CS344: Artificial Intelligence

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CS344 Artificial Intelligence

• Pushpak BhattacharyyaCSE Dept., IIT Bombay
• Lecture 12, 13 Predicate Calculus and Knowledge
  Representation

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Logic and inferencing
Vision
NLP

• Search
• Reasoning
• Learning
• Knowledge

Expert Systems
Robotics
Planning
Obtaining implication of given facts and rules --
Hallmark of intelligence
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• Inferencing through
• Deduction (General to specific)
• Induction (Specific to General)
• Abduction (Conclusion to hypothesis in absence of
  any other evidence to contrary)

Deduction Given All men are mortal
(rule) Shakespeare is a man (fact) To
prove Shakespeare is mortal (inference)
Induction Given Shakespeare is mortal
Newton is mortal (Observation) Dijkst
ra is mortal To prove All men are mortal
(Generalization)
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If there is rain, then there will be no
picnic Fact1 There was rain Conclude There was
no picnic
Deduction
Fact2 There was no picnic Conclude There was no
rain (?)
Induction and abduction are fallible forms of
reasoning. Their conclusions are susceptible to
retraction
Two systems of logic 1) Propositional
calculus 2) Predicate calculus
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• Propositions
• Stand for facts/assertions
• Declarative statements
• As opposed to interrogative statements
  (questions) or imperative statements (request,
  order)
• Operators
• gt and form a minimal set (can express other
  operations)
• - Prove it.
• Tautologies are formulae whose truth value is
  always T, whatever the assignment is

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• Model
• In propositional calculus any formula with n
  propositions has 2n models (assignments)
• - Tautologies evaluate to T in all models.
• Examples
• 1)
• 2)
• e Morgan with AND

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Semantic Tree/Tableau method of proving tautology
Start with the negation of the formula
- a - formula
a-formula
ß-formula
- ß - formula
a-formula
- a - formula
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Example 2
X
(a - formula)
(a - formulae)
a-formula
(ß - formulae)
B
C
B
C
Contradictions in all paths
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Exercise Prove the backward implication in the
previous example
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Inferencing in PC
Backward chaining
Forward chaining
Resolution
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Knowledge
Procedural
Declarative

• Declarative knowledge deals with factoid
  questions (what is the capital of India? Who won
  the Wimbledon in 2005? Etc.)
• Procedural knowledge deals with How
• Procedural knowledge can be embedded in
  declarative knowledge

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Example Employee knowledge base Employee
record Emp id 1124 Age 27 Salary 10L /
annum Tax Procedure to calculate tax from basic
salary, Loans, medical factors, and of children
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Text Knowledge Representation
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A Semantic Graph
The student bought a new computer in June.
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UNL representation
Representation of Knowledge
Ram is reading the newspaper
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UNL a United Nations project
Dave, Parikh and Bhattacharyya, Journal of
Machine Translation, 2002

• Started in 1996
• 10 year program
• 15 research groups across continents
• First goal generators
• Next goal analysers (needs solving various
  ambiguity problems)
• Current active language groups
• UNL_French (GETA-CLIPS, IMAG)
• UNL_Hindi (IIT Bombay with additional work on
  UNL_English)
• UNL_Italian (Univ. of Pisa)
• UNL_Portugese (Univ of Sao Paolo, Brazil)
• UNL_Russian (Institute of Linguistics, Moscow)
• UNL_Spanish (UPM, Madrid)

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Knowledge Representation
UNL Graph - relations
read
agt
obj
Ram
newspaper
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Knowledge Representation
UNL Graph - UWs
read(iclgtinterpret)
obj
agt
newspaper(iclgtprint_media)
Ram(iofgtperson)
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Knowledge Representation
UNL graph - attributes
_at_entry _at_present _at_progress
read(iclgtinterpret)
obj
agt
_at_def
newspaper(iclgtprint_media)
Ram(iofgtperson)
Ram is reading the newspaper
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The boy who works here went to school
Another Example
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Predicate Calculus
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Predicate Calculus well known examples

• Man is mortal rule
• ?xman(x) ? mortal(x)
• shakespeare is a man
• man(shakespeare)
• To infer shakespeare is mortal
• mortal(shakespeare)

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Forward Chaining/ Inferencing

