Transcript Knowledge Representation - Cognitive Science at Northwestern

Knowledge Representation
Praveen Paritosh
CogSci 207: Fall 2003: Week 1
Thu, Sep 30, 2004
Some
Representations
Elements of a Representation
•
•
•
•
Represented world: about what?
Representing world: using what?
Representing rules: how to map?
Process that uses the representation: conventions
and systems that use the representations resulting
from above.
• Analog versus Symbolic
Marr’s levels of description
• Computational: What is the goal of the
computation, why is it appropriate, and what is the
logic of the strategy by which it can be carried
out?
• Algorithmic: How can this computational theory
be implemented? In particular, what is the
representation for the input and output, and what
is the algorithm for the transformation?
• Implementation: How can the representation and
algorithm be realized physically?
Marr’s levels of description – 2
• Computational: a lot of cognitive psychology
• Algorithmic: a lot of cognitive science
• Implementation: neuroscience
A closer look
Overview
• How knowledge representation works
– Basics of logic (connectives, model theory, meaning)
• Basics of knowledge representation
– Why use logic instead of natural language?
– Quantifiers
– Organizing large knowledge bases
• Ontology
• Microtheories
• Resource: OpenCyc tutorial materials
How Knowledge Representation
Works
• Intelligence requires knowledge
• Computational models of intelligence require
models of knowledge
• Use formalisms to write down knowledge
– Expressive enough to capture human knowledge
– Precise enough to be understood by machines
• Separate knowledge from computational
mechanisms that process it
– Important part of cognitive model is what the organism
knows
How knowledge representations are used in
cognitive models
• Contents of
KB is part of
cognitive
model
• Some models
hypothesize
multiple
knowledge
bases.
Questions,
requests
Answers,
analyses
Inference
Mechanism(s)
Examples,
Statements
Learning
Mechanism(s)
Knowledge
Base
What’s in the knowledge base?
• Facts about the specifics of the world
– Northwestern is a private university
– The first thing I did at the party was talk to John.
• Rules (aka axioms) that describe ways to infer new
facts from existing facts
– All triangles have three sides
– All elephants are grey
• Facts and rules are stated in a formal language
– Generally some form of logic (aka predicate calculus)
Propositional logic
• A step towards understanding predicate calculus
• Statements are just atomic propositions, with no
structure
– Propositions can be true or false
• Statements can be made into larger statements via
logical connectives.
• Examples:
– C = “It’s cold outside” ; C is a proposition
– O = “It’s October” ; O is a proposition
– If O then C ;if it’s October then it’s cold outside
Symbols for logical connectives
Negation: not, , ~
Conjunction: and, 
Disjunction: or, 
Implication: implies, ,
Biconditional: iff, 
-----------------------------------------------------------• Universal quantifier: forall, 
• Existential quantifier: exists, 
•
•
•
•
•
Semantics of connectives
• For propositional logic, can define in terms of
truth tables
A
F
F
T
T
B
F
T
F
T
AB
F
F
F
T
A
F
F
T
T
B
F
T
F
T
AB
Implication and biconditional
A
F
F
B
F
T
AB
T
T
A
F
F
B
F
T
T
T
F
T
F
T
T
T
F
T
AB  AB
AB
AB  (AB)(BA)
Rules of inference
• There are many rules that enable new propositions
to be derived from existing propositions
– Modus Ponens: PQ, P, derive Q
– deMorgan’s law: (AB), derive AB
• Some properties of inference rules
– Soundness: An inference rule is sound if it always
produces valid results given valid premises
– Completeness: A system of inference rules is complete
if it derives everything that logically follows from the
axioms.
Predicate calculus
• Same connectives
• Propositions have structure: Predicate/Function +
arguments.
–
–
–
–
R, 2 ; Terms. Terms are not individuals, not propositions
Red(R), (Red R) ; A proposition, written in two ways
(southOf UnicornCafe UniHall) ;a proposition
(+ 2 2) ; Term, since the function + ranges over numbers
• Quantifiers enable general axioms to be written
– (forall ?x
(iff (Triangle ?x) (and (polygon ?x)
(numberOfSides ?x 3)))
Model Theory
• Meaning of a theory = set of models that satisfy it.
– Model = set of objects and relationships
– If statement is true in KB, then the corresponding
relationship(s) hold between the corresponding objects
in the modeled world
– The objects and relationships in a model can be formal
constructs, or pieces of the physical world, or whatever
• Meaning of a predicate = set of things in the
models for that theory which correspond to it.
– E.g., above means “above”, sort of
Caution: Meaning pertains to simplest
model
• There is usually an intended model, i.e., what one
is representing.
