INDUCTIVE REASONING IN MATHEMATICS
F. M a l l o y Brown
Department o f A r t i f i c i a l I n t e l l i g e n c e
U n i v e r s i t y o f Edinburgh
Scotland
a:
Sten-Ake T a r n l u n d
Dept. of I n f o r m a t i o n P r o c e s s i n g & Computer Science
U n i v e r s i t y o f Stockholm,
Royal I n s t i t u t e o f Technology, Sweden.
Abstract
We i n v e s t i g a t e s e v e r a l methods of i n d u c t i v e
r e a s o n i n g i n the domain o f d i f f e r e n c e e q u a t i o n s ,
i n c l u d i n g the method o f g e n e r a l i z a t i o n w i t h b e l i e f s ,
t h e method of successive r e f i n e m e n t , and temporal
methods based on comparisons w i t h p r e v i o u s l y s o l v e d
problems.
1.
Introduction
We b e g i n a study of i n d u c t i v e reasoning in
mathematics.
I n d u c t i v e r e a s o n i n g i n mathematics
d i f f e r s from i n d u c t i v e r e a s o n i n g i n the e m p i r i c a l
sciences i n t h a t t h e r e i s a n u l t i m a t e t e s t a l t h o u g h
n o t n e c e s s a r i l y a d e c i s i o n p r o c e d u r e , which can be
used t o determine what i s a c o r r e c t i n d u c t i o n .
Namely, t h a t i f what i s induced i s t r u e , t h a t i s :
i f i t can b e deduced b y o n e ' s d e d u c t i v e system
then i t must b e c o r r e c t .
Because o f t h i s r e l a t i o n ­
s h i p between i n d u c t i o n and d e d u c t i o n one of our
g o a l s i s t o c l a s s i f y what i n s t a n c e s o f r e a s o n i n g
are i n d u c t i v e and what are d e d u c t i v e .
Our main g o a l however, is to c r e a t e a taxonomy
o f f e a s i b l e i n d u c t i v e methods.
Such a g o a l i s n o t
u n r e l a t e d to t h e work of the m a t h e m a t i c i a n : G. Polya,
[1967, 1968A, 1968B], i t i s j u s t more d e t a i l e d .
In
p a r t i c u l a r , w e t r y t o c a r r y o u t the a n a l y s i s o f
each method to a l e v e l d e t a i l e d enough so as to be
programmed.
A l s o , where necessary, we r e l a t e the
i n d u c t i v e methods t o the l e v e l and c a p a b i l i t i e s o f
contemporary d e d u c t i v e systems.
Due t o t h i s r e ­
quirement f o r d e t a i l w e s h a l l r e s t r i c t our a t t e n ­
t i o n t o a p a r t i c u l a r , b u t s i g n i f i c a n t domain.
1.1 Our Problem Domain
We c o n s i d e r t h e problem of t r y i n g to f i n d by
i n d u c t i v e reasoning a closed-form s o l u t i o n , t h a t
is an algebraic s o l u t i o n , to a recursive f u n c t i o n .
That i s , from a s e t K o f equations o f t h e f o r m * :
ψ k ( n , f n , f (n-h) ,f (n-2h) . . .) =0 or r a t h e r :
fn = ψk ( n , f ( n - h ) , f ( n - 2 h ) , . . . )
we w i s h to f i n d an e q u a t i o n of t h e f o r m :
Vn fn - øn where f does n o t occur in ø.
For example, g i v e n the r e c u r s i v e e q u a t i o n f o r
the F i b o n a c c i f u n c t i o n :
F(n+2) = F(n+1)+Fn
Fl = 1
Fo = 0
We would l i k e to f i n d a theorem of the f o r m :
Vn Fn = øm
where øn is a sentence c o n s t r u c t e d from a l g e b r a i c
symbols such as numerals, p l u s ( + ) , times ( * ) ,
power ( ) , minus ( - ) , d i v i s i o n ( / ) , l o g a r i t h m ( I n ) ,
* Our c o n v e n t i o n
expressions i s t o
fore the f u n c t i o n
Thus we w r i t e (F
(a) + ( b ) .
f o r p a r e n t h e s i z i n g complex sub­
place the l e f t parenthesis be­
symbol as is done in EVAL-LISP.
n) and (a + b) n o t F(n) and
Socialized
s i n e ( s i n ) and c o s i n e ( c o s ) .
