Graphical Conjunctive Queries A completeness theorem for Cartesian - PowerPoint PPT Presentation

Graphical Conjunctive Queries A completeness theorem for Cartesian bicategories Filippo Bonchi, Jens Seeber , Pawe� l Soboci´ nski IMT School for Advanced Studies Lucca Birmingham - 21 st September, 2018

Contents 1 Cartesian bicategories 2 Conjunctive queries 3 Completeness 4 Summary

String diagrams • A graphical way of reasoning about monoidal categories

String diagrams • A graphical way of reasoning about monoidal categories • 2-dimensional diagrams manipulated according to algebraic rules – hot research topic

String diagrams • A graphical way of reasoning about monoidal categories • 2-dimensional diagrams manipulated according to algebraic rules – hot research topic • ZX calculus (Coecke, Duncan)

String diagrams • A graphical way of reasoning about monoidal categories • 2-dimensional diagrams manipulated according to algebraic rules – hot research topic • ZX calculus (Coecke, Duncan) • Signal flow graphs (Bonchi, Sobocinski, Zanasi)

String diagrams • A graphical way of reasoning about monoidal categories • 2-dimensional diagrams manipulated according to algebraic rules – hot research topic • ZX calculus (Coecke, Duncan) • Signal flow graphs (Bonchi, Sobocinski, Zanasi) • Monoidal computer (Pavlovic)

String diagrams • A graphical way of reasoning about monoidal categories • 2-dimensional diagrams manipulated according to algebraic rules – hot research topic • ZX calculus (Coecke, Duncan) • Signal flow graphs (Bonchi, Sobocinski, Zanasi) • Monoidal computer (Pavlovic) • . . .

Relations with string diagrams The category Rel of sets with relations as morphisms

Relations with string diagrams The category Rel of sets with relations as morphisms • forms a symmetric monoidal category:

Relations with string diagrams The category Rel of sets with relations as morphisms • forms a symmetric monoidal category: R 1 ⊗ R 2 = { (( a, b ) , ( c, d )) | ( a, c ) ∈ R 1 , ( b, d ) ∈ R 2 } a c R 1 b d R 2

Relations with string diagrams The category Rel of sets with relations as morphisms • forms a symmetric monoidal category: R 1 ⊗ R 2 = { (( a, b ) , ( c, d )) | ( a, c ) ∈ R 1 , ( b, d ) ∈ R 2 } a c R 1 b d R 2 • Composition: R 1 ; R 2 = { ( x, z ) | ∃ y : ( x, y ) ∈ R 1 , ( y, z ) ∈ R 2 } y x z R 1 R 2

Relations with string diagrams • Relations are ordered by inclusion

Relations with string diagrams • Relations are ordered by inclusion • Every object:

Relations with string diagrams • Relations are ordered by inclusion • Every object: • Copying and discarding

Relations with string diagrams • Relations are ordered by inclusion • Every object: • Copying and discarding ,

Relations with string diagrams • Relations are ordered by inclusion • Every object: • Copying and discarding , • Equality and “spawn”

Relations with string diagrams • Relations are ordered by inclusion • Every object: • Copying and discarding , • Equality and “spawn” ,

Observations = = = = Comonoid

Observations = = = = Comonoid = = = = Monoid

Observations = = = = Comonoid = = = = Monoid = = Frobenius

Observations = = ≤ = = ≤ Comonoid ≤ ≤ = Adjointness = = = Monoid = = Frobenius

Observations = = ≤ = = ≤ Comonoid ≤ ≤ = Adjointness = R ≤ R = = R ≤ Monoid R Lax Comonoid homomorphism = = Frobenius

Cartesian bicategories Definition (Carboni & Walters) A Cartesian bicategory

Cartesian bicategories Definition (Carboni & Walters) A Cartesian bicategory is a locally ordered

Cartesian bicategories Definition (Carboni & Walters) A Cartesian bicategory is a locally ordered symmetric monoidal category

Cartesian bicategories Definition (Carboni & Walters) A Cartesian bicategory is a locally ordered symmetric monoidal category where every object is equipped with

Cartesian bicategories Definition (Carboni & Walters) A Cartesian bicategory is a locally ordered symmetric monoidal category where every object is equipped with • a comonoid

Cartesian bicategories Definition (Carboni & Walters) A Cartesian bicategory is a locally ordered symmetric monoidal category where every object is equipped with • a comonoid • a monoid

Cartesian bicategories Definition (Carboni & Walters) A Cartesian bicategory is a locally ordered symmetric monoidal category where every object is equipped with • a comonoid • a monoid satisfying coherence

Cartesian bicategories Definition (Carboni & Walters) A Cartesian bicategory is a locally ordered symmetric monoidal category where every object is equipped with • a comonoid • a monoid satisfying coherence and the laws on the last slide.

