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

The sheaf-theoretic description of contextuality Part II: contextuality and valuation algebras

Samson Abramsky & Giovanni Car` u

Quantum Group Department of Computer Science University of Oxford

Winer Memorial Lectures Purdue University, 10 November 2018

QUANTUM GROUP

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

Introduction

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

Introduction

The high level of generality of the sheaf theoretic description of contextuality led to unexpected connections with fields unrelated to quantum mechanics: contextual behaviour has been observed in

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

Introduction

The high level of generality of the sheaf theoretic description of contextuality led to unexpected connections with fields unrelated to quantum mechanics: contextual behaviour has been observed in

◮ Relational databases (Abramsky 2013) Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Introduction

The high level of generality of the sheaf theoretic description of contextuality led to unexpected connections with fields unrelated to quantum mechanics: contextual behaviour has been observed in

◮ Relational databases (Abramsky 2013) ◮ Constraint satisfaction problems (Abramsky, Gottlob, Kolaitis 2013) Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Introduction

The high level of generality of the sheaf theoretic description of contextuality led to unexpected connections with fields unrelated to quantum mechanics: contextual behaviour has been observed in

◮ Relational databases (Abramsky 2013) ◮ Constraint satisfaction problems (Abramsky, Gottlob, Kolaitis 2013) ◮ Logic (Abramsky, Barbosa, Kishida, Lal, Mansfield 2015) Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Introduction

The high level of generality of the sheaf theoretic description of contextuality led to unexpected connections with fields unrelated to quantum mechanics: contextual behaviour has been observed in

◮ Relational databases (Abramsky 2013) ◮ Constraint satisfaction problems (Abramsky, Gottlob, Kolaitis 2013) ◮ Logic (Abramsky, Barbosa, Kishida, Lal, Mansfield 2015)

This leads to the idea of developing a contextual semantics, an all-comprehensive theory which captures the essence of all such contextual phenomena.

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

Introduction

The high level of generality of the sheaf theoretic description of contextuality led to unexpected connections with fields unrelated to quantum mechanics: contextual behaviour has been observed in

◮ Relational databases (Abramsky 2013) ◮ Constraint satisfaction problems (Abramsky, Gottlob, Kolaitis 2013) ◮ Logic (Abramsky, Barbosa, Kishida, Lal, Mansfield 2015)

This leads to the idea of developing a contextual semantics, an all-comprehensive theory which captures the essence of all such contextual phenomena. All the different instances of contextuality share a common trait: they concern pieces of information, which agree locally, but disagree globally.

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

Valuation algebras

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

Valuation algebras

Valuation algebras are a general framework to model concepts such as information and knowledge.

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

Valuation algebras

Valuation algebras are a general framework to model concepts such as information and knowledge.

Definition

Let V be a set of variables. A valuation algebra over V is a set Φ equipped with three

• perations:

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

Valuation algebras

Valuation algebras are a general framework to model concepts such as information and knowledge.

Definition

Let V be a set of variables. A valuation algebra over V is a set Φ equipped with three

• perations:

1

Labelling: Φ → P(V) :: φ → d(φ)

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

Valuation algebras

Valuation algebras are a general framework to model concepts such as information and knowledge.

Definition

Let V be a set of variables. A valuation algebra over V is a set Φ equipped with three

• perations:

1

Labelling: Φ → P(V) :: φ → d(φ)

2

Combination: Φ×Φ → Φ :: (φ,ψ) → φ ⊗ψ

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

Valuation algebras

Valuation algebras are a general framework to model concepts such as information and knowledge.

Definition

Let V be a set of variables. A valuation algebra over V is a set Φ equipped with three

• perations:

1

Labelling: Φ → P(V) :: φ → d(φ)

2

Combination: Φ×Φ → Φ :: (φ,ψ) → φ ⊗ψ

3

Projection: Φ×P(V) → Φ :: (φ,S) → φ↓S, for all S ⊆ d(φ),

such that axioms (A1)–(A6) are satisfied:

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

Axioms for a valuation algebra

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

Axioms for a valuation algebra

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

Axioms for a valuation algebra

(A1) Commutative Semigroup: (Φ,⊗) is associative and commutative.

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

Axioms for a valuation algebra

(A1) Commutative Semigroup: (Φ,⊗) is associative and commutative. (A2) Labelling: For all φ,ψ ∈ Φ, d(φ ⊗ψ) = d(φ)∪d(ψ)

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

Axioms for a valuation algebra

(A1) Commutative Semigroup: (Φ,⊗) is associative and commutative. (A2) Labelling: For all φ,ψ ∈ Φ, d(φ ⊗ψ) = d(φ)∪d(ψ) (A3) Projection: Given φ ∈ Φ and S ⊆ d(φ), d

• φ ↓S

= S

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

Axioms for a valuation algebra

(A1) Commutative Semigroup: (Φ,⊗) is associative and commutative. (A2) Labelling: For all φ,ψ ∈ Φ, d(φ ⊗ψ) = d(φ)∪d(ψ) (A3) Projection: Given φ ∈ Φ and S ⊆ d(φ), d

• φ ↓S

= S (A4) Transitivity: Given φ ∈ Φ and S ⊆ T ⊆ d(φ),

• φ ↓T↓S

= φ ↓S

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

Axioms for a valuation algebra

(A1) Commutative Semigroup: (Φ,⊗) is associative and commutative. (A2) Labelling: For all φ,ψ ∈ Φ, d(φ ⊗ψ) = d(φ)∪d(ψ) (A3) Projection: Given φ ∈ Φ and S ⊆ d(φ), d

• φ ↓S

= S (A4) Transitivity: Given φ ∈ Φ and S ⊆ T ⊆ d(φ),

• φ ↓T↓S

= φ ↓S (A5) Combination: For φ,ψ ∈ Φ, with d(φ) = S, d(ψ) = T and U ⊆ V such that S ⊆ U ⊆ S∪T, (φ ⊗ψ)↓U = φ ⊗ψ↓U∩T

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

Axioms for a valuation algebra

(A1) Commutative Semigroup: (Φ,⊗) is associative and commutative. (A2) Labelling: For all φ,ψ ∈ Φ, d(φ ⊗ψ) = d(φ)∪d(ψ) (A3) Projection: Given φ ∈ Φ and S ⊆ d(φ), d

• φ ↓S

= S (A4) Transitivity: Given φ ∈ Φ and S ⊆ T ⊆ d(φ),

• φ ↓T↓S

= φ ↓S (A5) Combination: For φ,ψ ∈ Φ, with d(φ) = S, d(ψ) = T and U ⊆ V such that S ⊆ U ⊆ S∪T, (φ ⊗ψ)↓U = φ ⊗ψ↓U∩T (A6) Domain: Given φ ∈ Φ, φ ↓d(φ) = φ

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

Axioms for a valuation algebra

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.