Or as another example, g i v e n t h e r e c u r s i v e
e q u a t i o n s f o r the minimum number of moves t h a t must
be made in the Tower of Hanoi p u z z l e of n d i s c s (a
d e s c r i p t i o n of t h i s p u z z l e may be found in Luger
[19761.)
H(n+1) = 2 ( H n ) + l
HO = 0
we would l i k e to f i n d a c l o s e d form s o l u t i o n :
Vn Hn = øn.
1.2 A d e f i n i t i o n of " I n d u c t i v e Reasoning"
O f course i f we, o r r a t h e r our program a l ­
ready "knew" a c l o s e d form s o l u t i o n øn f o r a r e ­
c u r s i v e f u n c t i o n such as F or H then we would n o t
want to c a l l t h e method by which t h a t øn was p r o ­
duced:
"inductive reasoning".
The problem then
o f d e f i n i n g j u s t which methods o f p r o d u c i n g ø n are
examples of i n d u c t i v e r e a s o n i n g , and which are n o t
- r e s i d e s i n t h e q u e s t i o n a s t o what i t means f o r
a system to "know" something.
For example if the
system had;
Vn Hn = øn
e x p l i c i t l y s t o r e d as an axiom we would c l e a r l y say
t h a t the system knew t h a t Vn Hn = øn. F u r t h e r ­
more, if the system c o u l d apply a sequence of items
( i . e . axioms, lemmas, axiom schemas and lemma
schemas, w r i t t e n say in LISP) which t r a n s f o r m e d Hn
i n t o ø n , then i f the system knew t h a t each a p p l i c ­
a t i o n of an i t e m in t h a t sequence produced a c o r ­
r e c t r e s u l t , t h e n again we would say t h a t the
system knew t h a t (Vn Hn = øn) .
So f a r t h i s is an omnipotent sense of knowledge:
We know whatever is d e d u c i b l e .
We would l i k e to
modify i t b y t h e f u r t h e r r e q u i r e m e n t o f f e a s i b i l i t y .
That i s , any sequence o f a p p l i c a t i o n o f items must
n o t be so l o n g as to exhaust the system's resources
o r our p a t i e n c e .
I n regard t o f e a s i b i l i t y , i t i s
p o i n t e d o u t t h a t w e d o n ' t s t r i c t l y speaking have
t o "know" t h a t each a p p l i c a t i o n o f a n i t e m i s c o r ­
r e c t , ( f o r a f t e r a l l , p r o o f s from axioms o n l y are
v e r y l o n g ) , b u t r a t h e r a l l t h a t i s needed i s the
p o s s i b i l i t y o f "knowing" t h a t our items are c o r r e c t
in the sense of an e x t e n s i b l e d e d u c t i v e system
(Brown [ 1 9 7 6 A ] ) , coupled w i t h t h e " b e l i e f " * t h a t
our items are c o r r e c t i n a t l e a s t t h e sense o f never
h a v i n g been r e f u t e d ( i . e . shown t o b e i n c o r r e c t )
(Popper[1968]) .
In summary then we w i l l c a l l any method of
p r o d u c i n g a c l o s e d - f o r m s o l u t i o n øn, which is n o t
"known" t o b e the c l o s e d - f o r m s o l u t i o n , i n d u c t i v e
reasoning.
For example, enumeration o f a l l p o s s ­
i b l e a l g e b r a i c f u n c t i o n s o r random guessing would
count a s i n d u c t i v e reasoning ( a l b e i t poor i n d u c t i v e
* We speak of "knowledge" as t r u e " b e l i e f " .
The
p o i n t i s t h a t a l t h o u g h w e hope t h a t our d e d u c t i v e
system is sound, t h e r e does n o t seem to be anyway
t o a b s o l u t e l y guarantee t h i s f a c t .
S stems-4:
8 4
Brown
reasoning) because as items of a d e d u c t i v e system
they would q u i c k l y b e r e f u t e d b y p r o d u c i n g f a l s e
results.
to the elements of each sequence.
For e x a m p l e , i n
t h e case of Hn we m i g h t s u b t r a c t from each element
o f the b e l i e f sequence, t h e corresponding element
generated by Hn, o b t a i n i n g
2. Elementary I n d u c t i v e Methods
We f i r s t d e s c r i b e two b a s i c methods of i n ­
d u c t i v e r e a s o n i n g , and t h e n compare them.