• A sparse set of axioms can be satisfied by
dramatically simpler worlds than those intended
– Example: Classic blocks world axioms have ordered
pairs of integers as a model
• (<position on table> <height>)  block
• (on A B)  p(A) = p(B) & h(A) = h(B)+1
• (above A B)  p(A) = p(B) & h(A) > h(B)
• Moral: Use dense, rich set of axioms
Misconceptions about meaning
• “Predicates have definitions”
– Most don’t. Their meaning is constrained by the sum
total of axioms that mention them.
• “Logic is too discrete to capture the dynamic
fluidity of how our concepts change as we learn”
– If you think of the set of axioms that constrain the
meaning of a predicate as large, then adding (and
removing) elements of that set leads to changes in its
models.
– Sometimes small changes in the set of axioms can lead
to large changes in the set of models. This is the logical
version of a discontinuity.
Representations as Sculptures
• How does one make a statue of an elephant?
– Start with a marble block. Carve away everything that
does not look like an elephant.
• How does one represent a concept?
– Start with a vocabulary of predicates and other axioms.
Add axioms involving the new predicate until it fits
your intended model well.
• Knowledge representation is an evolutionary
process
– It isn’t quick, but incremental additions lead to
incremental progress
– All representations are by their nature imperfect
Introduction to Cyc’s KR system
• These materials are based on tutorial materials
developed by Cycorp, for training knowledge
entry people and ontological engineers
• For this class, we have simplified them somewhat.
• In examinations, you will only be responsible for
the simplified versions
NL vs. Logic: Expressiveness
NL:
Jim’s injury resulted from his falling.
Jim’s falling caused his injury.
Jim’s injury was a consequence of his falling.
Jim’s falling occurred before his injury.
NL: Write the
rule for every
expression?
Logic: identify the common concepts, e.g.
the relation: x caused y
Write rules about the common concepts, e.g.
x caused y  x temporally precedes y
NL vs. Logic:
Ambiguity and Precision
NL:
Ambiguous
•x is at the bank.
•x is running.
•river bank?
•changing location?
•financial institution?
•operating?
•a candidate for office?
Logic:
Precise
x is running-InMotion  x is changing location
x is running-DeviceOperating x is operating
x is running-AsCandidate  x is a candidate
Reasoning: Figuring out what must be true, given what is
known. Requires precision of meaning.
NL vs. Logic:Calculus of Meaning
Logic: Well-understood operators enable reasoning:
Logical constants: not, and, or, all, some
Not (All men are taller than all women).
All men are taller than 12”.
Some women are taller than 12”.
Not (All A are F than all B).
All A are F than x.
Some B are F than x.
Syntax: Terms (aka Constants)
Terms denote specific individuals or collections
(relations, people, computer programs, types of cars . . . )
Each Terms is a character string prefixed by
• A sampling of some constants:
– Dog, SnowSkiing,
PhysicalAttribute
These denote collections
– BillClinton,Rover, DisneyLandTouristAttraction
– likesAsFriend, bordersOn,
objectHasColor, and, not, implies,
forAll
These denote individuals :
•Partially Tangible
Individuals
•Relations
– RedColor, Soil-Sandy
•Attribute Values
Syntax: Propositions
Propositions: a relation applied to some
arguments, enclosed in parentheses
– Also called formulas, sentences…
• Examples:
– (isa GeorgeWBush Person)
– (likesAsFriend GeorgeWBush AlGore)
– (BirthFn JacquelineKennedyOnassis)
Syntax: Non-Atomic Terms
• New terms can be made by applying functions to other
things
– In the Cyc system, functions typically end in “Fn”
• Examples of functions:
– BirthFn, GovernmentFn, BorderBetweenFn
• Examples of Non-Atomic Terms:
– (GovernmentFn France)
– (BorderBetweenFn France Switzerland)
– (BirthFn JacquelineKennedyOnassis)
Non-atomic Terms can be used in statements like any other term
• (residenceOfOrganization (GovernmentFn France)
CityOfParisFrance)
Why Use NATs?
• Uniformity
– All kinds of fruits, nuts, etc., are represented in the
same, compositional way:
(FruitFn PLANT) *
• Inferential Efficiency
– Forward rules can automatically conclude many useful
assertions about NATs as soon as they are created,
based on the function and arguments used to create the
NAT.