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.

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.

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) = {x1,...xn} ⊆ 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) = {x1,...xn} ⊆ V, which constitutes the domain of φ. For any finite set of variables S ⊆ V, we denote by ΦS := {φ ∈ Φ | d(φ) = S} the set of valuations with domain S. Thus, we have Φ =

• S⊆V

ΦS.

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 eS ∈ ΦS such that φ ⊗eS = eS ⊗φ = φ for all φ ∈ ΦS. Such neutral elements must satisfy the following identity: eS ⊗eT = eS∪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 zS ∈ ΦS such that φ ⊗zS = zS ⊗φ = zS. Moreover, for all S,T ⊆ V such that S ⊆ T, we have, for each φ ∈ ΦT, φ↓S = zS ⇐ ⇒ φ = zT.

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 zS ∈ ΦS such that φ ⊗zS = zS ⊗φ = zS. Moreover, for all S,T ⊆ V such that S ⊆ T, we have, for each φ ∈ ΦT, φ↓S = zS ⇐ ⇒ φ = zT.

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 zS ∈ ΦS such that φ ⊗zS = zS ⊗φ = zS. Moreover, for all S,T ⊆ V such that S ⊆ T, we have, for each φ ∈ ΦT, φ↓S = zS ⇐ ⇒ φ = zT.

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 zS ∈ ΦS such that φ ⊗zS = zS ⊗φ = zS. Moreover, for all S,T ⊆ V such that S ⊆ T, we have, for each φ ∈ ΦT, φ↓S = zS ⇐ ⇒ φ = zT.

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 Ωx 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 Ωx its frame, i.e. the set of possible values for x. A tuple with finite domain S ⊆ V is an element x of ΩS := ∏

x∈S

Ωx

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 Ωx its frame, i.e. the set of possible values for x. A tuple with finite domain S ⊆ V is an element x of ΩS := ∏

x∈S

Ωx We will denote by x↓T the cartesian projection of a tuple x ∈ ΩS to ΩT, 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 : ΩS − → R. such that

∑

x∈ΩS

φ(x) = 1.

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 : ΩS − → R. such that

∑

x∈ΩS

φ(x) = 1.

◮ Labelling: Given φ : ΩS → 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 : ΩS − → R. such that

∑

x∈ΩS

φ(x) = 1.

◮ Labelling: Given φ : ΩS → R, define d(φ) := S. ◮ Combination: For all distributions φ ∈ ΦS, ψ ∈ ΦT, define, for all x ∈ ΩS∪T,

(φ ⊗ψ)(x) :=

• ∑

y∈ΩS∪T

φ(y↓S)·ψ(y↓T ) −1 φ(x↓S)·ψ(x↓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 : ΩS − → R. such that

∑

x∈ΩS

φ(x) = 1.

◮ Labelling: Given φ : ΩS → R, define d(φ) := S. ◮ Combination: For all distributions φ ∈ ΦS, ψ ∈ ΦT, define, for all x ∈ ΩS∪T,

(φ ⊗ψ)(x) :=

• ∑

y∈ΩS∪T

φ(y↓S)·ψ(y↓T ) −1 φ(x↓S)·ψ(x↓T ).

◮ Projection: For all φ ∈ ΦS, T ⊆ S and x ∈ ΩT, define

φ↓T(x) := ∑

y∈ΩS\T

φ(x,y).

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 : ΩS − → R. such that

∑

x∈ΩS

φ(x) = 1.

◮ Labelling: Given φ : ΩS → R, define d(φ) := S. ◮ Combination: For all distributions φ ∈ ΦS, ψ ∈ ΦT, define, for all x ∈ ΩS∪T,

(φ ⊗ψ)(x) :=

• ∑

y∈ΩS∪T

φ(y↓S)·ψ(y↓T ) −1 φ(x↓S)·ψ(x↓T ).

◮ Projection: For all φ ∈ ΦS, T ⊆ S and x ∈ ΩT, define

φ↓T(x) := ∑

y∈ΩS\T

φ(x,y).

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 Ωx 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 Ωx to be the set of possible values for x. A valuation over S ⊆ V is a set of tuples R ⊆ ΩS, thus ΦS = P(ΩS).

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 Ωx to be the set of possible values for x. A valuation over S ⊆ V is a set of tuples R ⊆ ΩS, thus ΦS = P(ΩS).

◮ Labelling: For all R ∈ ΦS, 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 Ωx to be the set of possible values for x. A valuation over S ⊆ V is a set of tuples R ⊆ ΩS, thus ΦS = P(ΩS).

◮ Labelling: For all R ∈ ΦS, define d(R) := S. ◮ Combination given by the natural join: let R1 ∈ ΦS, R2 ∈ ΦT,

R1 ⊗R2 := R1 ⋊ ⋉ R2 = {x ∈ ΩS∪T | x↓S ∈ R1 ∧ x↓T ∈ R2}, 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 Ωx to be the set of possible values for x. A valuation over S ⊆ V is a set of tuples R ⊆ ΩS, thus ΦS = P(ΩS).

