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Structured Knowledge Chapter 7 Logic Notations Does logic represent well knowledge in structures? 2 Logic Notations Frege’s Begriffsschrift (concept writing) - 1879: assert P P not P P if P then Q Q P for every x, P(x) x P(x) 3 Logic Notations Frege’s Begriffsschrift (concept writing) - 1879: “Every ball is red” x red(x) ball(x) “Some ball is red” x red(x) ball(x) 4 Logic Notations Algebraic notation - Peirce, 1883: Universal quantifier: xPx Existential quantifier: xPx 5 Logic Notations Algebraic notation - Peirce, 1883: “Every ball is red”: x(ballx —< redx) “Some ball is red”: x(ballx • redx) 6 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)) 7 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 ~ 8 Logic Notations “If a farmer owns a donkey, then he beats it”: farmer owns donkey beats 9 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. 10 Existential Graphs Prove: ((p r) (q s)) ((p q) (r s)) is valid 11 Existential Graphs Prove: ((p r) (q s)) ((p q) (r s)) is valid p r q s p q r s 12 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. 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 p r q s 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. p r q s 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. p r q s Double negation: two negations with nothing between them may be erased or inserted. p r 15 Existential Graphs Prove: ((p r) (q s)) ((p q) (r s)) Erasure: in a positive context, any graph may be erased. p r q s Insertion: in a negative context, any graph may be inserted. Iteration: a copy of a graph may p r 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 p r q s context. Double negation: two negations with nothing between them may p q r be erased or inserted. 16 Existential Graphs Prove: ((p r) (q s)) ((p q) (r s)) Erasure: in a positive context, any graph may be erased. p r q s Insertion: in a negative context, any graph may be inserted. p q r 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 p r q s context. Double negation: two negations with nothing between them may pq r q s be erased or inserted. 17 Existential Graphs Prove: ((p r) (q s)) ((p q) (r s)) Erasure: in a positive context, any graph may be erased. p r q s Insertion: in a negative context, any graph may be inserted. Iteration: a copy of a graph may be written in the same context or pq r q s any nested context. Deiteration: any graph may be erased if a copy of its occurs in p r q s the same context or a containing context. Double negation: two negations with nothing between them may p q r s be erased or inserted. 18 Existential Graphs Prove: ((p r) (q s)) ((p q) (r s)) p r q s p r q s p r p r q s p r q s p r q s p q r s p q r pq r q s 19 Existential Graphs • a-graphs: propositional logic • b-graphs: first-order logic • -graphs: high-order and modal logic 20 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 21 Semantic Nets Mammal isa has-part Person Nose uniform instance color team Red Owen Liverpool person(Owen) instance(Owen, Person) team(Owen, Liverpool) 22 Semantic Nets John Bill height height greater-than H1 H2 value 1.80 23 Semantic Nets “John gives Mary a book” Give Book instance instance agent object John g b beneficiary give(John, Mary, book) Mary 24 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 25 Frames Animals Alive T Flies F isa Birds Mammals Legs 2 Legs 4 Flies T isa Penguins Cats Bats Flies F Legs 2 Flies T instance Opus Bill Pat Name Opus Name Bill Name Pat Friend Friend 26 Frames Hybrid systems: Frame component: to define terminologies (predicates and terms) Predicate calculus component: to describe individual objects and rules 27 Conceptual Graphs • Sowa, J.F. 1984. Conceptual Structures: Information Processing in Mind and Machine. • CG = a combination of Perice’s EGs and semantic networks. 28 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/ 29 Simple Conceptual Graphs concept concept relation relation type type (class) 1 2 CAT: tuna On MAT: * individual referent generic referent 30 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) 31 Ontology top-level categories domain-specific 32 Ontology Aristotle's categories Being Substance Accident Property Relation Inherence Directedness Containment Quality Quantity Movement Intermediacy Spatial Temporal Activity Passivity Having Situated 33 Ontology Geographical categories Geographical-Feature Area Point Line Block Dam On-Land On-Water Terrain Town Road Country Bridge Border River Wetland Airstrip Railroad Mountain Heliport Power-Line 34 Ontology Relation 35 Ontology Relation 36 Ontology ANIMAL Eat FOOD PERSON: john Eat CAKE: * 37 CG Projection 1 2 PERSON: john Has-Relative PERSON: * 1 2 PERSON: john Has-Wife WOMAN: mary 38 Nested Conceptual Graphs Neg CAT: tuna On MAT: * CAT: tuna On MAT: * It is not true that cat Tuna is on a mat. 39 Nested Conceptual Graphs CAT: * CAT: * On MAT: * coreference link Every cat is on a mat. 40 Nested Conceptual Graphs Poss PERSON: julian Fly-To PLANET: mars Past Julian could not fly to Mars. 41 Nested Conceptual Graphs Poss PERSON: julian Fly-To PLANET: mars Past Tom believes that Mary wants to marry a sailor. 42 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. 43