Notes

Outline

Finite Model Theory Lecture 3 Ehrenfeucht-Fraisse Games

Outline Proof of the Ehrenfeucht-Fraisse theorem

Notation If A is a structure over vocabulary s and a1, …, am 2 A then (A,a1, …, am) denotes the structure over vocabulary sn = s [ {c1, …, cm} s.t. the interpretation of each ci is ai In particular, (A,a) ' (B,b) means that there is an isomorphism A ' B that maps a to b

Types In classical model theory an m-type for m ¸ 0 is a set t of formulas with m free variables x1, …, xm s.t. there exists a structure A and m constants a = (a1, …, am) s.t. t = {f |  A ² f(a) } In finite model theory this is two strong: (A,a) and (B,b) have the same type iff they are isomorphic (A,a) ' (B,b)

Rank-k m-Types FO[k] = all formulas of quantifier rank · k Definition Let A be a structure and a be an m-tuple in A.  The rank-k m-type of a over A is tpk(A,a) = {f 2 FO[k] with m free vars | A ² f(a) } How any distinct rank-k types are there ? [finitely or infinitely many ?]

Rank-k m-Types For m ¸ 0, there are only finitely many formulas up to logical equivalence over m variables x1, …, xm in FO[0]   [why ?] For m ¸ 0, there are only finitely many formulas up to logical equivalence over m variables x1, …, xm in FO[k+1]   [why ?]

Rank-k m-Types For each rank-k m-type t there exists a unique rank-k formula f s.t. A ² f(a) iff tpk(A,a) = t In other words, if M = {f1, …, fn} are all formulas in FO[k] with n free variables, then for every subset M0 µ M there exists a f 2 M s.t. f = (Æy 2 M0) y Æ (Æy Ï M0 : y)

The Back-and-Forth Property The k-back-and-forth equivalence relation 'k is defined as follows: A '0 B iff the substructures induced by the constants in A and B are isomorphic A 'k+1 B iff the following hold: Forth: 8 a 2 A 9 b 2 B s.t. (A,a) 'k (B,b) Back: 8 b 2 B 9 a 2 A s.t. (A,a) 'k (B,b)

The Back-and-Forth Property What does A 'k B say ? If we have a partial isomorphism from (A, a1, …, ai) to (B,b1, …, bi), where i < k, and ai+1 2 A, then there exists bi+1 2 B s.t. there exists a partial isomorphism from (A, a1, …, ai, ai+1) to (B, b1, …, bi, bi+1); and vice versa

Ehrenfeucht-Fraisse Games Theorem The following two are equivalent:  A and B agree on FO[k]  A ºk B  A 'k B Proof  2 , 3 is straightforward 1 , 3 in class

More EF Games (informally) Prove, informally, the following:

More EF Games (informally)

Hanf’s Lemma One of several combinatoric methods for proving EF games formally Definition.  Let A be a structure.  The Gaifman graph G(A) = (A, EA) is s.t. (a,b) 2 EA iff 9 tuple t in A containing both a and b Definition.  The r-sphere, for r > 0, is:                 S(r,a) := {b 2 A | d(a,b) · r}

Hanf’s Lemma Theorem [Hanf’s lemma; simplified form]  Let A, B be two structures and there exists m > 0 s.t. 8 n · 3m and for each isomorphism type t of an n-sphere, A and B have the same number of elements of n-sphere type t.  Then A ºm B. Applications: previous examples.

Summary on EF Games Complexity: examples in class are simple; but in general the proofs get quite complex Informal arguments: We are all gamblers:  “If you play like this […] you will always win”.  We usually accept such statements after thinking about […] “here is a property not expressible in FO !”.  We don’t accept that until we see a formal proof. Logics v.s. games: Each logic corresponds to a certain kind of game.