[PPT] - Logic and Knowledge Representation R e i n f o r c e m e PowerPoint Presentation

SLIDE 1

Logic and Knowledge Representation

G i

v

a n n i S i l e n

g

s i l e n

@e

n s t . f r

T é l é c

m

P a r i s T e c h , P a r i s

D

a u p h i n e U n i v e r s i t y

R e i n f

r

c e m e n t l e a r n i n g , I n d u c t i v e L

g

i c P r

g

r a m m i n g D e s c r i p t i

n

C

m

p l e x i t y

1 5 J u n e 2 1 8

SLIDE 2

Induction (again) – after Pierce

I n d u c t i

n

F a c t : T h e s e b e a n s a r e f r

m

t h i s b a g . F a c t : T h e s e b e a n s a r e w h i t e .  H y p . r u l e : A l l t h e b e a n s f r

m

t h i s b a g a r e w h i t e .

SLIDE 3

Induction (again) – after Pierce

I n d u c t i

n

F a c t : T h e s e b e a n s a r e f r

m

t h i s b a g . F a c t : T h e s e b e a n s a r e w h i t e .  H y p . r u l e : A l l t h e b e a n s f r

m

t h i s b a g a r e w h i t e .

I

n d u c t i

n

e n a b l e s p r e d i c t i

n

t h r

u

g h t h e s e t t l e d m

d

e l .

SLIDE 4

Induction (again)

3 , 4 , 6 , 8 , 1 2 , 1 4 , 1 8 , 2 , 2 4 , 3 , 3 2 , 3 8 , 4 2 , … ? ? P

s

s i b l e m

d

e l s ?

SLIDE 5

Induction (again)

3 , 4 , 6 , 8 , 1 2 , 1 4 , 1 8 , 2 , 2 4 , 3 , 3 2 , 3 8 , 4 2 , … ? ? P

s

s i b l e m

d

e l s :

n

u m b e r s n + 1 , n p r i m e n u m b e r → 4 4 , 4 8 , 5 4 , . .

SLIDE 6

Induction (again)

3 , 4 , 6 , 8 , 1 2 , 1 4 , 1 8 , 2 , 2 4 , 3 , 3 2 , 3 8 , 4 2 , … ? ? P

s

s i b l e m

d

e l s :

n

u m b e r s n + 1 , n p r i m e n u m b e r → 4 4 , 4 8 , 5 4 , . .

n

u m b e r s n s u c h t h a t f

r

a l l k w i t h g c d ( n , k ) = 1 a n d n > k

2

, n

k

2

i s p r i m e . → 4 8 , 5 4 , 6 , . .

L

k
n

h t t p s : / /

e

i s .

r

g / f

r
t

h e r s .

F

u r t h e r

b

s e r v a t i

n

s e n a b l e t h e c

r

r e c t i

n
f

t h e m

d

e l .

SLIDE 7

Alien environment problem

S

u p p

s

e a r

b
t

l a n d s

n

a n u n k n

w

n p l a n e t .

– i

n

r

d e r t

a

c c

m

p l i s h i t s m i s s i

n

, i t h a s t

a

c q u i r e a n

p

e r a t i

n

a l k n

w

l e d g e

f

:

w

h a t ( m i g h t )

c

c u r

w

h a t i t s a c t i

n

s ( m i g h t ) a c h i e v e f r

m

i t s

b

s e r v a t i

n

s ! ! !

I n d u c t i

n

SLIDE 8

Reinforcement Learning

SLIDE 9

Nim game

T

w

p

l a y e r s g a m e

E

a c h p l a y e r m a y t a k e a s m a n y i t e m s f r

m

a s i n g l e r

w

i n t u r n

T

h e

n

e w h

t

a k e s t h e l a s t i t e m l

s

e s .

SLIDE 10

Nim game

H
w
n

e c a n l e a r n t

w

i n w i t h

u

t k n

w

i n g t h e r u l e s ?

– r

e c

r

d i n g s t a t e s e n c

u

n t e r e d d u r i n g t h e p l a y

– u

p d a t i n g v a l u e

f

s t a t e s w i t h fj n a l r e s u l t s ( w

n
r

l

s

t )

– s

e l e c t i n g a c t i

n

s b r i n g i n g t

w

i n n i n g s t a t e s v e r y s i m p l e e x a m p l e

f

r e i n f

r

c e m e n t l e a r n i n g !

