Bertram Lud ä scher [email protected]

CSE-291: Ontologies in Data Integration

Department of Computer Science & Engineering Department of Computer Science & Engineering University of California, San DiegoUniversity of California, San Diego

CSE-291: Ontologies in Data IntegrationCSE-291: Ontologies in Data IntegrationSpring 2003Spring 2003

Bertram LudBertram Ludää[email protected]@SDSC.EDU

• Recap: First-Order Logic (FO)• Formal Ontologies (Guarino)• BREAK• Guest Lecture by Dr. Gully Burns (USC) on NeuroScholar


Syntax of First-Order Logic (FO)Syntax of First-Order Logic (FO)

• Logical symbols:Logical symbols: , , , , , , (“for all”), (“exists”), ...

• Non-logical symbols: A FO Non-logical symbols: A FO signature signature consists of consists of– constant symbols: a,b,c, ...

– function symbols: f, g, ...

– predicate (relation) symbols: p,q,r, ....

function and predicate symbols have an associated arity;

– we can write, e.g., p/3, f/2 to denote the ternary predicate p and the function f with two arguments

• First-order First-order variables variables VarsVars: x, y, ...: x, y, ...

• Formation rules for Formation rules for terms terms TermTerm : :– constants and variables are terms

– if t_1,...t_k are terms and f is a k-ary function symbols then f(t_1,...,t_k) is a term


Syntax of First-Order Logic (FO)Syntax of First-Order Logic (FO)

• Formation rules for Formation rules for formulas formulas FmlFml : :

– if t1,...tk are terms and p/k is a predicate symbol (of arity k) then p(t1,...tk ) is an atomic formula At (short: atom)

• all variable occurrences in p(t1,...tk ) are free

– if F,G are formulas and x is a variable, then the following are formulas:

– FG, F G, F, FG , FG, F , x: F (“for all x: F(x,...) is true”) x: F (“there exists x such that F(x,...) is true”)

– the occurrences of a variable x within the scope of a quantifier are called bound occurrences.


ExamplesExamples

x malePerson(x) x malePerson(x) person(x). person(x).

malePerson(bill).malePerson(bill).

child(marriage(bill,hillary),chelsea).child(marriage(bill,hillary),chelsea).

Variable: xVariable: x

Constants (0-ary function symbols): bill/0, hillary/0, chelsea/0Constants (0-ary function symbols): bill/0, hillary/0, chelsea/0

Function symbols: marriage/2Function symbols: marriage/2

Predicate symbols: malePerson/1, person/1, child/2Predicate symbols: malePerson/1, person/1, child/2


Semantics of Predicate LogicSemantics of Predicate Logic

• Let Let DD be a non-empty be a non-empty domaindomain (a.k.a. (a.k.a. universe of discourseuniverse of discourse). A ). A structurestructure is a pair is a pair II = (= (DD,I), with an ,I), with an interpretationinterpretation I that maps ... I that maps ...– each constant symbols c to an element I(c) D– each predicate symbol p/k to a k-ary relation I(p) Dk,– each function symbol f/k to a k-ary function I(f): DkD

• Let Let II be a structurebe a structure, , : : VarsVars D D a a variable assignmentvariable assignment. A . A valuation valuation valvalI,I, maps maps TermTerm to to D D and and FmlFml to {to {truetrue, , falsefalse}}

– valI, (x) = (x) ; for x Vars

– valI, (f(t1,...,tk)) = I(f)( valI, (t1),..., valI, (tk) ); for f(t1,...,tk) Term

– valI, (p(t1,...,tk)) = I(p)( valI, (t1),..., valI, (tk) ); for p(t1,...,tk) At

– valI, (F G) = valI, (F) and valI, (G) ; for F,G Fml

– .... for Fml over , , , , , , in the obvious way


ExampleExample

Formula F = Formula F = x malePerson(x) x malePerson(x) person(x). person(x).

Domain D = {b, h, c, d, e}Domain D = {b, h, c, d, e}

Let’sLet’s pick an interpretation I: pick an interpretation I: I(bill) = b, I(hillary) = h, I(chelsea) = c

I(person) = {b, h, c}

I(malePerson) = {b}

Under this I, the formula F evaluates to Under this I, the formula F evaluates to truetrue..

