Probabilistic Team Semantics Jonni Virtema Distributions Probabilistic Team Semantics Probabilistic atoms Connectives and quantifiers Examples Jonni Virtema Benchmark logic Characterisation of Hasselt University, Belgium expressivity jonni.virtema@gmail.com Joint work with Arnaud Durand (Universit´ e Paris Diderot), Miika Hannula (University of Helsinki), Juha Kontinen (University of Helsinki), and Arne Meier (Leibniz Universit¨ at Hannover) August 3, 2018 1/ 16
Probabilistic Team Teams as collections of measurements Semantics Jonni Virtema Distributions Probabilistic atoms ◮ Multiteams (multisets of assignments) vs. Connectives and quantifiers Examples x y z Benchmark logic x y z # x y z prob. Characterisation of s 1 a a b expressivity 1 s 1 a a b 2 s 1 a a b 2 s 2 a a b 1 b c c 1 b c c s 2 s 2 4 b c c s 3 1 s 3 a b c 1 s 3 a b c 4 s 4 a b c 2/ 16
Probabilistic Team Teams as collections of measurements Semantics Jonni Virtema Distributions ◮ Multiteams (multisets of assignments) vs. probabilistic teams (distributions Probabilistic atoms Connectives and over assignments) quantifiers Examples x y z Benchmark logic x y z # x y z prob. Characterisation of expressivity s 1 a a b 1 s 1 a a b 2 s 1 a a b 2 a a b s 2 1 b c c 1 b c c s 2 s 2 4 s 3 b c c 1 s 3 a b c 1 s 3 a b c 4 s 4 a b c 2/ 16
Probabilistic Team Distributions of data Semantics Jonni Virtema Distributions Probabilistic atoms Connectives and Consider: quantifiers ◮ A collection of data from some repetitive science experiment. Examples Benchmark logic ◮ Data obtained from a poll. Characterisation of ◮ Any collection of data, that involves meaningful duplicates of data. expressivity One natural way to represent the data is to use multisets (sets with duplicates). Claim: Often the multiplicities themselves are not important; the distribution of data is. 3/ 16
Probabilistic Team Distributions of data Semantics Jonni Virtema Distributions Probabilistic atoms Connectives and Consider: quantifiers ◮ A collection of data from some repetitive science experiment. Examples Benchmark logic ◮ Data obtained from a poll. Characterisation of ◮ Any collection of data, that involves meaningful duplicates of data. expressivity One natural way to represent the data is to use multisets (sets with duplicates). Claim: Often the multiplicities themselves are not important; the distribution of data is. 3/ 16
Probabilistic Team Distributions Semantics Jonni Virtema Distributions Definition Probabilistic atoms A distribution is a mapping f : A → Q [0 , 1] from a set A of values to the closed Connectives and quantifiers interval [0 , 1] of rational numbers such that the probabilities sum to 1, i.e., Examples Benchmark logic � f ( a ) = 1 . Characterisation of expressivity a ∈ A ◮ A multiteam is a pair ( X , m ), where X is a set of assignments and m : X → N > 0 is a multiplicity function (a database with duplicates). ◮ A probabilistic team is a pair ( X , p ), where X is a set of assignments and p : X → Q [0 , 1] is a distribution (distribution of data). 4/ 16
Probabilistic Team Distributions Semantics Jonni Virtema Distributions Definition Probabilistic atoms A distribution is a mapping f : A → Q [0 , 1] from a set A of values to the closed Connectives and quantifiers interval [0 , 1] of rational numbers such that the probabilities sum to 1, i.e., Examples Benchmark logic � f ( a ) = 1 . Characterisation of expressivity a ∈ A ◮ A multiteam is a pair ( X , m ), where X is a set of assignments and m : X → N > 0 is a multiplicity function (a database with duplicates). ◮ A probabilistic team is a pair ( X , p ), where X is a set of assignments and p : X → Q [0 , 1] is a distribution (distribution of data). 4/ 16
Probabilistic Team Probabilistic teams Semantics Jonni Virtema Distributions ◮ Modelling of data that is inherently a probability distribution. Probabilistic atoms ◮ Abstraction of data with duplicates. Connectives and quantifiers ◮ There is close connection between multiteams and probabilistic teams. Examples Benchmark logic We introduce a logic that describe properties of probabilistic teams. Characterisation of expressivity We consider the expansion of first-order logic with the marginal identity atoms ( x 1 , . . . , x n ) ≈ ( y 1 , . . . , y n ) and with the probabilistic conditional independence atoms y ⊥ ⊥ x z . 5/ 16
