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Finite Model TheoryLecture 16
L1 Summary
and 0/1 Laws
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Outline
• Summary on L1
– All you need to know in 5 slides !
• Start 0/1 Laws: Fagin’s theorem– Will continue next time
Infinitary Logics and 0-1 Laws, Kolaitis&Vardi, 1992
New paper:
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Summary on L1
Notation Comes from in classical logic
• L = formulas where:
– Conjunctions/disjunctions of ordinal < Çi 2 i, Æ2, where <
– Quantifier chains of ordinal < 9i 2 xi. , where <
• Hence, L1 = [ L
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Summary on L1
Motivation• Any algorithmic computation that applies FO formulas
is expressible in L1
• Relational machines• While-programs with statements R := • Fixpoint logics: LFP, IFP, PFP, etc, etc
Consequence: cannot express EVEN, HAMILTONEAN
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Summary on L1
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=PFPBut note that DTC TC yet L ? NL [ why ?]
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Summary on L1
Pebble Games: with k pebbles• Notation: A 1
k B if duplicator wins
Theorem 1. For any two structures A, B:• A, B are Lk
1 equivalent iff• A 1
k BTheorem 2. If A, B are finite:• A, B are FOk equivalent iff• A, B are Lk
1 equivalent iff• A 1
k B
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Summary on L1
Definability of FOk types
• FOk types are the same as Lk1 types [ why ?]
Theorem [Dawar, Lindell, Weinstein] The type of A (or of (A, a)) can be expressed by some 2 FOk
B ² [b] iff Tpk(A,a) = Tpk(B,b)
Difficult result: was unknown to Kolaitis&Vardi
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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.
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Definitions
• Let = a vocabulary
• Let n ¸ 0, and An µ STRUCT[] be all models over domain {0, 1, …, n-1}
• Uniform probability distribution on An
• Given sentence , denote n() its probability
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Definition
• Denote () = limn ! 1 n() if it exists
Definition A logic L has a convergence law if for every sentence , () exists
Definition A logic L has a 0/1 law if for every sentence , () exists and is 0 or 1
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Theorems
• Suppose has no constants
Theorem [Fagin 76, Glebskii et al. 69] FO admits a 0/1 law
Theorem [Kolaitis and Vardi 92] L
1 admits a 0/1 law
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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
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Examples [ in class ]
• Compute n(), then ():
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))
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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
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Types
The type t(x) says:
• For each i, j whether xi = xj or xi xj
• For each R and each xi1, …, xip
whether
R(xi1, …, xip
) or : R(xi1, …, xip
)
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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
t,s = 8 x1…8 xk (t(x) ) 9 z.s(x, z))
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Example of Extension Axiom
t(x1, x2, x3) =x1 x2 Æ x2 x3 Æ x1 x3 ÆR(x1,x2) Æ R(x2,x3) Æ R(x2,x2) Æ : R(x1, x1) Æ : R(x2, x1) Æ …
x1
x2
x3
s(x1, x2, x3, z) =t(x1, x2, x3) Æ z x1 Æ z x2 Æ z x3 ÆR(z,x1) Æ R(x3,z) Æ R(z,z) Æ: R(x1, z) Æ : (z, x2) Æ …
z
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Example of Extension Axiom
8 x1.8 x2.8 x3. (t(x1, x2, x3) ) 9 z. s(x1, x2, x3, z))
t,s =
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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 ?
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The Theory T
• Let k be the conjunction of all extension axioms for types with up to k variables
• There exists a finite model for k [why ?]
• Hence any finite subset of T has a model
• Hence T has a model. [can it be finite ?]
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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
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The Theory T
Theorem T is -categorical
Proof: let A, B be two countable model.
Idea: use a back-and-forth argument to find an isomorphism f : A ! B
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The Theory T
Theorem T is -categoricalProof: (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
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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
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0/1 Law for FO
Lemma For every extension axiom , () = limn n() = 1
Proof: later
Corollary For any m extension axioms 1, …, m: (1 Æ … Æ m) = 1
Proof n(:(1 Æ … Æ m))
= n(: 1 Ç … Ç : m) · n(: 1) + … + n(: m) ! 0
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Fagin’s 0/1 Law for FO
Theorem For every 2 FO, either () = 0 or () = 1.
Proof. Case 1: R ² . Then there exists m extension
axioms s.t. 1, …, m ² . Then n() ¸ n(1 Æ … Æ m) ! 1
Case 2: R 2 . Then R ² : , hence (: ) = 1, and () = 0
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Proof for the Extension Axioms
• Let = 8 x. t(x) ) 9 z.s(x, z)• Assume wlog that t asserts xi xj forall i j.
Denote (x) the formula Æi < j xi xj
– Hence t(x) = (x) Æ t’(x)
• Similarly, s asserts z xi forall i.Denote (x, z) = Æi xi z– Hence s(x, z) = t(x) Æ (x, z) Æ s’(x, z)
where all atomic predicates in s’(x, z) contain z
• Hence: = 8 x.((x) Æ t’(x) ) 9 z. (x,z) Æ s’(x, z))
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Proof for the Extension Axioms
: = 9 x.((x) Æ t’(x) Æ 8 z.((x, z) ) : s’(x, z)))
n(: ) · n(9 x.((x) Æ 8 z.((x, z) ) : s’(x, z))))
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Proof for the Extension Axioms
n(: ) · n(9 x.((x) Æ 8 z.((x, z) ) :s’(x, z))))
· a1, ... , ak 2 {1, …, n} n(8 z. ((x, z) ) :s’(a1, …, ak, z)))
= n(n-1)…(n-k+1) n(8 z. (x, z) ) :s’(1, 2, …, k, z))
· nk n(8 z. (x, z) ) :s’(1, 2, …, k, z)) =
= nk z=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
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Complexity
Theorem [Grandjean] The problem whether () = 0 or 1 is PSPACE complete
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Discussion
• Old way to think about formulas and models: finite satsfiability/ validity
FO
unsatisfiable
valid
Undecidable
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Discussion
• New way to think about formulas and models: probability
FO
unsatisfiable
valid
PSPACE
)=0)=1