1 / 13

Finite Model Theory Lecture 15

Understand the concepts of type computing, IFP formulas, structure computations, and canonical structures in finite model theory. Learn about orderings on types and the relationship between PTIME and PSPACE.

loftism
Download Presentation

Finite Model Theory Lecture 15

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related