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

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)

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

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

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)]

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

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] }

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)].

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

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

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

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

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 )}

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

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

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

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

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

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

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

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

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

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 )

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