1. AI in logic perspective



AI is the study of mental faculties through the
use of computational models.
It is on the premise that what brain does may
be thought of as a kind of computation.
Though what brain does easily takes enormous
efforts to be done by a machine. Eg: vision.
12/23/13
1

2. Internal representation





In order to act intelligently, a computer must
have the knowledge about the domain of
interest.
Knowledge is the body of facts and principles
gathered or the act, fact, or state of knowing.
This knowledge needs to be presented in a
form, which is understood by the machine.
This
unique
format
is
called
internal
representation.
Thus plain English sentences could be translated
into an internal representation and they could
be used to answer based on the given
sentences.
12/23/13
2

3. Properties of internal representation




Internal representation must remove all referential
ambiguity.
Referential ambiguity is the ambiguity about what
the sentence refers to.
Eg: ‘ Raj said that Ram was not well. He must be
lying.’
Who does ‘he ‘ refers to…?.
12/23/13
3

4. Properties of internal representation..






Internal representation should avoid word-sense
ambiguity.
Word-sense ambiguity arise because of multiple
meaning of words.
Eg:
‘Raj caught a pen.
Raj caught a train.
Raj caught fever.’
12/23/13
4

5. Properties of internal representation..




Internal representation must explicitly mention
functional structure
Functional structure is the word order used in the
language to express an idea.
Eg: ‘Ram killed Ravan. Ravan was killed by Ram.’
Thus internal representation may not use the
order of the original sentence.
12/23/13
5

6. Properties of internal representation..

Internal representation should be able handle
complex sentence without losing meaning
attached with it.
12/23/13
6

7. Predicate Calculus




Predicate Calculus is an internal representation
methodology which help us in deducing more
results from the given propositions (statements).
Predicate calculus accesses individual
components of a proposition and represent the
proposition.
For example, the sentence ‘ Raj came late on
Sunday’ can be represented in predicate calculus
as
(came-late Raj Sunday)
Here ‘came-late’ is a predicate that describes the
relation between a person and a day.
12/23/13
7

8. ‘ Raj came late on a rainy Sunday’ can be
represented as

(came-late Raj Sunday)
(inst Sunday rainy)
Predicate permits us to break a statement down
into component parts namely, objects, a
characteristic of the object, or some assertion
about the object.

12/23/13
8

9. Syntax of Predicate calculus
1. Predicate and Arguments
In predicate calculus, a proposition is divided
into two parts:

Arguments (or objects)
Predicate (or assertion)
The arguments are the individual or objects an
assertion is made about. The predicate is the
assertion made about them.

12/23/13
9

10. 






In an English language sentence, objects are
nouns that serve as subject and object of the
sentence and predicate would be the verb or part
of the verb.
For example the proposition:
‘Vinod likes apple’
would be stated as:
(likes Vinod apple)
Where ‘likes’ is the predicate and Vinod and
apple are the arguments.
In some cases, the proposition may not have
any predicates. For example:
Anita is a woman.
12/23/13 i.e. (inst Anita woman).
10

11. 2. Constants
 Constants are fixed value terms that belong to
a given domain.
 They are denoted by numbers and words. Eg:
123,abc.
12/23/13
11

12. 




3.Variables
In predicate calculus, letters may be substituted for
the arguments.
The symbols x or y could be used to designate some
object or individual.
The example “Vinod likes apple “ could be expressed
in variable form if x = Vinod and y = apple. Then the
proposition becomes:
(likes x,y)
If variables are used, then the stated proposition
must be true for any names substituted for the
variables.
12/23/13
12

13. 


Instantiation
Instantiation is the process of assigning the
name of a specific individual or object to a
variable.
That object or individual becomes an
“
instance“ of that variable.
In the previous example, supplying Vinod and
apple for x and y is a case of instantiation.
12/23/13
13

14. 4. Connectives




There are four connectives used in predicate
calculus.
The are ‘not’, ‘and’, ‘or’ and ‘if’.
If p and q are formulas then
(and p, q),
(or p, q), (not p) and
(if p, q) are also
formulas.
They can be expressed in truth tables.
12/23/13
14

15. (not p)



12/23/13
p
T
F
(not p)
F
T
15

16. (and p, q)

p

T

T

F

F
12/23/13
q
T
F
T
F
(and p, q)
T
F
T
F
16

17. 





