Probability for a First-Order Language

Ken Presting

University of North Carolina

at Chapel Hill

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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

x ~x

y

~y

~x·yx·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]∀

[ x(Fx∀ Gx)]

[ x~Fx]∀

[ xFx]∀

[ 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 extensionsNf = |[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)]

=

=fg

f

N

N

2

2

fg

f

N N

N N

2 2

2 2

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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)]| =

=

• Probability

f fg((N - N ) + N )2fgf

N-NN(2 )(2 )(2 )

fgf

N-NN

N

(2 )(2 )(2 )P x(Fx Gx)

2

fg

f

N

N

2

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 setFxi|a1,…,an-1 = { ωΩ | 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) = { ωΩ | 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 x1

Value of x2

Value of x3

Value of 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)

aΩ

• Component-wise operation