Finite Model Theory/FO EFM

The method for employing Ehrenfeucht-Fraisse-Games for (in-)expressibility-proofs is given by the following:

Theorem

Let P be a property of finite σ-structures. Then the following are equivalent


 * P is not expressible in FO
 * for every k $$\in \mathbb{N}$$ there exist two finite σ-structures $$\mathfrak{A}_k$$ and $$\mathfrak{B}_k$$, such that the following are both satisfied
 * $$\mathfrak{A}_k\equiv_k \mathfrak{B}_k$$
 * $$\mathfrak{A}_k$$ has P and $$\mathfrak{B}_k$$ does not have P

Remarks


 * Thus using the EFM works roughly as follows:
 * choose a k
 * construct two structures - one with the property, one without - that are big enough s.t. the duplicator wins the k-ary EFG
 * show that this can be expanded with k
 * So, a non-expressible property (i.e. the effort to check it) must be somehow 'expandable' with k

Examples


 * To begin pick two linear orders say A ={1, 2, 3, 4} and B ={1, 2, 3, 4, 5}. For a two-move Ehrenfeucht game D is to win, obviously. This gives us two structures that satisfy the above conditions for k = 2 and the Property having even cardinality (that A has and B doesn't). Now we have to expand this over all k $$\in \mathbb{N}$$. From the above example we adopt that in a linear order of cardinality $$2^k$$ or higher D has a winning strategy. Thus we choose the cardinalities depending on k as |A| = $$2^k$$ and |B| = $$2^k$$+1. So we have found an even A and an odd B for every k, where D has a winning strategy. Thus (by the corollary) having even/odd cardinality is a property that can not be expressed in FO for finite σ-structures of linear order.