(or p, q)
p
T
T
F
F
12/23/13
q
T
F
T
F
(or p, q)
T
T
T
F
17

18. (if p, q)





12/23/13
p
T
T
F
F
q
T
F
T
F
(if p, q)
T
F
T
T
18

19. 



5. Quantifiers
A quantifier is a symbol that permits us to state
the range or scope of the variables in a
predicate logic expression.
Two quantifiers are used in logic:
The universal quantifier –’for all’.
i.e (forall (x) f) for a formula f.
The existential quantifier – ‘exists’.
i.e. (exists (x) f) for a formula f.
12/23/13
19

20. 


6. Function applications
It consists of a function which takes zero or
more arguments.
Eg: friend-of(x).
12/23/13
20

21. 



“All Maharastrians are Indian citizens” could be
expressed as:
(forall (x) (if Maharastrian(x) Indiancitizen(x)).
“ Every car has a wheel” could be expressed as:
(forall (x) (if (Car x) (exists (y) wheel-of (x y))).
12/23/13
21

22. The predicate calculus consists of:






A set of constant terms.
A set of variables.
A set of predicates, each with a specified
number of arguments.
A set of functions, each with a specified
number of arguments.
The connectives- ‘if’, ‘and’, ‘or’ and ‘not’.
The quantifiers- ‘exists’ and ‘forall’.
12/23/13
22

23. 
The terms used in predicate calculus are:
 Constant terms.
 Variables.
 Functions applied to the correct number of
terms.
12/23/13
23

24. 
The formulas used in predicate calculus are:
 A predicate applied to the correct number of
terms.
 If p and q are formulas then (if p, q), (and
p, q), or(p, q) and (not p).
 If x is a variable, and p is a formula, then
(exists(x) p), and (forall(x) p).
12/23/13
24

25. 


In predicate calculus, the initial facts from
which we can derive more facts are called
axioms.
The facts we deduce from the axioms are called
theorems.
The set of axioms are not stable and in fact
change over time as new information (axioms)
come.
12/23/13
25

26. Inference Rules





From a given set of axioms, we can deduce more
facts using inference rules. The important
inference rules are:
Modus ponens: From p and (if p q ) infer q.
Chain rule: From (if p q ) and (if q r )
infer (if p r ).
Substitution: if p is a valid axiom, then a
statement derived using consistent substitution of
propositions is also valid.
Simplification: From (and p q) infer p.
12/23/13
26

27. 




Conjunction: From p and q infer (and p q).
Transposition: From (if p q ) infer (if (not q )
(not p))
Universal instantiation: if something is true
of everything, then it is true for any particular
thing.
Abduction: From q and (if p q ) infer p.
(Abduction can lead to wrong conclusions. Still,
it is very important as it gives lot explanation.
For example: medical diagnosis.)
Induction: From (P a), (P, b),…. infer (forall
(x) (P x)).( Induction leads to learning.)
12/23/13
27

28. Express the following in predicate calculus:

Roses are red.
(if (inst x rose) (color x red)).
Violets are blue.
(if (inst x violet) (color x blue)).
Every chicken hatched from an egg.

(forall (x) (if (chicken x) (exists (y) hatched-from(x y))).

Some language is spoken by everyone in this class.

(forall (x) (if (belong-to-class x) (exists (y) speaklanguage(x y))).

If you push anything hard enough, it will fall over.

(forall (x) (if (push-hard x) (fall-over x)).

Everybody loves somebody sometime.

(forall (x) ((exists (y) loves-sometime(x y))).

Anyone with two or more spouses is a bigamist.





(forall (x) ((inst x have-more-spouse) (inst x bigamist(x)))
12/23/13
28

29. 





Arun likes all kinds of food.
Apples are food.
Chicken is a food.
Anything anyone eats and is not killed
by is food.
Varun eats peanuts and is still alive.
Kavita eats everything Varun eats.
12/23/13
29

30. 




The members of The Club are Anil,
Sangita, Ajit and Vanita.
Anil is married to Sangita.
Ajit is Vanita’s brother.
The spouse of every married person
in the club is also in the club.
The last meeting of the club was at
Anil’s house.
12/23/13
30

31. Alternative notations
12/23/13
31

32. 