◮ Labelling: For all R ∈ ΦS, define d(R) := S. ◮ Combination given by the natural join: let R1 ∈ ΦS, R2 ∈ ΦT,

R1 ⊗R2 := R1 ⋊ ⋉ R2 = {x ∈ ΩS∪T | x↓S ∈ R1 ∧ x↓T ∈ R2}, 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 Ωx to be the set of possible values for x. A valuation over S ⊆ V is a set of tuples R ⊆ ΩS, thus ΦS = P(ΩS).

◮ Labelling: For all R ∈ ΦS, define d(R) := S. ◮ Combination given by the natural join: let R1 ∈ ΦS, R2 ∈ ΦT,

R1 ⊗R2 := R1 ⋊ ⋉ R2 = {x ∈ ΩS∪T | x↓S ∈ R1 ∧ x↓T ∈ R2}, 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 Ωx to be the set of possible values for x. A valuation over S ⊆ V is a set of tuples R ⊆ ΩS, thus ΦS = P(ΩS).

◮ Labelling: For all R ∈ ΦS, define d(R) := S. ◮ Combination given by the natural join: let R1 ∈ ΦS, R2 ∈ ΦT,

R1 ⊗R2 := R1 ⋊ ⋉ R2 = {x ∈ ΩS∪T | x↓S ∈ R1 ∧ x↓T ∈ R2}, 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 eS := ΩS.

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 Ωx to be the set of possible values for x. A valuation over S ⊆ V is a set of tuples R ⊆ ΩS, thus ΦS = P(ΩS).

◮ Labelling: For all R ∈ ΦS, define d(R) := S. ◮ Combination given by the natural join: let R1 ∈ ΦS, R2 ∈ ΦT,

R1 ⊗R2 := R1 ⋊ ⋉ R2 = {x ∈ ΩS∪T | x↓S ∈ R1 ∧ x↓T ∈ R2}, 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 eS := ΩS.The null element is zS := / 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

• f 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

• f 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

• perations 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 ⊆ TQ := {t↓Q : t ∈ T}, where Q ⊆ V. Thus ΦQ := P(TQ).

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 ⊆ TQ := {t↓Q : t ∈ T}, where Q ⊆ V. Thus ΦQ := P(TQ).

◮ Labelling: For all S ∈ ΦQ, 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 ⊆ TQ := {t↓Q : t ∈ T}, where Q ⊆ V. Thus ΦQ := P(TQ).

◮ Labelling: For all S ∈ ΦQ, define d(S) := Q. ◮ Combination given by the natural join: let S1 ∈ ΦQ, S2 ∈ ΦU,

S1 ⊗S2 := S1 ⋊ ⋉ S2 = {t ∈ TQ∪U | t↓S ∈ S1 ∧ t↓U ∈ S2}, 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 ⊆ TQ := {t↓Q : t ∈ T}, where Q ⊆ V. Thus ΦQ := P(TQ).

◮ Labelling: For all S ∈ ΦQ, define d(S) := Q. ◮ Combination given by the natural join: let S1 ∈ ΦQ, S2 ∈ ΦU,

S1 ⊗S2 := S1 ⋊ ⋉ S2 = {t ∈ TQ∪U | t↓S ∈ S1 ∧ t↓U ∈ S2}, 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 ⊆ TQ := {t↓Q : t ∈ T}, where Q ⊆ V. Thus ΦQ := P(TQ).

◮ Labelling: For all S ∈ ΦQ, define d(S) := Q. ◮ Combination given by the natural join: let S1 ∈ ΦQ, S2 ∈ ΦU,

S1 ⊗S2 := S1 ⋊ ⋉ S2 = {t ∈ TQ∪U | t↓S ∈ S1 ∧ t↓U ∈ S2}, 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 eQ := TQ := {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 ⊆ TQ := {t↓Q : t ∈ T}, where Q ⊆ V. Thus ΦQ := P(TQ).

◮ Labelling: For all S ∈ ΦQ, define d(S) := Q. ◮ Combination given by the natural join: let S1 ∈ ΦQ, S2 ∈ ΦU,

S1 ⊗S2 := S1 ⋊ ⋉ S2 = {t ∈ TQ∪U | t↓S ∈ S1 ∧ t↓U ∈ S2}, 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 eQ := TQ := {t ∈ T : d(T) = Q}. The null element is zQ := / 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 ⊆ TQ := {t↓Q : t ∈ T}, where Q ⊆ V. Thus ΦQ := P(TQ).

◮ Labelling: For all S ∈ ΦQ, define d(S) := Q. ◮ Combination given by the natural join: let S1 ∈ ΦQ, S2 ∈ ΦU,

S1 ⊗S2 := S1 ⋊ ⋉ S2 = {t ∈ TQ∪U | t↓S ∈ S1 ∧ t↓U ∈ S2}, 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 eQ := TQ := {t ∈ T : d(T) = Q}. The null element is zQ := /

• 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 = {φ1,...,φn} ⊆ Φ be a knowledgebase, with D :=

n

• i=1

d(φi).

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 = {φ1,...,φn} ⊆ Φ be a knowledgebase, with D :=

n

• i=1

d(φi). 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 = {φ1,...,φn} ⊆ Φ be a knowledgebase, with D :=

n

• i=1

d(φi). 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 φi only concerns a set of the variables d(φi) ⊆ D.

Definition

We say that φ1,...,φn agree (or agree globally) if there exists a (global) truth valuation γ ∈ ΦD such that, for all 1 ≤ i ≤ n, γ↓d(φi) = φi.

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

Local disagreement

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

Local disagreement

A necessary condition for global agreement is that each pair of information sources agree at their common variables.

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

Local disagreement

A necessary condition for global agreement is that each pair of information sources agree at their common variables. This property is captured by the notion of local agreement:

Definition

Let K = {φ1,...,φn} ⊆ Φ be a knowledgebase. We say that K agrees locally if φ

↓d(φi)∩d(φj) i

= φ

↓d(φi)∩d(φj) j

for all 1 ≤ i,j ≤ n.

