Description logic (DL)¶

• Dominant formalism for KR today
• Basis for Web Ontology Language (OWL)
• Computationally tractable subset of first-order logic
• Basic entities:
  • concepts (classes)
  • roles (relations)
  • subsumptions (containments) of concepts/roles

DL examples¶

Syntax and semantics of a few concept constructors:

C⊓DFOL⇝(C⊓D)(x) ↔ C(x)∧D(x)∀R.CFOL⇝(∀R.C)(x) ↔ ∀y.R(x,y)∧C(y)∃R.⊤FOL⇝(∃R.⊤)(x) ↔ ∃y.R(x,y)

Syntax and semantics of concept subsumption:

C⊑DFOL⇝∀x.C(x)→D(x)

Ontology logs (ologs)¶

• Spivak and Kent (2012) introduce ontology logs
• Ologs are a categorical framework for KR
• A (functional) olog C is just a finitely presented category:
  • Objects are types in the subject-matter domain
  • Morphisms are aspects or properties
  • Commutative diagrams encode facts about the domain
• Instance data for an olog is a functor C→Set
• Further expressivity is achieved via
  • Limits aka layouts
  • Colimits aka grouping

Example olog¶

Advantages of relational ologs¶

In contrast to DL, relational ologs provide

1. An explicit type system, via objects
2. A flexible notion of instance data that cleanly separates universal and particular knowledge, via functors
3. An intuitive graphical syntax, via string diagrams

All these features emerge automatically from the categorical framework.

Rel as a monoidal category¶

Dagger: Given a relation R:X→Y,

:={(y,x):yRx}

Diagonals: For every set X,

:={(x,(x′,x″)):x=x′∧x=x″}
:={(x,∗):x∈X}

Together they define a family of commutative special †-Frobenius monoids.

Rel as a monoidal category¶

Rel is also a dagger compact category with units and counits defined by

:= ={(∗,(x,x′)):x=x′}
:= ={((x,x′),∗):x=x′}

Rel as a 2-category¶

Rel is a locally posetal 2-category: given relations R,S:X→Y,

R⟹SiffR⊆S.

2-morphisms correspond to subsumptions in DL.

Examples from description logic¶

Intersection of R,S:X→Y:

R⊓S=

Limited existential quantification of R:X→Y:

∃R.⊤=

Bicategories of relations¶

A bicategory of relations is a locally posetal 2-category B that is also a symmetric monoidal category (B,⊗,I) with diagonals (X,ΔX,◊X)X∈B, such that

• every morphism R:X→Y is a lax comonoid homomorphism:

⟹

⟹

• ΔX and ◊X have right adjoints, ∇X:=Δ∗X and ◻X:=◊∗X
• the Frobenius equation holds, making (X,ΔX,◊X,∇X,◻X) into a Frobenius monoid

Relational ologs¶

A relational olog is a finitely presented bicategory of relations.

"Finitely presented" means generated by

• a finite set of basic types or object generators
• a finite set of basic relations or morphism generators
• a finite set of subsumbtion axioms or 2-morphisms generators

Instance data¶

Instance data for a relational olog B is a functor B→D in BiRel, where the data category D is, e.g.,

• Rel or FinRel (the "default")

R=XY122133

• Mat(B), the category of boolean matrices

R=(010100001)

• VectRelk, the category of linear relations

Classifying category¶

Every regular theory T has a classifying category Cl(T), a bicategory of relations with

• objects = (equivalence classes of) contexts [x:A]=[x1:A1,…,xn:An]
• morphisms = (equivalence classes of) formulas in context [x:A;y:B | φ]

Internal language¶

Every (small) bicategory of relations B has an internal language Lang(B), a regular theory with

• types = objects of B
• relation symbols = morphisms of B
• theorems = 2-morphisms (subsumptions) of B

Theorem¶

For every (small) bicategory of relations B, there is an equivalence of categories Cl(Lang(B))≃BinBiRel.

Thanks!¶

Paper: E. Patterson, "Knowledge Representation in Bicategories of Relations", 2017 [arXiv]

Background reading:

• Ologs: D.I. Spivak & R.E. Kent, "Ologs: A Categorical Framework for Knowledge Representation", 2011 [arXiv, DOI]
• Description logic: M. Krötzsch, F. Simancik, I. Horrocks, "A Description Logic Primer", 2012 [arXiv]
• Rel: B. Coecke & E.O. Paquette, "Categories for the Practicising Physicist", 2010 [arXiv]
• Bicategories of relations: A. Carboni & R.F.C. Walters, "Cartesian Bicategories I", 1987