$\quad$ $\quad\implies\quad$
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: $$ \begin{align} C \sqcap D \qquad&\stackrel{\mathrm{FOL}}{\leadsto}\qquad (C \sqcap D)(x) \leftrightarrow (C(x) \wedge D(x)) \\ \forall R.C \qquad&\stackrel{\mathrm{FOL}}{\leadsto}\qquad (\forall R.C)(x) \leftrightarrow (\forall y. R(x,y) \wedge C(y)) \\ \exists R.\top \qquad&\stackrel{\mathrm{FOL}}{\leadsto}\qquad (\exists R.\top)(x) \leftrightarrow (\exists y. R(x,y)) \end{align} $$
Rel as a monoidal category, via the graphical language of string diagrams
A relation $R: X \to Y$ is a subset of $X \times Y$:
$\quad$
Composition: Given $R: X \to Y$ and $S: Y \to Z$,
$\quad$ $\quad:=\quad \{(x,z): \exists y \in Y. xRy \wedge ySz\}$
Cartesian product: Given $R: X \to Y$ and $S: Z \to W$,
$\quad$ $\quad:=\quad \{((x,z),(y,w)): xRy \wedge zSw\}$
Dagger: Given $R: X \to Y$,
$\quad$ $\quad:=\quad \{(y,x): yRx\}$
Diagonals: For every set $X$,
$\quad$ $\quad:=\quad \{(x,(x',x'')): x=x' \wedge x=x''\}$
$\quad$ $\quad:=\quad \{(x,*): x \in X\}$
Intersection of $R,S: X \to Y$:
$\quad R \sqcap S \quad=\quad$
Limited existential quantification of $R:X \to Y$:
$\quad \exists R.\top \quad=\quad$
Rel is a locally posetal 2-category: given relations $R,S: X \to Y$,
$$ R \implies S \qquad\text{iff}\qquad R \subseteq S. $$
2-morphisms correspond to subsumption in description logic.
A bicategory of relations is a locally posetal 2-category $\mathcal{B}$ that is also a symmetric monoidal category $(\mathcal{B},\otimes,I)$ with diagonals $(X,\Delta_X,\lozenge_X)_{X \in \mathcal{B}}$ subject to several axioms.
Axioms ensure that
A relational olog is a finitely presented bicategory of relations.
"Finitely presented" means generated by
$\quad$ $\quad\implies\quad$
Theorem. For every (small) bicategory of relations $\mathcal{B}$, there is an equivalence of categories $$ \mathrm{Cl}(\mathrm{Lang}(\mathcal{B})) \simeq \mathcal{B} \qquad\text{in}\qquad \mathbf{BiRel}. $$