Finite Model Theory
Lecture 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 s
and a1, …, an 2 A
then (A,a1, …, an) denotes the structure over
vocabulary sn = s [ {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 = {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)
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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) = {f 2 FO[k] with m free vars | A ² f(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 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  M0 : y)
[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) 11
...
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 nsphere 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.
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