• If we choose IIf we choose I’ like I but I’(malePerson) = {b,d}, then F ’ like I but I’(malePerson) = {b,d}, then F evaluates to evaluates to falsefalse

• Thus, I is a Thus, I is a model model of F, while I’ is not:of F, while I’ is not:– I |= F I’ |=/= F


FO Semantics (cont’d)FO Semantics (cont’d)

• F F entailsentails G (G is a G (G is a logical consequencelogical consequence of F) if every model of F) if every model of F is also a model of G: F |= Gof F is also a model of G: F |= G

• F is F is consistent consistent or or satisfiablesatisfiable if it has at least one model if it has at least one model

• F is F is valid valid or a or a tautology tautology if every interpretation of F is a model if every interpretation of F is a model

Proof TheoryProof Theory: :

Let F,G, ... be FO Let F,G, ... be FO sentences sentences (no free variables). (no free variables).

Then the following are equivalent:Then the following are equivalent:

1.1. F_1, ..., F_k |= GF_1, ..., F_k |= G

2.2. F_1 F_1 ... ... F_k F_k G is valid G is valid

3.3. F_1 F_1 ... ... F_k F_k G is unsatisfiable (inconsistent) G is unsatisfiable (inconsistent)


Proof TheoryProof Theory

• A A calculus calculus is formal proof system to establish is formal proof system to establish – F_1, ..., F_k |= G

• via formal (syntactic) via formal (syntactic) derivationsderivations – F_1, ..., F_k |– ... |– G, where the “|–” denotes allowed proof steps

• Examples: Examples: – Hilbert Calculus, Gentzen Calculus, Tableaux Calculus, Natural

Deduction, Resolution, ...

• First-order logic is “semi-decidable”:First-order logic is “semi-decidable”:– the set of valid sentences is recursively enumerable, but not recursive

(decidable)

• Some inference engines:Some inference engines:– http://www.semanticweb.org/inference.html


Flashback: Ontologies–Attempt at a Flashback: Ontologies–Attempt at a Definition Definition

• An ontology is ...An ontology is ...– an explicit specification of a conceptualization [Gruber93]

– a shared understanding of some domain of interest [Uschold, Gruninger96]

• Some aspects and parameters:Some aspects and parameters:– a formal specification (reasoning and “execution”)– ... of a conceptualization of a domain (community)– ... of some part of world that is of interest (application)

• Provides:Provides:– A common vocabulary of terms– Some specification of the meaning of the terms (semantics)– A shared understanding for people and machines


Human and machine communication (I)Human and machine communication (I)

• ... MachineAgent 1

Things

HumanAgent 2

Ontology Description

MachineAgent 2

exchange symbol,e.g. via nat. language

‘‘JAGUAR“

Internalmodels

Concept

Formalmodels

exchange symbol,e.g. via protocols

MA1HA1 HA2

MA2

Symbol

commit commit

a specific domain, e.g.animals

commitcommitOntology

Formal Semantics

HumanAgent 1

MeaningTriangle

[Maedche et al., 2002]


syntactic level

Some different uses of the word “Ontology” Some different uses of the word “Ontology” [Guarino’95][Guarino’95]

http://ontology.ip.rm.cnr.it/Papers/KBKS95.pdf

semantic level

1. 1. OntologyOntology as a philosophical discipline as a philosophical discipline2. 2. anan ontology as an informal conceptual system ontology as an informal conceptual system3. 3. anan ontology as a formal semantic account ontology as a formal semantic account4. 4. anan ontology as a specification of a “conceptualization” ontology as a specification of a “conceptualization”5. 5. anan ontology as a representation of a conceptual system ontology as a representation of a conceptual systemvia a logical theoryvia a logical theory

5.1 characterized by specific formal properties5.1 characterized by specific formal properties5.2 characterized only by its specific purposes5.2 characterized only by its specific purposes

6. 6. anan ontology as the vocabulary used by a logical theory ontology as the vocabulary used by a logical theory7. 7. anan ontology as a (meta-level) specification of a logical ontology as a (meta-level) specification of a logical

theorytheory


Ontology as a philosophical disciplineOntology as a philosophical discipline

• OntologyOntology: : – branch of philosophy dealing with the nature and organization

of reality (in contrast: Epistemology: deals with the nature and sources of our knowledge; “theory of knowledge”)

– Aristotle: Science of being; dealing with questions such as: • What exists? What are the features common to all beings?


What is a Conceptualization?What is a Conceptualization?

