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Algorithmic Computation and 0/1 Laws in Finite Model Theory

A concise summary on Lw1w with Fagin's theorem, canonical structure, pebble games, FOk types, and 0/1 laws in logic. Learn about the convergence laws, 0/1 laws, and their applications in database query processing. Explore definability, types, extension axioms, and the uniqueness of countable models. Discover the relationship between PTIME, PSPACE, IFP, and PFP in algorithmic computation. Dive into the theoretical aspects and applications in random graphs and probabilistic databases.

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Algorithmic Computation and 0/1 Laws in Finite Model Theory

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  1. Finite Model TheoryLecture 16 Lw1w Summary and 0/1 Laws

  2. Outline • Summary on Lw1w • All you need to know in 5 slides ! • Start 0/1 Laws: Fagin’s theorem • Will continue next time New paper: Infinitary Logics and 0-1 Laws, Kolaitis&Vardi, 1992

  3. Summary on Lw1w Notation Comes from in classical logic • Lab = formulas where: • Conjunctions/disjunctions of ordinal < aÇi 2gfi, Æi 2 g, where g < a • Quantifier chains of ordinal < b 9i 2g xi. f, where g < b • Hence, L1w = [a Law

  4. Summary on Lw1w Motivation • Any algorithmic computation that applies FO formulas is expressible in Lw1w • Relational machines • While-programs with statements R := f • Fixpoint logics: LFP, IFP, PFP, etc, etc Consequence: cannot express EVEN, HAMILTONEAN

  5. Summary on Lw1w Canonical Structure Any algorithmic computation on A can be decomposed • Compute the ¼k equivalence relation on k-tuples, and order the equivalence classes ) in LFP[how do we choose k ???] • Then compute on ordered structure ) any complexity Consequence: PTIME=PSPACE iff IFP=PFP But note that DTC ¹ TC yet L ¹? NL [ why ?]

  6. Summary on Lw1w Pebble Games: with k pebbles • Notation: A 1wk B if duplicator wins Theorem 1. For any two structures A, B: • A, B are Lk1w equivalent iff • A 1wk B Theorem 2. If A, B are finite: • A, B are FOk equivalent iff • A, B are Lk1w equivalent iff • A 1wk B

  7. Summary on Lw1w Definability of FOk types • FOk types are the same as Lk1w types [ why ?] Theorem [Dawar, Lindell, Weinstein] The type of A (or of (A, a)) can be expressed by some f2 FOk B ²f[b] iff Tpk(A,a) = Tpk(B,b) Difficult result: was unknown to Kolaitis&Vardi

  8. 0/1 Laws in Logic Motivation: random graphs • 0/1 law for FO proven by Glebskii et al., then rediscovered by Fagin (and with nicer proof) • Only for constant probability distribution • Later extended to other logics, and other probability distributions Why we care: applications in degrees of belief, probabilistic databases, etc.

  9. Definitions • Let s = a vocabulary • Let n ¸ 0, and Anµ STRUCT[s] be all models over domain {0, 1, …, n-1} • Uniform probability distribution on An • Given sentence f, denote mn(f) its probability

  10. Definition • Denote m(f) = limn !1mn(f) if it exists Definition A logic L has a convergence law if for every sentence f, m(f) exists Definition A logic L has a 0/1 law if for every sentence f, m(f) exists and is 0 or 1

  11. Theorems • Suppose s has no constants Theorem [Fagin 76, Glebskii et al. 69] FO admits a 0/1 law Theorem [Kolaitis and Vardi 92] Lw1w admits a 0/1 law

  12. Application • What does this tell us for database query processing ? • Don’t bother evaluating a query: it’s either true or false, with high probability 

  13. Examples [ in class ] • Compute mn(f), then m(f): R(0,1) /* I’m using constants here */ R(0,1) Æ R(0,3) Æ: R(1,3) 9 x.R(2,x) : (9 x.9 y.R(x,y)) 8 x.8 y.(9 z.R(x,z) Æ R(z,y))

  14. Types • We only need rank-0 types (i.e. no quantifiers) • Recall the definition Definition A type t(x) over variables (x1, …, xm) is conjunction of a maximally consistent set of atomic formulas over x1, …, xm

  15. Types The type t(x) says: • For each i, j whether xi = xj or xi¹ xj • For each R and each xi1, …, xip whether R(xi1, …, xip) or : R(xi1, …, xip)

  16. Extension Axioms Definition Type s(x, z) extends the type t(x) if {s, t} is consistent; Equivalently: every conjunct in t occurs in s Definition The extension axiom for types t, s is the formula tt,s = 8 x1…8 xk (t(x) )9 z.s(x, z))