SLIDE 11

Example of reinforcement learning algorithm

p* ( s ) = a r g m a x

a

( R ( s , a ) + V * ( ( s , a ) ) )

best strategy in s state of the world expected immediate reinforcement in performing a in s action transition function: a performed in s leads to s' discount factor expected (long-term) gain from a state s'

SLIDE 12

Example of reinforcement learning algorithm

p* ( s ) = a r g m a x

a

( R ( s , a ) + V * ( ( s , a ) ) )

best strategy in s state of the world expected immediate reinforcement in performing a in s action transition function: a performed in s leads to s' discount factor expected (long-term) gain from a state s'

Q ( s , a ) = R ( s , a ) + V * ( ( s , a ) ) U t i l i t y f u n c t i

n

expected gain

SLIDE 13

Q-learning

Q ( s , a ) = R ( s , a ) +  V * ( ( s , a ) ) = R ( s , a ) +  m a x

a '

( Q ( s ' , a ' ) ) = R ( s , a ) +  m a x

a '

( Q ( ( s , a ) , a ' ) ) p* ( s ) = a r g m a x

a

Q ( s , a )

SLIDE 14

Q-learning algorithm

Q ( s , a ) = R ( s , a ) +  m a x

a '

( Q ( ( s , a ) , a ' ) ) p* ( s ) = a r g m a x

a

Q ( s , a )

initialize the table Q(s,a) to zero

bserve the current state s.

repeat choose an action and execute it receive the reward r

bserve the new state s'

update the table Q(s,a) as: Q(s,a) := r + max Q(s’,a’) s := s’

a’

SLIDE 15

This was about behaviour, but what about knowledge?

I n d u c t i

n

a s g e n e r a l i z a t i

n

. . .

SLIDE 16

Version space learning

[ D u b

i

s , V i n c e n t ; Q u a f a f

u

, M

h

a m e d ( 2 2 ) . " C

n

c e p t l e a r n i n g w i t h a p p r

x

i m a t i

n

: R

u

g h v e r s i

n

s p a c e s " . R S C T C 2 2 . S v e r d l i k , W. ; R e y n

l

d s , R . G . ( 1 9 9 2 ) . " D y n a m i c v e r s i

n

s p a c e s i n m a c h i n e l e a r n i n g " . T A I ' 9 2 . ]

L
g

i c a l a p p r

a

c h t

b

i n a r y c l a s s i fj c a t i

n
S

e a r c h

n

a p r e d e fj n e d s p a c e

f

h y p

t

h e s e s : H

1

H ∨

2

. . . H ∨ ∨

n

Y
u

d

n
t

n e e d t

m

a i n t a i n e x e m p l a r s !

positive example counter-example specific boundary general boundary pessimistic

ptimistic

SLIDE 17

Using a version space

represent E predict from the representation of H whether or not E exemplifies H if correct then retain H if incorrect then identify the differences between E and H use the selected differences to

– generalize H if it is a positive instance – specialize H if it is a negative instance

c a n d i d a t e e l i m i n a t i

n

a l g

r

i t h m c u r r e n t e x a m p l e c u r r e n t h y p

t

h e s i s , t a k e n f r

m

a v e r s i

n

s p a c e

SLIDE 18

Machine learning

SLIDE 19

Machine learning

M a c h i n e l e a r n i n g i s a p r

c

e s s t h a t e n a b l e s a r t i fj c i a l s y s t e m s t

i

m p r

v

e w i t h e x p e r i e n c e .

what are the criteria?

SLIDE 20

Machine learning

M a c h i n e l e a r n i n g i s a p r

c

e s s t h a t e n a b l e s a r t i fj c i a l s y s t e m s t

i

m p r

v

e w i t h e x p e r i e n c e .

E

l e m e n t s

f

a l e a r n i n g t a s k

– I

t e m s

f

E x p e r i e n c e , i I ∈

– A

v a i l a b l e A c t i

n

s : a A ∈

– E

v a l u a t i

n

: v ( a , I )

