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Finite Model Theory Lecture 15

Finite Model Theory Lecture 15. FO k Types. Computing Types. Recall: tp FO k (A, a ) = the set of all FO k formulas that are true at (A, a ) First question: given a , b 2 A m , do they have the same type ? Notation: a ¼ FO k b. Intuition about ¼ FO k.

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Finite Model Theory Lecture 15

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  1. Finite Model TheoryLecture 15 FOk Types

  2. Computing Types • Recall: tpFOk(A, a) = the set of all FOk formulas that are true at (A, a) • First question: given a, b2 Am, do they have the same type ? • Notation: a¼FOkb

  3. Intuition about ¼FOk • Let k be larger then all m’s below (e.g. k=10) • Which implies what ?(a,b,c) ¼ (a’,b’,c’) (a,c) ¼ (a’,c’)(a,b) ¼ (a’,b’) (a,a,b) ¼ (a’,a’,b’) • Conclusion: if m · k, we may take m = k

  4. Computing Types Theorem There exists an IFP formula f(x, y) s.t. 8a, b2 Ak, a¼b iff A ²f(a, b) Proof Will compute the negation, aÀb as an IFP formula (should be not ¼)

  5. Proof • Let a1(x), …, as(x) be all quantifier free types with k variables (i.e. in FOk[0]) • y0(x, y) = Çi ¹ j (ai(x) Æaj(y)) • y(R, x, y) = y0(x,y) Ç ((Çi=1,k9 xi8 yi R(x, y)) Æ (Çi=1,k9 yi8 xi R(x, y))

  6. Proof What does IFP(y)(x, y) say ? • The n’th unfolding says that the spoiler can win the pebble game after at most n moves, if starting at x, y • The IFP says that the spoiler wins if starting at x, y

  7. Ordering the Types • An order on the FOk types of A is a total preorder a¹b s.t. a¼b iff a¹b and b¹a • There are many possible orders of types… Theorem There exists an IFP formula f(x, y) that computes an order on types Proof [ in class ]

  8. The Canonical Structure Given A 2 STRUCT[s] and a formula f in some logic with iteration, we can compute f in two steps: • First, compute a “canonical” structure Ck(A) = A/¼k over s’ • s’ = <, U, U1, …, Up, S1, …, Sk, P1, …, Pt • Where: < is order on types, Ui(a1, …, ak) iff Ri(a1, …, am) (for m · k); the others will be explained • Second, compute some modified formula f0 on Ck(A)

  9. Canonical Structure • Let’s construct f0, and discover what we need in s’ • f: xi = xjf0: 9 y.(Pp(x,y) Æ U(y)) where p(1) = i, p(2) = j • f: R(xi1, …, xim) f0: 9 y.(Pp(x,y) Æ Ui(y)) • :f:f0 • f1Æf2f10Æf20 • 9 xif9 y.(Si(x,y) Æf0(y))

  10. Abiteboul&Vianu’s Theorem TheoremPTIME=PSPACE ) IFP = PFP Proof. Supposes PTIME = PSPACE. Consider a PFP formula f. It can be expressed in two stages: first compute a canonical structure, using IFP, then compute f0 (still a PFP) on the canonical structure. The latter is PSPACE problem, hence in PTIME, and, since it is ordered, f0 can be expressed as IFP.

  11. The Paper • Extends this theorem to other forms of iterations and other complexity classes

  12. Computing Types • Slightly harder question: • Given (A, a), derive a formula f(x) s.t. forall (B, b):B ²f(b) iff tpFOk(B,b) = tpFOk(A,a) Theoremf can be expressed in FOk Proof in the book

  13. Structure of Lk1w Corollary Every formula in Lk1w is equivalent to:Çi 2 Nfiwhere f0, f1, … 2 FOk Comments on loose Generic Machines in class

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