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CSNB234 ARTIFICIAL INTELLIGENCE

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Title: CHAPTER 3 CALCULUS Author: ALICIA TANG Last modified by: Aliciat Created Date: 3/6/2001 6:40:06 AM Document presentation format: On-screen Show (4:3) – PowerPoint PPT presentation

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Title: CSNB234 ARTIFICIAL INTELLIGENCE


1
CSNB234ARTIFICIAL INTELLIGENCE
Chapter 3 Propositional Logic Predicate Logic
(Chapter 2, pp. 45-76, Textbook) (Chapter 8, pp.
240-253, Ref. 3) Read online supplementary
slides
Instructor Alicia Tang Y. C.
2
Early Development of Symbolic Logic
  • English mathematician DeMorgan criticised
    traditional logic because it was written in
    natural language.
  • He thought that the formal meaning of a
    syllogistic statement was confused by the
    semantics of natural language.
  • DeMorgan and Boole both contributed to the
    development of Propositional Logic (or
    Propositional Calculus).
  • Using familiar algebraic symbols, they showed how
    certain algebraic rules were equally applicable
    to numbers, set and truth values of propositions.

3
Propositional Logic (I)
  • Definition
  • Propositional Logic Sentences
  • Every propositional symbol and truth symbol is a
    sentence.
  • For example true, P, Q, and R are four sentences
  • The negation of a sentence is a sentence
  • For example ?P and ? false are sentences
  • The conjunction (and) of two sentences is a
    sentence
  • For example P ?? P is a sentence

4
Propositional Logic (II)
  • Propositional Logic Sentences
  • The disjunction (or) of two sentence s is a
    sentence
  • For example P ? ?P is a sentence
  • The implication of one sentence for another is a
    sentence
  • For example P ? Q is a sentence
  • The equivalence of two sentences is a sentence
  • for example P ? Q R is a sentence

5
Propositional Logic (III)
  • Propositional Logic Semantics
  • An interpretation of a set of propositions is the
    assignment of a truth value, either T of F, to
    each propositional symbol.
  • The interpretation or truth value for sentences
    is determined by
  • The truth assignment of negation, ? P, where P is
    any propositional symbol, is F if the assignment
    to P is T and T if the assignment to P is F.
  • The truth assignment of conjunction, ?, is T only
    when both conjuncts have truth value T otherwise
    it is F.

6
Propositional Logic (IV)
  • Propositional Calculus Semantics
  • The truth assignment of disjunction, ?, is F only
    when both conjuncts have truth value F otherwise
    it is T.
  • The truth assignment of implication, ?, is F
    only when the premise or symbol before the
    implication is T and the truth value of the
    consequent or symbol after the implication is F
    otherwise it is always T.
  • The truth assignment of equivalence, , is T only
    when both expressions have the same truth
    assignment for all possible interpretations
    otherwise it is F.

7
A Worked Example
  • Prove that ((P?Q?R) ?P ? ?Q ? R is a
    well-formed sentence in the propositional
    calculus.
  • Answer. Since
  • P, Q and R are propositions and thus sentences
  • P ? Q, the conjunction of two sentences, is a
    sentence
  • (P ? Q) ?R, the implication of a sentence for
    another, is a sentence

8
A Worked Example ..cont
  • P, Q and R are propositions and thus sentences
  • ?P and ?Q , the negation of two sentences, are
    sentences
  • ?P ? ?Q, the disjunction of two sentences, is a
    sentence
  • ?P ? ?Q ? R, the disjunction of two sentences, is
    a sentence
  • ((P ? Q) ?R) ?P ? ?Q ? R, the equivalence of
    two sentences, is a sentence

We get back the original sentence
9
Conclusion for the worked example
The above is our original sentence, which has
been constructed through a series of applications
of legal rules and is therefore well-formed.
10
Constant Compound Sentences in Propositional
Logic
  • Constants refer to atomic propositions.
  • raining snowing eating hungry wet
  • Compound sentences capture relationships among
    propositions.
  • raining ? snowing ? wet

11
Compound Sentences
  • Negations raining
  • The argument of a negation is called the target .
  • Conjunctions (raining ? snowing )
  • The arguments of a conjunction are called
    conjuncts .
  • Disjunctions (raining ? snowing )
  • The arguments of a disjunction are called
    disjuncts .