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

Local disagreement

A necessary condition for global agreement is that each pair of information sources agree at their common variables. This property is captured by the notion of local agreement:

Definition

Let K = {φ1,...,φn} ⊆ Φ be a knowledgebase. We say that K agrees locally if φ

↓d(φi)∩d(φj) i

= φ

↓d(φi)∩d(φj) j

for all 1 ≤ i,j ≤ n. Clearly, agreement implies local agreement.

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

Valuations algebras and sheaf theory

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

Valuations algebras and sheaf theory

Remarkably, many properties of valuations algebras can be described by sheaf theory.

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

Valuations algebras and sheaf theory

Remarkably, many properties of valuations algebras can be described by sheaf theory. Let Φ be a valuation algebra on a set of variables V. Consider the powerset P(V) as a discrete topological space.

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

Valuations algebras and sheaf theory

Remarkably, many properties of valuations algebras can be described by sheaf theory. Let Φ be a valuation algebra on a set of variables V. Consider the powerset P(V) as a discrete topological space. We can describe Φ as a presheaf: Φ : P(V)op − → Set by letting Φ(S) := ΦS for all S ⊆ V and Φ(S ⊆ T) := ρT

S : ΦT −

→ ΦS :: φ − → φ ↓T.

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

Valuations algebras and sheaf theory

Remarkably, many properties of valuations algebras can be described by sheaf theory. Let Φ be a valuation algebra on a set of variables V. Consider the powerset P(V) as a discrete topological space. We can describe Φ as a presheaf: Φ : P(V)op − → Set by letting Φ(S) := ΦS for all S ⊆ V and Φ(S ⊆ T) := ρT

S : ΦT −

→ ΦS :: φ − → φ ↓T. Functoriality is guaranteed by axioms (A4) and (A6), indeed, for all S ⊆ V and for all φ ∈ ΦS, ρS

S(φ) = φ ↓S = φ ↓d(φ) (A6)

= φ,

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

Valuations algebras and sheaf theory

Remarkably, many properties of valuations algebras can be described by sheaf theory. Let Φ be a valuation algebra on a set of variables V. Consider the powerset P(V) as a discrete topological space. We can describe Φ as a presheaf: Φ : P(V)op − → Set by letting Φ(S) := ΦS for all S ⊆ V and Φ(S ⊆ T) := ρT

S : ΦT −

→ ΦS :: φ − → φ ↓T. Functoriality is guaranteed by axioms (A4) and (A6), indeed, for all S ⊆ V and for all φ ∈ ΦS, ρS

S(φ) = φ ↓S = φ ↓d(φ) (A6)

= φ, and, by (A4), for all S ⊆ T ⊆ U ⊆ V and φ ∈ ΦU, ρT

S ◦ρU T (φ) =

• φ ↓T↓S (A4)

= φ ↓S = ρU

S (φ).

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

Valuations algebras and sheaf theory

Remarkably, many properties of valuations algebras can be described by sheaf theory. Let Φ be a valuation algebra on a set of variables V. Consider the powerset P(V) as a discrete topological space. We can describe Φ as a presheaf: Φ : P(V)op − → Set by letting Φ(S) := ΦS for all S ⊆ V and Φ(S ⊆ T) := ρT

S : ΦT −

→ ΦS :: φ − → φ ↓T. Functoriality is guaranteed by axioms (A4) and (A6), indeed, for all S ⊆ V and for all φ ∈ ΦS, ρS

S(φ) = φ ↓S = φ ↓d(φ) (A6)

= φ, and, by (A4), for all S ⊆ T ⊆ U ⊆ V and φ ∈ ΦU, ρT

S ◦ρU T (φ) =

• φ ↓T↓S (A4)

= φ ↓S = ρU

S (φ).

In general, this description does not capture composition.

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

Disagreement and sheaf theory

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

Disagreement and sheaf theory

We can rephrase the definitions of local and global disagreement in sheaf theoretic terms:

Definition

Let U ⊆ P(V), with D := U .

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

Disagreement and sheaf theory

We can rephrase the definitions of local and global disagreement in sheaf theoretic terms:

Definition

Let U ⊆ P(V), with D := U . A set of local sections {sU ∈ Φ(U)}U∈U of Φ agrees locally if it is compatible.

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

Disagreement and sheaf theory

We can rephrase the definitions of local and global disagreement in sheaf theoretic terms:

Definition

Let U ⊆ P(V), with D := U . A set of local sections {sU ∈ Φ(U)}U∈U of Φ agrees locally if it is compatible. It agrees globally if there exists a global section γ ∈ Φ(D) such that γ |U= sU for all U ∈ U .

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

Contextuality and disagreement

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

Contextuality and disagreement

Recall that an empirical model on a scenario X,M ,(Om)m∈X is a compatible family e = {eC}C∈M for the presheaf DRE .

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

Contextuality and disagreement

Recall that an empirical model on a scenario X,M ,(Om)m∈X is a compatible family e = {eC}C∈M for the presheaf DRE . This can be seen as a locally agreeing knowledgebase of the valuation algebra

• f R-distributions.

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

Contextuality and disagreement

Recall that an empirical model on a scenario X,M ,(Om)m∈X is a compatible family e = {eC}C∈M for the presheaf DRE . This can be seen as a locally agreeing knowledgebase of the valuation algebra

• f R-distributions.

We say that e is non-contextual if there exists a global R-distribution g ∈ DRE (X) such that g |C= eC, for all C ∈ M .

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

Contextuality and disagreement

Recall that an empirical model on a scenario X,M ,(Om)m∈X is a compatible family e = {eC}C∈M for the presheaf DRE . This can be seen as a locally agreeing knowledgebase of the valuation algebra

• f R-distributions.

We say that e is non-contextual if there exists a global R-distribution g ∈ DRE (X) such that g |C= eC, for all C ∈ M . Therefore, contextuality simply arises as an instance of a locally agreeing knowledgebase that disagrees globally.

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

Contextuality and disagreement

Recall that an empirical model on a scenario X,M ,(Om)m∈X is a compatible family e = {eC}C∈M for the presheaf DRE . This can be seen as a locally agreeing knowledgebase of the valuation algebra

• f R-distributions.

