What Is an Ontology?

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1 What Is an Ontology? Vytautas ČYRAS Vilnius University Faculty of Mathematics and Informatics Vilnius, Lithuania Based on: N. Guarino, D. Oberle, S. Staab (2009) What is an Ontology? In Handbook on Ontologies, S. Staab, R. Studer (eds.) pp.1-17, Springer. Lecture slides for computer science students, 2015

2 1. Introduction PDF Handbook on Ontologies by Staab and Studer (2009) see ologies%20se%20-%20s.staab,%20r.studer.pdf 2

3 Definitions (informal) [Gruber 1993]: Ontology is an explicit specification of a conceptualization. [Studer, Benjamins, Fensel 1998]: An ontology is a formal, explicit specification of a shared conceptualization. (bold added Čyras) specification formal explicit of conceptualization shared 3

4 Intension and extension Not to be confused with intention or intentionality; see 4

5 Semiotics: sign and meaning See also 5

6 Graphical signs 6

7 Intensional definition and extensional definition An intensional definition gives the meaning of a term by specifying all the properties required to come to that definition, that is, the necessary and sufficient conditions for belonging to the set being defined. E.g., an intensional definition of bachelor is 'unmarried man'. Being an unmarried man is an essential property. It is a necessary condition. It is also a sufficient condition: any unmarried man is a bachelor. [1] The extensional definition defines by listing everything that falls under that definition an extensional definition of bachelor would be a listing of all the unmarried men in the world. [1] See [1] Roy T. Cook (2009) Intensional Definition. In A Dictionary of Philosophical Logic. Edinburgh University Press, p

8 2. What is a Conceptualization 8

9 Extensional relational structure (D, R) Definition 2.1. An extensional relational structure, (or a conceptualization according to Genesereth and Nilsson), is a tuple (D, R) where D is a set called the universe of discourse R is a set of relations on D (end) 9

10 Running example. Set D Example 2.1. D = {I1, I2, I3, I4} 10

11 Running example. Set R 11

12 Set R formally R = {Person, Manager, Researcher, reports-to, cooperates-with} where Person = { (I1), (I2), (I3), (I4) } Manager = { (I1) } Researcher = { (I2), (I3) } reports-to = { (I2, I1), (I3, I1) } cooperates-with = { (I2, I3) } 12

13 Another example Example 2.2. D = D ={I1, I2, I3, I4} R = {Person, Manager, Researcher, reports-to, cooperates-with} where reports-to = reports-to {(I1, I4)} (D, R ) (D, R). Different extensional relational structures (conceptualizations according to Genesereth and Nilsson). 13

14 World W = {w 1, w 2, w 3 } Definition 2.2. A world is a totally ordered set of world states, corresponding to the system s evolution in time. W = {w 1, w 2, w 3 } Comment. With respect to a specific system S we want to model, a world state for S is a maximal observable state of affairs, i.e., a unique assignment of values to all the observable variables that characterize the system. 14

15 Intensional relation ρ (n) (or conceptual relation) Definition 2.3. Let S be a system D a set of distinguished elements of S W the set of world states for S (also called worlds, or possible worlds). The tuple (D, W) is called a domain space for S, as it intuitively fixes the space of variability of the universe of discourse D with respect to the possible states of S. An intensional relation (or conceptual relation) ρ (n) of arity n on (D, W) is a total function W 2 Dn from the set W into the set of all n-ary (extensional) relations on D. Comment by Čyras. ρ (n) : w i ρ (n) (w i ) a subset of D n, i.e. a set of tuples, i.e. an extensional relation, a table. 15

16 Intensional relation structure C = (D, W,R) (or conceptualization) Definition 2.4. An intensional relational structure (or a conceptualization) is a triple C = (D, W,R) with D a universe of discourse W a set of possible worlds R a set of intentional relations on the domain space (D, W) 16

17 Example 2.3 Example 2.3. Examples 2.1 and 2.2 describe 2 worlds compatible with the following conceptualization C = (D, W,R): D = {I1, I2, I3, I4} the universe of discourse W = {w 1, w 2, w 3 } the set of possible world states R = {Person (1), Manager (1), Researcher (1), reports-to (2), cooperates-with (2) } the set of intentional relations i=1,2,3 Person (1) (w i ) = {(I1), (I2), (I3), (I4)} i=1,2,3 Manager (1) (w i ) = {(I1)} i=1,2,3 Researcher (1) (w i ) = {(I2), (I3)} i=1,2,3 reports-to (2) (w 2i 1 ) = {(I2, I1), (I3, I1)} 2 edges in odd states of the world i=1,2,3 reports-to (2) (w 2i ) = {(I2, I1), (I3, I1), (I1, I4)} 3 edges in even states of the world i=1,2,3 cooperates-with (2) (w i ) = {(I2, I3)} 17