• man(x) ? mortal(x)
• Dropping the quantifier, implicitly Universal
  quantification assumed
• man(shakespeare)
• Goal mortal(shakespeare)
• Found in one step
• x shakespeare, unification

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Backward Chaining/ Inferencing

• man(x) ? mortal(x)
• Goal mortal(shakespeare)
• x shakespeare
• Travel back over and hit the fact asserted
• man(shakespeare)

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Resolution - Refutation

• man(x) ? mortal(x)
• Convert to clausal form
• man(shakespeare) mortal(x)
• Clauses in the knowledge base
• man(shakespeare) mortal(x)
• man(shakespeare)
• mortal(shakespeare)

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Resolution Refutation contd

• Negate the goal
• man(shakespeare)
• Get a pair of resolvents

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Resolution Tree

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Search in resolution

• Heuristics for Resolution Search
• Goal Supported Strategy
• Always start with the negated goal
• Set of support strategy
• Always one of the resolvents is the most recently
  produced resolute

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Inferencing in Predicate Calculus

• Forward chaining
• Given P, , to infer Q
• P, match L.H.S of
• Assert Q from R.H.S
• Backward chaining
• Q, Match R.H.S of
• assert P
• Check if P exists
• Resolution Refutation
• Negate goal
• Convert all pieces of knowledge into clausal form
  (disjunction of literals)
• See if contradiction indicated by null clause
  can be derived

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• P
• converted to
• Draw the resolution tree (actually an inverted
  tree). Every node is a clausal form and branches
  are intermediate inference steps.

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Terminology

• Pair of clauses being resolved is called the
  Resolvents. The resulting clause is called the
  Resolute.
• Choosing the correct pair of resolvents is a
  matter of search.

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Predicate Calculus

• Introduction through an example (Zohar Manna,
  1974)
• Problem A, B and C belong to the Himalayan club.
  Every member in the club is either a mountain
  climber or a skier or both. A likes whatever B
  dislikes and dislikes whatever B likes. A likes
  rain and snow. No mountain climber likes rain.
  Every skier likes snow. Is there a member who is
  a mountain climber and not a skier?
• Given knowledge has
• Facts
• Rules

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Predicate Calculus Example contd.

• Let mc denote mountain climber and sk denotes
  skier. Knowledge representation in the given
  problem is as follows
• member(A)
• member(B)
• member(C)
• ?xmember(x) ? (mc(x) ? sk(x))
• ?xmc(x) ? like(x,rain)
• ?xsk(x) ? like(x, snow)
• ?xlike(B, x) ? like(A, x)
• ?xlike(B, x) ? like(A, x)
• like(A, rain)
• like(A, snow)
• Question ?xmember(x) ? mc(x) ? sk(x)
• We have to infer the 11th expression from the
  given 10.
• Done through Resolution Refutation.

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Club example Inferencing

• member(A)
• member(B)
• member(C)
• Can be written as

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• Negate

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• Now standardize the variables apart which results
  in the following

• member(A)
• member(B)
• member(C)

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10
7
12
5
4
13
14
2
11
15
16
13
2
17
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Assignment

• Prove the inferencing in the Himalayan club
  example with different starting points, producing
  different resolution trees.
• Think of a Prolog implementation of the problem
• Prolog Reference (Prolog by Chockshin Melish)

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Problem-2

• From predicate calculus

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A department environment

• Dr. X is the HoD of CSE
• Y and Z work in CSE
• Dr. P is the HoD of ME
• Q and R work in ME
• Y is married to Q
• By Institute policy staffs of the same department
  cannot marry
• All married staff of CSE are insured by LIC
• HoD is the boss of all staff in the department

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Diagrammatic representation
CSE
ME
Dr. P
Dr. X
Z
Y
R
Q
married
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Questions on department

• Who works in CSE?
• Is there a married person in ME?
• Is there somebody insured by LIC?

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Problem-3 (Zohar Manna, Mathematical Theory of
Computation, 1974)

• From Propositional Calculus

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Tourist in a country of truth-sayers and liers

• Facts and Rules In a certain country, people
  either always speak the truth or always lie. A
  tourist T comes to a junction in the country and
  finds an inhabitant S of the country standing
  there. One of the roads at the junction leads to
  the capital of the country and the other does
  not. S can be asked only yes/no questions.
• Question What single yes/no question can T ask
  of S, so that the direction of the capital is
  revealed?