Probabilistic Team Probabilistic teams Semantics Jonni Virtema Distributions ◮ Modelling of data that is inherently a probability distribution. Probabilistic atoms ◮ Abstraction of data with duplicates. Connectives and quantifiers ◮ There is close connection between multiteams and probabilistic teams. Examples Benchmark logic We introduce a logic that describe properties of probabilistic teams. Characterisation of expressivity We consider the expansion of first-order logic with the marginal identity atoms ( x 1 , . . . , x n ) ≈ ( y 1 , . . . , y n ) and with the probabilistic conditional independence atoms y ⊥ ⊥ x z . 5/ 16
Probabilistic Team Probabilistic atoms Semantics Jonni Virtema Distributions Probabilistic atoms Connectives and quantifiers Examples We define that Benchmark logic Characterisation of A | = X � x ≈ � y iff the distribution of values for � x and � y in X coincide. expressivity 6/ 16
Probabilistic Team Probabilistic atoms Semantics Jonni Virtema Distributions Probabilistic atoms Connectives and We define that quantifiers Examples A | = X � x ≈ � y iff the distribution of values for � x and � y in X coincide. Benchmark logic Characterisation of expressivity We define that A | = X y ⊥ ⊥ x z iff for every fixed value for � x , the value distribution of � y remains unchanged if any value for � z is given. 6/ 16
Probabilistic Team Probabilistic atoms Semantics Jonni Virtema Let X = ( X , p ) be a probablistic team and � x , � a be tuples of variables and values Distributions of length k . We define Probabilistic atoms � a := | X | � p ( s ) . x = � Connectives and quantifiers s ∈ X s ( � x )= � a Examples Benchmark logic We define that Characterisation of expressivity a ∈ A k . A | = X � x ≈ � y iff | X | � a = | X | � a , for each � x = � y = � We define that A | = X y ⊥ ⊥ x z iff for all assignments s for � x , � y , � z | X | � y ) × | X | � z ) = | X | � z ) × | X | � x ) . x � y = s ( � x � x � z = s ( � x � x � y � z = s ( � x � y � x = s ( � 6/ 16
Probabilistic Team Semantics of complex formulae Semantics Jonni Virtema Distributions Definition Probabilistic atoms Let A be a structure over a finite domain A , and X : X → Q [0 , 1] a probabilistic Connectives and quantifiers team of A . The satisfaction relation | = X for first-order logic is defined as follows: Examples Benchmark logic A | = X x = y ⇔ for all s ∈ X : if X ( s ) > 0, then s ( x ) = s ( y ) Characterisation of expressivity A | = X x � = y ⇔ for all s ∈ X : if X ( s ) > 0, then s ( x ) � = s ( y ) = X R ( x ) ⇔ for all s ∈ X : if X ( s ) > 0, then s ( x ) ∈ R A A | = X ¬ R ( x ) ⇔ for all s ∈ X : if X ( s ) > 0, then s ( x ) �∈ R A A | A | = X ( ψ ∧ θ ) ⇔ A | = X ψ and A | = X θ 7/ 16
Probabilistic Team Semantics of complex formulae Semantics Jonni Virtema Distributions Probabilistic atoms Definition Connectives and quantifiers Let A be a structure over a finite domain A , and X : X → Q [0 , 1] a probabilistic Examples team of A . The satisfaction relation | = X for first-order logic is defined as follows: Benchmark logic Characterisation of A | = X ( ψ ∨ θ ) ⇔ A | = Y ψ and A | = Z θ for some Y , Z s . t . Y ⊔ Z = X expressivity A | = X ∀ x ψ ⇔ A | = X [ A / x ] ψ A | = X ∃ x ψ ⇔ A | = X [ F / x ] ψ holds for some F : X → p A . Above p A denote the set those distributions that have domain A . 7/ 16
Probabilistic Team Intuition of the quantifiers Semantics Jonni Virtema Distributions s i ( a / x ) s i ( a / x ) Probabilistic atoms Connectives and A → { 1 s 2 | A | } s 2 F ( s 2 ) quantifiers Examples A → { 1 Benchmark logic s 1 | A | } s 1 F ( s 1 ) Characterisation of expressivity A → { 1 s 0 | A | } s 0 F ( s 0 ) ◮ Universal quantification (i.e., the set X [ A / x ]) is depicted on left. ◮ Existential quantification (i.e., the set X [ F / x ]) is depicted on right. ◮ Height of a box corresponds to the probability of an assignment. 8/ 16
Probabilistic Team Intuition behind the disjunction Semantics Jonni Virtema Distributions Question: How do we split distributions? Probabilistic atoms Answer: We rescale. Connectives and quantifiers Let X : X → Q [0 , 1] and Y : Y → Q [0 , 1] be probabilistic teams and k ∈ Q [0 , 1] be a Examples rational number. Benchmark logic We denote by X ⊔ k Y the k -scaled union of X and Y , that is, the probabilistic Characterisation of expressivity team X ⊔ k Y : X ∪ Y → Q [0 , 1] defined s.t. for each s ∈ X ∪ Y , k · X ( s ) + (1 − k ) · Y ( s ) if s ∈ X and s ∈ Y , ( X ⊔ k Y )( s ) := k · X ( s ) if s ∈ X and s / ∈ Y , (1 − k ) · Y ( s ) if s ∈ Y and s / ∈ X . We then write that Z = X ⊔ Y if Z = X ⊔ k Y , for some k . 9/ 16
Recommend
More recommend