– P

e r f

r

m e r S y s t e m : b : I A →

– L

e a r n i n g S y s t e m : L : ( i

1

, a

1

, v

1

) . . . ( i

n

, a

n

, v

n

) b →

SLIDE 21

Types of learning problems

b

a t c h

r
ffmi

n e v s

n

l i n e l e a r n i n g

t r a i n i n g p h a s e a n d t e s t i n g v s l e a r n i n g w h i l e d

i

n g

c
m

p l e t e v s p a r t i a l v s p

i

n t w i s e f e e d b a c k

f e e d b a c k c

n

c e r n s a l l v s s

m

e v s

n

e p e r f

r

m e r s y s t e m

p

a s s i v e v s a c t i v e l e a r n i n g

b

s e r v a t i

n

v s e x p e r i m e n t a t i

n
a

c a u s a l

r

c a s u a l s e t t i n g

p r e s e n c e

r

n

t
f

s i d e

e

fg e c t s : e . g . r a i n p r e d i c t i

n

v s b e h a v i

u

r a l c

n

t r

l
s

t a t i

n

a r y v s n

n
s

t a t i

n

a r y e n v i r

n

m e n t

e v a l u a t i

n

d

e

s

r

d

e

s n

t

c h a n g e i n t i m e

SLIDE 22

Learning a function from examples

d

m

a i n X : d e s c r i p t i

n

s d

m

a i n Y : p r e d i c t i

n

s H : h y p

t

h e s i s s p a c e h : t a r g e t h y p

t

h e s i s ( x

1

, y

1

) ( x

2

, y

2

) . . ( x

n

, y

n

)

h ∈ H , h : X → Y

l e a r n e r

examples function

SLIDE 23

Learning a function from examples

d

m

a i n X : d e s c r i p t i

n

s d

m

a i n Y : p r e d i c t i

n

s H : h y p

t

h e s i s s p a c e h : t a r g e t h y p

t

h e s i s ( x

1

, y

1

) ( x

2

, y

2

) . . ( x

n

, y

n

)

h ∈ H , h : X → Y

l e a r n e r

examples function

M

a n y l e a r n i n g m e t h

d

s a r e a v a i l a b l e , b u t s t u d i e d a n d u s e d b y d i fg e r e n t c

m

m u n i t i e s !

A

f e w e x a m p l e s . . .

SLIDE 24

Learning a function from examples

d

m

a i n X : d e s c r i p t i

n

s d

m

a i n Y : p r e d i c t i

n

s H : h y p

t

h e s i s s p a c e h : t a r g e t h y p

t

h e s i s ( x

1

, y

1

) ( x

2

, y

2

) . . ( x

n

, y

n

)

h ∈ H , h : X → Y

l e a r n e r

examples function

M

e t h

d

1 : t r a d i t i

n

a l s t a t i s t i c s ( r e g r e s s i

n

a n a l y s i s ) h : R

n

→ R h i s a l i n e a r f u n c t i

n

s q u a r e d p r e d i c t i

n

e r r

r

SLIDE 25

Learning a function from examples

d

m

a i n X : d e s c r i p t i

n

s d

m

a i n Y : p r e d i c t i

n

s H : h y p

t

h e s i s s p a c e h : t a r g e t h y p

t

h e s i s ( x

1

, y

1

) ( x

2

, y

2

) . . ( x

n

, y

n

)

h ∈ H , h : X → Y

l e a r n e r

examples function

M

e t h

d

2 : t r a d i t i

n

a l p a t t e r n r e c

g

n i t i

n

h : R

n

→ { , 1 , … , m } h i s a d i s c r i m i n a n t b

u

n d a r y r i g h t / w r

n

g p r e d i c t i

n

e r r

r

SLIDE 26

Learning a function from examples

d

m

a i n X : d e s c r i p t i

n

s d

m

a i n Y : p r e d i c t i

n

s H : h y p

t

h e s i s s p a c e h : t a r g e t h y p

t

h e s i s ( x

1

, y

1

) ( x

2

, y

2

) . . ( x

n

, y

n

)

h ∈ H , h : X → Y

l e a r n e r

examples function

M

e t h

d

3 : “ s y m b

l

i c ” m a c h i n e l e a r n i n g h : { a t t r i b u t e

v

a l u e v e c t

r

s } → { , 1 } h i s a b

l

e a n f u n c t i

n

( e . g . a d e c i s i

n

t r e e )

