artificial intelligence · artificial intelligence 2012 lecture 03 delivered by zahid iqbal 24...
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ARTIFICIAL
INTELLIGENCE
LECTURE # 03
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Review of Last Lecture
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Today’s Lecture
• Review of last lecture
• Reasoning
• Types of Reasoning
• Logic
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Reasoning
• Reasoning is the process of deriving logical conclusions
from given facts.
• Durkin defines reasoning as ‘the process of working with
knowledge, facts and problem solving strategies to draw
conclusions’.
• Throughout this section, you will notice how representing
knowledge in a particular way is useful for a particular
kind of reasoning.
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Deductive reasoning
• As the name implies, is based on deducing new
information from logically related known information.
• A deductive argument offers assertions that lead
automatically to a conclusion, e.g.
• –If there is dry wood, oxygen and a spark, there will be a
fire
• Given: There is dry wood, oxygen and a spark
• We can deduce: There will be a fire.
• –All men are mortal. Socrates is a man.
• We can deduce: Socrates is mortal
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Inductive Reasoning
• Inductive reasoning is based on forming, or inducing a
‘generalization’ from a limited set of observations, e.g.
–Observation: All the crows that I have seen in my life are
black.
–Conclusion: All crows are black
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Comparison of deductive and inductive
reasoning
• –The inductive reasoning is as follows: By experience,
every time I have let a ball go, it falls downwards.
Therefore, I conclude that the next time I let a ball go, it
will also come down.
• –The deductive reasoning is as follows: I know Newton's
Laws. So I conclude that if I let a ball go, it will certainly
fall downwards.
• Thus the essential difference is that inductive reasoning is
based on experience,
• while deductive reasoning is based on rules, hence the
latter will always be correct.
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Analogical Reasoning
• Analogical reasoning works by drawing analogies
between two situations, looking for similarities and
differences, e.g.
• when you say driving a truck is just like driving a car, by
analogy you know that there are some similarities in the
driving mechanism,
• But you also know that there are certain other distinct
characteristics of each.
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Common-sense Reasoning
• Common-sense reasoning is an informal form of
reasoning that uses rules gained through experience or
what we call rules-of-thumb.
• It operates on heuristic knowledge and heuristic rules.
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Non-Monotonic Reasoning
• Non-Monotonic reasoning is used when the facts of the
case are likely to change after some time, e.g.
• Rule:
• IF the wind blows
• THEN the curtains sway
• When the wind stops blowing, the curtains should sway
no longer.
• However, if we use monotonic reasoning, this would not
happen. The fact that the curtains are swaying would be
retained even after the wind stopped blowing.
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Logic
• Algebra is a type of formal logic deals with number
• PROPOSITIONAL LOGIC
• PREDICATE CALCULUS/LOGIC
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Proposition
• A proposition (p, q, r, …) is simply a statement (i.e., a declarative sentence) with a definite meaning, having a truth value that’s either true (T) or false (F)
• Normally, a proposition is named e.g. P, Q, R etc.
• Propositional Logic is the logic of compound statements built from simpler statements using Boolean connectives.
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Proposition
• A proposition is a statement about the world that may be either true or false.
• Examples of propositions (“properly formed statements”): Ali’s car is blue. Seven plus six equals twelve. (7 + 6 = 12) Amjad is Ali’s uncle.
• Each of the sentences is a proposition - not to be broken
down into its constituent parts. i. e., we simply assign true, say, to “Amjad is Ali’s uncle.”
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Examples of non- propositions • Ali’s uncle • Seven plus four • “Who’s there?” (interrogative, question) • “Just do it!” (imperative, command) • “1 + 2” (expression with a non-true/false value)
Because we cannot assign truth value to them.
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Propositional Symbols
• Propositions are denoted by propositional symbols such as: P, Q, R, S,.
• Truth symbols are: true (or T), false (or F).
• Single propositions by themselves are not very interesting.
• We need to express complex propositions/compound propositions: • The book is on the table or it is on the chair.
• If Socrates is a man then he is mortal.
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Propositional Symbols
We can use logical connecters such as: ...and [conjunction] ...or [disjunction] ...implies [implication / conditional] ..is equivalent [biconditional] ...not [negation]
Sentences in the propositional calculus are formed from these atomic
symbols according to the syntax rules.
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Operators / Connectives
• An operator or connective combines one or more operand expressions into a larger expression. (E.g., “+” in numeric exprs.)
• Unary operators take 1 operand (e.g., -3);
• Binary operators take 2 operands (eg 3 4).
• Propositional or Boolean operators operate on propositions or truth values instead of on numbers.
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The Negation Operator
• The unary negation operator “¬” (NOT) transforms a prop. into its logical negation.
• E.g. If p = “I have brown hair.” then ¬p = “I do not have brown hair.” • Truth table for NOT:
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p p
T F
F T
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The Conjunction Operator
• The binary conjunction operator “” (AND) combines two propositions to form their logical conjunction.
• E.g. If p=“I will have salad for lunch.” and q=“I will have steak for dinner.”, then pq=“I will have salad for lunch and I will have steak for dinner.”
• Conjunction Truth Table
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p q pq
F F F
F T F
T F F
T T T
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The Disjunction Operator
• The binary disjunction operator “” (OR) combines two propositions to form their logical disjunction.
• Example:
• p=“That car has a bad engine.” • q=“That car has a bad carburetor.”
• pq=“Either that car has a bad engine, or
that car has a bad carburetor.”