2 . 1 The Method o f G e n e r a l i z a t i o n w i t h B e l i e f s
One p o p u l a r t h e o r y of i n d u c t i v e r e a s o n i n g
(Meltzer [ 1 9 6 9 ] , P l o t k i n [ 1 9 6 9 , 1971],Popplestone
[ 1 9 6 9 ] , Feldman[1969], Hardy[1976]) suggests t h a t
it is basically generalization.*
Consider f o r
example the Tower of Hanoi p r o b l e m .
Using t h e
r e c u r s i v e e q u a t i o n s we can deduce c l o s e d - f o r m s o l u ­
t i o n s f o r n u m e r i c a l i n s t a n c e s o f Hn.
HO = 0
HI = HO+1 = 2.04-1 = 1
H2 - 2H1+1 = 2.1+1 = 3
H3 - 2H2+1 = 2.3+1 = 7
H4 = 2H3+1 = 2 . 7 + 1 - 1 5
H5 - 2H4+1 = 2.15+1 = 31
H6 - 2H5+1 = 2.31+1 = 63
I suspect the reader has a l r e a d y induced a c l o s e d form s o l u t i o n f o r Hn, b u t l e t u s i n v e s t i g a t e how
t h i s m i g h t be done.
From
H0=0 H l = 1 H2=3 H3=7 H4=15 H5=31 H6=63
we w i s h to f i n d * *
F i r s t l e t u s note t h a t a l e a s t g e n e r a l g e n e r a l i x a t i o n method such as P l o t k i n [1969,1971] does n o t
work because the l e a s t g e n e r a l g e n e r a l i z a t i o n :
Furthermore,even if we i n c l u d e an a l g e b r a i c
matching f a c i l i t y s i m i l a r t o t h a t used i n Feldman
[1969] and Hardy [ I 9 7 6 ] * * * w h i c h a l l o w s u s t o t r y t o
r e l a t e t h e arguments o f t h e H n f u n c t i o n t o i t s v a l ­
ues b y a d d i n g , m u l t i p l y i n g , s u b t r a c t i n g , a n d d i v i d i n g ,
we s t i l l would n o t o b t a i n a s o l u t i o n . For example,
adding one to t h e sequence of v a l u e s of Hn g i v e s :
T h i s would g i v e us t h e know-
and v i a our b e l i e f , we induce t h a t :
Hn = 2 n - 1
which i n f a c t i s t r u e .
There i s , however, another i n d u c t i v e method
w h i c h , o t h e r t h a n f o r p o s s i b l y the t r i g g e r i n g o f
t h e b e l i e f t h a t t h e c l o s e d form s o l u t i o n might i n ­
v o l v e an e x p e n e n t i a l f u n c t i o n of base 2, does n o t
r e a l l y need to produce those i n s t a n c e s of t h e Hn
function.
Nor does it need to go t h r o u g h any
generalization steps.
We c a l l t h i s method, t h e
Method of Successive Refinement.
2.2 The .Method of Successive Refinement
The b a s i c i d e a of t h e Method of Success Re­
finement i s t o make a n i n i t i a l ( i n c o r r e c t ) guess
a s t o what i s t h e c l o s e d form s o l u t i o n , l e t a
theorem p r o v e r t r y t o prove t h i s guess, and when
i t f a i l s t o prove i t , g o back and t r y t o modify
each branch of t h e p r o t o c o l on which n had been
produced.
For example, l e t us suppose t h a t we
b e l i e v e a s o l u t i o n to Hn m i g h t i n v o l v e an exponent­
i a l f u n c t i o n t o a power o f t w o :
( B e l i e f Hn i n v o l v e s 2 n )
I n o t h e r words, l e t u s f i r s t form t h e h y p o t h e s i s
that;
Hn - 2 n
and see what a d e d u c t i v e system w i l l do to t h i s
expression.
I n d u c t i n g on n we g e t * :
HO+1=1 Hl+1=2 H2+l=4 H3+l=8 H4+l=16 H5+l=32 H6+1=64
But s t i l l the r e m a i n i n g l e a s t g e n e r a l g e n e r a l i z a t i o n
is false.
