1

Ontology as a logic of Ontology as a logic of intensionsintensions

Marie Duží, Martina Číhalová, Marek Menšík

VSB–Technical University Ostrava, Department of Computer Science FEI,

Silesian University in Opava, FPF, Institute of Computer Science

Czech RepublicCzech Republic

2

ContentContent

Ontology and Knowledge Representation Languages for Ontology Specification Ontology Content Logic of Intensions (TIL in Brief)

Requisite Relation Part-whole Relation Integrity Constraints – inference rules

(presupposition vs. mere entailment)

4

Ontology and Ontology and KKnowledge nowledge RRepresentationepresentation

Why do we need an ontologyontology? To make hidden knowledge explicit and

logically tractable. How do we build an ontologyontology?

By applying an expressiveexpressive semantic framework in order to make all the semantically salient features of knowledge specification explicitexplicit and logically tractablelogically tractable.

5

Languages for Languages for OOntologyntology S Specificationpecification

F-calculi DL – description logic RDF – Resource Description Framework

• OIL, DAML-OIL, DAML+OIL OWL – Ontology Web Language based on

DL SKIF (Possibility to mention properties) SWRL – Semantic Web Rule Language

• OWL and RuleML

6

Well-defined ontology should serve asWell-defined ontology should serve as::

universal library, universal library, thethe backdrop work of computational agentsbackdrop work of computational agents integrating a knowledge base and integrating a knowledge base and procesproces development development

However, current ontology languages do not make it possible to express modalitiesmodalities (what is necessary and what is contingent), to distinguish three kinds of contextthree kinds of context, viz.

extensional levelextensional level of objects like individuals, numbers, functions (-in-extension),

intensional levelintensional level of properties, propositions, offices and roles, and finally

hyperintensional level of conceptshyperintensional level of concepts (i.e. algorithmically structured procedures).

Concepts of n-ary relations are unreasonably modelled by properties.

Ontology language should be universal, highly expressive, with Ontology language should be universal, highly expressive, with transparent semanticstransparent semantics and and meaning driven axiomatisationmeaning driven axiomatisation..

7

Procedural Semantics of Hyperintensional Logic (TIL)

Procedural semanticsProcedural semantics contrasts with set-theoretical denotational semantics.

denotational approachdenotational approach the meaning of ‘E’ = the extra-linguistic entity denoted (or referred to) by ‘E’.

hyperintensional procedural semanticshyperintensional procedural semantics expressions encode algorithmically structured procedures producing either extensional or intensional entities or lower-order procedures as their products.

Algorithmic Algorithmic or or computational turncomputational turn: : the early 1970s, Tichý introduced his notion of construction as abstract procedure (see also Moschovakis, 1994).

8

TIL constructions (are structured procedures)

a) Atomic constructions (consisting of just one constituent: itself): supply objects on which molecular constructions operate

VariablesVariables x, w, t, … v(aluation)-construct entities

Trivialization Trivialization 00XX constructs Xb) Molecular constructions (consisting of other constituents than

themselves)

Composition Composition [[X XX X11……XXnn] ] v-v-constructs the value of constructs the value of f f at at aa

f af a otherwise (otherwise (v-v-)improper)improper

Closure Closure [[xx11……xxnn X X]] v-constructs a function f

Double Execution Double Execution 2X: X Y, Y Z, then 2X Z; otherwise (v-)improperimproper

9

TIL types: ramified hierarchy

1.1. Types of order 1Types of order 1 (non-constructions) BaseBase of atomic types: {, , , } Functional typesFunctional types: ( 1… n), i.e. the set of partialpartial

functions (1 … n)

2.2. Constructions of order Constructions of order nn: v-construct objects of types of order n

3.3. Types of order Types of order n+n+11 (constructions and functions involving constructions in their domain or range)

The collection of constructions of order n, n, is the type type

of order of order nn+1+1 ( 1… n) involving n is the type of order type of order nn+1+1

10

Example

‘‘Dividing any number by 0 is improper’Dividing any number by 0 is improper’Improper/(1) – the class of constructions of order 1 that are v-improper for any valuation vDivide/() – the function of dividing; x ; 0/:[0Improper 00[0Divide x 00]]/2, v-constructs True

‘‘Tom knows that dividing any number by 0 is Tom knows that dividing any number by 0 is improper’; improper’;

Know/(((3))), (3)

wt [0Knowwt 0Tom 00[0Improper 0[0Divide x

00]]]

11

Ontology Ontology CContentontent

1.1. Conceptual (terminological) dictionaryConceptual (terminological) dictionary primitive concepts compound concepts (ontological definitions of entities) the most important descriptive attributes, in particular

identification of entities 2.2. Conceptual RelationsConceptual Relations

contingent empirical relations between entities, in particular the part-whole relation

analytical relations between intensions, i.e., requisites and essence, which give rise to ISA hierarchy

3.3. Integrity constraints (inference rules)Integrity constraints (inference rules) Analytically necessary rules Nomologically necessary rules Common rules of ‘necessity by convention’

