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					    4.0 Knowledge Representation
    Outline
      4.1 Introduction
      4.2 Knowledge Representation schemes.
      4.3 Propositional Calculus (PC)
      4.3.1 PC Symbols and Sentences
      4.3.2 Syntax of the propositional calculus
      4.3.3 Semantics of PC
      4.3.3.1 logic Laws and inference rules
      4.3.3.2 Classification of Compound Propositions
      4.3.4 Exercises
      4.4 Predicate Calculus
      4.4.1          Syntax of predicate calculus
      4.4.2          Some predicate calculus equivalences
      4.4.3          Inference rules
      4.5 Semantic Networks
      4.5.1          Representation of Some Relations
      4.5.2          Inheritance in Semantic Networks
      4.6 Conceptual Graphs
      4.6.1          Types and individuals
1     4.7 Frame Structure
      4.7.1          Examples
                    4.1 Introduction
                                AI System



        Knowledge Base (KB)               Inference Mechanism


    Facts about objects:   Rules,Procedures,   Set of procedures to
    properties and             Theories        examine the KB in an
    relations                                  orderly manner to
                                               answer questions,
                                               solve problems, or
                                               make decisions within
                                               the KB domain

2
    4.2 Knowledge Representation schemes


                Logic            Network                 Structural
            Representation    Representation           Representation




Propositional    Predicate   Semantic     Conceptual          Frame
  Calculus       Calculus    Networks      Graphs           Structures




   3
     4.3 Propositional Calculus
•   Proposition:
    is a statement about the world that is either true or false.
    It is a declarative sentence.
    Examples : P= “It is hot”& Q =“The earth orbits the sun” are
    propositions. R=“can you play tennis” is not a proposition.
•   Premise:
    is a proposition which is assigned a logical value( true or false).
    Example : P is true       Interpretation of P = assertion of its truth value

•   Rules are used to draw new propositions from              known premises
•   Example Q=“We are in summer”, P=“It is hot”
•   Rule: QèP
                                conclusion or consequent
    Premise or antecedent
4
       4.3.1 Propositional Calculus Symbols and
                       Sentences
 4.3.1.1 Symbols:         P Q P^Q PvQ PàQ P↔Q
 •   Propositional
     symbols:
     P,Q,R,S,…..          T   T    T        T        T          T
 •   Truth symbols:
     true, false
 •   Connective           T   F    F        T        F          F
     symbols:
    Not(┐,!,~), AND(^)
    OR(v), Imply(è)       F   T    F        T        T          F
    Equivalence
    (↔)
 The shown truth table
    gives their meaning
                          F   F    F        F        T          T

A sentence in PC can be a symbol or a collection of symbols
connected legally by the connective symbols (^,v,!,à, ↔). Symbols ( ) and
  ]
[ 5 are used to group symbols into subexepressions.
4.3.1.2 Examples for using connectives
•  1-Although Ali and Samy are not young, Ali has a better chance of
   winning the next tennis match, despite Samy’s considerable
   experience.
• The above is a complex sentence that can be broken into the
   following propositions:
• S =“Ali is young”
• G =“Samy is young”
• B =“Ali has a better chance of winning the next tennis match”
• E =“Samy has a considerable experience in tennis”
• Formalization is done using connectives as follows:
• (~S) ^(~G) ^B^E
• 2- If Ali is not in a hospital then Ali is well
C =“Ali is in a hospital”
W =“Ali is well”
Formalization:
(~C)èW
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4.3.2 Syntax of the propositional calculus
  •   The syntax provides us with a set of rules for generating complex
      formulae as follows:
  •   1- An identifier is a proposition (e.g. A,B, p, q).
  •   2-If p and q are propositions then so are ~P, (P), p^q, pvq, pèq,
      póq.

  •   Example: if A, B and C are propositions then
  •   Aè C is a grammatically correct formula and, therefore
  •   AèC v B is also correct.