We say that e is non-contextual if there exists a global R-distribution g ∈ DRE (X) such that g |C= eC, for all C ∈ M . Therefore, contextuality simply arises as an instance of a locally agreeing knowledgebase that disagrees globally. This is a very general concept, which has meaningful realisations in many fields captured by the valuation algebraic framework.

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

Local agreement vs global disagreement: Examples

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

Local agreement vs global disagreement: Examples

A first example in relational databases, taken from real sources. Breast cancer guidelines from 3 different medical associations:

◮ Screening with mammography annually, clinical breast exam annually or biannually ◮ Women aged 50 to 54 years should get mammograms. Women aged 55 years and

• lder should switch to clinical breast exams

◮ Women aged 50 to 54 years should undergo an exam every year. Women aged 55

years and older should be examined every 2 years.

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

Local agreement vs global disagreement: Examples

A first example in relational databases, taken from real sources. Breast cancer guidelines from 3 different medical associations:

◮ Screening with mammography annually, clinical breast exam annually or biannually ◮ Women aged 50 to 54 years should get mammograms. Women aged 55 years and

• lder should switch to clinical breast exams

◮ Women aged 50 to 54 years should undergo an exam every year. Women aged 55

years and older should be examined every 2 years.

Exam Frequency Age M CBE Y 2Y 54+ 54−

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

Local agreement vs global disagreement: Examples

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

Local agreement vs global disagreement: Examples

For an example in CSPs, consider a graph colorability problem:

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

Local agreement vs global disagreement: Examples

For an example in CSPs, consider a graph colorability problem: Consider the problem of coloring a political map with 3 colors.

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

Local agreement vs global disagreement: Examples

For an example in CSPs, consider a graph colorability problem: Consider the problem of coloring a political map with 3 colors. We focus on the geographical region surrounding Malawi:

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

Local agreement vs global disagreement: Examples

For an example in CSPs, consider a graph colorability problem: Consider the problem of coloring a political map with 3 colors. We focus on the geographical region surrounding Malawi:

Zambia Mozambique Tanzania Malawi Zimbabwe

MWI TZA ZMB MOZ ZWE

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

Local agreement vs global disagreement: Examples

For an example in CSPs, consider a graph colorability problem: Consider the problem of coloring a political map with 3 colors. We focus on the geographical region surrounding Malawi:

Zambia Mozambique Tanzania Malawi Zimbabwe

MWI TZA ZMB MOZ ZWE

One can easily show that it is impossible to color the map using only 3 colors.

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

Local agreement vs global disagreement: Examples

For an example in CSPs, consider a graph colorability problem: Consider the problem of coloring a political map with 3 colors. We focus on the geographical region surrounding Malawi:

Zambia Mozambique Tanzania Malawi Zimbabwe

MWI TZA ZMB MOZ ZWE

One can easily show that it is impossible to color the map using only 3 colors. This can be seen as an instance of local agreement (LA) vs global disagreement (GD) both for the algebra of CSP-information sets, and the algebra of CSP-formulae.

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

Local agreement vs global disagreement: Examples

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

Local agreement vs global disagreement: Examples

For an example from logic, consider the liar’s cycle

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

Local agreement vs global disagreement: Examples

For an example from logic, consider the liar’s cycle S1 : S2 is true, S2 : S3 is true, . . . Sn−1 : Sn is true, Sn : S1 is false.

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

Local agreement vs global disagreement: Examples

For an example from logic, consider the liar’s cycle S1 : S2 is true, S2 : S3 is true, . . . Sn−1 : Sn is true, Sn : S1 is false. Also in this case, this is an instance of LA vs GD both for the algebra of propositional information sets, and the algebra of propositional formulae.

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

Local agreement vs global disagreement: Examples

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

Local agreement vs global disagreement: Examples

Finally, an example concerning linear equations.

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

Local agreement vs global disagreement: Examples

Finally, an example concerning linear equations. Consider the following system

• f equations in Z2:

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

Local agreement vs global disagreement: Examples

Finally, an example concerning linear equations. Consider the following system

• f equations in Z2:

e1 := (x1 ⊕x2 ⊕x3 = 1) e2 := (x1 ⊕y2 ⊕y3 = 0) e3 := (y1 ⊕x2 ⊕y3 = 0) e4 := (y1 ⊕y2 ⊕x3 = 0)

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

Local agreement vs global disagreement: Examples

Finally, an example concerning linear equations. Consider the following system

• f equations in Z2:

e1 := (x1 ⊕x2 ⊕x3 = 1) e2 := (x1 ⊕y2 ⊕y3 = 0) e3 := (y1 ⊕x2 ⊕y3 = 0) e4 := (y1 ⊕y2 ⊕x3 = 0) The equations are locally consistent (i.e. every pair of equations admit solutions for their common variables), yet if we sum them all we obtain 0 = 1, which means that there is no global solution, i.e. the knowledgebase {e1,2,3,4} disagrees globally.

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

Local agreement vs global disagreement: Examples

Finally, an example concerning linear equations. Consider the following system

• f equations in Z2:

e1 := (x1 ⊕x2 ⊕x3 = 1) e2 := (x1 ⊕y2 ⊕y3 = 0) e3 := (y1 ⊕x2 ⊕y3 = 0) e4 := (y1 ⊕y2 ⊕x3 = 0) The equations are locally consistent (i.e. every pair of equations admit solutions for their common variables), yet if we sum them all we obtain 0 = 1, which means that there is no global solution, i.e. the knowledgebase {e1,2,3,4} disagrees globally. These equations are exactly those used by Mermin to prove strong contextuality

• f the GHZ model.

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

A (new!) dictionary

Valuation algebras Empirical models variables measurements frame Ωx

• utcome set Ox

knowledgebase domains measurement scenario domain of valuation context tuple event local agreement no-signalling locally-agreeing knowledgebase empirical model projection restriction (marginalisation) combination glueing truth valuation global section disagreement contextuality

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

Translating results...

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

Translating results...

The valuation algebraic definition of disagreement allows us to translate definitions, methods, results and algorithms from one field to the other.

