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 Complexity Joint work with Arnaud Durand (Universit´ e Paris Diderot), Miika Hannula (University of Auckland), Juha Kontinen (University of Helsinki), and Arne Meier (Leibniz Universit¨ at Hannover) May 17, 2018 1/ 19
Probabilistic Team Distributions Semantics Jonni Virtema Distributions Consider: Probabilistic atoms ◮ A collection of data from some repetitive science experiment. Connectives and quantifiers ◮ Data obtained from a poll. Examples ◮ Any collection of data, that involves meaningful duplicates of data. Benchmark logic Characterisation of One natural way to represent the data is to use multisets (sets with duplicates). expressivity Complexity Often the multiplicities themselves are not important; the distribution of data is: ◮ The locations of the electrons of an atom. ◮ Pre-election poll of party support. ◮ Distribution of a population with attributes like education, salary, and age. 2/ 19
Probabilistic Team Distributions Semantics Jonni Virtema Distributions Consider: Probabilistic atoms ◮ A collection of data from some repetitive science experiment. Connectives and quantifiers ◮ Data obtained from a poll. Examples ◮ Any collection of data, that involves meaningful duplicates of data. Benchmark logic Characterisation of One natural way to represent the data is to use multisets (sets with duplicates). expressivity Complexity Often the multiplicities themselves are not important; the distribution of data is: ◮ The locations of the electrons of an atom. ◮ Pre-election poll of party support. ◮ Distribution of a population with attributes like education, salary, and age. 2/ 19
Probabilistic Team Distributions Semantics Jonni Virtema Definition Distributions Probabilistic atoms A distribution is a mapping f : A → Q [0 , 1] from a set A of values to the closed Connectives and interval [0 , 1] of rational numbers such that the probabilities sum to 1, i.e., quantifiers Examples � f ( a ) = 1 . Benchmark logic Characterisation of a ∈ A expressivity Complexity ◮ A team is a set of first-order assignments (a database without duplicates). ◮ A multiteam is a pair ( X , m ), where X is a team 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 team and p : X → Q [0 , 1] is a distribution (distribution of data). 3/ 19
Probabilistic Team Distributions Semantics Jonni Virtema Definition Distributions Probabilistic atoms A distribution is a mapping f : A → Q [0 , 1] from a set A of values to the closed Connectives and interval [0 , 1] of rational numbers such that the probabilities sum to 1, i.e., quantifiers Examples � f ( a ) = 1 . Benchmark logic Characterisation of a ∈ A expressivity Complexity ◮ A team is a set of first-order assignments (a database without duplicates). ◮ A multiteam is a pair ( X , m ), where X is a team 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 team and p : X → Q [0 , 1] is a distribution (distribution of data). 3/ 19
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 Complexity ( x 1 , . . . , x n ) ≈ ( y 1 , . . . , y n ) and with the probabilistic conditional independence atoms y ⊥ ⊥ x z . 4/ 19
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 Complexity ( x 1 , . . . , x n ) ≈ ( y 1 , . . . , y n ) and with the probabilistic conditional independence atoms y ⊥ ⊥ x z . 4/ 19
Probabilistic Team Probabilistic atoms Semantics Jonni Virtema Distributions Probabilistic atoms Connectives and quantifiers The semantics are inherited from multiteam semantics. Examples Let X = ( X , p ) be a probablistic team and � x , � a be tuples of variables and values Benchmark logic of length k . We define Characterisation of expressivity � | X | � a := p ( s ) . x = � Complexity s ∈ X s ( � x )= � a 5/ 19
Probabilistic Team Probabilistic atoms Semantics Jonni Virtema Distributions Probabilistic atoms Let X = ( X , p ) be a probablistic team and � x , � a be tuples of variables and values Connectives and of length k . We define quantifiers � Examples | X | � a := p ( s ) . x = � Benchmark logic s ∈ X Characterisation of s ( � x )= � a expressivity Complexity We define that a ∈ A k , A | = X � x ≈ � y iff | X | � a = | X | � a , for each � x = � y = � 5/ 19
Probabilistic Team Probabilistic atoms Semantics Jonni Virtema Distributions Probabilistic atoms Let X = ( X , p ) be a probablistic team and � x , � a be tuples of variables and values Connectives and of length k . We define quantifiers � Examples | X | � a := p ( s ) . x = � Benchmark logic s ∈ X Characterisation of s ( � x )= � a expressivity Complexity 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 ( � 5/ 19
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 ) Complexity = 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 6/ 19
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 Complexity 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 . 6/ 19
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 ) Complexity ◮ 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. 7/ 19
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 , Complexity 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 . 8/ 19
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