Super-Exponential Complexity of Presburger Arithmetic

Abstract

Lecturer: Richard Ladner Notes: Recall that Presburger Arithmetic is the theory of the natural numbers with addition. More speciically, let P = (N; s; +; 0. Proof: The idea is to build a formula of linear size that describes the computation of a nonde-terministic Turing machine that runs in double exponential time. There are a number of building blocks needed in order to build our formula. We begin by deening a formula M n (x; y; z) if length O(n) with the property for all nonnegative integers a; b; and c. P j = M n (a; b; c) if and only if a < 2 2 n and ab = c: As can be seen the formula M n deenes a limited form of multiplication. We build M n inductively on n. M 0 (x; y; z) = (x = 0 ^ z = 0) _ (x = 1 ^ z = y): In order to understand the formula M n for n > 0 we use the following fact about nonnegative integers: The key thing to observe is that the length of M n is linear in n. 75