Post on 05-Jan-2016
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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
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 Negate the goal
~man(shakespeare) Get a pair of resolvents
)(~ eshakespearmortal )()(~ eshakespearmortaleshakespearman
)(~ eshakespearman )(~ eshakespearman
Resolution Tree
1Re solvent 2Re solvent
soluteRe
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
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
QP
QP
1. P
2. converted to
3.
Draw the resolution tree (actually an inverted tree). Every node is a clausal form and branches are intermediate inference steps.
QP QP ~
Q~
Q~
QP ~
P~ P
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
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. Let mc denote mountain climber and sk denotes skier.
Knowledge representation in the given problem is as follows:
1. member(A)2. member(B)3. member(C)4. ∀x[member(x) → (mc(x) ∨ sk(x))]5. ∀x[mc(x) → ~like(x,rain)]6. ∀x[sk(x) → like(x, snow)]7. ∀x[like(B, x) → ~like(A, x)]8. ∀x[~like(B, x) → like(A, x)]9. like(A, rain)10. like(A, snow)11. 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: Inferencing1. member(A)
2. member(B)
3. member(C)
4. – Can be written as –
5. –
6. –
7. –
))]()(()([ xskxmcxmemberx
))]()(()([ xskxmcxmember )()()(~ xskxmcxmember )],()([ snowxlkxskx
),()(~ snowxlkxsk )],(~)([ rainxlkxmcx
),(~)(~ rainxlkxmc )],(~),([ xBlkxAlikex
),(~),(~ xBlkxAlike
8. –
9.
10.
11. – Negate–
)],(),([~ xBlkxAlkx
),(),( xBlkxAlk
),( rainAlk
),( snowAlk
)](~)()([ xskxmcxmemberx
)]()(~)([~ xskxmcxmemberx
Now standardize the variables apart which results in the following
1. member(A)
2. member(B)
3. member(C)
4.
5.
6.
7.
8.
9.
10.
11.
)()()(~ 111 xskxmcxmember
),()(~ 22 snowxlkxsk ),(~)(~ 33 rainxlkxmc
),(~),(~ 44 xBlkxAlike
),(),( 55 xBlkxAlk
),( rainAlk
),( snowAlk)]()(~)([~ 666 xskxmcxmemberx
),(~),(~ 44 xBlkxAlike ),( snowAlk
),(~ snowBlk ),()(~ 22 snowxlkxsk
)()()(~ 111 xskxmcxmember )(~ Bsk
)()(~ BmcBmember )(Bmember
)(Bmc)]()(~)([~ 666 xskxmcxmemberx
)()(~ BskBmember )(~ Bsk
)(~ Bmember )(Bmember
710
12 5
13 4
14 2
1115
16 13
17 2
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
From predicate calculus
A “department” environment1. Dr. X is the HoD of CSE2. Y and Z work in CSE3. Dr. P is the HoD of ME4. Q and R work in ME5. Y is married to Q6. By Institute policy staffs of the same
department cannot marry7. All married staff of CSE are insured by LIC8. HoD is the boss of all staff in the
department
Diagrammatic representation
Dr. X
Y Z
Dr. P
Q R
married
CSE ME
Questions on “department”
Who works in CSE? Is there a married person in ME? Is there somebody insured by LIC?
Text Knowledge Representation
A Semantic Graph
in: modifiera: indefinite
the: definite
student
past tense
agent
bought
objecttime
computer
new
June
modifier
The student bought a new computer in June.
UNL representation
Ram is reading the newspaper
Representation of Knowledge
UNL: a United Nations project
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)
Dave, Parikh and Bhattacharyya, Journal of Machine Translation, 2002
Knowledge Representation
Ram
read
newspaper
agt obj
UNL Graph - relations
Knowledge Representation
Ram(iof>person)
read(icl>interpret)
newspaper(icl>print_media)
UNL Graph - UWs
agt obj
Knowledge Representation
Ram(iof>person)
read(icl>interpret)
newspaper(icl>print_media)
@entry@present@progress
@def
Ram is reading the newspaper
UNL graph - attributes
agt obj
The boy who works here went to school
plt
agt@ entry @ past
school(icl>institution)
go(icl>move)
boy(icl>person)
work(icl>do)
here
@ entry
agt plc
:01
Another Example
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
About the SA/GA assignments
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 accurate
2. 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
5. It may be OK to get an ALMOST sorted sequence QUICKLY; so see if SA and GA can be used
6. SO what is coming out strongly is TIME vs. ACCURACY TRADE-OFF
7. ***THE ABOVE HAS TO COME OUT IN YOUR ASSIGNMENT
8. What about 8 puzzle? Optimal path is not needed.
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.