Structured Knowledge

1. Structured Knowledge Chapter 7

2. Logic Notations Does logic represent well knowledge in structures?

3. P(x) x Logic Notations Frege’s Begriffsschrift (concept writing) - 1879: assertP notP if PthenQ for every x,P(x) P P Q P

4. Logic Notations Frege’s Begriffsschrift (concept writing) - 1879: “Every ball is red” “Some ball is red” red(x) x ball(x) red(x) x ball(x)

5. Logic Notations Algebraic notation - Peirce, 1883: Universal quantifier: xPx Existential quantifier: xPx

6. Logic Notations Algebraic notation - Peirce, 1883: “Every ball is red”: x(ballx —< redx) “Some ball is red”: x(ballx • redx)

7. Logic Notations Peano’s and later notation: “Every ball is red”: (x)(ball(x)  red(x)) “Some ball is red”: (x)(ball(x)  red(x))

8. Logic Notations Existential graphs - Peirce, 1897: Existential quantifier: a link structure of bars, called line of identity, represents  Conjunction: the juxtaposition of two graphs represents  Negation: an oval enclosure represents ~

9. Logic Notations “If a farmer owns a donkey, then he beats it”: farmer owns donkey beats

10. Logic Notations EG’s rules of inferences: Erasure: in a positive context, any graph may be erased. Insertion: in a negative context, any graph may be inserted. Iteration: a copy of a graph may be written in the same context or any nested context. Deiteration: any graph may be erased if a copy of its occurs in the same context or a containing context. Double negation: two negations with nothing between them may be erased or inserted.

11. Existential Graphs Prove: ((p  r)  (q  s))  ((p q)  (r s)) is valid

12. Existential Graphs Prove: ((p  r)  (q  s))  ((p q)  (r s)) is valid p q r s p q r s

13. Existential Graphs Prove: ((p  r)  (q  s))  ((p q)  (r s)) Erasure: in a positive context, any graph may be erased. Insertion:in a negative context, any graph may be inserted. Iteration: a copy of a graph may be written in the same context or any nested context. Deiteration: any graph may be erased if a copy of its occurs in the same context or a containing context. Double negation: two negations with nothing between them may be erased or inserted.

14. Existential Graphs Prove: ((p  r)  (q  s))  ((p q)  (r s)) Erasure: in a positive context, any graph may be erased. Insertion:in a negative context, any graph may be inserted. Iteration: a copy of a graph may be written in the same context or any nested context. Deiteration: any graph may be erased if a copy of its occurs in the same context or a containing context. Double negation: two negations with nothing between them may be erased or inserted. p q r s

15. Existential Graphs Prove: ((p  r)  (q  s))  ((p q)  (r s)) Erasure: in a positive context, any graph may be erased. Insertion:in a negative context, any graph may be inserted. Iteration: a copy of a graph may be written in the same context or any nested context. Deiteration: any graph may be erased if a copy of its occurs in the same context or a containing context. Double negation: two negations with nothing between them may be erased or inserted. p q r s p q r s p r

16. Existential Graphs Prove: ((p  r)  (q  s))  ((p q)  (r s)) Erasure: in a positive context, any graph may be erased. Insertion:in a negative context, any graph may be inserted. Iteration: a copy of a graph may be written in the same context or any nested context. Deiteration: any graph may be erased if a copy of its occurs in the same context or a containing context. Double negation: two negations with nothing between them may be erased or inserted. p q r s p r p q r s p q r

17. Existential Graphs Prove: ((p  r)  (q  s))  ((p q)  (r s)) Erasure: in a positive context, any graph may be erased. Insertion:in a negative context, any graph may be inserted. Iteration: a copy of a graph may be written in the same context or any nested context. Deiteration: any graph may be erased if a copy of its occurs in the same context or a containing context. Double negation: two negations with nothing between them may be erased or inserted. p q r s p q r p q r s r p q q s

18. Existential Graphs Prove: ((p  r)  (q  s))  ((p q)  (r s)) Erasure: in a positive context, any graph may be erased. Insertion:in a negative context, any graph may be inserted. Iteration: a copy of a graph may be written in the same context or any nested context. Deiteration: any graph may be erased if a copy of its occurs in the same context or a containing context. Double negation: two negations with nothing between them may be erased or inserted. p q r s r p q q s p q r s p q r s

