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Ehrenfeucht-Fraisse Games |
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Proof of the Ehrenfeucht-Fraisse theorem |
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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 |
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In particular, (A,a) ' (B,b) means that there is
an isomorphism A ' B that maps a to b |
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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) } |
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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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FO[k] = all formulas of quantifier rank · k |
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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) } |
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How any distinct rank-k types are there ?
[finitely or infinitely many ?] |
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For m ¸ 0, there are only finitely many formulas
up to logical equivalence over m variables x1, …, xm
in FO[0] [why ?] |
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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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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 |
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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) |
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The k-back-and-forth equivalence relation 'k
is defined as follows: |
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A '0 B iff the substructures induced
by the constants in A and B are isomorphic |
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A 'k+1 B iff the following hold: |
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Forth: 8 a 2 A 9 b 2 B s.t. (A,a) 'k
(B,b) |
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Back: 8 b 2 B 9 a 2 A s.t. (A,a) 'k (B,b) |
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What does A 'k B say ? |
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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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Theorem The following two are equivalent: |
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A and B
agree on FO[k] |
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A ºk
B |
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A 'k
B |
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Proof 2 ,
3 is straightforward |
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1 , 3 in class |
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Prove, informally, the following: |
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One of several combinatoric methods for proving
EF games formally |
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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 |
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Definition.
The r-sphere, for r > 0, is: |
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S(r,a) := {b 2 A | d(a,b) · r} |
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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. |
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Applications: previous examples. |
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Complexity: examples in class are simple; but in
general the proofs get quite complex |
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Informal arguments: We are all gamblers: |
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“If you
play like this […] you will always win”.
We usually accept such statements after thinking about […] |
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“here is a property not expressible in FO
!”. We don’t accept that until we
see a formal proof. |
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Logics v.s. games: Each logic corresponds to a
certain kind of game. |
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