  •   Terminology
  •   Formula that contains only symbols of the agreed alphabet and
      obey grammatical rules given in the definition are called well-formed
      formulae (wffs).
  •   Note that when mixing connectives, the rules of precedence should
      be taken into consideration.
  •   Precedence from higher to lower :(),~,^,v, è ,ó
  •   Hence A^BvC means (A^B)vC not A^(BvC).
  7
4.3.3 Semantics of PC
• Semantics deal with the meaning of compound
   propositions.
• Meaning of a propositional sentence is
   The determination of its truth value
 ≡ Mapping it into T or F ≡Interpretation of it
• This is done by using truth tables or by applying the
   laws and rules of propositional logic listed in the
   following section.
4.3.3.1 Logic laws and inference rules
Cumulative laws: (P^Q) ≡(Q^P) and (PvQ) ≡(QvP)
Associative Laws:      ( (PvQ)vR) ≡(Pv(QvR)) and
                       ( (P^Q)^R) ≡(P^(Q^R))
Distributive laws:     Pv(Q^R) ≡(PvQ)^(PvR) and
                       P^(QvR) ≡(PvQ)v(P^R)
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   ~(~P) ≡ P
   (PvQ) ≡(~PàQ) and
   (~PvQ) ≡ PàQ
Contrapositive law: (PàQ)
   ≡(~Qà~P)
de Morgan’s law:
~(PvQ) ≡~P^~Q and
~(P^Q) ≡~Pv~Q

An example of an inference rule:
Modus Ponens rule:
if P is true and
PàQ                                 Prime              Premises Conclusion
                                    propositions
Then Q is true
The validity (correctness or        p         q pèq         p   q
   soundness) of this rule can be
   proven using truth tables as     T    T         T        T   T
   shown.                           T    F         F        T   F
                                    F    T         T        F   T
9                                   F    F         T        F   F
List of inference rules: each introduces or eliminates a connective
Note that : _I means introduction of a connective.
           _E means Elimination of a connective.
• Conjunction: ^_I                   • Modus ponens: è_E
p                                    Pèq
q                                    p
p^q                                  q
• Simplification: ^_E                • Modus tollens: è_E
p^q                                  Pèq
p                                    ~q
• Addition: v_I                      ~p
p                                    • Double negation: ~_E
pvq                                  ~~p
• Disjunctive syllogism: v_E         p
pvq                                  • Transivity of equivalence: ó_E
~p                                   p óq            p óq
q                                    pèq             qèp
  10
4.3.3.2 Classification of Compound Propositions
•    Compound propositions can be classified based on the
     truth value each proposition can take under all possible
     interpretations of its constituent prime proposition as
     follows:

•    Tautology: is true under all possible assignments of
     truth values to its prime propositions èalways true
     –   Example p v ~p is a tautology.

•    Contradiction: èalways false
     –   Example: p ^ ~p is a contradiction.

•    Contingent: sometimes true and sometimes false.
     –   Example: pè~p


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4.3.4 Exercises
1- Build the truth table for the following expression and
    deduce an expression equivalent to it:
  (PèQ)^(QèP) ≡ P….Q

     P   Q   PàQ     QàP    (PèQ)^(QèP)




 2-Determine the type of the following expressions:
 a- p v qóp ^ q            b- pèqóqèp
 c-~p v ~qó ~(p^q)         d-~p ^ ~q v p ^ q
 3-Demonestrate the validity of modes tollens’ rule:
 If pèq and ~q then ~p
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           4.4 Predicate Calculus
•   In Propositional Calculus, each symbol denotes a proposition. There
    is no way to access the components of an individual assertion.
    Predicate calculus provides this ability.
•   For Example in Propositional Calculus the symbol P may represent
    the proposition:
    P=“Ali is the father of Salem”
•   In Predicate calculus, the same proposition can be represented
    as
    father (ali , salem).
    By dividing the proposition into its internal components, predicate
       calculus allows inference of another information such as :
    Ali is a father,
    Salem is the son of Ali,
    And many other conclusions can be derived if combined with other
       predicates.
• Predicate calculus also allows expressions to
   contain variables.
Ex: "X
13           alive(X)àdrinks(X, water).
     4.4.1 Syntax of predicate calculus
• Constatnsà specific names of objects, properties, or
   relations.
Must begin with small letters:
Ex: ali, book, red, father, above.
• Variables à unknown or general classes of objects.
Represented by symbols beginning with capital letters.
X, Y, Z, Man…
• Functions:à maps its arguments from the domain into
   a unique object in the range (co-domain):
Examples:                                       1-arity function
   fatherof(ali): maps ali to his father.
   plus(3,5): maps 3,5à8                  2-arity function
   price (banans): maps banans to price value.

