolution we can prove a space lower bound just by proving a lower bound on the widest clause in the proof, that is showing that any proof must have a large clause. We prove a similar relation between resolution width and polynomial calculus space that let us derive space lower bounds, and we use it to separate degree and space. For cutting planes we show length-space trade-offs. This is, there are formulas that have a proof in small space and a proof in small length, but there is no proof that can optimize both measures at the same time. We introduce a new measure of space, cumulative space, that accounts for the space used throughout a proof rather than only its maximum. This is mostly exploratory work, but we can also prove new results for the usual space measure, for instance that there are trade-off results for resolution where every short proof not only needs to use large space, but it needs to do so most of the time. We define a new proof system that aims to capture the power of current SAT solvers based on conflict-driven clause learning (CDCL), and we show a rich landscape of length-space trade-offs comparable to those in resolution. To prove these results we build and use tools from other areas of computational complexity. One area is pebble games, which are very simple computational models that are useful for modelling space. In addition to results with direct applications to proof complexity, we show that pebble game cost is PSPACE-hard to approximate. Another area is communication complexity, which is the study of the amount of communication that is needed to solve a problem when its description is shared by multiple parties. We prove a simulation theorem that relates the query complexity of a function with the communication complexity of a composed function.