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Probability for a First-Order Language Ken Presting University of North Carolina at Chapel Hill

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2 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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3 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 · y x·y x · ~y~x · ~y

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4 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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5 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 = ⊕ = [∀ 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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6 Definitions The Domain space - –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 P 0 (S) = |S|/N The Sample Space - –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|/2 N

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7 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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8 Theorem Let L be a first-order language Probability P and P 0 as above If ‘Fx’, ‘Gx’ are open formulas of L, then P[ x(Fx Gx)] = P[ x(Gx) | x(Fx)].

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9 Proof Define values for predicate extensions N f = |[Fx]| N g = |[Gx]| N fg = |[Fx & Gx]| Calculate sentence extensions |[ x(Fx)]| = 2 Nf |[ x(Gx)]| = 2 Ng |[ x(Fx & Gx)]| = 2 Nfg

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10 Conditional Probability P[ x(Gx) | x(Fx)] = P[ x(Fx & Gx)] P[ x(Fx)] =

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11 Probability of the Conditional Extension of open material conditional |[Fx Gx]| = |[~Fx] v [Fx & Gx]| = (N-N f ) + N fg Extension of its generalization |[ x(Fx Gx)]| = = Probability

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12 Relations on a Domain Domain is an arbitrary set, Ω Relations are subsets of Ω n All examples used today take Ω n as ordered tuples of natural numbers, Ω n = {(a i ) 1≤i≤n | a i N } All definitions and proofs today can extend to arbitrary domains, indexed by ordinals

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13 Hyperplanes and Lines Take an n-dimensional Cartesian product, Ω n, as an abstract coordinate space. Then an n-1 dimensional subspace, Ω n-1, is an abstract hyperplane in Ω n. For each point (a 1,…,a n-1 ) in the hyperplane Ω n-1, there is an abstract “perpendicular line,” Ω x {(a 1,…,a n-1 )}

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14 Decomposition of a Relation Hyperplane, Perpendicular Line, Graph and Slice

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15 Slices of the Graph Let F(x 1,…,x n ) be an n-ary relation Let the plain symbol F denote its graph: F = {(x 1,…,x n )| F(x 1,…,x n )} Let a 1,…,a n-1 be n-1 elements of Ω Then for each variable x i there is a set F x i |a1,…,an-1 = { ω Ω | F(a 1,…,a i-1,ω,a i,…, a n-1 } This set is the x i ’s which satisfy F(…x i …) when all the other variables are fixed

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16 The Matrix of Slices Every n-ary relation defines n set-valued functions on n-1 variables: F x i (v 1,…,v n-1 ) = { ω Ω | F(v 1,…,v i-1,ω,v i,…,v n-1) } The n-tuple of these functions is called the “matrix of slices” of the relation F

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17 Example: x 2 < x 3 IndexValue of x 1 Value of x 2 Value of x 3 Value of x 4 0,0,0ΩØ{1,2,3,…}Ω 0,0,1ΩØ{1,2,3,…}Ω 0,0, …ΩØ{1,2,3,…}Ω 0,1,0Ω{0}{2,3,4,…}Ω 0,1,1Ω{0}{2,3,4,…}Ω …Ω……Ω

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18 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 Ω n Matrix columns are indexed by variables in the relation

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19 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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20 The Diagonal Relations Matrix images of an identity relation, x i = x j Example. In four dimensions, x 2 = x 3 maps to: IndexValue of x 1 Value of x 2 Value of x 3 Value of x 4 0,0,0Ω{0} Ω 0,0,1Ω{0} Ω 0,0, …Ω{0} Ω 0,1,0Ω{1} Ω 0,1,1Ω{1} Ω …Ω……Ω

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21 Cylindrical Identity Elements 1 is the matrix with all components Ω, i.e. the image of a universal relation such as x i =x i 0 is the matrix with all components Ø, i.e. the image of the empty relation

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22 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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23 Instantiation Take an n-ary relation, F = F(x 1,…,x n ) Fix x i = a, that is, consider the n-1-ary relation F |x i =a = F(x 1,…,x i-1,a,x i+1,…,x n ) Each column in the matrix of F |x i =a is: F x j |x i =a (v 1,…,v n-2 ) = F(v 1,…,v j-1,x j,v j,…,v i-1,a,v i+1,…,v n )

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24 Cylindrification as Union Cylindrification affects all slices in every non-maximal column Each slice in F |x i is a union of slices from instantiations: F x j |x i (v 1,…,v n-2 ) = U F x j |x i =a (v 1,…,v n-2 ) a Ω Component-wise operation

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