A Unifying Framework for Primitive Ontological ...

A Unifying Framework for Primitive Ontological Relations in Dependent Type Theory Richard Dapoigny1 1

Patrick Barlatier1

LISTIC/Polytech'Savoie

University of Savoie, Annecy, (FRANCE)

Dapoigny, Barlatier (University of Savoie)A Unifying Framework for Primitive Ontological Relations in Dependent Type 1 / 24 Theo

Contents 1

Introduction

2

Motivations

3

4

The Unifying Framework Representing is a and part-whole Relations Representing the Lesniewskian ontology in Coq Representing the Lesniewskian mereology in Coq Conclusion-Perspectives


Introduction

The Whole Picture A lot of searchers are striving for putting more and more intelligence on computers. The means for that: representing information as knowledge that computers are able to deal with. Two basic problems: I How to formalize knowledge? e.g., ontologies, Description Logics I How to reason with that knowledge? e.g., FOL, HOL In many applications these solutions are not suciently expressive (e.g., in biological ontologies). We will show how a Dependent Type Theory (DTT) such as the Calculus of Inductive Constructions (implemented in Coq) can be a fruitful candidate.


Introduction

Ontologies

In Computer Science, an ontology formally represents knowledge as: I a collection of concepts within a domain, I the relationships between pairs of concepts. Ontologies are now under development and/or in use in diverse areas (e.g., geography, defense, the automotive and aerospace industries, and life sciences to cite a few).


Introduction

Kinds of ontologies Ontologies are (at least) divided in two abstraction levels, i.e., core (Foundational) ontologies and Domain ontologies.

Core ontologies vs Domain ontologies Core ontologies (e.g., SUMO, DOLCE, BFO) contain the basic knowledge specifying what are kinds, relations, etc. (similar to set theory for classical mathematics) Domain ontologies (e.g., SNOMED-RT) focus on a particular area such as Biology, Law, etc. and include a single core ontology. Ontologies can be used to model a domain and support reasoning about concepts and relations.


Introduction

The DOLCE taxonomy


Introduction

The Language for Ontology Modelling

Language Requirements We are seeking for a representation language able to: to check for conceptual errors during design (well-formed terms), to infere concepts or relations from an existing model, and to build requests over the ontology. So far, current answers use ontologies and involve multiple languages which rely on distinct theories (Modal Logic, FOL, Description Logics, etc.) not necessarily compatibles between each other which means: waste of time during translation, information lost, and so forth.


Introduction

The Proposal


Motivations

Core Relations Reasoning in knowledge representation deals with the generalization/specialization relation (i.e., the is a relation) and the mereological relation (part-whole relation). In usual knowledge representation theories the is a relation is well understood while the part-whole relation is clearly not. A challenging issue is the co-existence of these two relations within a single framework. As shown by Lesniewski, the part-whole relation departs from the standard set-theoretical assumptions. In the Lesniewskian ontology, a single axiom (expressible as three sub-axioms) introduces the copula is upon which the mereology can be constructed.


Motivations

Translating the Lesniewskian Theory

Protothetic is higher-order by essence, as is the dependent type theory of Coq, and as shown in Henkin's paper [1], prototethics can be formally expressed within a typed framework which incorporates the abstractor λ. The Lesniewski's system is based on two categories, i.e., S and N which denote resp. propositions and names which translate to Prop and a Type universe of Coq. We argue that protothetic can be substituted with type theory, but care should be given to the extensionality.


Motivations

Translating the Lesniewskian Theory Diculty: the extensionality of Lesniewski theory should be expressed coherently in an intensional type theory (i.e., in Coq). I Protothetic uses propositions p , q , ... that can be combined, using unary, binary, ... , functors, to form new propositions. I Theses of extensionality follow: ∀pq, (p ≡ p) ≡ ∀f , f (p) ≡ f (q) ∀fg , (∀p, f (p) ≡ g (p)) ≡ (∀F , F (f ) ≡ F (g ))

In type theory there are two equalities: I the intensional one is called "judgmental" or "denitional equality". I the extensional one is known as "propositional equality", i.e., a propositional predicate for proving that two terms are equal.


