1. Finite Model Theory Lecture 4
Ordered Structures, Gaifman’s Theorem

2. Outline Order-invariant queries Gaifman’s theorem (time permitting)
Gurevitch’s theroem
Gaifman’s theorem (time permitting)

3. Invariant Queries Idea:
Use some innocuous additional information to express a query
“Forget” the extra information
Need to check that the query is invariant w.r.t. to the extra info

4. Example Recall: EVEN(A) is not expressible in FO
Let s<,+ = {<,+}
Consider A 2 STRUCT[s<,+] s.t.
< is interpreted as a linear order a1 < … < an
(ai, aj, ak) iff i+j=k
Then, we can write a sentence f that checks EVEN(A) [ HOW ?? ]

5. Example (cont’d) f = 9 x. 9 y.(x+x=y Æ : 9 z(y < z))
For any vocabulary s and A 2 STRUCT[s], if A’ is any structure over the same universe and vocabulary s<,+ then (A, A’) ² f iff EVEN(A)

6. C-Invariance
Let s, s’ be two disjoint vocabularies, and C a class of s’-structures (C µ STRUCT[s’])
Definition A sentence f over s [ s’ is called C-invariant if for any A 2 STRUCT[s] and A’, A’’ 2 C, s.t. A=A’=A” (i.e. same universes)
(A,A’) ² f iff (A,A’’) ² f
Notation f 2 (FO+C)inv

7. C-Invariance (cont’d)
We have seen that EVEN 2 (FO + (<,+))inv
Main question for today: Is (FO + <)inv more powerful than FO ?

8. Order-Invariant FO
Theorem (Gurevich) There are properties in (FO + <)inv that are not in FO.
FO & (FO + <)inv

9. The Proof
Idea: recall that we can express EVEN in MSO(<). Let S(x,y) = :(9 z.x<z Æ z<y) Æ x < y
We can’t say 9 P in FO. Instead, we will use a vocabulary s that allows to assert the existence of all 2n sets P.
9 P.( x.y.(S(x,y) ) (P(x)Æ: P(y) Ç : P(x)ÆP(y)) Æ P(min) Æ : P(max))
EVEN=

10. The Proof
3 Steps
Design s, fB s.t. A ² fB iff A is an atomic boolean algebra, A = (2X, µ) Let EVENB = (EVEN(|X|) Æ fB
Show that EVENB 2 (FO + <)inv
Prove (using EF-games) that EVENB Ï FO

11. Boolean Algebras Background: two alternative definitions
Definition 1. A boolean algebra is A = (A, µ) s.t.:
µ is a partial order
8 x, y 2 A 9 inf(x,y); will denote it x Å y
8 x,y 2 A 9 sup(x,y); will denote it x [ y
There are minimal, maximal elements: ?, > 2 A
8 x, there exists x 2 A s.t. x Å x = ?, x [ x = >.

12. Boolean Algebras (con’t)
Definition 2. A BA is an algebra (A,[, Å, ?, >, ) s.t.
[, Å = commutative, associative, idempotent, distributive
x Å ? = ?, x [ ? = x, x Å > = x, x [ > = >
x Å x = ?, x [ x = >

13. Boolean Algebra’s (cont’d)
Proposition. The two definitions are equivalent
Proof:
From µ to Å, [: define x Å y = inf(x,y), x [ y = sup(x,y)
From Å, [ to µ: define x µ y to mean x Å y = x

14. Boolean Algebras (cont’d)
Define: atom(x) = (x ¹ ?) Æ 8 y. (y µ x ) y=? Ç y = x)
Proposition. Let A be finite. Let X = {x | x 2 A, atom(x)}. Then (A,µ) is isomorphic to (2X, µ)
[why ?]

15. Step 1
Define fB to consists of the axioms of a boolean algebra (using Definition 1)

16. Step 2 Show that EVENB 2 (FO + <)inv
Recall:
9 P.( x.y.(S(x,y) ) (P(x)Æ: P(y) Ç : P(x)ÆP(y)) Æ P(min) Æ : P(max))
EVEN =
9 p.( x.y.(atom(x) Æ atom(y) Æ S(x,y) ) (x µ p Æ: (y µ p) Ç : (x µ p) Æ y µ p) Æ minatom µ p Æ : (maxatom µ p)
EVENB =

17. Step 3 Now we prove that EVENB Ï FO
Proposition. Let |X|, |Y| ¸ 2k. Then: (2X, µ) k (2Y, µ)
We prove it using two lemmas.

18. Step 3
Lemma. Assume (2X, µ) k (2Y, µ). Then the duplicator has a winning strategy where he responds to ; with ;, to Y with X, to X with Y.
[Why ?]
Lemma. Assume X1 Å X2 = ;, Y1 Å Y2 = ; and that (2X1, µ) k (2Y1, µ), (2X2, µ) k (2Y2, µ). Then: (2X1 [ X2, X1, X2, µ) k (2Y1 [ Y2, Y1, Y2, µ)
[How do prove the proposition ?]

19. Summary Having < seems to add lots of extra power
Will see this later, for fixpoint logics
In FO(<) it is even more surprising
The extra structure (boolean algebra) seems quite artificial. Can we get rid of it ?
Open problem: does < add extra power over strings ?

20. Gaifman’s Theorem Let A 2 STRUCT[s]
Recall the r-ball (also called r-sphere): Br(x) = {y | d(x,y) · r}
Br(x) can be expressed in FO (obviously)

21. Gaifman’s Theorem
Definition A formula f(x) is r-local if all its quantifiers are bounded to Br(x):
9 z.y  9 z. (z 2 Br(x)) Æ y
8 z.y  8 z.(z 2 Br(x)) ) y)
We write f(r) to emphasize that f is r-local.

22. Gaifman’s Theorem
Theorem [Gaifman] Let s be relational. Then every FO sentence over s is equivalent to a Boolean combination of sentences of the form:
9 x1 … 9 xs(Æi=1,sa(r)(xi)) Æ (Æ1 · i < j · s d>2r(xi, xj))