Chapter 10 Complexity theoretic model theory and algebra

Abstract

This chapter explores recent results on complexity theoretic model theory and algebra. There are results in recursive model theory and algebra for which the natural complexity theoretic analogue is true, but it requires a more delicate proof that incorporates the resource bounds. There are also results in recursive model theory and algebra for which the natural complexity theoretic analogue is false because the proof of the recursive result uses the unbounded resources allowed in recursive constructions in a crucial way. However, there are a number of interesting new phenomena that arise because of the fact that not all infinite polynomial time sets are polynomial time isomorphic or the complexity theoretic results do not relativize as is the case for most recursion theoretic results. The chapter gives the basic complexity theoretic definitions, establishes notations, and gives a series of lemmas that are useful for building models. A survey of the main existence theorems for feasible models and various feasible categoricity results is presented in the chapter. The chapter discusses the complexity theoretic algebra, focusing on the structure of the binary and tally representation of an infinite dimensional vector space over a polynomial time field.