Motivations

Translating the Lesniewskian Theory

The equivalence relation of intensional type theories becomes undecidable in extensional type theories (and so is type checking). While an extensional system is conservative over the intensional one for rst-order sytems [2], it is more complicated in type theories with dependent types. Coq includes proof-irrelevance and functional extensionality, then with user-dened equalities through setoids, we can recover function extensionality by construction. Notice that this is a provisional solution, a more satisfying solution will be investigated as soon as HoTT will be part of the Coq kernel.


Motivations

Expressing extensionality from Protothetic to Coq

Axioms Axiom

prop extens

Hypothesis

: ∀

:Prop, (A ↔ B ) → A = B . ∀ ( A:Prop )(P :Type →Prop), A↔

A B

Assertion :

P A

.

Lemmas Lemma Assert eq : ∀ (A:Prop), ∀ (P :Type→Prop), (A ↔

P A)

→

.

A = P A

Lemma ProtoT1 : ∀ (p q :Prop)(f :Type→Prop ), (p ↔q ) ↔ (f

p

↔

f

Lemma ProtoT2 : ∀ p

p

↔

f

q:

Prop, (p ↔ q ) ↔ (∀ f :Type →Prop, f

). q ).

q

Note: The code is intended to be part of the future library.


The Unifying Framework

Representing is a and part-whole Relations

The unifying Framework

Using the type kind as base type in a root module, we construct a hierarchy of concept names according to the DOLCE taxonomy of particulars. I All concept names are described by Type Classes. I Coercions are used to construct the hierarchy. The domain ontology can be built by adding domain-based names to the hierarchy in a module working with subsumption. Alternatively, a mereology is built in another module using N for generic names, together with all required axioms and theorems.



Representing is a and part-whole Relations

The unifying Framework



Representing the Lesniewskian ontology in Coq

The ontology of names Three kinds of names are available without making them explicit, i.e., singular names plural names and empty names.

For readability, singular names are capitalized while plural names are given in lower cases.




The ontology of names In the subsumption module, an object, e.g., John Doe has the type Person while Person is subsumed by kind . At the same time it denotes a name in N through the corresponding predicate. In the mereology module, an object does not make use of the subsumption hierarchy but rather denotes a name through the type class: Class Class

Denote

(a: N )(o : kind ): Prop. (A:N )(o :kind ): Prop := { d001 :> Denote A o ; d002 : ∀ o' :kind , Denote

DenoteS

A o'

→

o' = o

}.




The ontological axiom

Denition of the copula 'is' usually called epsilon: Definition epsilon :=

fun

(A a:N ) ⇒ { o :kind |

DenoteS A o

∧

Denote a o

}.

The single axiom of the ontology: Axiom

isEpsilon :

∀

(∀

A a

, epsilon A a ↔ ((∃ B, epsilon B A) ∧ , (epsilon C A ∧ epsilon D A → epsilon C D )) ∧ (∀ C , epsilon C A → epsilon C a)).

C D




Some ontological theorems ... ... Lemma ... Lemma ... Lemma Lemma

OntoT5 : ∀

A b

OntoT19 : ∀A OntoT27 : ∀ OntoT28 : ∀

,

epsilon A b

B a

,

→

epsilon A B

epsilon A A

∧

.

epsilon B a

→

epsilon A a

.

: , singular equality A B → singular equality B A. B C :N, singular equality A B ∧ singular equality B C → singular equality A C.

A B N A

...



Representing the Lesniewskian mereology in Coq

Basic denitions Name forming function: pt describing a part (pt A).

