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Finite Model TheoryLecture 3

Ehrenfeucht-Fraisse Games

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Outline

• Proof of the Ehrenfeucht-Fraisse theorem

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Notation

If A is a structure over vocabulary

and a1, …, an 2 A

then (A,a1, …, an) denotes the structure over vocabulary n = [ {c1, …, cn} 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

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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 = { | 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)

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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) = { 2 FO[k] with m free vars | A ² (a) }

How any distinct rank-k types are there ? [finitely or infinitely many ?]

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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 ?]

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Rank-k m-Types

• For each rank-k m-type t there exists a unique rank-k formula s.t. A ² (a) iff tpk(A,a) = t

• In other words, if M = {1, …, n} are all formulas in FO[k] with n free variables, then for every subset M0 µ M there exists a 2 M s.t. = (Æ 2 M0

) Æ (Æ M0 : )

[WAIT ! Isn’t this a contradiction ?]

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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)

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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

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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

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More EF Games (informally)

Prove, informally, the following:

. . .

(N,S) (N,S) [ (Z,S)

k

. . .

. . .. . .

(Perfectly balanced binary trees are not expressible in FO)

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More EF Games (informally)

k

CONN is not expressible in FO

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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}

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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 m B.

Applications: previous examples.

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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.