Probability for a First-Order Language

1. 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 ~x·y x·y y ~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 = <Fx> ⊕ <~Fx> = [∀xFx] ⊕ [∀x~Fx] • Conditional [∀x(FxGx)] = [∀x(Fx&Gx)] ⊕ [∀x~Fx] [∀xFx] [∀x(Fx&Gx)] [∀x~Fx] [∀x(FxGx)]

6. 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 inΣ, 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

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 P0 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 Nf = |[Fx]| Ng = |[Gx]| Nfg = |[Fx & Gx]| • Calculate sentence extensions |[x(Fx)]| = 2Nf |[x(Gx)]| = 2Ng |[x(Fx & Gx)]| = 2Nfg

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-Nf) + Nfg • Extension of its generalization |[x(Fx  Gx)]| = = • Probability

12. 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

13. 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 hyperplaneΩn-1, there is an abstract “perpendicular line,” Ω x {(a1,…,an-1)}

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

15. 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

16. 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

17. Example: x2 < x3

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 Ωn • 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, xi = xj • Example. In four dimensions, x2 = x3 maps to:

21. 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

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(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)

24. 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