• what kind of thing that NAT represents
• how to refer to the NAT in English
•…
Well-formedness: Arity
• Arity constraints are represented in CycL with the predicate
arity:
• (arity performedBy 2)
Represents the fact that performedBy takes two arguments, e.g.:
(performedBy AssassinationOfPresidentLincoln
JohnWilkesBooth)
• (arity BirthFn 1)
Represents the fact that BirthFn takes one arguments, e.g.:
(BirthFn JacquelineKennedyOnassis)
Well-Formedness: Argument Type
Argument type constraints are represented in CycL with the following
2 predicates:
1 argIsa
(argIsa performedBy 1 Action) means that the first argument
of performedBy must be an individual Action, such as the assassination
of Lincoln in:
(performedBy AssassinationOfPresidentLincoln
JohnWilkesBooth)
2 argGenl
(argGenl penaltyForInfraction 2 Event) means that the
second argument of penaltyForInfraction must be a type of Event, such
as the collection of illegal equipment use events in:
(penaltyForInfraction SportsEvent
IllegalEquipmentUse Disqualification)
Why constraints are important
• They guide reasoning
– (performedBy PaintingTheHouse Brick2)
– (performedBy MarthaStewart CookingAPie)
• They constrain learning
Compound propositions
• Connectives from propositional logic can be used
to make more complex statements
(and
(performedBy GettysburgAddress Lincoln)
(objectHasColor Rover TanColor))
(or (objectHasColor Rover TanColor)
(objectHasColor Rover BlackColor))
(implies (mainColorOfObject Rover TanColor)
(not (mainColorOfObject Rover RedColor)))
(not (performedBy GettysburgAddress BillClinton))
Variables and Quantifiers
• General statements can be made by using variables and quantifiers
– Variables in logic are like variables in algebra
• Sentences involving concepts like “everybody,” “something,” and
“nothing” require variables and quantifiers:
Everybody loves somebody.
Nobody likes spinach.
Some people like spinach and some people like broccoli, but no one
likes them both.
Quantifiers
• Adding variables and quantifiers, we can represent more
general knowledge.
• Two main quantifiers:
1. Universal Quantifer -- forAll
Used to represent very general facts, like:
All dogs are mammals
Everyone loves dogs
2. Existential Quantifier -- thereExists
Used to assert that something exists, to state facts like:
Someone is bored
Some people like dogs
Quantifiers
• Universal Quantifier
(forAll ?THING
(isa ?THING Thing))
• Existential Quantifier:
(thereExists ?JOE
(isa ?JOE Poodle))
Everything is a thing.
Something is a poodle.
• Others defined in CycL:
(thereExistsExactly 12 ?ZOS (isa
?ZOS ZodiacSign))
(thereExistsAtLeast 9 ?PLNT (isa
?PLNT Planet))
There are exactly
12 zodiac signs
There are at least
9 planets
Implicit Universal Quantification
All variables occurring “free” in a formula are understood by
Cyc to be implicitly universally quantified.
So, to CYC, the following two formulas represent the same
fact:
(forAll ?X
(implies
(isa ?X Dog)
(isa ?X Animal))
(implies
(isa ?X Dog)
(isa ?X Animal))
Pop Quiz #1
• What does this formula mean?
(thereExists ?PLANET
(and
(isa ?PLANET Planet)
(orbits ?PLANET Sun)))
Pop Quiz #1
• What does this formula mean?
(thereExists ?PLANET
(and
(isa ?PLANET Planet)
(orbits ?PLANET Sun)))
“There is at least one planet orbiting the Sun.”
Pop Quiz #2
• What does this formula mean?
(forAll ?PERSON1
(implies
(isa ?PERSON1 Person)
(thereExists ?PERSON2
(and
(isa ?PERSON2 Person)
(loves ?PERSON1 ?PERSON2)))
Pop Quiz #2
• What does this formula mean?
(forAll ?PERSON1
(implies
(isa ?PERSON1 Person)
(thereExists ?PERSON2
(and
(isa ?PERSON2 Person)
(loves ?PERSON1 ?PERSON2)))
“Everybody loves somebody.”
Pop Quiz #3
• How about this one?
(implies
(isa ?PERSON1 Person)
(thereExists ?PERSON2
(and
(isa ?PERSON2 Person)
(loves ?PERSON2 ?PERSON1))))
Pop Quiz #3
• How about this one?
(implies
(isa ?PERSON1 Person)
(thereExists ?PERSON2
(and
(isa ?PERSON2 Person)
(loves ?PERSON2 ?PERSON1))))
“Everyone is loved by someone.”
Pop Quiz #4
And this?
(implies
(isa ?PRSN Person)
(loves ?PRSN ?PRSN))
Pop Quiz #4
And this?
(implies
(isa ?PRSN Person)
(loves ?PRSN ?PRSN))
“Everyone loves his (or her) self.”