• Conceptualization [Genesereth, Nilsson]: Conceptualization [Genesereth, Nilsson]: – universe of discourse (domain) D = {a,b,c,d,e}– relations = {on/2, above/2, clear/1, table/1}

• Compare (A) and (B): Compare (A) and (B): – world_A: {on(a,b), on(b,c), on(d,e), table(c), table(e)}– world_B: {on(a,b), on(c,d), on(d,e), table(b), table(e)}– two different conceptualizations? – or rather two different states of the same conceptualization?

(A)(A) (B)(B)

Meaning is NOTcaptured by

extensional relations / a single state of

affairs


Intensional StructuresIntensional Structures

• Meaning is NOT in a single state of affairs Meaning is NOT in a single state of affairs (extensional relations) but can be captured by (extensional relations) but can be captured by intensional relationsintensional relations

• An n-ary An n-ary intensional relation R intensional relation R over domain over domain D D is a is a functionfunction

R : W Powerset(Dn)

– W set of possible worlds {w1, w2, w3, ...} (a possible world is one state of affairs, or a situation)

– Powerset(Dn) = set of all subsets of Dn (= D x ... x D)

– So for each w W we have R(w) = a subset of Dn, i.e., with each world we associate the interpretation of R in that world


ExampleExample

• Syntax: Syntax: signature (vocabulary) signature (vocabulary) = = – constant symbols: {a,b}

– relation symbols: {on/2, table/1}

• Semantics: domain D = {“a_block”, “b_block”} Semantics: domain D = {“a_block”, “b_block”} Structure Structure I = I = (D,I) with some interpretation I:(D,I) with some interpretation I:– I(a) = “a_block”, I(b) = “b_block”

– I(on) = {(I(a),I(b)), (I(b),I(c)), (I(d),I(e))} = {(“a_block”,”b_block”), ...}

– I(table) = {c, e}


How can we capture (some of!) the How can we capture (some of!) the meaning of “on-ness”? meaning of “on-ness”?

• Many things can be said about “on-ness” (physics of gravity, Many things can be said about “on-ness” (physics of gravity, pressure and deformation, etc.)pressure and deformation, etc.)

• What is common among all possible states of on/2 over a What is common among all possible states of on/2 over a certain domain D?certain domain D?

• That is, if we look at all possible worlds W, and the values that That is, if we look at all possible worlds W, and the values that I(on)(w) can take, what is common among all those states?I(on)(w) can take, what is common among all those states?

• What is What is alwaysalways true (in all possible worlds) about on/2 is (part true (in all possible worlds) about on/2 is (part of) the meaning of on/2. of) the meaning of on/2. – (� x: on(x,x)) ; in all possible worlds: x is not on x– � (x,y: (on(x,y) on(y,x))) ; in all possible worlds: no x is on y while

y is on x– Good enough? what about on(a,b), on(b,c), on(c,a) ? – Even worse: What is someone sees “on” and interprets it as “below”? we only capture some aspects using the above ontological theory


Another ExampleAnother Example

a logical theorya logical theory

attempt as isolation some ofattempt as isolation some ofthe intrinsic propertiesthe intrinsic properties

ontological theoryontological theory

ontological commitment ontological commitment of T1:of T1:what should be true in all what should be true in all possible worldspossible worlds

T3T3 as a as a meta-level theorymeta-level theory

“” is the traditional logician’s symbol for implication (“”)


Some different uses of the word “Ontology” Some different uses of the word “Ontology” [Guarino’95][Guarino’95]

http://ontology.ip.rm.cnr.it/Papers/KBKS95.pdf

1. 1. OntologyOntology as a philosophical discipline as a philosophical discipline2. 2. anan ontology as an informal conceptual system ontology as an informal conceptual system3. 3. anan ontology as a formal semantic account ontology as a formal semantic account4. 4. anan ontology as a specification of a “conceptualization” ontology as a specification of a “conceptualization”5. 5. anan ontology as a representation of a conceptual system ontology as a representation of a conceptual systemvia a logical theoryvia a logical theory

5.1 characterized by specific formal properties5.1 characterized by specific formal properties5.2 characterized only by its specific purposes5.2 characterized only by its specific purposes

6. 6. anan ontology as the vocabulary used by a logical theory ontology as the vocabulary used by a logical theory7. 7. anan ontology as a (meta-level) specification of a logical ontology as a (meta-level) specification of a logical

theorytheory

conceptualization

ontological theory

Bertram Lud ä scher [email protected]

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