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Diagrammatic representation
Capital
S (either always says the truth Or always lies)
T (tourist)
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Deciding the Propositions a very difficult step-
needs human intelligence

• P Left road leads to capital
• Q S always speaks the truth

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Meta Question What question should the tourist
ask

• The form of the question
• Very difficult needs human intelligence
• The tourist should ask
• Is R true?
• The answer is yes if and only if the left road
  leads to the capital
• The structure of R to be found as a function of P
  and Q

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A more mechanical part use of truth table
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Get form of R quite mechanical

• From the truth table
• R is of the form (P x-nor Q) or (P Q)

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Get R in English/Hindi/Hebrew

• Natural Language Generation non-trivial
• The question the tourist will ask is
• Is it true that the left road leads to the
  capital if and only if you speak the truth?
• Exercise A more well known form of this question
  asked by the tourist uses the X-OR operator
  instead of the X-Nor. What changes do you have to
  incorporate to the solution, to get that answer?

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Problem-4

• From Propositional Calculus

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Another tourist example this time in a
restaurant setting in a different country (Manna,
1974)

• Facts A tourist is in a restaurant in a country
  when the waiter tells him
• do you see the three men in the table yonder?
  One of them is X who always speaks the truth,
  another is Y who always lies and the third is Z
  who sometimes speaks the truth and sometimes
  lies, i.e., answers yes/no randomly without
  regard to the question.
• Question Can you (the tourist) ask three yes/no
  questions to these men, always indicating who
  should answer the question, and determine who of
  them is X, who y and who Z?

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Solution Most of the steps are doable by humans
only

• Number the persons 1, 2, 3
• 1 can be X/Y/Z
• 2 can be X/Y/Z
• 3 can be X/Y/Z
• Let the first question be to 1
• One of 2 and 3 has to be NOT Z.
• Critical step in the solution only humans can do?

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Now cast the problem in the same setting as the
tourist and the capital example

• Solving by analogy
• Use of previously solved problems
• Hallmark of intelligence

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Analogy with the tourist and the capital problem

• Find the direction to the capital
• ? Find Z who amongst 1, 2 and 3 is Z?
• Ask a single yes/no question to S (the person
  standing at the junction)
• ? Ask a single yes/no question to 1
• Answer forced to reveal the direction of the
  capital
• ? Answer forced to reveal who from 1,2,3 is Z

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Question to 1

• Ask Is R true and the answer is yes if and only
  if 2 is not Z
• Propositions
• P 2 is not Z
• Q 1 always speaks the truth, i.e., 1 is X

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Use of truth table as before
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Question to 1 the first question

• Is it true that 2 is not Z if and only if you are
  X?

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Analysis of 1s answer

• Ans yes
• Case 1 1 is X/Y (always speaks the truth or
  always lies)
• 2 is indeed not Z (we can trust 1s answer)
• Case 2 1 is Z
• 2 is indeed not Z (we cannot trust 1s answer
  but that does not affect us)

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Analysis of 1s answer (contd)

• Ans no
• Case 1 1 is X/Y (always speaks the truth or
  always lies)
• 2 is Z hence 3 is not Z
• Case 2 1 is Z
• 3 is not Z

Note carefully how cleverly Z is identified.
Can a machine do it?
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Next steps ask the 2nd question to determine X/Y

• Once Not Z is identified- say 2, ask him a
  tautology
• Is PP
• If yes, 2 is X
• If no, 2 is Y

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Ask the 3rd Question

• Ask 2 is 1 Z
• If 2 is X
• Ansyes, 1 is Z
• Ansno, 1 is Y
• If 2 is Y (always lies)
• Ansyes, 1 is X
• Ansno, 1 is Z
• 3 is the remaining person

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What do these examples show?

• Logic systematizes the reasoning process
• Helps identify what is mechanical/routine/automata
  ble
• Brings to light the steps that only human
  intelligence can perform
• These are especially of foundational and
  structural nature (e.g., deciding what
  propositions to start with)
• Algorithmizing reasoning is not trivial