Alternation-Trading Proofs, Linear Programming, and Lower Bounds

A fertile area of recent research has demonstrated concrete polynomial time lower bounds for solving natural hard problems on restricted computational models. Among these problems are Satisfiability, Vertex Cover, Hamilton Path, Mod6-SAT, Majority-of-Majority-SAT, and Tautologies, to name a few. The proofs of these lower bounds follow a certain proof-by-contradiction strategy that we call alternation-trading. An important open problem is to determine how powerful such proofs can possibly be. We propose a methodology for studying these proofs that makes them amenable to both formal analysis and automated theorem proving. We prove that the search for better lower bounds can often be turned into a problem of solving a large series of linear programming instances. Implementing a small-scale theorem prover based on this result, we extract new human-readable time lower bounds for several problems. This framework can also be used to prove concrete limitations on the current techniques.Comment: To appear in STACS 2010, 12 page

Probabilistic Simulations

The results of this paper concern the question of how fast machines with one type of storage media can simulate machines with a different type of storage media. Most work on this question has focused on the question of how fast one deterministic machine can simulate another. In this paper we shall look at the question of how fast a probabilistic machine can simulate another. This approach should be of interest in its own right, in view of the great attention that probabilistic algorithms have recently attracted

Efficient On-Line Simulations of Tree Machines and Multidimensional Turing Machines by Random Access Machines

Coordinated Science Laboratory was formerly known as Control Systems LaboratoryOffice of Naval Research / N00014-85-K-0570Air Force Institute of Technolog

Minimizing Access Pointer into Trees and Arrays

Coordinated Science Laboratory was formerly known as Control Systems LaboratoryJoint Services Electronics Program / N00014-79-C-0424National Science Foundation / MCS-801070

Parallelism Always Helps

Coordinated Science Laboratory was formerly known as Control Systems LaboratoryNational Science Foundation / CCR-892200

What can we know about that which we cannot even imagine?

In this essay I will consider a sequence of questions. The first questions concern the biological function of intelligence in general, and cognitive prostheses of human intelligence in particular. These will lead into questions concerning human language, perhaps the most important cognitive prosthesis humanity has ever developed. While it is traditional to rhapsodize about the cognitive power encapsulated in human language, I will emphasize how horribly limited human language is -- and therefore how limited our cognitive abilities are, despite their being augmented with language. This will lead to questions of whether human mathematics, being ultimately formulated in terms of human language, is also deeply limited. I will then combine these questions to pose a partial, sort-of, sideways answer to the guiding concern of this essay: what we can ever discern about that we cannot even conceive?Comment: 38 pages, 9 pages are reference