# 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)

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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)

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Logic Notations

Algebraic notation - Peirce, 1883:
Universal quantifier: xPx

Existential quantifier: xPx

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Logic Notations

Algebraic notation - Peirce, 1883:
“Every ball is red”: x(ballx —< redx)

“Some ball is red”: x(ballx • redx)

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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))

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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 ~

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Logic Notations

“If a farmer owns a donkey, then he beats it”:

farmer         owns             donkey

beats

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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.

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

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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.
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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      p r      q s
the same context or a containing
context.
Double negation: two negations
with nothing between them may
be erased or inserted.
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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.               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
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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.            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.
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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.            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.
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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.
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.
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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

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Existential Graphs

•   a-graphs: propositional logic
•   b-graphs: first-order logic
•   -graphs: high-order and modal logic

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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)

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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

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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
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Frames

Hybrid systems:
Frame component: to define terminologies (predicates and terms)

Predicate calculus component: to describe individual objects and
rules

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Conceptual Graphs

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

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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/

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Simple Conceptual Graphs

concept         concept          relation   relation type
type (class)

1               2
CAT: tuna               On             MAT: *

individual referent                    generic referent

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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)

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Ontology
top-level
categories

domain-specific

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Ontology
Aristotle's categories              Being

Substance                    Accident

Property               Relation

Inherence                Directedness                     Containment

Quality    Quantity     Movement            Intermediacy         Spatial   Temporal

Activity        Passivity   Having      Situated

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Ontology
Geographical categories       Geographical-Feature

Area                    Point                Line

Block               Dam                         On-Land               On-Water

Country                Bridge               Border                    River

Mountain            Heliport                     Power-Line

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Ontology

Relation

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Ontology

Relation

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Ontology

ANIMAL
Eat
FOOD

PERSON: john         Eat          CAKE: *

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CG Projection

1                  2
PERSON: john       Has-Relative       PERSON: *

1                  2
PERSON: john        Has-Wife          WOMAN: mary

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Nested Conceptual Graphs

Neg          CAT: tuna      On         MAT: *


CAT: tuna      On         MAT: *

It is not true that cat Tuna is on a mat.
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Nested Conceptual Graphs



CAT: *           CAT: *      On       MAT: *

Every cat is on a mat.
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Nested Conceptual Graphs



Poss   PERSON: julian      Fly-To   PLANET: mars

Past

Julian could not fly to Mars.
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Nested Conceptual Graphs



Poss      PERSON: julian      Fly-To   PLANET: mars

Past

Tom believes that Mary wants to marry a sailor.
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Exercises

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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