Example

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

Translating results...

The valuation algebraic definition of disagreement allows us to translate definitions, methods, results and algorithms from one field to the other.

Example

A key result in contextuality is the characterisation of all scenarios that do not admit contextual behavior. The following result was proven by Barbosa, via an adaptation of Vorob’ev’s theorem:

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

Translating results...

The valuation algebraic definition of disagreement allows us to translate definitions, methods, results and algorithms from one field to the other.

Example

A key result in contextuality is the characterisation of all scenarios that do not admit contextual behavior. The following result was proven by Barbosa, via an adaptation of Vorob’ev’s theorem: Every empirical model on a scenario X,M ,O is non-contextual iff the simplicial complex described by M is acyclic.

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

Translating results...

The valuation algebraic definition of disagreement allows us to translate definitions, methods, results and algorithms from one field to the other.

Example

A key result in contextuality is the characterisation of all scenarios that do not admit contextual behavior. The following result was proven by Barbosa, via an adaptation of Vorob’ev’s theorem: Every empirical model on a scenario X,M ,O is non-contextual iff the simplicial complex described by M is acyclic. This theorem can be generalised to the level of valuation algebras:

Theorem

Every locally agreeing knowledgebase {φ1,...,φn} on a set of domains D := {d1,...,dn} agrees globally iff the simplicial complex described by D is acyclic.

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

Translating results...

Theorem

Every locally agreeing knowledgebase {φ1,...,φn} on a set of domains D := {d1,...,dn} agrees globally iff the simplicial complex described by D is acyclic.

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

Translating results...

Theorem

Every locally agreeing knowledgebase {φ1,...,φn} on a set of domains D := {d1,...,dn} agrees globally iff the simplicial complex described by D is acyclic. This theorem then specialises to results for each specific valuation algebra:

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

Translating results...

Theorem

Every locally agreeing knowledgebase {φ1,...,φn} on a set of domains D := {d1,...,dn} agrees globally iff the simplicial complex described by D is acyclic. This theorem then specialises to results for each specific valuation algebra:

◮ Probability distributions: Vorob’ev’s theorem. Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Translating results...

Theorem

Every locally agreeing knowledgebase {φ1,...,φn} on a set of domains D := {d1,...,dn} agrees globally iff the simplicial complex described by D is acyclic. This theorem then specialises to results for each specific valuation algebra:

◮ Probability distributions: Vorob’ev’s theorem. ◮ Relational databases: Every database instance on a database schema admits a

global relation iff the database schema is acyclic

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

Translating results...

Theorem

Every locally agreeing knowledgebase {φ1,...,φn} on a set of domains D := {d1,...,dn} agrees globally iff the simplicial complex described by D is acyclic. This theorem then specialises to results for each specific valuation algebra:

◮ Probability distributions: Vorob’ev’s theorem. ◮ Relational databases: Every database instance on a database schema admits a

global relation iff the database schema is acyclic

◮ CSPs – graph colorability: Every tree is k-colorable, for any k ≥ 2. Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Translating results...

Theorem

Every locally agreeing knowledgebase {φ1,...,φn} on a set of domains D := {d1,...,dn} agrees globally iff the simplicial complex described by D is acyclic. This theorem then specialises to results for each specific valuation algebra:

◮ Probability distributions: Vorob’ev’s theorem. ◮ Relational databases: Every database instance on a database schema admits a

global relation iff the database schema is acyclic

◮ CSPs – graph colorability: Every tree is k-colorable, for any k ≥ 2. ◮ ... Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Translating results...

Theorem

Every locally agreeing knowledgebase {φ1,...,φn} on a set of domains D := {d1,...,dn} agrees globally iff the simplicial complex described by D is acyclic. This theorem then specialises to results for each specific valuation algebra:

◮ Probability distributions: Vorob’ev’s theorem. ◮ Relational databases: Every database instance on a database schema admits a

global relation iff the database schema is acyclic

◮ CSPs – graph colorability: Every tree is k-colorable, for any k ≥ 2. ◮ ...

More generally speaking, we would like to apply the wide range of methods and algorithms of the valuation algebraic framework to study contextuality.

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

Inference problems

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

Inference problems

The classic problem of extracting relevant knowledge about a given query out of a certain set of information sources can be effectively modelled in the valuation algebraic framework:

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

Inference problems

The classic problem of extracting relevant knowledge about a given query out of a certain set of information sources can be effectively modelled in the valuation algebraic framework:

Definition

Given a valuation algebra Φ, a knowledgebase {φ1,...,φn} ⊆ Φ, and a set of domains x = {x1,...,xk}, with xi ⊆ d(φ1 ⊗···⊗φn), we call an inference problem the task of computing (φ1 ⊗···⊗φn)↓xi. The valuation φ = (φ1 ⊗···⊗φn) is called joint valuation or objective function, while each domain xi is called a query.

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

Inference problems

The classic problem of extracting relevant knowledge about a given query out of a certain set of information sources can be effectively modelled in the valuation algebraic framework:

Definition

Given a valuation algebra Φ, a knowledgebase {φ1,...,φn} ⊆ Φ, and a set of domains x = {x1,...,xk}, with xi ⊆ d(φ1 ⊗···⊗φn), we call an inference problem the task of computing (φ1 ⊗···⊗φn)↓xi. The valuation φ = (φ1 ⊗···⊗φn) is called joint valuation or objective function, while each domain xi is called a query. There is a vast class of algorithms designed to solve inference problems efficiently.

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

Inference problems

The classic problem of extracting relevant knowledge about a given query out of a certain set of information sources can be effectively modelled in the valuation algebraic framework:

Definition

Given a valuation algebra Φ, a knowledgebase {φ1,...,φn} ⊆ Φ, and a set of domains x = {x1,...,xk}, with xi ⊆ d(φ1 ⊗···⊗φn), we call an inference problem the task of computing (φ1 ⊗···⊗φn)↓xi. The valuation φ = (φ1 ⊗···⊗φn) is called joint valuation or objective function, while each domain xi is called a query. There is a vast class of algorithms designed to solve inference problems efficiently. Can we turn the problem of detecting disagreement in a inference problem?