18 Example 2.3: visualization 18

19 3. What is a proper formal, explicit specification? 19

20 3.1. Committing to a conceptualization 20

21 Extensional first-order structure M = (S, I) (also called model for language L) Language L is (a variant of) a first-order logical language Vocabulary V = constants predicate_symbols Definition 3.1. Let L be a first-order logical language with vocabulary V. Let S = (D, R) be an extensional relational structure. An extensional first order structure (also called model for L) is a tuple M = (S, I), where I (called extensional interpretation function) is a total function I: V D R that maps each vocabulary symbol of V to either an element of D or an extensional relation belonging to the set R. 21

22 Intensional first-order structure K = (C, I ) (also called ontological commitment) Definition 3.2. Let L be a first-order logical language with vocabulary V. Let C = (D, W,R) be an intensional relational structure (i.e. a conceptualization). An intensional first order structure (also called ontological commitment ) for L is a tuple K = (C, I ), where I (called intensional interpretation function) is a total function I : V D R that maps each vocabulary symbol of V to either an element of D or an intensional relation belonging to the set R. Recall that V = constants predicate_symbols 22

23 Example 3.1. Ontological commitment Example 3.1. V = {'I1', 'I2', 'I3', 'I4'} {'Person', 'Manager', 'Researcher', 'reports-to', 'cooperates-with'}. Our ontological commitment maps the relation symbol 'Person' to the intensional relation Person (1) ; analogically 'Manager', 'Researcher', 'reports-to' and 'cooperates-with': I : 'I1' I1, 'I2' I2, 'I3' I3, 'I4' I4, 'Person' Person (1), 'Manager' Manager (1), 'Researcher' Researcher (1), 'reports-to' reports-to (2), 'cooperates-with' cooperates-with (2) 23

24 Explaining ontological commitment The predicate symbol 'Person' has both 1. an extensional interpretation (through the notion of model, or extensional first-order structure) 2. an intensional interpretation (through the notion of an ontological commitment, or intensional first-order structure). See [Guarino et al. 2009] Fig. 3 24

25 3.2. Specifying a conceptualization 25

26 Intended models Definition 3.3. Let C = (D, W,R ) be an intensional relational structure (i.e. a conceptualization). Let L be a first-order logical language with vocabulary V. Let K = (C, I ) be an intensional first order structure (i.e. ontological commitment). A model M = (S, I), with S = (D, R), is called an intended model of L according to K iff 1. For all constant symbols c V we have I(c) = I (c) 2. There exists a world w W such that, for each predicate symbol v V there exists an intentional relation ρ (n) R such that I (v) = ρ (n) and I(v) = ρ (n) (w). The set I K (L) of all models of L that are compatible with K is called the set of intended models of L according to K. 26

27 Recall the mapping in Example 2.1 In Example 2.1, we have that for w 1 : I('Person') = { (I1), (I2), (I3), (I4) } = Person (1) (w 1 ) I('reports-to') = { (I2, I1), (I3, I1) } = reports-to (2) (w 1 ) 27

28 Ontology Definition 3.4. Let C = (D, W,R ) be an intensional relational structure (i.e. a conceptualization). Let L be a first-order logical language with vocabulary V. Let K = (C, I ) be an intensional first order structure (i.e. ontological commitment). An ontology O K for C with vocabulary V and ontological commitment K is a logical theory consisting of a set of formulas of L, designed so that the set of its models approximates as well as possible the set I K (L) of intended models of L according to K. 28

29 Example 3.2 Ontology O K consists of a set of logical formulae. Through O 1 to O 5 we specify our human resources domain. Taxonomic information. 'Researcher' and 'Manager' are subconcepts of 'Person': O 1 = { 'Researcher'(x) 'Person'(x), 'Manager'(x) 'Person'(x) } Domains and ranges O 2 =O 1 { 'cooperates-with'(x,y) 'Person'(x) & 'Person'(y), 'reports-to'(x,y) 'Person'(x) & 'Person'(y) } Symmetry O 3 = O 2 { 'cooperates-with'(x,y) 'cooperates-with'(y, x) } Transitivity O 4 = O 3 { 'reports-to'(x, z) 'reports-to'(x, y) & 'reports-to'(y, z) } Disjointness. There is no 'Person' who is both a 'Researcher and a 'Manager' : O 5 = O 4 { 'Manager'(x) 'Researcher'(x) } 29

30 The relationships between phenomena in reality, perception (at different times), abstracted conceptualization, the language, its intended models, and an ontology 30

31 Different approaches to the language L [Guarino et al. 2009] Fig. 4 31

32 Thank you 32

Demystifying Ontology

Demystifying Ontology International UDC Seminar 2011 Classification & Ontology: Formal Approaches and Access to Knowledge 19 Sept 2011 Emad Khazraee, Drexel University Xia Lin, Drexel University Agenda