12

1.1. Conceptual dictionaryConceptual dictionary

primitive concepts 0Car, 0Vehicle, 0Road, 0Junction, 0Driver, …

compound concepts (ontological definitions of entities) ‘driver is a person with a driving license’ 0Driver = wt x [[0Personwt x]

[0Havewt x 0Driving_License]]

the most important descriptive attributes, in particular identification of entities

13

22. . Conceptual RelationsConceptual Relations

analytical (necessary)analytical (necessary) relations between intensions, i.e., requisites and essencerequisites and essence, which give rise to ISA hierarchyISA hierarchy [0Requisite 0Vehicle 0Car]: necessarily,

if something is a car then it is a vehicle:wt x [[0Carwt x] [0Vehiclewt x]]

Requisite/(() ()); Vehicle, Car/()

contingent typical empirical relationscontingent typical empirical relations between entities, in particular the part-part-whole relation whole relation

14

33. . Integrity constraintsIntegrity constraints

Analytically necessaryAnalytically necessary rules Necessarily, no car is a ship wt [0No 0Carwt 0Shipwt]

Nomologically necessaryNomologically necessary rules No distinct physical objects can occur in the

same place (at the same time) wt xy [x y [0Locwt x] = [0Locwt x]]

Common rulesCommon rules of ‘necessity by convention’ wt x [C …x …] Use the right-hand side lane (if possible)

The degree of necessity decreasing top-down agents’ reasoning

15

Logic of Intensions: requisite relation

obtains between intensions of any types; the most important types:

Req1/(()()): an individual property is a requisite of another property.

Req2/(): an individual office is a requisite of another such office.

Req3/(()): an individual property is a requisite of an individual office.

Req4/(()): an individual office is a requisite of an individual property.

Definition: “Y is a requisiterequisite of X” iff “necessarily whatever occupies/instantiates X at w, t it also occupies/instantiates Y at this w, t.”

16

Requisite relations between properties

Req1 /(()()): basic relation that gives rise to ISA taxonomies explicitly record in ontology hierarchies of intensions based on

requisite relations establish inheritanceinheritance of attributes and possibly also of operations

Claim 1 Req1 is a quasi-orderquasi-order on the set of -properties.

Proof obvious

17

Requisite relations between properties

Due to partiality – not anti-symmetric (the property of having stopped smoking): X = wt x [0StopSmokewt x] Y = wt x [0Truewt wt [0StopSmokewt x]]

In order to obtain week partial order, we need antisymmetryantisymmetry; apply the usual “trick”: factor set of equivalent classes defined as follows:

0Eq = pq [x [[0Truewt wt [pwt x]] = [0Truewt wt [qwt x]]]].

[p]eq = q [0Eq p q] and [Req1’ [p]eq [q]eq] = [Req1 p q].Claim 2 Req1’ is a weak partial orderweak partial order on the factor set of

the set of -properties with respect to Eq. Proof obvious

18

Part-whole relationPart-whole relation ( (modest modest individual anti-essentialismindividual anti-essentialism)) If an individual i has a property P necessarily (in all worlds

and times), then P is a constant or partly constant function. In other words, the property has a non-empty non-empty essential core essential core EssEss, where EssEss is a set of individuals that have the property necessarily, and i is an element of Ess.

frequently voiced objectionobjection:If, for instance, Tom’s only car is disassembled into its elementary physical parts, then Tom’s car no longer exists; hence, the property of being a car is essential of the individual referred to by ‘Tom’s only car’.

First, what is denoted (as opposed to referred to) by ‘Tom’s only car’ is not an individual, but an individual office/role / .

Second, the individual referred to as ‘Tom’s only car’ does not cease to exist even after having been taken apart into its most elementary parts. It has simply lost some properties, among them the property of being a car, the property of being composed of its current parts, etc, while acquiring some other properties.

19

Part-whole relationPart-whole relation

Question Which parts are

essential for an individual in order to have a property P?

For instance, the property of having an engine is essential for the property of being a car

We have an instance of a requisite relation between intensionsintensions

20

Part-whole relationPart-whole relation

Part-whole relation obtains contingentlycontingently between individuals which consist of other individuals and thereby create a mereological sum.

Being a part of is a relation between individuals, not between intensions.

From a logical point of view a car is not a structured whole that organizes its parts in a particular manner.

There is no inheritance or implicative relationno inheritance or implicative relation between the respective properties ascribed to a whole and its individual parts.

21

Some other properties of intensions

Some higher-order properties of intensions are necessarily valid due to the way they are due to the way they are constructedconstructed.

