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CS621: Artificial Intelligence. Pushpak Bhattacharyya CSE Dept., IIT Bombay Lecture 34– Predicate Calculus and Himalayan Club example (lectures 32 and 33 were on HMM+Viterbi combined AI and NLP). Resolution - Refutation. man(x) → mortal(x) Convert to clausal form

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cs621 artificial intelligence

CS621: Artificial Intelligence

Pushpak BhattacharyyaCSE Dept., IIT Bombay

Lecture 34– Predicate Calculus and Himalayan Club example

(lectures 32 and 33 were on HMM+Viterbi combined AI and NLP)

resolution refutation
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)
resolution refutation contd
Resolution – Refutation contd
  • Negate the goal
    • ~man(shakespeare)
  • Get a pair of resolvents
search in resolution
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
inferencing in predicate calculus
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
slide7
P
  • converted to

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

terminology
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.
predicate calculus
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
predicate calculus example contd
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)
    • ∀x[member(x) → (mc(x) ∨ sk(x))]
    • ∀x[mc(x) → ~like(x,rain)]
    • ∀x[sk(x) → like(x, snow)]
    • ∀x[like(B, x) → ~like(A, x)]
    • ∀x[~like(B, x) → like(A, x)]
    • like(A, rain)
    • like(A, snow)
    • Question: ∃x[member(x) ∧ mc(x) ∧ ~sk(x)]
  • We have to infer the 11th expression from the given 10.
  • Done through Resolution Refutation.
club example inferencing
Club example: Inferencing
  • member(A)
  • member(B)
  • member(C)
    • Can be written as
slide13

member(A)

  • member(B)
  • member(C)
  • Now standardize the variables apart which results in the following
slide14

10

7

12

5

4

13

14

2

11

15

16

13

2

17

assignment
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)
problem 2

Problem-2

From predicate calculus

a department environment
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
diagrammatic representation
Diagrammatic representation

CSE

ME

Dr. P

Dr. X

Z

Y

R

Q

married

questions on department
Questions on “department”
  • Who works in CSE?
  • Is there a married person in ME?
  • Is there somebody insured by LIC?
a semantic graph

past tense

bought

time

agent

object

student

June

computer

the: definite

in: modifier

a: indefinite

modifier

new

A Semantic Graph

The student bought a new computer in June.

unl representation
UNL representation

Representation of Knowledge

Ram is reading the newspaper

unl a united nations project
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)
knowledge representation
Knowledge Representation

UNL Graph - relations

read

agt

obj

Ram

newspaper

knowledge representation1
Knowledge Representation

UNL Graph - UWs

read(icl>interpret)

obj

agt

newspaper(icl>print_media)

Ram(iof>person)

knowledge representation2
Knowledge Representation

UNL graph - attributes

@entry

@present

@progress

read(icl>interpret)

obj

agt

@def

newspaper(icl>print_media)

Ram(iof>person)

Ram is reading the newspaper

the boy who works here went to school

go(icl>move)

@ entry @ past

agt

plt

boy(icl>person)

@ entry

school(icl>institution)

here

agt

plc

:01

work(icl>do)

The boy who works here went to school

Another Example

what do these examples show
What do these examples show?
  • Logic systematizes the reasoning process
  • Helps identify what is mechanical/routine/automatable
  • 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
key points
Key points
  • 1. SA and GA are randomized search algorithms;      (a) why does one do randomized search?      (b) To QUICKLY find a solution even if the the solution is not FULLY accurate2. For example, TSP is NP hard; so any algorithm that purports to give the correct tour ALWAYS is going to take exponential amount of time.3. But it may be alright to get the solution certain percentage of time. Then one can use SA/GA.4. For sorting , consider getting the sorted sequences for any set of of numbers of any sequence length, say 200,000 numbers.
key points cntd
Key points cntd
  • 5. It may be OK to get an ALMOST sorted sequence QUICKLY; so see if SA and GA can be used6. SO what is coming out strongly is TIME vs. ACCURACY TRADE-OFF7. ***THE ABOVE HAS TO COME OUT IN YOUR ASSIGNMENT8. What about 8 puzzle? Optimal path is not needed.
key points cntd1
Key points cntd
  • 9. But you HAVE TO demonstrate randomness. That means Ther are times when the goal state will not be reached;10. The above will be the case when randomness is INTRODUCED in the system by making the tempearure HIGH.11. Thus a key point of the assignment is the EFFECT OF HIGH TEMPERATURE on the system.12. Another point about the next state: make sure you pick it up RANDOMLY and not deterministically.13. Think about the connection between BFS and random search. The former will guarantee finding the goal state, the latter not. But there may be gain in time complexity.