Expansions of pseudofinite structures and circuit and proof complexity

I shall describe a general model-theoretic task to construct expansions of pseudofinite structures and discuss several examples of particular relevance to computational complexity. Then I will present one specific situation where finding a suitable expansion would imply that, assuming a one-way permutation exists, the computational class NP is not closed under complementation. Consider the following situation: M is a nonstandard model of true arithmetic (in the usual language of arithmetic 0, 1,+, ·,≤), n is a nonstandard element of M, L is a finite language and W ∈ M is its interpretation on the universe [n] = {1, . . . , n}; W can be identified with a subset of [n] for some k ∈ N. We shall denote the resulting structure AW ; it is coded by an element of M that is ≤ 2 k . Without a loss of generality we shall assume that L contains constants 1, n, the ordering relation ≤ interpreted as in M, and ternary relation symbols ⊕ and ⊙ for the graphs of addition and multiplication inherited from M. Because M is a model of true arithmetic AW is pseudofinite: it satisfies the theory of all finite L-structures. Paris and Dimitracopoulos [19] studied the problem of for how large m > n does the theory of the arithmetic structure on [n] determine the theory of the arithmetic structure on [m] and proved that it does not for m = 2. They also pointed out various links between questions of this type and complexity theory problems around the collapse of the polynomial time hierarchy. Ajtai [1] showed (among other similar results) that if M is a countable nonstandard model of PA and L is finite then for any L-structure AW there are two sets U,U ′ ⊆ [n], both elements of M, such that M thinks that |U | is odd and |U | is even while, as structures, (AW , U) ∼= (AW , U ) (the isomorphism is not in M, of course). Kraj́ıček and Pudlák [17] showed that for any nonstandard t ≤ n ∈ M We shall discuss another example with parity later.