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• Term: is a constant, variable, or function:
 Examples: cat,        X, times(2,3),
• Predicate: names a property of an object or a
    relationship between zero or more objects in the world:
 Examples: tall, likes, equal, on, near , part_of,….etc.
• A predicate expression consists of a predicate symbol
    followed by n terms equal to its arity:
  predicate_symbol( term 1, term 2,… term n)
Examples:
likes (ali, football)
friends (ali, father_of(ibrahim))
likes (X,Y)
• Atomic sentence: includes truth symbols or predicate
    expression:
true , false
friends (ali, khaled)
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• Sentences:
     § Every atomic sentence is a sentence.
     § A legal combination of sentences using logical
       operators ( connectives of propositional calculus).
• Quantifiers:
     – constrains the meaning of a sentence containing a
       variableà to make it a proposition (can be T or F).
     – If X is a variable and S is a sentence then
        • "X S is a sentence, and $X S is a sentence.
     – The universal quantifier " , for all: the sentence is true
       for all values of its variable.
     – The existential quantifier $, there exist: the sentence
       is true for at least one value of its variable.
     – For complex expressions “()” are used to indicate the
       scope of quantification.
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     – Free variables are not allowed in quantified
       expressions.
• Examples:
     – Ali is a teacher and a good man
       • goodman (ali) ^ isa (ali, teacher)
     – Nobody likes taxes:
       • Ø $X likes (X , taxes)
     – Every dog is an animal:
       • "X ( dog (X) è animal (X))
     – Every student has a pen:
       • "X ($Y(student(X) è (pen (Y) ^ owns (X,Y))))

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                 More examples
• Everybody is an adult or a child
        • "X ( adult(X) v child (X))
• Everybody is an adult or there are children
        • ("X adult(X)) v ($X child (X))
• Everybody is tall but there are children
        • ("X tall(X)) ^ ($X child (X))
• Although all adults are tall, they are not clever.
        • "X (( adult(X)ètall(X)) ^ (adult (X)è~clever (X)))
• All first-year students are clever.
        • "X ( first_year(X)^student (X)èclever(X) )
• No one can be clever without being hardworking
        • ~$X (clever (X) ) ^ ~hardworking (X))
• Not being lazy is equivalent to being hardworking.
        • "X (~lazy(X)óhardworking (X))
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Examples for expressing predicate formulae in English
   • "X (child(X)^ clever(X)è$Y loves(Y,X) ):
      – Clever children are loved.
   • "X (child(X)^ clever(X)è$Y (adult (Y) ^loves(Y,X))):
      – Clever children are loved by adults.
   • "X (clever(X)è$Y cleverer(X,Y) ):
         – One can not be clever without being cleverer than someone.
     • Tall(samir) ^ ~$Y loves(Y,samir) ^ "X (respects( X,
       samir)) :
         – Samir is tall and nobody loves him although everybody
           respects him.
     • ~ $X ( person (X) ^ (X/=samir) ^ speaks_slowly(X):
         – Nobody, except Samir, speaks slowly.
     •

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    Important notes
•        Scope of the quantifier
         – the variable X may refer to different objects in the same argument as
           follows:
                 – "X distant(X)^ planet (X)è$X moon(X)

                           XàOne object         Xàanother object
•        Bound and free variables:
• "X .p(X,Y)

          bound variable    free variable

• Types of values
• When a variable is quantified, we should associate it with
  a type or domain as follows:
• "Y:person. mortal (Y)
    20
                  Semantics of
       4.4.2 of universal quantification predicate logic
   Interpretation
   •        Assuming we deal with finite domain of individuals
   • "X.p(X)ó p(x1)^p(x2)^….^p(Xn)
   •    Example: assume the human society H consist of the individuals samir, samy, ali, salma,
        amira, and let the following propositions hold true :
   • male(samir)         greedy (samir)        kind(ali)
   • male (samy)         greedy (samy)         kind (amira)
   • male (ali)
        Let these predicate be false for individuals not mentioned above.
   Is the following formula (Z) true in the given context?
   "X:H.male(X)è greedy(X) v kind (X)
   To validate this argument we build a truth table as follows:
                      (1)     (2)        (3)       (4)          (5)         (6)
                      X       male (X)   greedy      Kind (X)   (3) v (4)   (2)è(5)
                                         (X)
                      samir   T          T         F            T           T
                      samy    T          T         F            T           T
                      ali     T          F         T            T           T
                                                                                        Z is T
                      salma   F          F         F            F           T
                      amira   F          F         T            T           T

