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Finite Model Theory Lecture 4

Finite Model Theory Lecture 4. Ordered Structures, Gaifman’s Theorem. Outline. Order-invariant queries Gurevitch’s theroem Gaifman’s theorem (time permitting). Invariant Queries. Idea: Use some innocuous additional information to express a query “Forget” the extra information

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Finite Model Theory Lecture 4

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

  2. Outline • Order-invariant queries • 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 A2 STRUCT[s<,+] s.t. < is interpreted as a linear order a1 < … < an (ai, aj, ak) 2 + 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 A2 STRUCT[s], if A’ is any structure over the same universe and vocabulary s<,+ then (A, A’) ²f 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 A2 STRUCT[s] and A’, A’’ 2 C, s.t. A=A’=A” (i.e. same universes) (A,A’) ²f iff (A,A’’) ²f Notationf2 (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.(8 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. A²fB iff A is an atomic boolean algebra, A = (2X, µ)Let EVENB = (EVEN(|X|) ÆfB • Show that EVENB2 (FO + <)inv • Prove (using EF-games) that EVENBÏ 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 x2 A s.t. x Åx = ?, 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 Åx = ?, 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?) Æ8 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 EVENB2 (FO + <)inv 9 P.(8 x.y.(S(x,y) ) (P(x)Æ: P(y) Ç: P(x)ÆP(y)) Æ P(min) Æ: P(max)) Recall: EVEN = 9 p.(8 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 EVENBÏ 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 X1Å X2 = ;, Y1Å 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) [Why ?] [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 A2 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)) Æy 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))

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