Revised Research Proposal or Preliminary Thesis Proposal
The broad relevance and importance of bounded arithmetic and propositional proof complexity are well appreciated; these subjects are two of three which are interconnected in various and interesting ways, the third subject being computational complexity theory. This latter area is rife with old, well-established open problems and one good way to get at them is by studying the other two, whose problems are different yet connected, and many of which may be tractable. The purpose of this paper is to discuss some possible avenues of research related to bounded arithmetic and propositional proof complexity. Some of these are specific technical open questions related closely to recent research and are placed in their context. Others are more vague general directions along with some discussion of what fruit may be borne by efforts so directed. Yet others are larger issues which are not expected to be resolved but which should be kept in mind and toward which some incremental progress may be possible. This paper is a revised version of a previous research proposal and includes a report on progress towards solving some of the stated problems, as well as some additional research possibilities. Some necessary definitions and background will be presented first although this will mostly be kept to a minimum.