Notes

Outline

Finite Model Theory Lecture 16 Lw1w Summary and 0/1 Laws

Outline Summary on Lw1w All you need to know in 5 slides ! Start 0/1 Laws: Fagin’s theorem Will continue next time

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

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

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 ?]

Summary on Lw1w Pebble Games: with k pebbles Notation:  A º1wk B if duplicator wins Theorem 1.  For any two structures A, B: A, B are Lk1w equivalent iff A º1wk B Theorem 2.  If A, B are finite: A, B are FOk  equivalent iff A, B are Lk1w equivalent iff A º1wk B

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 f 2  FOk B ² f[b] iff Tpk(A,a) = Tpk(B,b) Difficult result: was unknown to Kolaitis&Vardi

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.

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

Definition Denote m(f) = limn ! 1 mn(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

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

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 J

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))

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

Types The type t(x) says: For each i, j whether xi = xj or xi ¹ xj For each R and each xi1, …, xip whether R(xi1, …, xip) or : R(xi1, …, xip)

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))

Example of Extension Axiom

Example of Extension Axiom

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 ?

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 ?]

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

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

The Theory T Theorem T is w-categorical Proof:  (cont’d) A = {a1, a2, a3, ….}     B = {b1, b2, b3, ….} Build partial isomorphisms f1 µ f2 µ f3 µ … such that: 8 n.9 m. an 2 dom(fm) and 8 n.9 m. bn 2 rng(fm) [in class] Then f = ([m ¸ 1 fm) : A ! B is an isomorphism

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

0/1 Law for FO Lemma  For every extension axiom t,  m(t) = limn mn(t) = 1 Proof: later Corollary For any m extension axioms t1, …, tm:         m(t1 Æ … Æ tm) = 1 Proof mn(:(t1 Æ … Æ tm)) =   mn(: t1 Ç … Ç : tm) ·   mn(: t1) + … + mn(: tm)   ! 0

Fagin’s 0/1 Law for FO Theorem   For every f 2 FO, either m(f) = 0 or m(f) = 1. Proof. Case 1:  R ² f.  Then there exists m extension axioms s.t. t1, …, tm ² f.  Then mn(f) ¸ mn(t1 Æ … Æ tm) ! 1 Case 2: R 2 f.  Then R ² : f, hence m(: f) = 1, and m(f) = 0

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

Proof for the Extension Axioms  : t  =  9 x.(¹(x) Æ t’(x) Æ 8 z.(¹(x, z) ) : s’(x, z)))  mn(: t)  ·  mn(9 x.(¹(x) Æ 8 z.(¹(x, z) ) : s’(x, z))))

Proof for the Extension Axioms mn(: t) · mn(9 x.(¹(x) Æ 8 z.(¹(x, z) ) :s’(x, z)))) · åa1, ... , ak 2 {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)) · nk mn(8 z. ¹(x, z) ) :s’(1, 2, …, k, z)) = = nk  Õz=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

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

Discussion Old way to think about formulas and models: finite satsfiability/ validity

Discussion New way to think about formulas and models: probability