So how can t h e c l o s e d form s o l u t i o n be induced?
L e t u s suppose t h a t t h e i n d u c t i v e system has a v a i l ­
a b l e a b e l i e f t h a t any f u n c t i o n which produces t h e
sequence o f v a l u e s 1 , 2 , 4 , 8 , 1 6 , 32, f o r t h e
arguments O , 1 , 2 , 3 , 4 , 5 , i s p r o b a b l y equal t o
t h e e x p o n e n t i a l f u n c t i o n o f base 2 :
( B e l i e f 1 , 2 , 4 , 8 , 16, 3 2 i s p r o b a b l y 2 n )
Then g i v e n t h i s b e l i e f a system m i g h t t r y t o com­
pare t h e sequence 1 , 2 , 4 , 8 , 1 6 , 3 2 t o t h e s e ­
quence o f v a l u e s g i v e n b y t h e r e c u r s i v e f u n c t i o n
by applying various algebraic operations pairwise
* We c o n s i d e r analogy to be an e s s e n t i a l component
o f t h e method o f g e n e r a l i z a t i o n a s here d e s c r i b e d ,
and hence do n o t d e f i n e a s e p a r a t e : method of
analogy.
* * V i s our symbol f o r a l l .
* * * In our case, however, we work d i r e c t l y on t h e
numbers themselves r a t h e r t h a n a p p l y i n g a l g e b r a i c
o p e r a t i o n s t o t h e number o f occurrences o f symbols
such a s :
t h e successor symbol.
A l r e a d y we are in t r o u b l e as t h e d e d u c t i v e system
shows t h a t our h y p o t h e s i s is f a l s e , on the base
case.
Can we m o d i f y our h y p o t h e s i s to induce a
b e t t e r one?
A n a l y s i n g our p r o o f we see t h a t if
the e x p r e s s i o n 0 = 1 were r e p l a c e d by 0 = 1 - 1,
then o and n o t n would be produced. This means t h a t
HO ■ 1 would have to be HO = 1 - 1,
HO = 2 would have to be HO = 2 - 1, and
Hn = 2 n would have to be hn = 2 n - 1.
G i v i n g o u r new h y p o t h e s i s to a d e d u c t i v e system we
get:
* ■ is our symbol f o r t r u e ;
a is our symbol f o r
f a l s e , and >- is our symbol f o r i m p l i c a t i o n .
The
l e f t branch i s t h e base o f t h e i n d u c t i o n , and t h e
r i g h t branch i s t h e i n d u c t i o n s t e p .
S p e c i a l i z e d Systems-U:
Brown
which we deduce to be t r u e .
We see t h e n t h a t we can o b t a i n a p r o o f by mod­
i f y i n g a p r e v i o u s unsuccessful attempt a t p r o v i n g
a theorem.
T h i s leads n a t u r a l l y t o the q u e s t i o n
a s t o what i s the space o f a l l such m o d i f i c a t i o n s ?
A l t h o u g h , p o t e n t i a l l y t h i s space may be q u i t e l a r g e ,
we w i l l o n l y c o n s i d e r two simple m o d i f i c a t i o n s :
1) a d d i t i o n of a c o n s t a n t to an e q u a t i o n in o r d e r
t o make i t t r u e
2) m u l t i p l i c a t i o n of a c o n s t a n t to an a d d i t i v e p a r t
o f a n e q u a t i o n i n order t o make i t t r u e .
We have j u s t seen an example of (1) in o b t a i n i n g a
s o l u t i o n to the Hanoi f u n c t i o n .
An example of
(2) a p p l i e d t o t h e f a l s e e q u a t i o n
1 - / 5 would b e t o m u l t i p l y / 5 b y 1//5
and an example of (2) a p p l i e d to an a d d i t i v e p a r t
o f the f a l s e e q u a t i o n
0 - 1 + /5
would be to m u l t i p l y t h e a d d i t i v e p a r t 1 by - 1//5
Note t h a t m o d i f i c a t i o n s are o n l y made to t h e
r i g h t hand s i d e of an e q u a t i o n .
The reason f o r
t h i s i s t h a t t h a t side c o n t a i n s t h e c l o s e d - f o r m
s o l u t i o n whereas the o t h e r side merely c o n t a i n s
the recursive f u n c t i o n .
