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Evan Patterson
Stanford University, Statistics Department
6th CSLI Workshop on Logic, Rationality, & Intelligent Interaction
June 3, 2017
"An ontology is a specification of a conceptualization."
—Thomas Gruber, co-founder of Siri, Inc.
New applications motivating KR research:
Example 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))
Rel as a monoidal category, via the graphical language of string diagrams
A relation R:X→Y is a subset of X×Y:
Composition: Given R:X→Y and S:Y→Z,
:={(x,z):∃y∈Y.xRy∧ySz}
Cartesian product: Given R:X→Y and S:Z→W,
:={((x,z),(y,w)):xRy∧zSw}
Dagger: Given R:X→Y,
:={(y,x):yRx}
Diagonals: For every set X,
:={(x,(x′,x″)):x=x′∧x=x″}
:={(x,∗):x∈X}
Intersection of R,S:X→Y:
R⊓S=
Limited existential quantification of R:X→Y:
∃R.⊤=
Rel is a locally posetal 2-category: given relations R,S:X→Y,
R⟹SiffR⊆S.
2-morphisms correspond to subsumption in description logic.
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 subject to several axioms.
Axioms ensure that
A relational olog is a finitely presented bicategory of relations.
"Finitely presented" means generated by
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Theorem. For every (small) bicategory of relations B, there is an equivalence of categories Cl(Lang(B))≃BinBiRel.