Artificial Intelligence - PowerPoint 8 by mWrNSE

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									Structured Knowledge
       Chapter 7
                Logic Notations

Does logic represent well knowledge in structures?




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                      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




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                     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

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             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




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                   Semantic Nets

“John gives Mary a book”

                           Give               Book
                                instance           instance
                   agent             object
            John            g                  b
                                beneficiary
give(John, Mary, book)
                           Mary




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                      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.




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