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

Ordered valuation algebras

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

Ordered valuation algebras

Given a valuation algebra Φ on a set of variables V, and a valuation φ ∈ ΦS for some S ⊆ V, one could ask whether it is possible to quantify the amount of information carried by φ and compare it to other valuations of ΦS.

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

Ordered valuation algebras

Given a valuation algebra Φ on a set of variables V, and a valuation φ ∈ ΦS for some S ⊆ V, one could ask whether it is possible to quantify the amount of information carried by φ and compare it to other valuations of ΦS. This idea is captured by the notion of ordered valuation algebras

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

Ordered valuation algebras

Given a valuation algebra Φ on a set of variables V, and a valuation φ ∈ ΦS for some S ⊆ V, one could ask whether it is possible to quantify the amount of information carried by φ and compare it to other valuations of ΦS. This idea is captured by the notion of ordered valuation algebras

Definition

Let Φ be a valuation algebra on V. We say that Φ is ordered if there exists a partial

• rder on Φ such that:

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

Ordered valuation algebras

Given a valuation algebra Φ on a set of variables V, and a valuation φ ∈ ΦS for some S ⊆ V, one could ask whether it is possible to quantify the amount of information carried by φ and compare it to other valuations of ΦS. This idea is captured by the notion of ordered valuation algebras

Definition

Let Φ be a valuation algebra on V. We say that Φ is ordered if there exists a partial

• rder on Φ such that:

(A10) Partial order: For all φ,ψ ∈ Φ, φ ψ implies d(φ) = d(ψ). Moreover, for every S ⊆ V and Ψ ⊆ ΦS, the infimum inf(Ψ) exists.

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

Ordered valuation algebras

Given a valuation algebra Φ on a set of variables V, and a valuation φ ∈ ΦS for some S ⊆ V, one could ask whether it is possible to quantify the amount of information carried by φ and compare it to other valuations of ΦS. This idea is captured by the notion of ordered valuation algebras

Definition

Let Φ be a valuation algebra on V. We say that Φ is ordered if there exists a partial

• rder on Φ such that:

(A10) Partial order: For all φ,ψ ∈ Φ, φ ψ implies d(φ) = d(ψ). Moreover, for every S ⊆ V and Ψ ⊆ ΦS, the infimum inf(Ψ) exists. (A11) Null element: For all S ⊆ V, we have inf(ΦS) = zS.

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

Ordered valuation algebras

Given a valuation algebra Φ on a set of variables V, and a valuation φ ∈ ΦS for some S ⊆ V, one could ask whether it is possible to quantify the amount of information carried by φ and compare it to other valuations of ΦS. This idea is captured by the notion of ordered valuation algebras

Definition

Let Φ be a valuation algebra on V. We say that Φ is ordered if there exists a partial

• rder on Φ such that:

(A10) Partial order: For all φ,ψ ∈ Φ, φ ψ implies d(φ) = d(ψ). Moreover, for every S ⊆ V and Ψ ⊆ ΦS, the infimum inf(Ψ) exists. (A11) Null element: For all S ⊆ V, we have inf(ΦS) = zS. (A12) Monotonicity of combination: For all φ1,φ2,ψ1,ψ2 ∈ Φ such that φ1 φ2 and ψ1 ψ2 we have φ1 ⊗ψ1 φ2 ⊗ψ2.

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

Ordered valuation algebras

Given a valuation algebra Φ on a set of variables V, and a valuation φ ∈ ΦS for some S ⊆ V, one could ask whether it is possible to quantify the amount of information carried by φ and compare it to other valuations of ΦS. This idea is captured by the notion of ordered valuation algebras

Definition

Let Φ be a valuation algebra on V. We say that Φ is ordered if there exists a partial

• rder on Φ such that:

(A10) Partial order: For all φ,ψ ∈ Φ, φ ψ implies d(φ) = d(ψ). Moreover, for every S ⊆ V and Ψ ⊆ ΦS, the infimum inf(Ψ) exists. (A11) Null element: For all S ⊆ V, we have inf(ΦS) = zS. (A12) Monotonicity of combination: For all φ1,φ2,ψ1,ψ2 ∈ Φ such that φ1 φ2 and ψ1 ψ2 we have φ1 ⊗ψ1 φ2 ⊗ψ2. (A13) Monotonicity of projection: For all φ,ψ ∈ Φ, if φ ψ then

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

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

Most valuations algebras have an order structure: relational databases, propositional information, algebra of sentences, any algebra related to general notions of language and models, etc. (not probability distributions!)

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

Most valuations algebras have an order structure: relational databases, propositional information, algebra of sentences, any algebra related to general notions of language and models, etc. (not probability distributions!) All of these algebras have a key common property.

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

Most valuations algebras have an order structure: relational databases, propositional information, algebra of sentences, any algebra related to general notions of language and models, etc. (not probability distributions!) All of these algebras have a key common property. Their composition operation is described by the same categorical construction:

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

Most valuations algebras have an order structure: relational databases, propositional information, algebra of sentences, any algebra related to general notions of language and models, etc. (not probability distributions!) All of these algebras have a key common property. Their composition operation is described by the same categorical construction: Φ(S) Φ(S)×Φ(T) Φ(T) Φ(S∪T)

π1 π2 ρS∪T

S

ρS∪T

T ρS∪T S ,ρS∪T T

• Proposition

Let Φ be an ordered algebra in the list above. The composition law ⊗ : Φ(S)×Φ(T) → Φ(S∪T) is uniquely characterised as the right adjoint of ρS∪T

S

,ρS∪T

T

. In other words, it is the unique map such that idΦ(S∪T) ≤ ⊗◦ρS∪T

S

,ρS∪T

T

, ρS∪T

S

,ρS∪T

T

• ⊗ ≤ idΦ(S)×Φ(T),

where ≤ is the pointwise order inherited by the partial order .

