1 Finite Model Theory Lecture 3 Ehrenfeucht-Fraisse Games

2 Outline Proof of the Ehrenfeucht-Fraisse theorem

3 Notation If A is a structure over vocabulary  and a 1, …, a n 2 A then ( A,a 1, …, a n ) denotes the structure over vocabulary  n =  [ {c 1, …, c n } s.t. the interpretation of each c i is a i In particular, ( A,a) ' ( B,b) means that there is an isomorphism A ' B that maps a to b

4 Types In classical model theory an m-type for m ¸ 0 is a set t of formulas with m free variables x 1, …, x m s.t. there exists a structure A and m constants a = (a 1, …, a m ) s.t. t = {  | A ²  (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)

5 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 tp k ( A,a) = {  2 FO[k] with m free vars | A ²  (a) } How any distinct rank-k types are there ? [finitely or infinitely many ?]

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

7 Rank-k m-Types For each rank-k m-type t there exists a unique rank-k formula  s.t. A ²  (a) iff tp k ( A,a) = t In other words, if M = {  1, …,  n } are all formulas in FO[k] with n free variables, then for every subset M 0 µ M there exists a  2 M s.t.  = ( Æ  2 M 0 )  Æ ( Æ   M 0 :  ) [WAIT ! Isn’t this a contradiction ?]

8 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)

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

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

11 More EF Games (informally) Prove, informally, the following:... (N,S)  (N,S) [ (Z,S) kk...  (Perfectly balanced binary trees are not expressible in FO)

12 More EF Games (informally) kk CONN is not expressible in FO

13 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, E A ) is s.t. (a,b) 2 E A 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}

14 Hanf’s Lemma Theorem [Hanf’s lemma; simplified form] Let A, B be two structures and there exists m > 0 s.t. 8 n · 3 m 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  m B. Applications: previous examples.

15 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.