SLIDE 27

Learning a function from examples

d

m

a i n X : d e s c r i p t i

n

s d

m

a i n Y : p r e d i c t i

n

s H : h y p

t

h e s i s s p a c e h : t a r g e t h y p

t

h e s i s ( x

1

, y

1

) ( x

2

, y

2

) . . ( x

n

, y

n

)

h ∈ H , h : X → Y

l e a r n e r

examples function

M

e t h

d

4 : N e u r a l n e t w

r

k s

– h

: R

n

→ R

– h

i s a f e e d f

r

w a r d n e u r a l n e t

SLIDE 28

Learning a function from examples

d

m

a i n X : d e s c r i p t i

n

s d

m

a i n Y : p r e d i c t i

n

s H : h y p

t

h e s i s s p a c e h : t a r g e t h y p

t

h e s i s ( x

1

, y

1

) ( x

2

, y

2

) . . ( x

n

, y

n

)

h ∈ H , h : X → Y

l e a r n e r

examples function

M

e t h

d

5 : I n d u c t i v e L

g

i c P r

g

r a m m i n g

– h

: { t e r m s t r u c t u r e } → { , 1 }

– h

i s a “ s i m p l e ” l

g

i c p r

g

r a m .

SLIDE 29

Inductive Logic Programming

SLIDE 30

Symbolic induction

B a c k g r

u

n d k n

w

l e d g e E x e m p l a r

f

c l a s s X

E1 = square(A) & circle(B) & above(A, B) E2 = triangle(C) & square(D) & above(C, D)

Wh a t i s X ? a g r

u

p

f

g e

m

e t r i c s h a p e s ? a g r

u

p

f

2 g e

m

e t r i c s h a p e s ? a g r

u

p

f

2 g e

m

e t r i c s h a p e s w i t h a s q u a r e ?

I

n d u c t i

n

a s l e a s t g e n e r a l g e n e r a l i z a t i

n
f

e x e m p l a r s .

E x e m p l a r

f

c l a s s X

SLIDE 31

Inductive Logic Programming

E

x a m p l e s :

cute(X) :- dog(X), small(X), fluffy(X). cute(X) :- cat(X), fluffy(X).

G

e n e r a l i s a t i

n

:

cute(X) :- fluffy(X).

SLIDE 32

Inductive Logic Programming

E

x a m p l e s :

cute(X) :- dog(X), small(X), fluffy(X). cute(X) :- cat(X), fluffy(X).

G

e n e r a l i s a t i

n

:

cute(X) :- fluffy(X).

B

a c k g r

u

n d k n

w

l e d g e :

pet(X) :- cat(X). pet(X) :- dog(X). small(X) :- cat(X).

G

e n e r a l i s a t i

n

:

cute(X) :- pet(X), small(X), fluffy(X).

SLIDE 33

Inductive Logic Programming

E

x a m p l e s E a r e e x p e c t e d t

r

e s u l t f r

m

b a c k g r

u

n d k n

w

l e d g e B a n d h y p

t

h e s i s H : B H ∧ ⊨ E

I

n v e r s e r e s

l

u t i

n

:

– f

r

m

e x a m p l e :

cute(X) :- cat(X), fluffy(X).

– f

r

m

k n

w

l e d g e :

pet(X) :- cat(X). small(X) :- cat(X).

i n d u c e :

cute(X) :- pet(X), small(X), fluffy(X).

SLIDE 34

Explanation-Based Generalization

SLIDE 35

Explanation-based Generalization

telephone(T) :- connected(T), partOf(T, D), dialingDevice(D), emitsSound(T). connected(X) :- hasWire(X, W), attached(W, wall). connected(X) :- feature(X, bluetooth). connected(X) :- feature(X, wifi). connected(X) :- partOf(X, A), antenna(A), hasProtocol(X, gsm). dialingDevice(DD) :- rotaryDial(DD). dialingDevice(DD) :- frequencyDial(DD). dialingDevice(DD) :- touchScreen(DD), hasSoftware(DD,DS), dialingSoftware(DS). emitsSound(P) :- hasHP(P). emitsSound(P) :- feature(P, bluetooth).

SLIDE 36

Explanation-based Generalization

example(myphone, Features) :- Features = [silver(myphone), belongs(myphone, jld), partOf(myphone, tc), touchScreen(tc), partOf(myphone, a), antenna(a), hasSoftware(tc, s1), game(s1), hasSoftware(tc, s2), dialingSoftware(s2), feature(myphone,wifi), feature(myphone,bluetooth), hasProtocol(myphone, gsm), beautiful(myphone)].