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

Disagreement as an inference problem

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

Disagreement as an inference problem

We call algebras with this structure lossy. They have the following key property:

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

Disagreement as an inference problem

We call algebras with this structure lossy. They have the following key property:

Proposition

Let Φ be a lossy valuation algebra on a set of variables V. Let K = {φ1,...,φn} ⊆ Φ be a knowledgebase. Let γ =

n

• i=1

φi. (1) Then φ1,...,φn agree globally if and only if γ↓d(φi) = φi. In this case, γ is the most informative of all the possible truth valuations.

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

Disagreement as an inference problem

We call algebras with this structure lossy. They have the following key property:

Proposition

Let Φ be a lossy valuation algebra on a set of variables V. Let K = {φ1,...,φn} ⊆ Φ be a knowledgebase. Let γ =

n

• i=1

φi. (1) Then φ1,...,φn agree globally if and only if γ↓d(φi) = φi. In this case, γ is the most informative of all the possible truth valuations. In other words, a truth valuation can only be obtained by combining all of the valuations in a knowledgebase.

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

Disagreement as an inference problem

We call algebras with this structure lossy. They have the following key property:

Proposition

Let Φ be a lossy valuation algebra on a set of variables V. Let K = {φ1,...,φn} ⊆ Φ be a knowledgebase. Let γ =

n

• i=1

φi. (1) Then φ1,...,φn agree globally if and only if γ↓d(φi) = φi. In this case, γ is the most informative of all the possible truth valuations. In other words, a truth valuation can only be obtained by combining all of the valuations in a knowledgebase. Consequently, in order to determine whether a knowledgebase {φ1,...φn} disagrees globally, all we have to do is to solve the inference problem (φ1 ⊗···⊗φn)↓d(φi) for all 1 ≤ i ≤ n.

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

Strong disagreement

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

Strong disagreement

Let Φ be a lossy information algebra.

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

Strong disagreement

Let Φ be a lossy information algebra. By the previous proposition, we know that γ =

n

• i=1

φi is the most informative of all the possible candidate truth valuations.

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

Strong disagreement

Let Φ be a lossy information algebra. By the previous proposition, we know that γ =

n

• i=1

φi is the most informative of all the possible candidate truth valuations. Even when the knowledgebase disagrees, γ represents the portion of the information on which the sources do agree.

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

Strong disagreement

Let Φ be a lossy information algebra. By the previous proposition, we know that γ =

n

• i=1

φi is the most informative of all the possible candidate truth valuations. Even when the knowledgebase disagrees, γ represents the portion of the information on which the sources do agree. In extreme cases, it can happen that the information sources disagree completely.

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

Strong disagreement

Let Φ be a lossy information algebra. By the previous proposition, we know that γ =

n

• i=1

φi is the most informative of all the possible candidate truth valuations. Even when the knowledgebase disagrees, γ represents the portion of the information on which the sources do agree. In extreme cases, it can happen that the information sources disagree completely.In this case, the truth valuation is the least informative valuation, i.e. the null element of the information algebra.

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

Strong disagreement

Let Φ be a lossy information algebra. By the previous proposition, we know that γ =

n

• i=1

φi is the most informative of all the possible candidate truth valuations. Even when the knowledgebase disagrees, γ represents the portion of the information on which the sources do agree. In extreme cases, it can happen that the information sources disagree completely.In this case, the truth valuation is the least informative valuation, i.e. the null element of the information algebra.

Definition

We say that a knowledgebase {φ1,...,φn}, with D := n

i=1 d(φi), disagrees strongly

if γ = zD.

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

Strong disagreement

Let Φ be a lossy information algebra. By the previous proposition, we know that γ =

n

• i=1

φi is the most informative of all the possible candidate truth valuations. Even when the knowledgebase disagrees, γ represents the portion of the information on which the sources do agree. In extreme cases, it can happen that the information sources disagree completely.In this case, the truth valuation is the least informative valuation, i.e. the null element of the information algebra.

Definition

We say that a knowledgebase {φ1,...,φn}, with D := n

i=1 d(φi), disagrees strongly

if γ = zD. Strong contextuality is an instance of strong disagreement, where the information algebra in question is the one of boolean distributions.

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

Algorithms:

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

Algorithms:

The valuation algebraic framework provides a wide range of algorithms to solve inference problems,

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

Algorithms:

The valuation algebraic framework provides a wide range of algorithms to solve inference problems, In particular, the paradigm of local computation has proved particularly useful to efficiently solve single query inference problems:

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

Algorithms:

The valuation algebraic framework provides a wide range of algorithms to solve inference problems, In particular, the paradigm of local computation has proved particularly useful to efficiently solve single query inference problems:

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

Algorithms:

The valuation algebraic framework provides a wide range of algorithms to solve inference problems, In particular, the paradigm of local computation has proved particularly useful to efficiently solve single query inference problems:

◮ Fusion algorithm ◮ Bucket elimination algorithm Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Algorithms:

The valuation algebraic framework provides a wide range of algorithms to solve inference problems, In particular, the paradigm of local computation has proved particularly useful to efficiently solve single query inference problems:

◮ Fusion algorithm ◮ Bucket elimination algorithm ◮ Collect algorithm ◮ ... Samson Abramsky & Giovanni Car` u (Oxford CS) Contextuality and valuation algebras Winer Memorial Lectures 2018

Algorithms:

The valuation algebraic framework provides a wide range of algorithms to solve inference problems, In particular, the paradigm of local computation has proved particularly useful to efficiently solve single query inference problems:

◮ Fusion algorithm ◮ Bucket elimination algorithm ◮ Collect algorithm ◮ ...

One can use these efficient algorithm to detect contextuality in measurement scenarios.

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

Algorithms:

The valuation algebraic framework provides a wide range of algorithms to solve inference problems, In particular, the paradigm of local computation has proved particularly useful to efficiently solve single query inference problems:

◮ Fusion algorithm ◮ Bucket elimination algorithm ◮ Collect algorithm ◮ ...

One can use these efficient algorithm to detect contextuality in measurement scenarios. In particular, faster algorithms for non-locality detection can be implemented in specific scenarios.

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