1. John Baldwin and Michael Benedikt.
   Stability theory, permutations of indiscernibles, and embedded finite models.
   Trans. Amer. Math. Soc.,
   352(11):4937−4969 (electronic), 2000.

Definition 3 [John Baldwin and Michael Benedikt, 1] Let K be a class of first-order (L,P)-formulas. (M,I) is called (P,K)-reducible iff for every K-formula φ(x,y), there is an order formula ψ(w,y) such that for every m there is a cm∈I such that (∀ y∈P) (ψ( cm,y)≡ φ(m,y)). If K is the class of all the L-formulas, (P,K)-reducibility is called P-reducibility. If K is the class of all the (L,P)-formulas, (P,K)-reducibility is called strong P-reducibility.

Definition 4 [John Baldwin and Michael Benedikt, 1] A first-order (L,P)-formula is called P-restricted iff it contains no P or it has one of the forms (∀x∈P)Ψ or (∃x∈P)Ψ where Ψ is a P-restricted formula. A structure (M,I) is called P-restricted iff any (L,P)-formula is equivalent over (M,I) to a P-restricted one.

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