  17. Example of Extension Axiom t(x1, x2, x3) = x1¹ x2Æ x2¹ x3Æ x1¹ x3Æ R(x1,x2) Æ R(x2,x3) Æ R(x2,x2) Æ: R(x1, x1) Æ: R(x2, x1) Æ … x1 x2 z s(x1, x2, x3, z) = t(x1, x2, x3) Æ z ¹ x1Æ z ¹ x2Æ z ¹ x3Æ R(z,x1) Æ R(x3,z) Æ R(z,z) Æ: R(x1, z) Æ: (z, x2) Æ … x3

  18. Example of Extension Axiom tt,s = 8 x1.8 x2.8 x3. (t(x1, x2, x3) )9 z. s(x1, x2, x3, z))

  19. The Theory T • Let T be the set of all extension axioms • Studied by Gaifman • Is T consistent ? • In a model of T the duplicator always wins [ why ? ] • Does it have finite models ? • Does it have infinite models ?

  20. The Theory T • Let qk be the conjunction of all extension axioms for types with up to k variables • There exists a finite model for qk [why ?] • Hence any finite subset of T has a model • Hence T has a model. [can it be finite ?]

  21. The Model(s) of T • T has no finite models, hence it must have some infinite model • By Lowenheim-Skolem, it has a countable model

  22. The Theory T Theorem T is w-categorical Proof: let A, B be two countable model. Idea: use a back-and-forth argument to find an isomorphism f : A ! B

  23. The Theory T Theorem T is w-categorical Proof: (cont’d) A = {a1, a2, a3, ….} B = {b1, b2, b3, ….} Build partial isomorphisms f1µ f2µ f3µ …such that: 8 n.9 m. an2 dom(fm)and 8 n.9 m. bn2 rng(fm) [in class] Then f = ([m ¸ 1 fm) : A ! B is an isomorphism

  24. The Theory T Corollary T has a unique countable model R • R = the Rado graph = the “random” graph Corollary The theory Th(T) is complete

  25. 0/1 Law for FO LemmaFor every extension axiom t, m(t) = limnmn(t) = 1 Proof: later Corollary For any m extension axioms t1, …, tm: m(t1Æ … Ætm) = 1 Proofmn(:(t1Æ … Ætm)) = mn(:t1Ç … Ç:tm) ·mn(:t1) + … + mn(:tm) ! 0

  26. Fagin’s 0/1 Law for FO Theorem For every f2 FO, either m(f) = 0 or m(f) = 1. Proof. Case 1: R²f. Then there exists m extension axioms s.t. t1, …, tm²f. Then mn(f) ¸mn(t1Æ … Ætm) ! 1 Case 2: R2f. Then R²:f, hence m(:f) = 1, and m(f) = 0

  27. Proof for the Extension Axioms • Let t = 8x. t(x) )9 z.s(x, z) • Assume wlog that t asserts xi¹ xj forall i ¹ j. Denote ¹(x) the formula Æi < j xi¹ xj • Hence t(x) = ¹(x) Æ t’(x) • Similarly, s asserts z ¹ xi forall i.Denote ¹(x, z) = Æi xi¹ z • Hence s(x, z) = t(x) ƹ(x, z) Æ s’(x, z)where all atomic predicates in s’(x, z) contain z • Hence:t = 8x.(¹(x) Æ t’(x) ) 9 z. ¹(x,z) Æ s’(x, z))

  28. Proof for the Extension Axioms :t = 9x.(¹(x) Æ t’(x) Æ8 z.(¹(x, z) ): s’(x, z))) mn(:t) ·mn(9x.(¹(x) Æ8 z.(¹(x, z) ): s’(x, z))))

  29. Proof for the Extension Axioms mn(:t) ·mn(9x.(¹(x) Æ8 z.(¹(x, z) ):s’(x, z)))) ·åa1, ... , ak2 {1, …, n}mn(8 z. (¹(x, z) ):s’(a1, …, ak, z))) = n(n-1)…(n-k+1) mn(8 z. ¹(x, z) ):s’(1, 2, …, k, z)) · nkmn(8 z. ¹(x, z) ):s’(1, 2, …, k, z)) = = nkÕz=k+1, n: s’(1,2,…,k,z) /* by independence !! */ = nk ( 1 - 1 / 22k+1 )n-k /* since s’ is about 2k+1 edges */ ! 0 when n !1

  30. Complexity Theorem [Grandjean] The problem whether m(f) = 0 or 1 is PSPACE complete

  31. Discussion • Old way to think about formulas and models: finite satsfiability/ validity FO f valid f unsatisfiable Undecidable

  32. Discussion • New way to think about formulas and models: probability m(f)=1 FO m(f)=0 f valid f unsatisfiable PSPACE

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