1. Modelling Conceptual Knowledge using Logic - Week 6 Lee McCluskey Department of Computing and Mathematical Sciences University of Huddersfield

2. School of Computing and Mathematics, University of Huddersfield Recap RDF Schema: Extended RDF with classes and properties, property domains and ranges Definition of Ontology: a precise, structured representation of a ‘conceptualisation’ Reality Conceptualisation C subset of X U Y D&Y => Z Ontology X Y “an abstract, simplified view of the world”

3. School of Computing and Mathematics, University of Huddersfield Top down View of the Semantic Web Up to now we’ve worked from the bottom up.. XML.. XMLS.. RDF.. RDFS => ….. But what are the requirements of the SW from the top down? Here are some example areas: - scientists: accurate searching for all of the available information about each sub-area of research. Standardise scientific terminology. - Knowledge Management - managing the ‘knowledge assets of a business’. This may require the explicit representation of business models or application data - Expert System/KBS engineers: sharing and re-using knowledge. Knowledge Acquisition and encoding is the most prohibitive aspect of ‘intelligent’ systems The requirement is to represent diverse, rich, highly structured information and allow efficient reasoning with it

4. School of Computing and Mathematics, University of Huddersfield Knowledge on the Web? We want to represent CONCEPTUAL KNOWLEDGE on the WEB. How is this to be done?

5. School of Computing and Mathematics, University of Huddersfield Bigger Picture Knowledge Representation Formalisms Description Logic First Order Predicate Logic Scientific and Linguistic Models Object - Oriented Models (Software Development) Conceptual Models (Database) FORMAL / GENERAL LANGUAGES SPECIFIC / INFORMAL/ DIAGRAMATIC

6. School of Computing and Mathematics, University of Huddersfield Problems with Diagrammatic ‘Formalisms’ n Ambiguities.. Multiple inheritance, arcs and node semantics etc President Nixon RepublicanQuaker Pacifist Non-Pacifist Cat Black Has-colour

7. School of Computing and Mathematics, University of Huddersfield First - order predicate logic FOL (FOPL) is a notation used widely in computing for - giving meaning to systems eg relational calculus in data bases - model of computation (with computational forms such as Prolog) - to help prove program correctness - modelling intelligent software agents - expressing and manipulating knowledge … FOL can be used to represent conceptual knowledge

8. School of Computing and Mathematics, University of Huddersfield FOPL – grammar Syntax Classes Example Syntax constants.. a,b,c... functions.. f,g,h... - apply to constants/vars predicates.. p,q,r...- unary, binary,.. variables.. x,y,z... quantifiers.. A, E connectives.. V, &, =>,, <= Other bits.. Brackets Wffs – well formed formulae Eg Ax(p(x) => Ey q(x,y)&p(y))

9. School of Computing and Mathematics, University of Huddersfield FOPL – grammar WFF ::= ATOM | ~ ATOM | (WFF) | WFF connective WFF | quantifier variable WFF ATOM ::= predicate | predicate(ARG-LIST) ARG-LIST ::= TERM | TERM, ARG-LIST TERM ::= constant | variable | function(ARG-LIST)

10. School of Computing and Mathematics, University of Huddersfield Ontologies and FOPL: Specifying the Conceptualisation n Let constants denote object names and date type values in the world n Let unary predicates represent properties/classes eg cat(x), person(x),.. n Let binary predicates represent relations between objects, and values of attributes eg brother(bill,ben) status(tank, full) n Let the Wffs represent the logical structure of the conceptualisation

11. School of Computing and Mathematics, University of Huddersfield FOPL: interpretation Given Wffs in FOPL, an interpretation I is given by mapping the constants, function and predicate symbols to elements in the conceptualisation We say that Wff W is true in an interpretation I if W evaluates to true under I. The evaluation uses the well known meaning of connectives and quantifiers

12. School of Computing and Mathematics, University of Huddersfield FOPL: interpretation - example Universe = persons Wffs Ax,Ax,Az g(x,y) <= (f(x,z)&p(z,y)) Ax,Ay,Az u(x,y) <= (p(z,y)&b(x,z)) Ax m(x) => p(x) Ax f(x) => p(x) Example Interpretation is: g = grandfather, f = father, p = parent, b = brother, u = uncle, m = mother Are the wffs true?

13. School of Computing and Mathematics, University of Huddersfield FOPL – reasoning IN fopl we are fundamentally interested in if a wff w LOGICALLY FOLLOWS from another wff W (usually written as “..a set of WFFs”) W |= w Definition: w logically follows from W if and only if every interpretation that makes W true also makes w true

14. School of Computing and Mathematics, University of Huddersfield FOPL – more definitions A Wff is Satisfiable – at least one interpretation makes it true Unsatisfiable – no interpretation makes it true Tautological or Valid – all interpretations makes it true

15. School of Computing and Mathematics, University of Huddersfield Conclusion: n We’ve introduced how Logic can be used to specify a conceptualisation n Note the (logic) definitions of “Interpretation” and “Logically Follows”, “(Un)satisfiable”