Since we explicate conceptsconcepts as closed constructions modulo modulo - and - and -transformation-transformation, i.e., procedurally isomorphic constructionsprocedurally isomorphic constructions, we can also speak about mutual relations betweenrelations between and properties of conceptsproperties of concepts which define particular intensions, in particular:

22

Relations between concepts Incompatibility of concepts;Incompatibility of concepts; the populations of the defined properties are

necessarily disjoint; Example: bachelor vs. married man

Equivalence of concepts;Equivalence of concepts; the defined properties are one and the same property (in particular ontological definitions); Example: bachelor is an unmarried man

Week-equivalence of conceptsWeek-equivalence of concepts, the defined properties are ‘almost the same’; Example: we echo the relation Eq between individual properties defined

above Functionality of a relation-in-intension;Functionality of a relation-in-intension; necessarily, in each w, t-pair, a

given relation R Awt Bwt is a mapping fR: Awt Bwt assigning to each element of A at most one element of B Example: Each person has at most one driving license

Inverse functionality of a relation-in-intensionInverse functionality of a relation-in-intension; necessarily, in each w, t-pair, a given relation-in-extension R Awt Bwt is a mapping fR–1: Bwt Awt assigning to each element of Bwt at most one element of Awt.

23

Reasoning of agents based on ontology

It is useful to include into ontology important inference inference rulesrules, in particular the relations between hyper-propositions of (mere) entailment and presupposition(mere) entailment and presupposition

P is a presupposition of SS |= P and non-S |= P

Corollary: If non-P then neither S nor non-S is true truth-value truth-value gapgap

S merely entails PS merely entails PS |= P and neither (non-S |= P) nor (non-S |= non-P)(entailement: necessarily, P is true whenever S is true)

24

Topic-focus articulation

Sentences communicate something (focus Ffocus F) about something (topic topic TT). schematic structure: FF(T)(T).

The topic topic TT of a sentence S is often associated with a presuppositionpresupposition PP of S P P is entailed both by is entailed both by S S and and non-Snon-S.

(1)(1) “ “The critical situationThe critical situation on the highway D1 was caused by the agent a”. presupposes that there be a critical situation on D1

wt [if 0Crisiswt then [0Causewt 0a 0Crisis] else Fail]

(2)(2) ““The agent aThe agent a caused the critical situation on the highway D1”. merely entails that there be a critical situation on D1

25

If-then-else (the strict definition)

Non-adequate analysis: [(Crisis Caused-by-a) & (Crisis Fail)] The whole Composition fails even if it is the case of crisis

Mechanism of lazy evaluation: The procedural semanticsprocedural semantics of TIL operates smoothly

even at the hyper-intensional levelhyper-intensional level of constructions: The analysis of “IfIf P P thenthen CC, , elseelse DD” is a procedureprocedure that

decomposes into two phasestwo phases:

1. on the basis of the condition P v , select one of C, D as the procedure to be executed.

2. execute the selected procedure.

26

If-then-else (the strict definition)

1. The selectionselection is realized by the Composition [0the_only c [[P [c=0C]] [P [c=0D]]]]2. the chosen construction c is executedexecuted (Double

Execution) The schematic analysis of “If P then C else D”:

2[0the_only c [[P [c=0C]] [P [c=0D]]]].

“If P then C else Fail”: 2[0the_only c [P [c=0C]]]IfIf Crisis thenthen Caused by a elseelse Failwt 2[0the_only c

[0Crisiswt [c = 0[0Causewt 0a 0Crisis]]]]

27

Analytic schema of a sentence with a presupposition P

“If P then S else Fail.” The corresponding schematic TIL construction wt 2[0c [Pwt [c=0Swt]]].

In general, logic cannot disambiguate a sentence. Yet our logical analysis can substantially contribute to the contribute to the disamiguationdisamiguation by making all the possible readings explicit and logically tractable.

Thus the agent can ask: “What do you mean? This or that?”

28

ConclusionConclusion‘‘Logic and AI for Multi-Agent SystemsLogic and AI for Multi-Agent Systems’’

(http://labis.vsb.cz/)(http://labis.vsb.cz/)Development of FIPA compliant computational variant of TIL,

the TIL-ScriptTIL-Script languagecontinue development into its full-fledged version

equivalent to TIL calculus.Implementation of a method that decides a subset of the TIL-

Script language computable by Prolognow the subset equivalent to standard FOL.

We developed an extension of the editor Protégé-OWLProtégé-OWL so that to create an interface between OWLOWL and and TIL-ScriptTIL-Script.

Sample test: 5 mobile agents (cars), 3 car parks and a GIS agent. The GIS agentGIS agent provided the mobile agents with ‘visibility’. Communicated in TIL-Script and started with minimal (but not overlapping) ontologies. During the test they learned new conceptslearned new concepts and enriched their ontology. The agents’ goal was to find a vacant parking lot (out of 3 available) and park the car – succeeded.

29

Reference

Duží, M., Jespersen, B., Materna, P. Duží, M., Jespersen, B., Materna, P. ((20102010):):

Procedural Semantics for Hyperintensional Procedural Semantics for Hyperintensional

Logic,Logic, Berlin, SpringerBerlin, Springer..

30

Thank you for your attentionThank you for your attention

IfIf questions thenthen answers elseelse Fail