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Title: CSNB234 ARTIFICIAL INTELLIGENCE


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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.
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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.

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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

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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

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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.

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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.

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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

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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
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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.
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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

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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 .

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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

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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)

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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
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Semantics of Logical Operators

• Disjunction

P Q P ? Q T T T T F T F T
T F F F
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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
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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.
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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.
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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
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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)
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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

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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
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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.

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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
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Predicate Calculus(Predicate Logic)
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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.

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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

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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

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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

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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).

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Rule All purple mushrooms are poisonous. ?X
(purple(X) ? mushroom(X) ? poisonous(X)) Fact
Tom loves Jerry. loves(tom, Jerry).
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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.
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Exercise 1
Everybody likes something. There is something
whom everybody likes.
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Answers to Exercise 1

• Everybody likes something.
• "x.y. likes(x,y)
• There is something whom everybody likes.
• y."x. likes(x,y)

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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)
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Answers to Exercise 2
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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)

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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

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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
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• 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)

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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

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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)
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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