Normal forms for second-order logic over finite structures, and classification of NP optimization problems

Abstract

We start with a simple proof of Leivant's normal form theorem for ∑ 1 1 formulas over finite successor structures. Then we use that normal form to prove the following: 1. (i) over all finite structures, every ∑ 2 1 formula is equivalent to a ∑ 2 1 formula whose first-order part is a Boolean combination of existential formulas, and 2. (ii) over finite successor structures, the Kolaitis-Thakur hierarchy of minimization problems collapses completely and the Kolaitis-Thakur hierarchy of maximization problems collapses partially. The normal form theorem for ∑ 2 1 fails if ∑ 2 1 is replaced with ∑ 1 1 or if infinite structures are allowed.