F

e a t u r e s a c t i v a t e d d u r i n g t h e p r

f

:

[ feature(myphone, bluetooth), partOf(myphone, tc), touchScreen(tc), hasSoftware(tc, s2), dialingSoftware(s2), feature(myphone, bluetooth) ]

SLIDE 37

Explanation-based Generalization

F

e a t u r e s a c t i v a t e d d u r i n g t h e p r

f

:

[ feature(myphone, bluetooth), partOf(myphone, tc), touchScreen(tc), hasSoftware(tc, s2), dialingSoftware(s2), feature(myphone, bluetooth) ]

SLIDE 38

Explanation-based Generalization

F

e a t u r e s a c t i v a t e d d u r i n g t h e p r

f

:

[ feature(myphone, bluetooth), partOf(myphone, tc), touchScreen(tc), hasSoftware(tc, s2), dialingSoftware(s2), feature(myphone, bluetooth) ]

F

r

m

t h e t r a c e , b y g e n e r a l i z i n g s h a r e d c

n

s t a n t s :

C001(X) :- feature(X, bluetooth), partOf(X, Y), touchScreen(Y), hasSoftware(Y, Z), dialingSoftware(Z).

SLIDE 39

Explanation-based Generalization

F

e a t u r e s a c t i v a t e d d u r i n g t h e p r

f

:

[ feature(myphone, bluetooth), partOf(myphone, tc), touchScreen(tc), hasSoftware(tc, s2), dialingSoftware(s2), feature(myphone, bluetooth) ]

F

r

m

t h e t r a c e , b y g e n e r a l i z i n g s h a r e d c

n

s t a n t s :

C001(X) :- feature(X, bluetooth), partOf(X, Y), touchScreen(Y), hasSoftware(Y, Z), dialingSoftware(Z).

B

y g r

u

p i n g p r e d i c a t e s t h a t d

n
t

d e p e n d

n

X :

C002(Y) :- touchScreen(Y), hasSoftware(Y, Z), dialingSoftware(Z).

SLIDE 40

Explanation-based Generalization

F

e a t u r e s a c t i v a t e d d u r i n g t h e p r

f

:

[ feature(myphone, bluetooth), partOf(myphone, tc), touchScreen(tc), hasSoftware(tc, s2), dialingSoftware(s2), feature(myphone, bluetooth) ]

F

r

m

t h e t r a c e , b y g e n e r a l i z i n g s h a r e d c

n

s t a n t s :

C001(X) :- feature(X, bluetooth), partOf(X, Y), touchScreen(Y), hasSoftware(Y, Z), dialingSoftware(Z).

B

y g r

u

p i n g p r e d i c a t e s t h a t d

n
t

d e p e n d

n

X :

C002(Y) :- touchScreen(Y), hasSoftware(Y, Z), dialingSoftware(Z).

C

1 t h e n b e c

m

e s

C001(X) :- feature(X, bluetooth), partOf(X, Y), C002(Y).

SLIDE 41

Description Complexity

SLIDE 42

Informal definition of Kolmogorov complexity

T

h e c

m

p l e x i t y

f

a n

b

j e c t c

r

r e s p

n

d s t

t

h e m i n i m a l l e n g t h

f

a c

m

p u t e r p r

g

r a m p r

d

u c i n g t h i s

b

j e c t .

[ i n f

r

m a l r e d u c t i

n
f

a p r e s e n t a t i

n

b y P i e r r e

A

l e x a n d r e M u r e n a ( T é l é c

m

P a r i s T e c h ) ]

SLIDE 43

Informal definition of Kolmogorov complexity

T

h e c

m

p l e x i t y

f

a n

b

j e c t c

r

r e s p

n

d s t

t

h e m i n i m a l l e n g t h

f

a c

m

p u t e r p r

g

r a m p r

d

u c i n g t h i s

b

j e c t .

A

fj n i t e s t r i n g l i k e “ a a a . . . ” i s n

t

v e r y c

m

p l e x : for i=1..n: print a

SLIDE 44

Informal definition of Kolmogorov complexity

T

h e c

m

p l e x i t y

f

a n

b

j e c t c

r

r e s p

n

d s t

t

h e m i n i m a l l e n g t h

f

a c

m

p u t e r p r

g

r a m p r

d

u c i n g t h i s

b

j e c t .

A

fj n i t e s t r i n g l i k e “ a a a . . . ” i s n

t

v e r y c

m

p l e x : for i=1..n: print a

I

s p c

m

p l e x ? p/ 4 = 1 – 1 / 3 + 1 / 5 – 1 / 7 + 1 / 9 – . . .

SLIDE 45

Informal definition of Kolmogorov complexity

T

h e c

m

p l e x i t y

f

a n

b

j e c t c

r

r e s p

n

d s t

t

h e m i n i m a l l e n g t h

f

a c

m

p u t e r p r

g

r a m p r

d

u c i n g t h i s

b

j e c t .

