Definition 1

Let M be a model of T and an infinite set I be an indiscernible sequence in M. A complete L-theory T has the (M,I)-Pseudo-finite Homogeneity Property iff whenever (N,J) is first-order equivalent to (M,I), A and B are pseudo-finite subsets of J in the model N, C and D are finite subsets of N, and h:(A∪C)→(B∪D) is an elementary map in N with ω-saturated (N,A,B,h), for any a∈N there is b∈N such that h∪{(a,b)} is an elementary map in N.

Theorem 2

Let M be first-order equivalent to U and an infinite set I be an indiscernible sequence in M. Suppose the first-order theory of a universe U has the (M,I)-Pseudo-finite Homogeneity Property. Let an extended query ψ be locally generic over finite states over U. Then ψ is equivalent over finite states over U to a restricted query.

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