       21
From the same table we can deduce that "X:H. greedy(X) v kind (X) is false.
                  Semantics of predicate logic
Interpretation of Existential quantification
• Assuming we deal with finite domain of individuals
• $X.p(X)ó p(x1) v p(x2) v….v p(Xn)
•        Example: assume the human society H mensioned in the previous example
         and consider the truth value of the formula Z

• $ X:H.male(X)^(~ greedy(X) è kind (X))
To validate this argument we build a truth table as
  follows:
          (1)      (2)        (3)      (4)          (5)    (6)       (7)
          X        male (X)   greedy     Kind (X)   ~(3)   (5)è(4)   (2) ^(6)
                              (X)
          samir    T          T        F            F      T         T
          samy     T          T        F            F      T         T          Z is T
          ali      T          F        T            T      T         T
          salma    F          F        F            T      F         F
          amira    F          F        T            T      T         F

    22
4.4.2 Some predicate calculus equivalences
     Assume p and q are predicates and X,Y are variables:
     Ø   Ø $X p(X) ó "X Ø p(X)
     Ø   Ø "X p(X) ó $X Ø p(X)
     Ø   $X p(X)      ó $Y p(Y)
     Ø   "X p(X)      ó "Y p(Y)
     Ø   "X ( p(X) ^ q(X) ) ó "X p(X) ^ "Y q(Y)
     Ø   $X ( p(X) v q(X) ) ó $X p(X) v $Y q(Y)




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             4.4.3 Inference rules
                                                     Predicate calculus
                                                     equivalence:
•    Modus Ponens:
Given :PèQ and P are true                            Given:
Conclusion: Q is true                                "X p(X) èq(X) and
Example:                                             p(a)
PèQ: “If I am in Jeddah then I’m in Saudi Arabia”
P=“I am in Jeddah “                                  Conclusion: q(a)
Q=“I’m in Saudi Arabia” added to PèQ and P           Using substitution
•    Modus Tolens:                                   X=a
Given :PèQ and Q is false                           Predicate calculus
Conclusion: P is false                              equivalence:
Example:
                                                    Given:
PèQ: “If I am in Jeddah then I’m in Saudi Arabia”
Ø Q=“I’m not in Saudi Arabia”                       "X p(X) èq(X) and
Ø P=“I am not in Jeddah “ added to PèQ and Ø Q      Ø q(a)
                                                    Conclusion: Ø p(a)
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                                                    Using substitution X=a
        4.5 Semantic Networks
• Network representation provides a means of structuring
   knowledge.
• Semantic networks are :
Directed graphs with labeled nodes and arcs ( arrows, or
   links).
• Nodes corresponds to objects (physical as book or
   nonphysical such as concepts, events, or actions)
• Links corresponds to relationships between objects.
   Examples: IS_A, MEMBER_OF, SUBSET_OF, AKO (a
   kind of), HAS_PARTS, INSTANCE_OF, SHAPED_LIKE,
   OWNS


25
      4.5. 1 Representation of some
                Relations
• In our course we will limit the use of these
  links to definite relations.
• IS_A : to represent inheritance :
subclass to superclass relationship:
Example: a lion is an animal
            lion       IS_A       animal

• Instance_of: To represent object to its class
  relationship:
Example :Ali is a student
                   Instance_of
           ali                   student
 26
• HAS_PARTS: to represent composition or
  aggregation
• Example: A computer is an electronic
  device consisting of CPU unit, screen,
  keyboard          Electronic Device


                               IS_A
                 HAS_PARTS               HAS_PARTS
        screen               computer                keyboard

                             HAS_PARTS


                             CPU UNIT




27
• HAS_ATTRIBUTE: to link an object to a
  property.
Example : A person named Ahmad has a red
  car       person

                instance_of
                              owns
                ahmad                  car