12
Compound Sentences
  • Implications (raining ? cloudy )
  • The left argument of an implication is the
    antecedent .
  • The right argument of an implication is called
    the consequent .
  • Reductions cloudy ? raining
  • The left argument of a reduction is the
    consequent .
  • The right argument of a reduction is called the
    antecedent .
  • Equivalences raining ? cloudy

13
Rules of Algebraic Manipulation
Some Laws for Logic Use
  • x ? y y ? x Commutativity
  • x ? y y ? x
  • x ? (y ? z) (x ? y) ? z Associativity
  • x ? (y ? z) (x ? y) ? z
  • x ? (y ? z) (x ? y) ? (x ? z)
    Distributivity
  • x ? (y ? z) (x ? y) ? (x ? z)

14
Semantics of Logical Operators
  • Negation
  • Conjunction

P ?P T F F T
P Q P ? Q T T T T F F F T
F F F F
15
Semantics of Logical Operators
  • Disjunction

P Q P ? Q T T T T F T F T
T F F F
16
More Semantics of Logical Operators
Implication Reverse Implication Equ
ivalence
P Q P ? Q T T T T F F F T
T F F T
P Q Q ? P T T T T F T F T
F F F T
P Q Q ? P T T T T F F F T
F F F T
17
Satisfaction
An interpretation i satisfies a sentence f
(written i f ) if and only if f i T . A
sentence is satisfiable if and only if there is
some interpretation that satisfies it. A
sentence is valid if and only if every
interpretation satisfies it. A sentence is
unsatisfiable if and only if there is no
interpretation that satisfies it.
18
Truth Tables
A truth table is a table of all possible values
for a set of propositional constants. p q r T
T T T T F T F T T F F F T T F T F F F T F F F
Each interpretation of a language is a row in the
truth table for that language. For a
propositional language with n logical
constants,there are 2 n interpretations.
19
Logical Equivalence
Two sentences are logically equivalent if and
only if they logically entail each
other. Examples (p) p (p ? q ) p ? q
de Morgans law (p ? q ) p ? q de
Morgans law (p ? q ) p ? q
20
Problems
  • There can be many, many interpretations for a
    propositional language.
  • Remember that, for a language with n constants,
    there are 2n possible interpretations.
  • Sometimes there are many constants among premises
    that are irrelevant to the conclusion. ----
    Much work wasted.

Solution use other kind of proof theory, such
as refutation proof (later part)
21
Truth Tables
  • The interpretation of any expression in
    propostional logic can be specified in a truth
    table. An example of a truth table is shown here

22
Example of validity Problem to solve
Problem (p ? q) ?(q ? r)?
Solution p q r (p ? q) (q ? r)
(p? q ) ? (q ? r ) T T T
T T T T T F T F T T F T F T T T F
F F T T F T T T T T F T F T F T F F
T T T T F F F T T T
It is a valid sentence!
All values are true
23
Clausal Form
  • Propositional resolution works only on
    expressions in clausal form.
  • Fortunately, it is possible to convert any set of
    propositional calculus sentences into an
    equivalent set of sentences in clausal form.

24
Conversion to Clausal Form
Implications Out P ? Q Ø P Ú Q P ? Q
P Ú Ø Q P ? Q (Ø P Ú Q) Ù (P Ú ØQ
) Negations In Ø Ø P P Ø (P Ù Q) Ø P
Ú Ø Q Ø (P ÚQ ) Ø P Ù Ø Q
25
Predicate Calculus(Predicate Logic)
26
Predicate Calculus (I)
  • In Proposition Logic, each atomic symbol (P, Q,
    etc) denotes a proposition of some complexity.
    There is no way to access the components of an
    individual assertion. Through inference rules we
    can manipulate predicate calculus expressions,
    accessing their individual components and
    inferring new sentences.