Knowledge, which is represented in the internal
representation technique predicate calculus,
could be represented in a number of alternative
notations.
The important representations are:



Semantic networks
Slot assertion notation.
Frame notation
12/23/13
32

33. Semantic network ( Associative networks)


One of the oldest and easiest to understand
knowledge representation schemes is the
semantic network.
They are basically graphical depictions of
knowledge that show hierarchical relationships
between objects.
12/23/13
33

34. 

For example ‘Sachin is a cricketer’
ie. ( inst Sachin cricketer), can be represented
in associative network as
Cricketer
inst
Sachin
12/23/13
34

35. 





A semantic network is made up of a number of
ovals or circles called nodes.
Nodes represent objects and descriptive
information about those objects.
Objects can be any physical item, concept,
event or an action.
The nodes are interconnected by links called
arcs.
These arcs show the relationships between the
various objects and descriptive factors.
The arrows on the lines point from an object to
its value along the corresponding arc.
12/23/13
35

36. 


From the viewpoint of predicate calculus,
associative networks replace terms with nodes
and relation with labeled directed arcs.
The semantic network is a very flexible
method of knowledge representation.
There are no hard rules about knowledge in
this form.
12/23/13
36

37. 

Semantic networks can show inheritances in the sense
that it can explain how elements of specific classes
inherit attributes and values from more general classes
in which they are included.
The isa relation is a subset relation. The cricketers is a
subset of the set of sportsman.
Cricketer
inst
isa
Sportsman
Sachin
12/23/13
37

38. 




Eg: (isa cricketer sportsman).
The instance relation corresponds to the
relation element-of.
Sachin is an element of the set of cricketers.
Thus he is an element of all the supersets of
Indian international cricketers.
The ‘isa’ relation corresponds to the relation
‘subset of’.
Cricketers is a subset of sportsmen and hence
cricketers
inherit
al the
properties of
sportsmen.
12/23/13
38

39. Example..
Is a
Boy
has a
Ravi
Child
Goes to
School
Is a
Anitha
owns
Maruti
White
is a
Anil
is a
S.E
a
Human
Is a
works for
plays
Is a
Color
Woman Is
Man
married to
Car
12/23/13
Is a
Belongs to
TATA
Cricket
made in
is a
India
Sport
TCS
39

40. 

The predicate calculus lacks a backward pointer
resulting a long search for retrieving information.
Thus the predicate calculus along with an
indexing (pointing) scheme is a much better
internal representation scheme than semantic
networks as it has connectives and quantifiers.
12/23/13
40

41. Slot assertion notation.






In a slot assertion notation various arguments ,
called slots, of predicate are expressed as
separate assertions.
Slot assertion notation is a special type of
predicate calculus representation.
For example (catch-object sachin ball) can be
expressed as
(inst catch1 catch-object)…. // catch1 is a one
type of catching.
(catcher catch1 sachin)….// sachin did the
catching.
(caught catch1 ball)…..// he caught the ball.
12/23/13
41

42. Frame ( Slot and Filler)notation.






Frame notation combines the different slots of
the slot assertion notation.
Thus we have,
(catch-object catch1
(catcher sachin)
(caught ball)).
Here we have constructed a single structure
called a frame that includes all the
information.
12/23/13
42

43. Convert the following to first-order predicate logic
using the predicates indicated:





swimming_pool(X)
steamy(X)
large(X)
unpleasant(X)
noisy(X)
place(X)
All large swimming pools are noisy and steamy
places.
All noisy and steamy places are unpleasant.
All noisy and steamy places except swimming
pools are unpleasant.
The swimming pool is small and quiet.
12/23/13
43

44. 







All large swimming pools are noisy and steamy places.
(forall (x) (if (and large(X) swimming_pool(X))
(and noisy(X) (and (steamy(X) place(X)))).
All noisy and steamy places are unpleasant.
(forall (x)(and noisy(X) (and (steamy(X) place(X))
unpleasant(X))).
All noisy and steamy places except swimming pools are
unpleasant.
(forall (x)((not swimming_pool(x)) and noisy(X) (and
(steamy(X) place(X)) unpleasant(X)))).
The swimming pool is small and quiet.
(and swimming_pool(x) and (not large(X)) (not noisy(X)))
12/23/13
44

45. Represent in predicate calculus and then in
semantic network
Circus elements are elephants.
Elephants have heads and trunks.
Heads have mouths.
Elephants are animals.
Animals have hearts.
Circus elephants are performers.
Performers have costumes.
Costumes are clothes.
12/23/13
45