Department of Computer Science, University of Toronto May 2008

In this thesis we explore a number of ways in which combinatorial games can be used to help prove results in the area of propositional proof complexity. The results in this thesis can be divided into two sets, the first being dedicated to the study of Resolution space (memory) requirements, whereas the second is centered on formalizing the notion of `dangerous' reductions. The first group of results investigate Resolution space measures by asking questions of the form, `Given a formula F and integer k, does F have a [Type of Resolution] proof with [Type of Resource] at most k?'. We refer to this as a proof complexity resource problem, and provide comprehensive results for several forms of Resolution as well as various resources. These results include the PSPACE-Completeness of Tree Resolution clause space (and the Prover/Delayer game), the PSPACE-Completeness of Input Resolution derivation total space, and the PSPACE-Hardness of Resolution variable space. This research has theoretical as well as practical motivations: Proof complexity research has focused on the size of proofs, and Resolution space requirements are an interesting new theoretical area of study. In more practical terms, the Resolution proof system forms the underpinnings of all modern SAT-solving algorithms, including clause learning. In practice, the limiting factor on these algorithms is memory space, so there is a strong motivation for better understanding it as a resource. With the second group of results in this thesis we investigate and formalize what it means for a reduction to be `dangerous'. The area of SAT-solving necessarily employs reductions in order to translate from other domains to SAT, where the power of highly-optimized algorithms can be brought to bear. Researchers have empirically observed that it is unfortunately possible for reductions to map easy instances from the input domain to hard SAT instances. We develop a non-Hamiltonicity proof system and combine it with additional results concerning the Prover/Delayer game from the first part of this thesis as well as proof complexity results for intuitionistic logic in order to provide the first formal examples of harmful and beneficial reductions, ultimately leading to the development of a framework for studying and comparing translations from one language to another.

Part I -Preliminaries

- Thesis Overview
- An Overview of Proof Complexity & Broad Definitions

- Introduction to Part II
- The PSPACE-Completeness of Tree Resolution Clause Space
- The PSPACE-Completeness of Input Resolution Total Space
- The PSPACE-Hardness of Resolution Variable Space
- The Complexity of Resolution Width

- Introduction to Part III
- A Non-Hamiltonicity Proof System
- Prover/Delayer Game Upper Bounds
- Formalizing Dangerous Reductions
- The Proof Complexity of Intuitionistic Propositional Logic