A

is to be a part of

Name forming function: el describing an element (el element of A). Name forming function: kl describing a class (kl the a). Parameter Parameter Parameter

a

A

is to be an

is to be the class of

: N → N. : N → N. Kl : N → N. pt el




Basic denitions

Then we introduce the following relations (epsilon' is the ip of epsilon): Class Class Class

isPartOf

(A

: ) := ispart001 :> epsilon' (pt B ) A. (A B :N ) := isElem001 :> epsilon' (el B ) A. (A a:N ) := eKlass001 :> epsilon' (Kl a) A. B N

isElementOf isKlassOf




Mereological axioms Axiom Axiom Axiom Axiom

: Transitive isPartOf . : Asymmetry isPartOf . uniqueness : ∀ A B a: N, (isKlassOf

transitive isPart

asymmetric isPart Klass

Klass existence

: ∀

A a

:

,

N

A a

∧

)→ . a.

isKlassOf B a

singular equality A B epsilon' a A

→∃

, isKlassOf B) A ↔ B

B

: ∀ A B, epsilon' (el (singular equality A B ∨ isPartOf A B ). Hypothesis P000 isKlassOf : ∀ A a:N, isKlassOf A a → epsilon' A A. Hypothesis P001 isKlassOf : ∀ A a:N, isKlassOf A a → ∃ B, epsilon' a B. Hypothesis P002 isKlassOf : ∀ (A B a:N ), isKlassOf A a → epsilon' a B → isElementOf B A. Hypothesis P003 isKlassOf : ∀ A B a:N, isKlassOf A a → isElementOf B A → (∃ C D, epsilon' a C ∧ isElementOf D C ∧ isElementOf D B ). Hypothesis

P001 isElementOf




Mereological theorems (some!) Lemma MereoT5 : ∀ A:N, Lemma MereoT6 : ∀

isPartOf A A

→¬

isPartOf A A

.

: , epsilon A a → isElementOf A A. Lemma MereoT7 : ∀ A:N, epsilon A A → isElementOf A A. Lemma MereoT9 : ∀ A B :N, isElementOf A B → epsilon B B. Lemma MereoT15 : ∀ B C, epsilon B C ∧ epsilon C B → (∀ A, epsilon A B ↔ epsilon A C ). Lemma MereoT17 : ∀ A B C, isPartOf A B ∧ singular equality B C → isPartOf A C. Lemma MereoT18 : ∀ A B C, isElementOf A B ∧ singular equality B C → isElementOf A C. Lemma MereoT19 : ∀ A B C, isPartOf A B ∧ singular equality A C → isPartOf C B. A a N




Working in the same framework




Working in the same framework


Conclusion-Perspectives

In summary

The dependently-typed framework is able to constrain the semantics of knowledge representation based on expressive typed structures. The higher order capabilities of the type-theoretical layer are a crucial advantage for substituting the protothetic and for meta-reasoning. The co-existence of distributive classes (subsumption hierarchies) and collective classes (i.e., mereology) has never been addressed (and implemented) under these hypotheses. Much remains to do!



On-going work And now? Show that such a mereological theory is a Heyting lattice with isElementOf as a partial ordering relation. Why a Heyting lattice, since Tarski has shown that every mereological structure can be transformed into a complete Boolean lattice without the zero element? I It is weaker than a boolean lattice but its (pseudo-)complement is best suited for the innite case [3]. I It enables us to make metaphysically important distinctions that disappear for Boolean systems [4] (e.g., the natural distinction between connected and non-connected objects that vanishes for Boolean algebras). I It provides an easier way of building point-free geometrical and topological systems based on mereology (locales).



Thanks for your attention.



[1] L. Henkin, A theory of propositional types, Fundamenta Mathematicae 52, 323334 (1963). [2] Q. Wang, and B. Barras, Semantics of Intensional Type Theory extended with Decidable Equational Theories, Computer Science Logic 2013 (CSL 2013), vol. 23, 653667 (2013). [3] P. Forrest, Nonclassical mereology and its application to sets, Notre Dame J. Formal Logic, 43(2) (2002), 7994. [4] T. Mormann, On the mereological structure of complex states of aairs, Synthese, 187(2), 403418, (2012) Springer.