Cartesian bicategories Definition (Carboni & Walters) A Cartesian bicategory is a locally ordered symmetric monoidal category where every object is equipped with • a comonoid • a monoid satisfying coherence and the laws on the last slide. A morphism

Cartesian bicategories Definition (Carboni & Walters) A Cartesian bicategory is a locally ordered symmetric monoidal category where every object is equipped with • a comonoid • a monoid satisfying coherence and the laws on the last slide. A morphism is a monoidal functor

Cartesian bicategories Definition (Carboni & Walters) A Cartesian bicategory is a locally ordered symmetric monoidal category where every object is equipped with • a comonoid • a monoid satisfying coherence and the laws on the last slide. A morphism is a monoidal functor preserving the ordering, the comonoid and the monoid.

Cartesian bicategories Definition (Carboni & Walters) A Cartesian bicategory is a locally ordered symmetric monoidal category where every object is equipped with • a comonoid • a monoid satisfying coherence and the laws on the last slide. A morphism is a monoidal functor preserving the ordering, the comonoid and the monoid. Idea: Do categorical logic with Cartesian bicategories.

Categorical logic with Cartesian bicategories Definition A model of B (in Rel ) is a morphism M : B → Rel

Categorical logic with Cartesian bicategories Definition A model of B (in Rel ) is a morphism M : B → Rel Problem (Completeness) For morphisms x, y in B such that M ( x ) ⊆ M ( y ) for all models M , is x ≤ y ?

Categorical logic with Cartesian bicategories Definition A model of B (in Rel ) is a morphism M : B → Rel Problem (Completeness) For morphisms x, y in B such that M ( x ) ⊆ M ( y ) for all models M , is x ≤ y ? Not to be confused with “functional completeness”!

The syntactic Cartesian bicategory Signature Σ

The syntactic Cartesian bicategory Signature Σ, each R ∈ Σ equipped with arity and coarity R : n → m .

The syntactic Cartesian bicategory Signature Σ, each R ∈ Σ equipped with arity and coarity R : n → m . Freely generated (syntactic) Cartesian bicategory CB Σ has objects N and morphisms

The syntactic Cartesian bicategory Signature Σ, each R ∈ Σ equipped with arity and coarity R : n → m . Freely generated (syntactic) Cartesian bicategory CB Σ has objects N and morphisms . . . . S 1 � � � . . � . . . Mor( CB Σ ) ::= ǫ . . . � � � � S 1 S 2 . . . . . � � � � . . S 2 . .

The syntactic Cartesian bicategory Signature Σ, each R ∈ Σ equipped with arity and coarity R : n → m . Freely generated (syntactic) Cartesian bicategory CB Σ has objects N and morphisms . . . . S 1 � � � . . � � . . . Mor( CB Σ ) ::= ǫ . . . � � � � � S 1 S 2 . . . . . � � � � � . . S 2 . . � � � � . . . . � � � � R . . � � � �

The syntactic Cartesian bicategory Signature Σ, each R ∈ Σ equipped with arity and coarity R : n → m . Freely generated (syntactic) Cartesian bicategory CB Σ has objects N and morphisms . . . . S 1 � � � . . � � . . . Mor( CB Σ ) ::= ǫ . . . � � � � � S 1 S 2 . . . . . � � � � � . . S 2 . . � � � � . . . . � � � � R . . � � � � modulo the laws of Cartesian bicategories.

Cartesian bicategories and logic CB Σ can emulate regular logic.

Cartesian bicategories and logic CB Σ can emulate regular logic. Example ∃ z 0 , z 1 : R ( x 0 , z 0 ) ∧ R ( x 1 , z 0 ) ∧ R ( x 0 , z 1 ) ∧ R ( x 1 , z 1 ) ,

Cartesian bicategories and logic CB Σ can emulate regular logic. Example ∃ z 0 , z 1 : R ( x 0 , z 0 ) ∧ R ( x 1 , z 0 ) ∧ R ( x 0 , z 1 ) ∧ R ( x 1 , z 1 ) , R R R R

Cartesian bicategories and logic CB Σ can emulate regular logic. Example ∃ z 0 , z 1 : R ( x 0 , z 0 ) ∧ R ( x 1 , z 0 ) ∧ R ( x 0 , z 1 ) ∧ R ( x 1 , z 1 ) , R R R R One-to-one correspondence between string diagrams and regular logic.

Contents 1 Cartesian bicategories 2 Conjunctive queries 3 Completeness 4 Summary

Conjunctive queries • Conjunctive queries: logical formulas made of ∃ , ∧ , ⊤ , = and symbols from the signature Σ.