The reader w i l l see in the method of success­
i v e r e f i n e m e n t an echo of M e l t z e r ' s [1969] hypo­
t h e s i s t h a t i n d u c t i v e reasoning i s i n v e r s e deduc­
tion.
But in our case, t h i s is not so much a
semantical hypothesis:
if A->B t h e n A may be i n ­
duced from B , a s i t i s a n o t i o n o f search s t r a t e g y
i n t h e sense t h a t i f B i s deduced from A b y a p p l y ­
i n g t h e i t e m * A -> B to A, then A is induced from
B b y a p p l y i n g t h e i t e m i n the o p p o s i t e d i r e c t i o n .
We have seen how the c l o s e d - f o r m s o l u t i o n f o r
Hn may be o b t a i n e d from the b e l i e f t h a t the s o l u ­
t i o n i n v o l v e s a n e x p o n e n t i a l f u n c t i o n o f base t w o .
Note t h a t t h i s b e l i e f i s s t r o n g e r than what i s
necessary, and t h a t the b e l i e f t h a t some a r b i t r a r y
exponential function a is involved s u f f i c e s :
That
i s , i f 2 i s r e p l a c e d b y a i n the p r e v i o u s p r o t o c o l s ,
e s s e n t i a l l y the same r e s u l t w i l l b e o b t a i n e d .
2.3 Comparison of the two Methods
So f a r , we have found t h a t simple g e n e r a l i z a ­
t i o n methods are n o t s u f f i c i e n t and t h a t s o p h i s t i ­
cated b e l i e f s seem to be used in i n d u c t i v e r e a s o n ­
ing.
We have c a s t doubt on t h e suggestion t h a t
i n d u c t i v e reasoning i s o n l y g e n e r a l i z a t i o n b y d e ­
s c r i b i n g another t e c h n i q u e , the method o f success­
i v e r e f i n e m e n t , which seems to be more p o w e r f u l
(because it needs a weaker b e l i e f to produce t h e
solution).
One advantage of the g e n e r a l i z a t i o n method is
the p o s s i b i l i t y o f t r i g g e r i n g the 2 n b e l i e f b y a n
a l g e b r a i c matching o f the 1 , 2 , 4 , 8 , 16, 32, s e ­
quence to the sequence generated by the Hanoi
function:
Hn.
A second advantage i s t h a t i t i s
*<-> i s our symbol f o r i f f ( i f and o n l y i f ) .
q u i t e easy t o e x p l a i n how the b e l i e f , t h a t a f u n c ­
t i o n whose i n i t i a l i n s t a n c e s are 1 , 2 , 4 , 8 , 16,
32, i s the e x p o n e n t i a l o f base 2 :
( B e l i e f 1 , 2 , 4 , 8 , 1 6 , 3 2 i s p r o b a b l y 2n) n
was produced by s i m p l y i n s t a n t i a t i n g n in 2 to
s u c c e s s i v e l y 0 , 1 , 2 , 3 , 4 , 5 whereas i t does n o t
seem q u i t e as easy to e x p l a i n the p r o d u c t i o n of
the i n i t i a l h y p o t h e s i s t h a t Hn i n v o l v e d some e x ­
ponential function:
( B e l i e f Hn i n v o l v e s a n )
Are we f o r example to b e l i e v e t h a t the c l o s e d - f o r m
s o l u t i o n o f every r e c u r s i v e f u n c t i o n i n v o l v e s a n
e x p o n e n t i a l f u n c t i o n or what?
Although i t i s e a s i e r t o e x p l a i n the o r i g i n s
o f a b e l i e f o f the f o r m :
( B e l i e f 1 , 2 , 4 , 8 , 16, 3 2 i s p r o b a b l y 2 n )
than o f the f o r m :
( B e l i e f Hn i n v o l v e s a )
w e note t h a t i f a r e c u r s i v e f u n c t i o n i s n o t a c t u ­
a l l y equal to a simple a l g e b r a i c f u n c t i o n such as
2
then i t i s q u i t e improbable t h a t the necessary
b e l i e f o f t h i s f i r s t form c o u l d b e a v a i l a b l e t o b e
used in o b t a i n i n g the c l o s e d - f o r m s o l u t i o n .
