Download Non-Trivial Conditional Probability

Probability for a
First-Order Language
Ken Presting
University of North Carolina
at Chapel Hill
A qualified homomorphism
• If A, B disjoint
P(A ∪ B) = P(A) + P(B)
• If A, B independent
P(A ∩ B) = P(A) · P(B)
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Quotient by a Subalgebra
• Let x, y, ~x, ~y be pairwise independent
• Direct product of factors = {x, ~x} x {y, ~y}
• Probability is area of rectangles in unit square
y
~y
x
~x
x·y
~x·y
x·~y
~x·~y
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Probability on Extensions
• A predicate is true-of an individual
– Set of individuals is the extension
– Measure of that set is probability
• A generalization is true-in a domain
– Set of domains is the extension
– Measure of that set is the probability
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Quotient by an Ideal
• If Fx is a predicate in L, then every sample is a disjoint
union, split by [Fx] and [~Fx]
• Sample space Σ is a direct sum of principal ideals,
Σ = <Fx> ⊕ <~Fx> = [∀xFx] ⊕ [∀x~Fx]
• Conditional [∀x(FxGx)] = [∀x(Fx&Gx)] ⊕ [∀x~Fx]
[∀xFx]
[∀x(Fx&Gx)]
[∀x~Fx]
[∀x(FxGx)]
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Definitions
• The Domain space - <Ω, Σ, P0>
– Ω is a domain of interpretation for L (with N members)
– Σ is generated by predicates of L
– For any S in Σ, we set P0(S) = |S|/N
• The Sample Space - <Σ, Ψ, P>
– Σ is the field of subsets from the space above
– Ψ is generated by closed sentences of L
– For any C in Ψ, we set P(C) = |C|/2N
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Sentences and Extensions
• Extensions of Formulas
– (only one free variable)
– [Fx] = { s in Ω | ‘Fs’ is true in L }
• Extensions of Sentences
– [x(Fx)] = { S in Σ | ‘x(Fx)’ is “true in S” }
–
= { S in Σ | S is a subset of [Fx] }
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Theorem
• Let L be a first-order language
• Probability P and P0 as above
• If ‘Fx’, ‘Gx’ are open formulas of L, then
P[x(Fx  Gx)] = P[x(Gx) | x(Fx)].
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Proof
• Define values for predicate extensions
Nf = |[Fx]|
Ng = |[Gx]|
Nfg = |[Fx & Gx]|
• Calculate sentence extensions
|[x(Fx)]| = 2Nf
|[x(Gx)]| = 2Ng
|[x(Fx & Gx)]| = 2Nfg
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Conditional Probability
•
P[x(Gx) | x(Fx)] = P[x(Fx & Gx)]
P[x(Fx)]
N
=
2 fg 2 N
2 Nf 2 N
N fg
=
2
2 Nf
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Probability of the Conditional
• Extension of open material conditional
|[Fx  Gx]| = |[~Fx] v [Fx & Gx]|
= (N-Nf) + Nfg
• Extension of its generalization
|[x(Fx  Gx)]| = 2((N - Nf ) + Nfg )
N
-N
N
= (2 )(2 )(2 )
f
fg
• Probability
N
-N f
(2 )(2 )(2
P x(Fx  Gx)  
2N
N fg
)
N
2 fg
 Nf
2
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Relations on a Domain
• Domain is an arbitrary set, Ω
• Relations are subsets of Ωn
• All examples used today take Ωn as ordered
tuples of natural numbers,
Ωn = {(ai)1≤i≤n | ai  N }
• All definitions and proofs today can extend to
arbitrary domains, indexed by ordinals
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Hyperplanes and Lines
• Take an n-dimensional Cartesian product, Ωn, as
an abstract coordinate space.
• Then an n-1 dimensional subspace, Ωn-1, is an
abstract hyperplane in Ωn.
• For each point (a1,…,an-1) in the hyperplane
Ωn-1, there is an abstract “perpendicular line,”
Ω x {(a1,…,an-1)}
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Decomposition of a Relation
Hyperplane, Perpendicular Line, Graph and Slice
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Slices of the Graph
• Let F(x1,…,xn) be an n-ary relation
• Let the plain symbol F denote its graph:
F = {(x1,…,xn)| F(x1,…,xn)}
• Let a1,…,an-1 be n-1 elements of Ω
• Then for each variable xi there is a set
Fxi|a1,…,an-1 = { ωΩ | F(a1,…,ai-1,ω,ai,…, an-1}
• This set is the xi’s which satisfy F(…xi…) when
all the other variables are fixed
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The Matrix of Slices
• Every n-ary relation defines n set-valued
functions on n-1 variables:
Fxi(v1,…,vn-1) = { ωΩ | F(v1,…,vi-1,ω,vi,…,vn-1) }
• The n-tuple of these functions is called the
“matrix of slices” of the relation F
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Example: x2 < x3
Index
Value of x1
Value of x2
Value of x3
Value of x4
0,0,0
Ω
Ø
{1,2,3,…}
Ω
0,0,1
Ω
Ø
{1,2,3,…}
Ω
0,0, …
Ω
Ø
{1,2,3,…}
Ω
0,1,0
Ω
{0}
{2,3,4,…}
Ω
0,1,1
Ω
{0}
{2,3,4,…}
Ω
…
Ω
…
…
Ω
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Boolean Operations on Matrices
• Matrices treated as vectors
– direct product of Boolean algebras
– Component-wise conjunction, disjunction, etc.
• Matrix rows are indexed by n-1 tuples from
Ωn
• Matrix columns are indexed by variables in
the relation
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Cylindrical Algebra Operations
• Diagonal Elements
– Images of identity relations: x = y
– Operate by logical conjunction with operand relation
• Cylindrifications
– Binding a variable with existential quantifier
• Substitutions
– Exchange of variables in relational expression
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The Diagonal Relations
• Matrix images of an identity relation, xi = xj
• Example. In four dimensions, x2 = x3 maps to:
Index
Value of Value of Value of Value of
x1
x2
x3
x4
0,0,0
Ω
{0}
{0}
Ω
0,0,1
Ω
{0}
{0}
Ω
0,0, …
Ω
{0}
{0}
Ω
0,1,0
Ω
{1}
{1}
Ω
0,1,1
Ω
{1}
{1}
Ω
…
Ω
…
…
Ω
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Cylindrical Identity Elements
• 1 is the matrix with all components Ω, i.e.
the image of a universal relation such as
xi=xi
• 0 is the matrix with all components Ø, i.e.
the image of the empty relation
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Diagonal Operations
• Boolean conjunction of relation matrix with
diagonal relation matrix
• Reduces number of free variables in
expression, ‘x + y > z’ & ‘x = y’
• Constructs higher-order relations from low
order predicates
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Instantiation
• Take an n-ary relation, F = F(x1,…,xn)
• Fix xi = a, that is, consider the n-1-ary
relation F|xi=a = F(x1,…,xi-1,a,xi+1,…,xn)
• Each column in the matrix of F|xi=a is:
Fxj|xi=a(v1,…,vn-2) =
F(v1,…,vj-1,xj,vj,…,vi-1,a,vi+1,…,vn)
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Cylindrification as Union
• Cylindrification affects all slices in every
non-maximal column
• Each slice in F|xi is a union of slices from
instantiations:
Fxj|xi(v1,…,vn-2) = U Fxj|xi=a(v1,…,vn-2)
aΩ
• Component-wise operation
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