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Lw1w Summary |
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and 0/1 Laws |
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Summary on Lw1w |
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All you need to know in 5 slides ! |
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Start 0/1 Laws: Fagin’s theorem |
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Will continue next time |
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Notation Comes from in classical logic |
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Lab = formulas where: |
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Conjunctions/disjunctions of ordinal < a
Çi
2 g fi, Æi 2 g,
where g < a |
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Quantifier chains of ordinal < b
9i
2 g xi. f, where g < b |
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Hence, L1w = [a
Law |
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Motivation |
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Any algorithmic computation that applies FO
formulas is expressible in Lw1w |
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Relational machines |
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While-programs with statements R := f |
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Fixpoint logics: LFP, IFP, PFP, etc, etc |
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Consequence: cannot express EVEN, HAMILTONEAN |
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Canonical Structure |
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Any algorithmic computation on A can be
decomposed |
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Compute the ¼k equivalence relation
on k-tuples, and order the equivalence classes ) in LFP
[how do we choose k ???] |
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Then compute on ordered structure ) any
complexity |
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Consequence: PTIME=PSPACE iff IFP=PFP |
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But note that DTC ¹ TC yet L ¹?
NL [ why ?] |
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Pebble Games: with k pebbles |
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Notation:
A º1wk
B if duplicator wins |
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Theorem 1.
For any two structures A, B: |
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A, B are Lk1w
equivalent iff |
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A º1wk B |
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Theorem 2.
If A, B are finite: |
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A, B are FOk equivalent iff |
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A, B are Lk1w
equivalent iff |
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A º1wk B |
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Definability of FOk types |
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FOk types are the same as Lk1w
types [ why ?] |
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Theorem
[Dawar, Lindell, Weinstein] The type of A (or of (A, a)) can be expressed by some f 2 FOk |
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B ² f[b] iff Tpk(A,a) = Tpk(B,b) |
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Difficult result: was unknown to
Kolaitis&Vardi |
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Motivation: random graphs |
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0/1 law for FO proven by Glebskii et al., then
rediscovered by Fagin (and with nicer proof) |
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Only for constant probability distribution |
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Later extended to other logics, and other
probability distributions |
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Why we care: applications in degrees of belief,
probabilistic databases, etc. |
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Let s = a vocabulary |
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Let n ¸ 0, and An µ STRUCT[s] be all
models over domain {0, 1, …, n-1} |
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Uniform probability distribution on An |
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Given sentence f, denote mn(f) its
probability |
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Denote m(f) = limn ! 1
mn(f) if it exists |
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Definition A logic L has a convergence law if
for every sentence f, m(f) exists |
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Definition A logic L has a 0/1 law if for every
sentence f, m(f) exists and is 0 or 1 |
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Suppose s has no constants |
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Theorem [Fagin 76, Glebskii et al. 69]
FO admits a 0/1 law |
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Theorem [Kolaitis and Vardi 92]
Lw1w
admits a 0/1 law |
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What does this tell us for database query
processing ? |
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Don’t bother evaluating a query: it’s either
true or false, with high probability J |
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Compute mn(f), then m(f): |
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R(0,1)
/* I’m using constants here */ |
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R(0,1) Æ R(0,3) Æ : R(1,3) |
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9
x.R(2,x) |
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: (9 x.9
y.R(x,y)) |
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8 x.8
y.(9 z.R(x,z) Æ R(z,y)) |
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We only need rank-0 types (i.e. no quantifiers) |
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Recall the definition |
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Definition A type t(x) over variables (x1,
…, xm) is conjunction of a maximally consistent set of atomic
formulas over x1, …, xm |
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The type t(x) says: |
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For each i, j whether xi = xj
or xi ¹ xj |
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For each R and each xi1,
…, xip whether R(xi1, …, xip)
or : R(xi1, …, xip) |
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Definition Type s(x, z) extends the type t(x) if
{s, t} is consistent; |
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Equivalently: every conjunct in t occurs in s |
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Definition The extension axiom for types t, s is
the formula
tt,s
= 8 x1…8 xk
(t(x) ) 9 z.s(x, z)) |
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Let T be the set of all extension axioms |
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Studied by Gaifman |
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Is T consistent ? |
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In a model of T the duplicator always wins [ why
? ] |
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Does it have finite models ? |
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Does it have infinite models ? |
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Let qk be the conjunction of all
extension axioms for types with up to k variables |
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There exists a finite model for qk [why ?] |
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Hence any finite subset of T has a model |
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Hence T has a model. [can it be finite ?] |
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T has no finite models, hence it must have some
infinite model |
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By Lowenheim-Skolem, it has a countable model |
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Theorem T is w-categorical |
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Proof: let A, B be two countable model. |
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Idea: use a back-and-forth argument to find an
isomorphism f : A ! B |
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Theorem T is w-categorical |
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Proof:
(cont’d) |
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A = {a1, a2, a3,
….} B = {b1, b2,
b3, ….} |
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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) |
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[in class] |
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Then f = ([m ¸ 1
fm) : A ! B is an isomorphism |
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Corollary T has a unique countable model R |
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R = the Rado graph
= the “random” graph |
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Corollary The theory Th(T) is complete |
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Lemma
For every extension axiom t, m(t) = limn mn(t) = 1 |
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Proof: later |
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Corollary For any m extension axioms t1,
…, tm: m(t1
Æ … Æ tm) = 1 |
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Proof
mn(:(t1 Æ
… Æ tm))
= mn(:
t1 Ç … Ç : tm)
· mn(: t1) + … + mn(:
tm) ! 0 |
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Theorem
For every f 2 FO, either m(f) = 0 or m(f) = 1. |
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Proof. |
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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 |
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Case 2: R 2 f.
Then R ² : f, hence m(: f) = 1, and m(f) = 0 |
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Let t = 8 x. t(x) ) 9 z.s(x, z) |
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Assume wlog that t asserts xi ¹ xj
forall i ¹ j. Denote ¹(x) the
formula Æi < j xi ¹ xj |
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Hence t(x) = ¹(x) Æ t’(x) |
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Similarly, s asserts z ¹ xi forall
i.
Denote ¹(x, z) = Æi
xi ¹ z |
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Hence s(x, z) = t(x) Æ ¹(x, z) Æ s’(x, z)
where all atomic predicates in s’(x,
z) contain z |
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Hence:
t = 8 x.(¹(x) Æ t’(x) ) 9 z. ¹(x,z)
Æ s’(x, z)) |
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: t =
9 x.(¹(x) Æ t’(x) Æ 8 z.(¹(x, z) ) : s’(x, z))) |
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mn(:
t) · mn(9 x.(¹(x) Æ 8 z.(¹(x, z) ) : s’(x, z)))) |
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mn(: t) · mn(9 x.(¹(x) Æ 8
z.(¹(x, z) ) :s’(x, z)))) |
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· åa1, ... , ak
2 {1, …, n} mn(8
z. (¹(x, z) ) :s’(a1, …, ak, z))) |
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= n(n-1)…(n-k+1) mn(8 z. ¹(x, z) ) :s’(1,
2, …, k, z)) |
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· nk mn(8 z. ¹(x, z) ) :s’(1,
2, …, k, z)) = |
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= nk
Õz=k+1, n : s’(1,2,…,k,z) /* by independence !! */ |
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= nk ( 1 - 1 / 22k+1 )n-k /* since s’ is about 2k+1 edges */ |
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! 0
when n ! 1 |
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Theorem [Grandjean] The problem whether m(f) = 0
or 1 is PSPACE complete |
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Old way to think about formulas and models:
finite satsfiability/ validity |
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New way to think about formulas and models:
probability |
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