We
w i l l now g i v e an example of such a f u n c t i o n and
suggest t h a t the c l o s e d - f o r m s o l u t i o n s o f most r e ­
c u r s i v e f u n c t i o n s are o f t h i s c h a r a c t e r .
This
example i s i n t e r e s t i n g because i t supports the view
t h a t i n d u c t i v e reasoning based o n l y on the method
o f g e n e r a l i z a t i o n even w i t h s o p h i s t i c a t e d b e l i e f s ,
i s n o t s u f f i c i e n t , and t h a t o t h e r methods are a l s o
needed.
Our example is the F i b o n a c c i f u n c t i o n .
Using
i t s r e c u r s i v e equations we deduce:
FO = 0
F1 = 1
F2 = F l + FO = 1 + 0 = 1
F3 = F2 + F l = 1 + 1 = 2
F4 = F3 + F2 = 2 + 1 = 3
F5 = F4 + F3 - 3 + 2 = 5
F6 = F5 + F4 = 5 + 3 = 8
F7 = F6 + F5 = 8 + 5 = 13
Thus from
F0=0 F l = l F2-1 F3=2 F4=3 F5=5 F6=8 F7=13
we wish to f i n d :
Fn = ?
But t h e r e is no simple a l g e b r a i c f u n c t i o n which
produces a n i n i t i a l sequence a n y t h i n g l i k e 0 , 1 , 1 ,
2, 3, 5, 8, 13.
In o t h e r words t h e r e c o u l d n o t
be any b e l i e f of the f o r m :
( B e l i e f a 1 . . . . a m is p r o b a b l y a )
which would b e h e l p f u l i n s o l v i n g t h i s problem.
For a f t e r a l l w e cannot expect t o have b e l i e f s
about every a l g e b r a i c f u n c t i o n i n our system; o n l y
the simple ones.
We can see t h a t the c l o s e d - f o r m
s o l u t i o n i s not simple, b y a c t u a l l y e x h i b i t i n g i t .
C o n c u r r e n t l y we w i l l take the o p p o r t u n i t y to show
how t h i s c l o s e d - f o r m s o l u t i o n c o u l d be o b t a i n e d
by the method of successive r e f i n e m e n t u s i n g t h e
b e l i e f t h a t t h e s o l u t i o n i n v o l v e s two e x p o n e n t i a l
functions:
( B e l i e f Fn i n v o l v e s a , b )
where a # 0 can b # 0.
From t h i s b e l i e f we form an i n i t i a l h y p o t h e s i s
that:
Fn = an + bn
and see what our d e d u c t i v e system w i l l do to t h i s
expression.
I n d u c t i n g on n t w i c e , because Fn
S p e c i a l i z e d S y s t e m s - 4 : Brown
846
does n o t r e c u r s e on Fn+1, we o b t a i n ;
Working upwards we see t h a t the f i v e p r e v i o u s l i n e s
in the p r o o f must have been something l i k e :
So f a r we have been able to induce a new hypothesii
which w i l l p r o b a b l y make the F l - b a s e branch of the
proof r e s u l t in
.
B u t , does t h i s change the r e ­
s u l t of
w e had p r e v i o u s l y o b t a i n e d i n the i n ­
d u c t i o n step branch o f t h e p r o o f ?
Since i n d u c t -
The i n d u c t i o n
two base cases are
t h e s i s to induce a
A n a l y s i n g the
s t e p seems to be t r u e , but the
false.
Can we modify our hypob e t t e r one?
Fl base case we see t h a t if the
Fn+2=aFn+l+(Fn+l-aFn)b
and since the t h i r d l i n e i s i d e n t i c a l t o our p r e v ­
ious t h i r d l i n e we see t h a t the i n d u c t i o n step
branch o f our p r o o f w i l l s t i l l r e s u l t i n ■ .
Of course we d o n ' t a c t u a l l y have to analyse
our p r o o f s t e p s downward, as we could simply apply
our d e d u c t i v e system to such a step and see what
i t does.
For example, t o see t h a t
Fn+1 = aFn + ( ( b - a ) / / 5 ) b n r e s u l t s in = a l l we need
d o i s t o apply our d e d u c t i v e system t o i t .
We have now produced a m o d i f i c a t i o n which r e ­
s u l t s i n a o n b o t h the F l base, and i n d u c t i o n step
branches of our p r o o f .
Only the FO base step r e ­
mains.