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The Disjunction Operator
• Note that pq means that p is true, or q is true, or both are true!
• So this operation is also called inclusive or, because it includes the possibility that both p and q are true.
• Disjunction Truth Table
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p q pq
F F F
F T T
T F T
T T T
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Examples
• Example: “BCS AI Class” P = Ali is the teacher Q = Saira is the student R= AI is a course teaching in BS
P ^ Q = Ali is the teacher and Saira is the student. Q ^ R= Saira is the student and tought AI in BS
The book is on the table or it is on the chair. If Socrates is a man then he is mortal.
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A Simple Exercise
Let p=“It rained last night”, q=“The sprinklers came on last night,” r=“The lawn was wet this morning.”
Translate each of the following into English:
• ¬p • ¬q ^ r • r ¬p • ¬ r p q
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The Exclusive Or Operator
• The binary exclusive-or operator “” (XOR) combines two propositions to form their logical “exclusive or”.
• p = “I will earn an A in this course,” • q = “I will drop this course,” • p q = “I will either earn an A for this course, or I will drop it
(but not both!)”
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Exclusive-Or Truth Table
• Note that pq means that p is true, or q is true, but not both!
• This operation is called exclusive or, because it excludes the possibility that both p and q are true.
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p q pq
F F F
F T T
T F T
T T F
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The Implication Operator
• The implication p q states that p implies q. • It is FALSE only in the case that p is TRUE but q is FALSE. • E.g., p=“I am elected.”
q=“I will lower taxes.” • p q = “If I am elected, then I will lower taxes”
• Its premise or antecedent is p and its conclusion or consequent
is q
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Implication Truth Table
• p q is false only when
p is true but q is not true.
• Examples:
• “If 1+1=2, then I am richer than Bill Gates.” True or False?
• “If the moon is made of green cheese, then I am richer than Bill Gates.” True or False?
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p q pq
F F T
F T T
T F F
T T T
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The Biconditional Operator
• The biconditional p q states that p is true if and only if (IFF) q is true.
• It is TRUE when both p q and q p are TRUE. p = “It is raining.” q = “The home team wins.” • p q = “If and only if it is raining, the home team wins.”
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Biconditional Truth Table
• p q means that p and q have the same truth value.
• Note this truth table is the exact opposite of ’s! p q means ¬(p q)
• p q does not imply p and q are true, or cause each other.
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p q p q
F F T
F T F
T F F
T T T
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Truth Table
30
p q p q p q p p q p q p q
F F F F T T T T
F T F T T T T F
T F F T F F F F
T T T T F T T T
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Precedence of Logical Operators
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Precedence of Logical Operators
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Operator Precedence
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Some Alternative Notations
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Name: not and or xor implies iffPropositional logic: Boolean algebra: p pq + C/C++/Java (wordwise): ! && || != ==C/C++/Java (bitwise): ~ & | ^Logic gates:
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Propositional Calculus Sentences (Syntax)
• Every propositional symbol and truth symbol is a sentence.
• e. g., true, P, R.
• The negation of a sentence is a sentence.
• e. g., ~P, ~false
• The conjunction of two sentences is a sentence.
• e. g., P Q, P Ù Q
• The disjunction of two sentences is a sentence.
• e. g., Q Ú R
• The implication of one sentence for another is a sentence.
• e. g., P ® Q
• The equivalence of two sentences is a sentence
• e. g., P Ú Q = R
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Exercise
• Fact 1: Saira likes cakes. = P
• Fact 2: Saira eats cakes. = Q
• P Q, PQ, Q, P Q, P Q ????????
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Exercise
• Fact 1: Saira likes cakes. = P
• Fact 2: Saira eats cakes. = Q
• P Q, PQ, Q, P Q, P Q ????????
• PQ : Saira Likes cakes or eats cakes.
• PQ : Saira likes cakes and eats cakes.
• Q : Saira does not eat cakes.
• PQ: If Saira likes cakes then he eats cakes.
• PQ:Saira eats cakes if and only if he likes cakes.
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Limitations of Propositional logic
We Can’t describe things in terms of their properties or relationships (very limited expressive power)
Propositional logic is declarative Propositional logic is compositional: meaning of B1,1 P1,2 is derived from meaning of B1,1 and of
P1,2 We can’t express rules or generalizations
If the train is late and there are no taxis, john is late for the meeting If trains are late and there are no taxis, anyone traveling by trains is late
for the meeting…
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Limitations…
• Propositions can only represent knowledge as complete sentences, e.g.
• a = the ball’s color is blue. • Cannot analyze the internal structure of the sentence.
• No quantifiers are available, e.g. for-all, there-exists • Propositional logic provides no framework for proving statements
such as: • All humans are mortal • All women are humans • Therefore, all women are mortals
• This is a limitation in its representational power.
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References
• Artificial Intelligence: Structures and Strategies for
Complex Problem Solving
• Internet
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End of Lecture
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Puzzle Game
• A farmer went to market and purchased a fox, a goose, and a bag of beans. On his way home, the farmer came to the bank of a river and hired a boat. But in crossing the river by boat, the farmer could carry only himself and a single one of his purchases - the fox, the goose, or the bag of the beans.
• If left alone, the fox would eat the goose, and the goose would eat the beans.
• The farmer's challenge was to carry himself and his purchases to the far bank of the river, leaving each purchase intact. How did he do it?
Artificial Intelligence 2012 Lecture 03 Delivered By Zahid Iqbal 41