27
Predicate Calculus (II)
  • In Predicate Calculus, there are two ways
    variables may be used or quantified. In the
    first, the sentence is true for all constants
    that can be substituted for the variable under
    the intended interpretation. The variable is said
    to be universal quantified. Variables may also be
    quantified existentially. In this case the
    expression containing the variable is said to be
    true for at least one substitution from the
    domain of definition. Several relationships
    between negation and the universal and
    existential quantifiers are given below

28
Predicate Calculus (III)
  • Predicate calculus sentences
  • Every atomic sentence is a sentence
  • if s is a sentence, then so is its negation, ?s
  • if s1 and s2 are sentences, then so is their
    conjunction, s1 ? s2
  • if s1 and s2 are sentences, then so is their
    disjunction, s1 ? s2
  • if s1 and s2 are sentences, then so is their
    implication, s1 ? s2
  • if s1 and s2 are sentences, then so is their
    equivalence, s1 s2

29
Predicate Calculus (IV)
  • If X is a variable and s is a sentence, then ?X s
    is a sentence
  • If X is a variable and s is a sentence, then ?X s
    is a sentence

30
English sentences represented in Predicate
Calculus
  • Some people like fried chicken.
  • ?X (people(X) ? likes(X, fried_chicken)).
  • Nobody likes income taxes.
  • ?? X likes(X, income_taxes).
  • ?X ? likes(X, income_taxes).

31
Rule All purple mushrooms are poisonous. ?X
(purple(X) ? mushroom(X) ? poisonous(X)) Fact
Tom loves Jerry. loves(tom, Jerry).
32
Quiz Translate the following English Statements
into Predicate Expressions
All people that are not poor and are intelligent
are happy. Students who like to read books are
not stupid. Batman is knowledgeable and he is
wealthy. Tweety can fly if it is not fried and
has wings.
33
Exercise 1
Everybody likes something. There is something
whom everybody likes.
34
Answers to Exercise 1
  • Everybody likes something.
  • "x.y. likes(x,y)
  • There is something whom everybody likes.
  • y."x. likes(x,y)

35
Exercise 2
For predicates p q, and variables X and Y
Write the following in English
? ? X p(X) ? X ? p(X) ?? Y q(Y) ? Y ? q(Y)
36
Answers to Exercise 2
37
Quiz Convert each of the following predicate
logic to English sentences
  • ?X loves(X, superman) ? loves(superman, X)
  • food(laksa)
  • ?X food(X) ? like(arul, X)
  • ?X ?Y eat(X, Y) ? alive(X) ? food(Y)
  • ?X eat(haswan, X) ? eat(hasman, X)

38
Stages involved in Proof Theory
  • Stage 1
  • convert all axioms into prenex form
  • i.e. all quantifiers are at the front
  • Stage 2
  • purge existential quantifiers
  • this process is known as skolemization
  • Stage 3
  • drop universal quantifiers
  • as they convey no information

39
An Example
Consider the argument All men are mortal
(given premise) Superman is a man (given
premise) Superman is mortal (goal to test)
The argument gets formalised as ?X man(X) ?
mortal(X) man(Superman)
mortal(Superman) (goal)
And has, as its conflict set in Clausal form ?
man(X) ? mortal(X) ---- (1)
man(Superman) ---- (2) ? mortal(Superman) ----
(3)
Negation of goal
40
  • Apply resolution to derive at a contradiction
  • We get
  • man(Superman) from (1) (3)
  • and,
  • direct contradiction from (2) (4)
  • The conclusion is that the goal is true
  • (i.e. superman is mortal)

41
Exercise 3
  • Convert each of the following into Predicate
    Calculus equivalence
  • Marcus was a man
  • Marcus was a Pompeian
  • All Pompeians were Romans
  • Caesar was a ruler
  • All Romans were either loyal to Caesar or hated
    him
  • Everyone is loyal to someone
  • people only try to assassinate rulers they are
    not loyal to
  • Marcus tried to assassinate Caesar

42
Predicate logic for the 8 facts in Exercise 3
1. man(Marcus) 2. pompeian(Marcus) 3. ? X.
pompeian(X) ? roman(X) 4. ruler(Caesar) 5. ? X.
roman(X) ? loyalto(X, Caesar) ? hate(X,
Caesar) 6. ? X. ?Y. loyalto(X,Y) 7. ? X. ?
Y. person(X) ? ruler(Y) ? tryassassinate(X,Y)
? ? loyalto(X,Y) 8. tryassasinate
(Marcus, Caesar) 9. ? X. man(X) ? person(X)
43
Answers to Exercise 3
?loyato(Marcus, Caesar)
(using 7, substitution, apply M.P)
person(Marcus) ? tryassassinate(Marcus,
Caesar) ? ruler(Caesar) using (4)
person(Marcus) ? tryassassinate(Marcus,
Caesar) using (8) person(Marcus)
(using 9, substitution apply M.P)
man(Marcus) using (1) nil
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