A

fj n i t e s t r i n g l i k e “ a a a . . . ” i s n

t

v e r y c

m

p l e x : for i=1..n: print a

I

s p c

m

p l e x ? p/ 4 = 1 – 1 / 3 + 1 / 5 – 1 / 7 + 1 / 9 – . . .

K

l

m

g
r
v

c

m

p l e x i t y i s i n c

m

p u t a b l e .

SLIDE 46

Randomness

SLIDE 47

Randomness

A

r e b

t

h t h e s e s e q u e n c e s e q u a l l y r a n d

m

? 00000000000001111111111111 10010011011000111010110010

SLIDE 48

Randomness and compression

A

r e b

t

h t h e s e s e q u e n c e s e q u a l l y r a n d

m

? 00000000000001111111111111 10010011011000111010110010

A

fj n i t e s e q u e n c e i s s a i d t

b

e r a n d

m

i f i t i s i n c

m

p r e s s i b l e , i . e . i f i t s s h

r

t e s t d e s c r i p t i

n

i s t h e s e q u e n c e i t s e l f .

SLIDE 49

Deduction

D

e d u c t i

n

g e n e r a l l y w

r

k s f r

m

t h e g e n e r a l t

t

h e p a r t i c u l a r g e n e r a l p r e m i s e a n d p a r t i c u l a r p r e m i s e p a r t i c u l a r c

n

c l u s i

n

→

all animals eat fido is an animal fido eats. →

g e n e r a l p r e m i s e a n d l e s s g e n e r a l p r e m i s e s l e s s g e n e r a l c

n

c l u s i

n

→

all animals eat cats are animals cats eat. →

SLIDE 50

Deduction and compression

D

e d u c t i

n

g e n e r a l l y w

r

k s f r

m

t h e g e n e r a l t

t

h e p a r t i c u l a r

I

n t u i t i

n

: A f

r

m a l s y s t e m i s a c

m

p r e s s i

n
f

t h e s e t

f

t h e

r

e m s i t c a n p r

v

e .

SLIDE 51

Deduction and compression

D

e d u c t i

n

g e n e r a l l y w

r

k s f r

m

t h e g e n e r a l t

t

h e p a r t i c u l a r

I

n t u i t i

n

: A f

r

m a l s y s t e m i s a c

m

p r e s s i

n
f

t h e s e t

f

t h e

r

e m s i t c a n p r

v

e .

Understanding is compressing.

SLIDE 52

Minimum Description Length as inductive principle

T

h e M D L p r i n c i p l e s t a t e s t h a t : t h e b e s t t h e

r

y t

d

e s c r i b e

b

s e r v e d d a t a i s t h e

n

e w h i c h m i n i m i z e s t h e s u m

f

t h e d e s c r i p t i

n

l e n g t h ( i n b i t s )

f

:

– t

h e t h e

r

y d e s c r i p t i

n

– t

h e d a t a e n c

d

e d f r

m

t h e t h e

r

y

SLIDE 53

Hofstadter’s problems

A B C : A B D : : I J K : x R S T : R S U : : R R S S T T : x A B C : A B D : : B C A : x A B C : A B D : : A A B A B C : x I J K : I J L : : I J J K K K : x

problems of analogy

SLIDE 54

Hofstadter’s problems

A B C : A B D : : I J K : x R S T : R S U : : R R S S T T : x A B C : A B D : : B C A : x A B C : A B D : : A A B A B C : x I J K : I J L : : I J J K K K : x

problems of analogy

L

e t u s a p p l y t h e M D L p r i n c i p l e t

d

e c i d e x :

– w

e n e e d t

s

e t t l e a d e s c r i p t i

n

l a n g u a g e w i t h a s e t

f
p

e r a t

r

s m a n i p u l a t i n g s t r i n g s .

– w

e i n t e r p r e t t h e d a t a t h r

u

g h t h e d e s c r i p t i

n

l a n g u a g e

– w

e c

m

p u t e t h e c

m

p l e x i t y

f

t h e h y p

t

h e t i c a l

r

g a n i z a t i

n

s

SLIDE 55

Hofstadter’s problems

A B C : A B D : : I J K : x R S T : R S U : : R R S S T T : x A B C : A B D : : B C A : x A B C : A B D : : A A B A B C : x I J K : I J L : : I J J K K K : x

problems of analogy

SLIDE 56