19. Existential Graphs Prove: ((p  r)  (q  s))  ((p q)  (r s)) p q r s p q r s p r p q r s p q p q r s r s p q r s p q r r p q q s

20. Existential Graphs • a-graphs: propositional logic • b-graphs: first-order logic • -graphs: high-order and modal logic

21. Semantic Nets • Since the late 1950s dozens of different versions of semantic networks have been proposed, with various terminologies and notations. • The main ideas: For representing knowledge in structures The meaning of a concept comes from the ways it is connected to other concepts Labelled nodes representing concepts are connected by labelled arcs representing relations

22. Semantic Nets Mammal isa has-part Person Nose instance uniform color team Red Owen Liverpool person(Owen)  instance(Owen, Person) team(Owen, Liverpool)

23. Semantic Nets John Bill height height greater-than H1 H2 value 1.80

24. Semantic Nets “John gives Mary a book” Give Book instance instance agent object John g b beneficiary give(John, Mary, book) Mary

25. Frames • A vague paradigm - to organize knowledge in high-level structures • “A Framework for Representing Knowledge” - Minsky, 1974 • Knowledge is encoded in packets, called frames (single frames in a film) Frame name + slots

26. Frames Animals T Alive F Flies isa Birds Mammals 2 4 Legs Legs T Flies isa Penguins Cats Bats F 2 Flies Legs T Flies instance Opus Bill Pat Opus Bill Pat Name Name Name Friend Friend

27. Frames Hybrid systems: Frame component: to define terminologies (predicates and terms) Predicate calculuscomponent: to describe individual objects and rules

28. Conceptual Graphs • Sowa, J.F. 1984. Conceptual Structures: Information Processing in Mind and Machine. • CG = a combination of Perice’s EGs and semantic networks.

29. Conceptual Graphs • 1968: term paper to Marvin Minsky at Harvard. • 1970's: seriously working on CGs • 1976: first paper on CGs • 1981-1982: meeting with Norman Foo, finding Peirce’s EGs • 1984: the book coming out • CG homepage: http://conceptualgraphs.org/

30. Simple Conceptual Graphs concept type (class) concept relation relation type 1 2 CAT: tuna On MAT: * individual referent generic referent

31. Ontology • Ontology: the study of "being" or existence • An ontology = "A catalog of types of things that are assumed to exist in a domain of interest" (Sowa, 2000) • An ontology = "The arrangement of kinds of things into types and categories with a well-defined structure" (Passin 2004)

32. Ontology top-level categories domain-specific

33. Ontology Aristotle's categories Being Substance Accident Property Relation Inherence Directedness Containment Spatial Temporal Quality Quantity Movement Intermediacy Activity Having Situated Passivity

34. Ontology Geographical categories Geographical-Feature Area Point Line On-Land On-Water Block Dam Terrain Town Road River Country Bridge Border Wetland Airstrip Railroad Power-Line Mountain Heliport

35. Relation Ontology

36. Relation Ontology

37. ANIMAL Eat FOOD PERSON: john Eat CAKE: * Ontology

38. 1 2 PERSON: john Has-Relative PERSON: * 1 2 PERSON: john Has-Wife WOMAN: mary CG Projection

39. Neg CAT: tuna On MAT: *  CAT: tuna On MAT: * Nested Conceptual Graphs It is not true that cat Tuna is on a mat.

40.   CAT: * CAT: * On MAT: * coreference link Nested Conceptual Graphs Every cat is on a mat.

41.  Poss PERSON: julian Fly-To PLANET: mars Past Nested Conceptual Graphs Julian could not fly to Mars.

42.  Poss PERSON: julian Fly-To PLANET: mars Past Nested Conceptual Graphs Tom believes that Mary wants to marry a sailor.

43. Exercises • Reading: Sowa, J.F. 2000. Knowledge Representation: Logical, Philosophical, and Computational Foundations (Section 1.1: history of logic). Way, E.C. 1994. Conceptual Graphs – Past, Present, and Future. Procs. of ICCS'94.