The sheaf-theoretic description of contextuality Part II: - PowerPoint PPT Presentation

Axioms for a valuation algebra The elements of Φ are called valuations . A set of valuations is called a knowledgebase . A set of variables D ⊆ V is called a domain . Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Axioms for a valuation algebra The elements of Φ are called valuations . A set of valuations is called a knowledgebase . A set of variables D ⊆ V is called a domain . The domain of a valuation φ is the set d ( φ ) . Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Axioms for a valuation algebra The elements of Φ are called valuations . A set of valuations is called a knowledgebase . A set of variables D ⊆ V is called a domain . The domain of a valuation φ is the set d ( φ ) . Intuitively, a valuation φ ∈ Φ represents information about the possible values of a finite set of variables d ( φ ) = { x 1 ,... x n } ⊆ V , which constitutes the domain of φ . Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Axioms for a valuation algebra The elements of Φ are called valuations . A set of valuations is called a knowledgebase . A set of variables D ⊆ V is called a domain . The domain of a valuation φ is the set d ( φ ) . Intuitively, a valuation φ ∈ Φ represents information about the possible values of a finite set of variables d ( φ ) = { x 1 ,... x n } ⊆ V , which constitutes the domain of φ . For any finite set of variables S ⊆ V , we denote by Φ S : = { φ ∈ Φ | d ( φ ) = S } the set of valuations with domain S . Thus, we have � Φ = Φ S . S ⊆ V Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Information algebras Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Information algebras It is often desirable to add additional postulates, which collectively give rise to the notion of information algebra Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Information algebras It is often desirable to add additional postulates, which collectively give rise to the notion of information algebra Definition Let Φ be a valuation algebra on V . Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Information algebras It is often desirable to add additional postulates, which collectively give rise to the notion of information algebra Definition Let Φ be a valuation algebra on V . We say that Φ has neutral elements if it satisfies Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Information algebras It is often desirable to add additional postulates, which collectively give rise to the notion of information algebra Definition Let Φ be a valuation algebra on V . We say that Φ has neutral elements if it satisfies (A7) Commutative monoid : For each S ⊆ V , there exists a neutral element e S ∈ Φ S such that φ ⊗ e S = e S ⊗ φ = φ for all φ ∈ Φ S . Such neutral elements must satisfy the following identity: e S ⊗ e T = e S ∪ T for all subsets S , T ⊆ V . Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Information algebras Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Information algebras We say that Φ has null elements if it satisfies Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Information algebras We say that Φ has null elements if it satisfies (A8) Nullity : For each S ⊆ V there exists a null element z S ∈ Φ S such that φ ⊗ z S = z S ⊗ φ = z S . Moreover, for all S , T ⊆ V such that S ⊆ T , we have, for each φ ∈ Φ T , φ ↓ S = z S ⇐ ⇒ φ = z T . Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Information algebras We say that Φ has null elements if it satisfies (A8) Nullity : For each S ⊆ V there exists a null element z S ∈ Φ S such that φ ⊗ z S = z S ⊗ φ = z S . Moreover, for all S , T ⊆ V such that S ⊆ T , we have, for each φ ∈ Φ T , φ ↓ S = z S ⇐ ⇒ φ = z T . We say that Φ is idempotent if it satisfies Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Information algebras We say that Φ has null elements if it satisfies (A8) Nullity : For each S ⊆ V there exists a null element z S ∈ Φ S such that φ ⊗ z S = z S ⊗ φ = z S . Moreover, for all S , T ⊆ V such that S ⊆ T , we have, for each φ ∈ Φ T , φ ↓ S = z S ⇐ ⇒ φ = z T . We say that Φ is idempotent if it satisfies (A9) Idempotency : For all φ ∈ Φ and S ⊆ d ( φ ) , it holds that φ ⊗ φ ↓ S = φ Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Information algebras We say that Φ has null elements if it satisfies (A8) Nullity : For each S ⊆ V there exists a null element z S ∈ Φ S such that φ ⊗ z S = z S ⊗ φ = z S . Moreover, for all S , T ⊆ V such that S ⊆ T , we have, for each φ ∈ Φ T , φ ↓ S = z S ⇐ ⇒ φ = z T . We say that Φ is idempotent if it satisfies (A9) Idempotency : For all φ ∈ Φ and S ⊆ d ( φ ) , it holds that φ ⊗ φ ↓ S = φ If Φ satisfies axioms (A7)–(A9) it is called an information algebra . Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Frames and tuples Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Frames and tuples For each variable x ∈ V , we denote by Ω x its frame , i.e. the set of possible values for x . Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Frames and tuples For each variable x ∈ V , we denote by Ω x its frame , i.e. the set of possible values for x . A tuple with finite domain S ⊆ V is an element x of Ω S : = ∏ Ω x x ∈ S Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Frames and tuples For each variable x ∈ V , we denote by Ω x its frame , i.e. the set of possible values for x . A tuple with finite domain S ⊆ V is an element x of Ω S : = ∏ Ω x x ∈ S We will denote by x ↓ T the cartesian projection of a tuple x ∈ Ω S to Ω T , where T ⊆ S . Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Examples Information algebra of R -distributions Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Examples Information algebra of R -distributions Let � R , + , · , 0 , 1 � be a commutative semiring and V a set of variables. Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Examples Information algebra of R -distributions Let � R , + , · , 0 , 1 � be a commutative semiring and V a set of variables. Define a valuation algebra Φ : Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Examples Information algebra of R -distributions Let � R , + , · , 0 , 1 � be a commutative semiring and V a set of variables. Define a valuation algebra Φ : Valuations : functions φ : Ω S − → R . such that ∑ φ ( x ) = 1 . x ∈ Ω S Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Examples Information algebra of R -distributions Let � R , + , · , 0 , 1 � be a commutative semiring and V a set of variables. Define a valuation algebra Φ : Valuations : functions φ : Ω S − → R . such that ∑ φ ( x ) = 1 . x ∈ Ω S ◮ Labelling : Given φ : Ω S → R , define d ( φ ) : = S . Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Examples Information algebra of R -distributions Let � R , + , · , 0 , 1 � be a commutative semiring and V a set of variables. Define a valuation algebra Φ : Valuations : functions φ : Ω S − → R . such that ∑ φ ( x ) = 1 . x ∈ Ω S ◮ Labelling : Given φ : Ω S → R , define d ( φ ) : = S . ◮ Combination : For all distributions φ ∈ Φ S , ψ ∈ Φ T , define, for all x ∈ Ω S ∪ T , � − 1 � ∑ ( φ ⊗ ψ )( x ) : = φ ( y ↓ S ) · ψ ( y ↓ T ) φ ( x ↓ S ) · ψ ( x ↓ T ) . y ∈ Ω S ∪ T Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Examples Information algebra of R -distributions Let � R , + , · , 0 , 1 � be a commutative semiring and V a set of variables. Define a valuation algebra Φ : Valuations : functions φ : Ω S − → R . such that ∑ φ ( x ) = 1 . x ∈ Ω S ◮ Labelling : Given φ : Ω S → R , define d ( φ ) : = S . ◮ Combination : For all distributions φ ∈ Φ S , ψ ∈ Φ T , define, for all x ∈ Ω S ∪ T , � − 1 � ∑ ( φ ⊗ ψ )( x ) : = φ ( y ↓ S ) · ψ ( y ↓ T ) φ ( x ↓ S ) · ψ ( x ↓ T ) . y ∈ Ω S ∪ T ◮ Projection : For all φ ∈ Φ S , T ⊆ S and x ∈ Ω T , define φ ↓ T ( x ) : = ∑ φ ( x , y ) . y ∈ Ω S \ T Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Examples Information algebra of R -distributions Let � R , + , · , 0 , 1 � be a commutative semiring and V a set of variables. Define a valuation algebra Φ : Valuations : functions φ : Ω S − → R . such that ∑ φ ( x ) = 1 . x ∈ Ω S ◮ Labelling : Given φ : Ω S → R , define d ( φ ) : = S . ◮ Combination : For all distributions φ ∈ Φ S , ψ ∈ Φ T , define, for all x ∈ Ω S ∪ T , � − 1 � ∑ ( φ ⊗ ψ )( x ) : = φ ( y ↓ S ) · ψ ( y ↓ T ) φ ( x ↓ S ) · ψ ( x ↓ T ) . y ∈ Ω S ∪ T ◮ Projection : For all φ ∈ Φ S , T ⊆ S and x ∈ Ω T , define φ ↓ T ( x ) : = ∑ φ ( x , y ) . y ∈ Ω S \ T The algebra has neutral elements and null elements, but it is idempotent only if R = B . Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Examples Relational databases Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Examples Relational databases Consider the following data table: branch-name account-no customer-name balance Cambridge 10991-06284 Newton 2,567.53 Hanover 10992-35671 Leibniz 11,245.75 ... ... ... ... Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Examples Relational databases Consider the following data table: branch-name account-no customer-name balance Cambridge 10991-06284 Newton 2,567.53 Hanover 10992-35671 Leibniz 11,245.75 ... ... ... ... Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Examples Relational databases Consider the following data table: branch-name account-no customer-name balance Cambridge 10991-06284 Newton 2,567.53 Hanover 10992-35671 Leibniz 11,245.75 ... ... ... ... Each column is labelled by an attribute . Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Examples Relational databases Consider the following data table: branch-name account-no customer-name balance Cambridge 10991-06284 Newton 2,567.53 Hanover 10992-35671 Leibniz 11,245.75 ... ... ... ... Each column is labelled by an attribute . Each entry of the table is a tuple specifying a value for each of the attributes. Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Examples Relational databases Consider the following data table: branch-name account-no customer-name balance Cambridge 10991-06284 Newton 2,567.53 Hanover 10992-35671 Leibniz 11,245.75 ... ... ... ... Each column is labelled by an attribute . Each entry of the table is a tuple specifying a value for each of the attributes. The full table is simply a set of tuples, i.e. a relation . Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Examples Relational databases Consider the following data table: branch-name account-no customer-name balance Cambridge 10991-06284 Newton 2,567.53 Hanover 10992-35671 Leibniz 11,245.75 ... ... ... ... Each column is labelled by an attribute . Each entry of the table is a tuple specifying a value for each of the attributes. The full table is simply a set of tuples, i.e. a relation . The set of attributes of a relation R is called its schema , denoted schema ( R ) Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Examples Relational databases Consider the following data table: branch-name account-no customer-name balance Cambridge 10991-06284 Newton 2,567.53 Hanover 10992-35671 Leibniz 11,245.75 ... ... ... ... Each column is labelled by an attribute . Each entry of the table is a tuple specifying a value for each of the attributes. The full table is simply a set of tuples, i.e. a relation . The set of attributes of a relation R is called its schema , denoted schema ( R ) A database instance is a family of relations. Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Examples Information algebra of relational databases Define a valuation algebra Φ such that Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Examples Information algebra of relational databases Define a valuation algebra Φ such that The set of variables V coincides with the set of all attributes . Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Examples Information algebra of relational databases Define a valuation algebra Φ such that The set of variables V coincides with the set of all attributes . For each x ∈ V , define the frame Ω x to be the set of possible values for x . Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Examples Information algebra of relational databases Define a valuation algebra Φ such that The set of variables V coincides with the set of all attributes . For each x ∈ V , define the frame Ω x to be the set of possible values for x . A valuation over S ⊆ V is a set of tuples R ⊆ Ω S , thus Φ S = P ( Ω S ) . Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Examples Information algebra of relational databases Define a valuation algebra Φ such that The set of variables V coincides with the set of all attributes . For each x ∈ V , define the frame Ω x to be the set of possible values for x . A valuation over S ⊆ V is a set of tuples R ⊆ Ω S , thus Φ S = P ( Ω S ) . ◮ Labelling : For all R ∈ Φ S , define d ( R ) : = S . Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Examples Information algebra of relational databases Define a valuation algebra Φ such that The set of variables V coincides with the set of all attributes . For each x ∈ V , define the frame Ω x to be the set of possible values for x . A valuation over S ⊆ V is a set of tuples R ⊆ Ω S , thus Φ S = P ( Ω S ) . ◮ Labelling : For all R ∈ Φ S , define d ( R ) : = S . ◮ Combination given by the natural join : let R 1 ∈ Φ S , R 2 ∈ Φ T , R 1 ⊗ R 2 : = R 1 ⋊ ⋉ R 2 = { x ∈ Ω S ∪ T | x ↓ S ∈ R 1 ∧ x ↓ T ∈ R 2 } , which is clearly idempotent. Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Examples Information algebra of relational databases Define a valuation algebra Φ such that The set of variables V coincides with the set of all attributes . For each x ∈ V , define the frame Ω x to be the set of possible values for x . A valuation over S ⊆ V is a set of tuples R ⊆ Ω S , thus Φ S = P ( Ω S ) . ◮ Labelling : For all R ∈ Φ S , define d ( R ) : = S . ◮ Combination given by the natural join : let R 1 ∈ Φ S , R 2 ∈ Φ T , R 1 ⊗ R 2 : = R 1 ⋊ ⋉ R 2 = { x ∈ Ω S ∪ T | x ↓ S ∈ R 1 ∧ x ↓ T ∈ R 2 } , which is clearly idempotent. ◮ Projection : Given a valuation R with domain d ( R ) = S , and a subset T ⊆ S , define R ↓ T : = { x ↓ T | x ∈ R } Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Examples Information algebra of relational databases Define a valuation algebra Φ such that The set of variables V coincides with the set of all attributes . For each x ∈ V , define the frame Ω x to be the set of possible values for x . A valuation over S ⊆ V is a set of tuples R ⊆ Ω S , thus Φ S = P ( Ω S ) . ◮ Labelling : For all R ∈ Φ S , define d ( R ) : = S . ◮ Combination given by the natural join : let R 1 ∈ Φ S , R 2 ∈ Φ T , R 1 ⊗ R 2 : = R 1 ⋊ ⋉ R 2 = { x ∈ Ω S ∪ T | x ↓ S ∈ R 1 ∧ x ↓ T ∈ R 2 } , which is clearly idempotent. ◮ Projection : Given a valuation R with domain d ( R ) = S , and a subset T ⊆ S , define R ↓ T : = { x ↓ T | x ∈ R } Let S ⊆ V . Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Examples Information algebra of relational databases Define a valuation algebra Φ such that The set of variables V coincides with the set of all attributes . For each x ∈ V , define the frame Ω x to be the set of possible values for x . A valuation over S ⊆ V is a set of tuples R ⊆ Ω S , thus Φ S = P ( Ω S ) . ◮ Labelling : For all R ∈ Φ S , define d ( R ) : = S . ◮ Combination given by the natural join : let R 1 ∈ Φ S , R 2 ∈ Φ T , R 1 ⊗ R 2 : = R 1 ⋊ ⋉ R 2 = { x ∈ Ω S ∪ T | x ↓ S ∈ R 1 ∧ x ↓ T ∈ R 2 } , which is clearly idempotent. ◮ Projection : Given a valuation R with domain d ( R ) = S , and a subset T ⊆ S , define R ↓ T : = { x ↓ T | x ∈ R } Let S ⊆ V . The neutral element is e S : = Ω S . Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Examples Information algebra of relational databases Define a valuation algebra Φ such that The set of variables V coincides with the set of all attributes . For each x ∈ V , define the frame Ω x to be the set of possible values for x . A valuation over S ⊆ V is a set of tuples R ⊆ Ω S , thus Φ S = P ( Ω S ) . ◮ Labelling : For all R ∈ Φ S , define d ( R ) : = S . ◮ Combination given by the natural join : let R 1 ∈ Φ S , R 2 ∈ Φ T , R 1 ⊗ R 2 : = R 1 ⋊ ⋉ R 2 = { x ∈ Ω S ∪ T | x ↓ S ∈ R 1 ∧ x ↓ T ∈ R 2 } , which is clearly idempotent. ◮ Projection : Given a valuation R with domain d ( R ) = S , and a subset T ⊆ S , define R ↓ T : = { x ↓ T | x ∈ R } Let S ⊆ V . The neutral element is e S : = Ω S .The null element is z S : = / 0. Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Examples Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Examples The algebra of relational databases can be generalised by elevating the concept of tuple to a higher level: Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Examples The algebra of relational databases can be generalised by elevating the concept of tuple to a higher level: Definition A tuple system over P ( V ) , where V is a set of variables, is a set T equipped with two operations d : T → P ( V ) and ↓ : T × P ( V ) → T satisfying the following axioms: (T1) If Q ⊆ d ( t ) , then d ( t ↓ Q ) = Q . � � (T2) If Q ⊆ U ⊆ d ( t ) , then t ↓ U ↓ Q = t ↓ Q . (T3) If d ( t ) = Q , then t ↓ Q = t . (T4) For d ( t ) = Q , d ( u ) = U such that t ↓ Q ∩ U = u ↓ Q ∩ U , there exists g ∈ T such that d ( g ) = Q ∪ U , g ↓ Q = t and g ↓ U = u . (T5) For d ( t ) = Q and Q ⊆ U , there exists g ∈ T such that d ( g ) = U and g ↓ Q = t . Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Examples General information sets Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Examples General information sets Given any tuple system T on a set of variables V , one can define an information algebra of information sets relative to it: Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Examples General information sets Given any tuple system T on a set of variables V , one can define an information algebra of information sets relative to it: Valuations are subsets S ⊆ T Q : = { t ↓ Q : t ∈ T } , where Q ⊆ V . Thus Φ Q : = P ( T Q ) . Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Examples General information sets Given any tuple system T on a set of variables V , one can define an information algebra of information sets relative to it: Valuations are subsets S ⊆ T Q : = { t ↓ Q : t ∈ T } , where Q ⊆ V . Thus Φ Q : = P ( T Q ) . ◮ Labelling : For all S ∈ Φ Q , define d ( S ) : = Q . Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Examples General information sets Given any tuple system T on a set of variables V , one can define an information algebra of information sets relative to it: Valuations are subsets S ⊆ T Q : = { t ↓ Q : t ∈ T } , where Q ⊆ V . Thus Φ Q : = P ( T Q ) . ◮ Labelling : For all S ∈ Φ Q , define d ( S ) : = Q . ◮ Combination given by the natural join : let S 1 ∈ Φ Q , S 2 ∈ Φ U , S 1 ⊗ S 2 : = S 1 ⋊ ⋉ S 2 = { t ∈ T Q ∪ U | t ↓ S ∈ S 1 ∧ t ↓ U ∈ S 2 } , which is clearly idempotent. Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Examples General information sets Given any tuple system T on a set of variables V , one can define an information algebra of information sets relative to it: Valuations are subsets S ⊆ T Q : = { t ↓ Q : t ∈ T } , where Q ⊆ V . Thus Φ Q : = P ( T Q ) . ◮ Labelling : For all S ∈ Φ Q , define d ( S ) : = Q . ◮ Combination given by the natural join : let S 1 ∈ Φ Q , S 2 ∈ Φ U , S 1 ⊗ S 2 : = S 1 ⋊ ⋉ S 2 = { t ∈ T Q ∪ U | t ↓ S ∈ S 1 ∧ t ↓ U ∈ S 2 } , which is clearly idempotent. ◮ Projection : Given a valuation S with domain d ( S ) = Q , and a subset U ⊆ Q , define S ↓ U : = { t ↓ U | t ∈ S } Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Examples General information sets Given any tuple system T on a set of variables V , one can define an information algebra of information sets relative to it: Valuations are subsets S ⊆ T Q : = { t ↓ Q : t ∈ T } , where Q ⊆ V . Thus Φ Q : = P ( T Q ) . ◮ Labelling : For all S ∈ Φ Q , define d ( S ) : = Q . ◮ Combination given by the natural join : let S 1 ∈ Φ Q , S 2 ∈ Φ U , S 1 ⊗ S 2 : = S 1 ⋊ ⋉ S 2 = { t ∈ T Q ∪ U | t ↓ S ∈ S 1 ∧ t ↓ U ∈ S 2 } , which is clearly idempotent. ◮ Projection : Given a valuation S with domain d ( S ) = Q , and a subset U ⊆ Q , define S ↓ U : = { t ↓ U | t ∈ S } Given any Q ⊆ V , the neutral element is e Q : = T Q : = { t ∈ T : d ( T ) = Q } . Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Examples General information sets Given any tuple system T on a set of variables V , one can define an information algebra of information sets relative to it: Valuations are subsets S ⊆ T Q : = { t ↓ Q : t ∈ T } , where Q ⊆ V . Thus Φ Q : = P ( T Q ) . ◮ Labelling : For all S ∈ Φ Q , define d ( S ) : = Q . ◮ Combination given by the natural join : let S 1 ∈ Φ Q , S 2 ∈ Φ U , S 1 ⊗ S 2 : = S 1 ⋊ ⋉ S 2 = { t ∈ T Q ∪ U | t ↓ S ∈ S 1 ∧ t ↓ U ∈ S 2 } , which is clearly idempotent. ◮ Projection : Given a valuation S with domain d ( S ) = Q , and a subset U ⊆ Q , define S ↓ U : = { t ↓ U | t ∈ S } Given any Q ⊆ V , the neutral element is e Q : = T Q : = { t ∈ T : d ( T ) = Q } . The null element is z Q : = / 0. Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Examples General information sets Given any tuple system T on a set of variables V , one can define an information algebra of information sets relative to it: Valuations are subsets S ⊆ T Q : = { t ↓ Q : t ∈ T } , where Q ⊆ V . Thus Φ Q : = P ( T Q ) . ◮ Labelling : For all S ∈ Φ Q , define d ( S ) : = Q . ◮ Combination given by the natural join : let S 1 ∈ Φ Q , S 2 ∈ Φ U , S 1 ⊗ S 2 : = S 1 ⋊ ⋉ S 2 = { t ∈ T Q ∪ U | t ↓ S ∈ S 1 ∧ t ↓ U ∈ S 2 } , which is clearly idempotent. ◮ Projection : Given a valuation S with domain d ( S ) = Q , and a subset U ⊆ Q , define S ↓ U : = { t ↓ U | t ∈ S } Given any Q ⊆ V , the neutral element is e Q : = T Q : = { t ∈ T : d ( T ) = Q } . The null element is z Q : = / 0. Thus, we obtain an information algebra . Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Examples General information sets Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Examples General information sets Any instance of a tuple system gives rise to a different intormation algebra: Example Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Examples General information sets Any instance of a tuple system gives rise to a different intormation algebra: Example Cartesian tuples (cartesian projection) � relational databases . Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Examples General information sets Any instance of a tuple system gives rise to a different intormation algebra: Example Cartesian tuples (cartesian projection) � relational databases . Probability distributions (marginalisation) � probability distribution information sets Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Examples General information sets Any instance of a tuple system gives rise to a different intormation algebra: Example Cartesian tuples (cartesian projection) � relational databases . Probability distributions (marginalisation) � probability distribution information sets Propositional truth valuations v : L → { 0 , 1 } (function restriction) � propositional information sets Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Examples General information sets Any instance of a tuple system gives rise to a different intormation algebra: Example Cartesian tuples (cartesian projection) � relational databases . Probability distributions (marginalisation) � probability distribution information sets Propositional truth valuations v : L → { 0 , 1 } (function restriction) � propositional information sets Propositional formulae (existential quantification) � algebra of propositional formulae Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Examples General information sets Any instance of a tuple system gives rise to a different intormation algebra: Example Cartesian tuples (cartesian projection) � relational databases . Probability distributions (marginalisation) � probability distribution information sets Propositional truth valuations v : L → { 0 , 1 } (function restriction) � propositional information sets Propositional formulae (existential quantification) � algebra of propositional formulae More generally, given any logical ‘context’ � L , M , | = � , one can define both an algebra of information sets, and an algebra of formulae, e.g. Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Examples General information sets Any instance of a tuple system gives rise to a different intormation algebra: Example Cartesian tuples (cartesian projection) � relational databases . Probability distributions (marginalisation) � probability distribution information sets Propositional truth valuations v : L → { 0 , 1 } (function restriction) � propositional information sets Propositional formulae (existential quantification) � algebra of propositional formulae More generally, given any logical ‘context’ � L , M , | = � , one can define both an algebra of information sets, and an algebra of formulae, e.g. ◮ – Predicate logic Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Examples General information sets Any instance of a tuple system gives rise to a different intormation algebra: Example Cartesian tuples (cartesian projection) � relational databases . Probability distributions (marginalisation) � probability distribution information sets Propositional truth valuations v : L → { 0 , 1 } (function restriction) � propositional information sets Propositional formulae (existential quantification) � algebra of propositional formulae More generally, given any logical ‘context’ � L , M , | = � , one can define both an algebra of information sets, and an algebra of formulae, e.g. ◮ – Predicate logic ◮ – Linear equations Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Examples General information sets Any instance of a tuple system gives rise to a different intormation