Knowledge Representation

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					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
                  Questions,   Answers,    Examples,
• Contents of     requests     analyses    Statements

  KB is part of
  cognitive
  model            Inference               Learning
                  Mechanism(s)
• Some models                             Mechanism(s)
  hypothesize
  multiple
                                   Knowledge
  knowledge                          Base
  bases.
        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      B     AB           A       B    AB
    F      F      F             F      F
    F      T      F             F      T
    T      F      F             T      F
    T      T      T             T      T
    Implication and biconditional



A    B   AB          A    B    AB
F    F    T           F    F
F    T    T           F    T
T    F       F        T     F
T    T       T        T     T


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.             NL: Write the
Jim’s falling caused his injury.                    rule for every
Jim’s injury was a consequence of his falling.      expression?
Jim’s falling occurred before his injury.




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

                 •x is at the bank.            •x is running.
NL:                  •river bank?                  •changing location?
Ambiguous
                     •financial institution?       •operating?
                                                   •a candidate for office?

Logic:          x is running-InMotion  x is changing location
                x is running-DeviceOperating x is operating
Precise
                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,                             These denote collections
     PhysicalAttribute
   – BillClinton,Rover, DisneyLand-               These denote individuals :
     TouristAttraction                            •Partially Tangible
   – likesAsFriend, bordersOn,                    Individuals
     objectHasColor, and, not, implies,
     forAll                                       •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
                                          Everything is a thing.
       (isa ?THING Thing))
• Existential Quantifier:
    (thereExists ?JOE
      (isa ?JOE Poodle))                  Something is a poodle.

• Others defined in CycL:
    (thereExistsExactly 12 ?ZOS (isa
      ?ZOS ZodiacSign))                   There are exactly
                                          12 zodiac signs
    (thereExistsAtLeast 9 ?PLNT (isa
      ?PLNT Planet))
                                          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.”

				
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