                                     has_color
     Important Not : links in
     semantic networks                 red

     can represent binary
     relations only.
28
        4.5.2 Inheritance in Semantic
                  Networks
Inheritance is represented with the IS_A And
  instance_of links. This allows unstated
  information to be inferred.
Example:
                  Has_parts              drinks
     hair                     mammal              milk

                                IS_A
     From the shown net,
     using inheritance, we    rodent
     can infer that a
     mouse has hair and           IS_A
     drinks milk.
                              mouse
29
         4.6 Conceptual Graphs
• It is a Network scheme in which both
  concepts and conceptual relations are
  represented by nodes in a directed graph.
  The links are not labelled. They indicate
  the direction of the relation only.
• Concepts are represented with labelled
  rectangles.
                                               concepts
• Relations are represented by labelled
  ellipsis.
                               pen    color red
• Example: the pen is red
30                            2-ary conceptual relation
• A concept may represent concrete or
   abstract objects.
• Concrete concepts are like cat, telephone ,
   or anything that we can imagine its shape.
• Abstract concepts include concepts such
   as love, beauty….that do not correspond
   to images in our minds.
• A conceptual relation has 1 to n- ary.
flies is a 1-ary relation   bird      flies

parents is a 3-ary relation             father
               child     parents
31                                   mother
      4.6.1 Types and individuals
• A concept symbol may be individual (as instance
   objects) or generic) as classes).
• Individual concepts have a type field followed by a
   referent field.
• Example: the concept Person: ali
 has type ‘person’ and referent ‘ali’
• Each conceptual graph represents a single proposition.
   The shown graph represents the proposition “ali gave
   ahmad the book”
      Person: ali     agent             object
                                give

     Person:ahmad   recipient           book
32
                4.7 Frame Structure
• Extends network by allowing each node to be a
  complex data structure.
• The main advantage is that knowledge about an
  object or event is stored together in memory as a
  unit.
• Each frame consists of named slots with
  attached values.
• The slot values may be
     –   Numeric data.
     –   Symbolic data
     –   Pointers to other frames
     –   Procedures for performing a particular task.

33
•    Frames are general record-like structures, consisting of a collection
     of slots.
•    The slots may be of any size and type
•    Slots also may have subfields called facets.
•    Facets may also have names and any number of values.
•    A general frame structure is as follows:
•    (<frame name>
         (<slot 1> (<facet 1> <value 1> ……<value k1>)
                   (<facet 2> <value 1> ……<value k2>)
                   --
         )
         (<slot 2> (<facet 1> <value 1> ……<value km>)
         --
         --
      )


34
                       4.7.1 Examples
   • To define a frame which describes the shown
      screen: Frame name
   ) screen
      (object 1 (shape (value pyramid))
                (color (value yellow))
slots          (supported_by (value object2)))
      (object 2 (shape (value cube))
                (color (value red))
               (supported_by (value object3)))
      (object 3 (shape (value cube))
                (color (value blue))
       slots
      facets    (supported_by (value nil)))).


   35
• Another example: to describe a type
(Cressida ( ako ( value car))
              ( color ( value white, blue, black))
Pointer to    ( speed (value 220 km/hr))
another frame
to allow      ( model ( value 4-doors)).
inheritance
(subclass-
superclass
relationship)

( Mycar ( instance_of (value Cressida ))
Pointer to
           ( color ( value white))).
another frame
to allow object
-class
relationship
36
•
                        Problem Set 2
     Construct truth tables for the compound proposition:
      – p v (q ^ r) è (p ^q) v r
•    Test Validity of the following rule
      – (p v q) ^ ~pèq
•    Express the following in predicate logic
      –   Although the tree is tall, it has no leaves.
      –   If x is the father of y or mother of y then x is a parent of y
      –   If y and x have same parent then they are sibling
      –   If x and y are parents of z then they are married.
      –   Every Muslim does prayer.
      –   Nobody likes taxes
      –   All students like AI
•    Evaluate the truth of the following, assuming the data is given in
     section 4.2
      – $ X:H.male(X)^(~ greedy(X) v ~kind (X))
      – ($ X:H.~male(X) ^ kind (X))è("X:H.male(X)è~kind (X))


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