Human Reasoning
2013-11-08
An ontology is a specification of a conceptualization. That is, an ontology is a description (like a formal specification of a program) of the concepts and relationships that can exist for an agent or a community of agents. [...] My colleagues and I have been designing ontologies for the purpose of enabling knowledge sharing and reuse. [Gruber, 1992]
Snomed CT är ett internationellt begreppssystem som syftar till att göra dokumentationen i vård och omsorg enhetlig, entydig och ändamålsenlig. [...] Målet är att Snomed CT ska användas inom vård och omsorg i hela landet. [Socialstyrelsen]
A concept \(C\) is satisfiable if there is a model \(M\) such that \(C^M \neq \emptyset\).
\(C\) is satisfiable iff \(\not\models C \sqsubseteq \bot\).
A TBox is a set \(O\) of formulas of the form \(A \equiv D\) or \(A \sqsubseteq D\) where \(A\) is an atomic concept and each atomic concept occurs at most once on the left hand side of a formula in \(O\).
Sometimes we need the TBox to be acyclic:
An atomic concept \(A\) directly uses an atomic concept \(B\) in \(O\) if \(B\) occurs in the right-hand side of any axiom with \(A\) on the left-hand side. The relation uses is then the transitive closure of the directly uses relation. \(O\) is acyclic if no atomic concept \(A\) uses \(A\).
An ABox consists of assertions of the form \(C(a)\) and \(R(a,b)\) where \(a\) and \(b\) are constants.
Not included in our simplified system.
An ABox is, in some sense, equivalent to a relational database where we only allow unary and binary relations.
Important: In the context of relational databases we have the closed world assumption. Not so in DL. DL is monotone.
\(\forall R_1. (C \sqcap \exists R_2 .C)\) is translated to \[\forall y (R_1(x,y) \rightarrow (C(y) \land \exists z (R_2(y,z) \land C(z))))\]
ALC is multi modal logic K:
An engineer developing an ontology \(O\) suddenly realizes that \(O \models \top \sqsubseteq \bot\). (debugging)
A medical doctor gets a recommended treatment relying on a medical ontology, such as SNOMED-CT. (safety)
Actually, it follows from SNOMED-CT that \(\mathtt{AmpOfFinger} \sqsubseteq \mathtt{AmpOfHand}\).
Why does an entailment \(O \models \varphi\) hold?
A justification of \(\varphi\) (wrt \(O\)) is a minimal subset \(I \subseteq O\) such that \(I \models \varphi\).
is a justification for \(\mathtt{RRated} \sqsubseteq \mathtt{Movie}\).
- \(\mathtt{RRated} \sqsubseteq \mathtt{Thriller}\)
- \(\mathtt{Thriller} \sqsubseteq \mathtt{Movie}\)
- \(\mathtt{Person} \sqsubseteq \lnot\mathtt{Movie}\)
- \(\mathtt{RRated} \sqsubseteq \mathtt{Movie}\)
- \(\mathtt{RRated} \equiv \mathtt{Thriller} \sqcup (\forall \mathtt{hasViolenceLevel}.\mathtt{High})\)
- \(\exists \mathtt{hasViolenceLevel}.\top \sqsubseteq \mathtt{Movie}\)
is a justification for \(\mathtt{Person} \sqsubseteq \bot\).
[Horridge, Parsia, Sattler]
The definition rests on a notion of complexity measure.
The complexity of an entailment \(I \models \varphi\) is the sum of a number of complexity scores.
The complexity scores are syntactic and semantic scores such as:
Justification | Success rate | Model complexity | Proof length |
---|---|---|---|
1 | 57% | 654 | 11 |
2 | 57% | 703 | 14 |
3 | 29% | 675 | 18 |
Why can't we extract explanations directly from the proofs?
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