algebra: Example Cartesian tuples (cartesian projection) � relational databases . Probability distributions (marginalisation) � probability distribution information sets Propositional truth valuations v : L → { 0 , 1 } (function restriction) � propositional information sets Propositional formulae (existential quantification) � algebra of propositional formulae More generally, given any logical ‘context’ � L , M , | = � , one can define both an algebra of information sets, and an algebra of formulae, e.g. ◮ – Predicate logic ◮ – Linear equations ◮ – Constraint satisfaction problems Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Examples General information sets Any instance of a tuple system gives rise to a different intormation algebra: Example Cartesian tuples (cartesian projection) � relational databases . Probability distributions (marginalisation) � probability distribution information sets Propositional truth valuations v : L → { 0 , 1 } (function restriction) � propositional information sets Propositional formulae (existential quantification) � algebra of propositional formulae More generally, given any logical ‘context’ � L , M , | = � , one can define both an algebra of information sets, and an algebra of formulae, e.g. ◮ – Predicate logic ◮ – Linear equations ◮ – Constraint satisfaction problems ◮ – . . . Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Disagreement Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Disagreement Disgreement between sources is a fundamental aspect of information. Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Disagreement Disgreement between sources is a fundamental aspect of information. Despite this, there is no general definition of disagreement in the valuation algebraic approach, which focuses more on the problem of extracting information (more on that later). Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Disagreement Disgreement between sources is a fundamental aspect of information. Despite this, there is no general definition of disagreement in the valuation algebraic approach, which focuses more on the problem of extracting information (more on that later). We propose a natural formulation: Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Disagreement Disgreement between sources is a fundamental aspect of information. Despite this, there is no general definition of disagreement in the valuation algebraic approach, which focuses more on the problem of extracting information (more on that later). We propose a natural formulation: Consider a valuation algebra Φ on a set of variables V , let K = { φ 1 ,..., φ n } ⊆ Φ be a knowledgebase, with n � D : = d ( φ i ) . i = 1 Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Disagreement Disgreement between sources is a fundamental aspect of information. Despite this, there is no general definition of disagreement in the valuation algebraic approach, which focuses more on the problem of extracting information (more on that later). We propose a natural formulation: Consider a valuation algebra Φ on a set of variables V , let K = { φ 1 ,..., φ n } ⊆ Φ be a knowledgebase, with n � D : = d ( φ i ) . i = 1 To say that the information sources agree is equivalent to say that there is a truth which is agreed upon by all the sources. Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Disagreement Disgreement between sources is a fundamental aspect of information. Despite this, there is no general definition of disagreement in the valuation algebraic approach, which focuses more on the problem of extracting information (more on that later). We propose a natural formulation: Consider a valuation algebra Φ on a set of variables V , let K = { φ 1 ,..., φ n } ⊆ Φ be a knowledgebase, with n � D : = d ( φ i ) . i = 1 To say that the information sources agree is equivalent to say that there is a truth which is agreed upon by all the sources. The truth valuation gives information about all the variables appearing in K , while each φ i only concerns a set of the variables d ( φ i ) ⊆ D . Definition We say that φ 1 ,..., φ n agree (or agree globally ) if there exists a (global) truth valuation γ ∈ Φ D such that, for all 1 ≤ i ≤ n , γ ↓ d ( φ i ) = φ i . Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018