The FO base branch r e s u l t e d in o, so we
know t h a t some s o r t of m o d i f i c a t i o n is what was
needed to make t h e FO base branch r e s u l t in ■:
We f i r s t check to see if the m o d i f i c a t i o n induced
from the F l base branch w i l l s u f f i c e t o make t h i s
branch r e s u l t in
:
in (1 = / 5 ) had been d i v i d e d by √ 5 then ■ would
have been produced.
T h i s means t h a t t h e base
case would have had to have looked something l i k e :
Fl=aFO+((b-a)//5)b
Fl=a-0+((b-a)/5)-l
Fl-0+(b-a)/√5
Fl=(b-a)//5
1= (b-a)/√5
1= ( ( l - √ 5 ) / 2 - ( l + √ 5 ) / 2 ) / √ 5
1= 2√5/2√5
1-1
D
* In contemporary d e d u c t i v e systems (Brown [1976B,
1 9 7 7 ] , Boyer [1975]) the e q u a l i t y o r u n i f i c a t i o n
items would end up r e p l a c i n g a l l occurrences of say
" a " by b - 1 .
A system used i n successive r e f i n e ­
ment should n o t do t h i s because if Fn = a n + b n is
f a l s e a might n o t equal b - 1, and then we would
n o t want t o propagate t h i s f a l s i t y i n t o the branch
o f the p r o o f which i s t h e i n d u c t i o n s t e p which may
a f t e r a l l b e t r u e a s i n t h i s example.
The reason
we p r e f e r to have n d e r i v e d on the branch of t h e
p r o o f which i s the i n d u c t i o n base r a t h e r than the
i n d u c t i o n step i s t h a t i t i s a s i m p l e r branch and
w i l l b e e a s i e r t o analyse why i t produced n , i n
o r d e r to induce a h y p o t h e s i s .
is true.
Replacing a and b by the a l g e b r a i c terms t h a t
we have found them to be in our p r o o f we f i n d t h a t
t h e c l o s e d form s o l u t i o n f o r the F i b o n a c c i f u n c ­
tion is:
S p e c i a l fzed S y s t e m s - 4 : Brown
847
Thus we see t h a t a l t h o u g h it is p l a u s i b l e t h a t we
might b e able t o induce t h i s s o l u t i o n b y the i n ­
d u c t i v e method o f successive r e f i n e m e n t , i t i s n o t
v e r y p l a u s i b l e t h a t we c o u l d have induced t h i s
s o l u t i o n by the g e n e r a l i z a t i o n method.
The reason
f o r t h i s , a s w i l l b e r e c a l l e d , i s because the gen­
e r a l i z a t i o n method would r e q u i r e p r i o r b e l i e f s
that:
And it s i m p l y does n o t seem p l a u s i b l e t h a t such
b e l i e f s would b e a v a i l a b l e .
T h i s technique o f u s i n g e x i s t e n t i a l f u n c t i o n s has
been used by B i b e l [ 1 9 7 6 ] ( i n our case c o n s t a n t s :
a,b,α,β).Because it is so powerful t h a t we s h a l l
r a i s e i t t o the s t a t u s o f a t h i r d i n d u c t i v e method
which we s h a l l c a l l : The supplementary method of
existential functions.
I t should b e noted t h a t
t h i s technique i s n o s u b s t i t u t e f o r t h e more gen­
e r a l method o f successive r e f i n e m e n t , f o r i t s power
depends on knowledge possessed by the d e d u c t i v e
system.
I n t h e p r o t o c o l s u s i n g t h e method o f
e x i s t e n t i a l f u n c t i o n s we have assumed the a b i l i t y
t o solve equations o f the f o r m :
f - 0
f o r an a r b i t r a r y e x i s t e n t i a l constant.
This
knowledge was n o t assumed when n o t u s i n g t h i s
method.
We s h a l l see more of t h i s method in sec­
t i o n 3.
2.5 Summary
In summary then we have d e s c r i b e d two b a s i c
i n d u c t i v e methods and one supplementary method; and
have shown t h a t t h e e f f e c t i v e n e s s of each method
depends o n the a v a i l a b i l i t y o f p a r t i c u l a r b e l i e f s .
We have suggested t h a t the b e l i e f s needed by t h e
g e n e r a l i z a t i o n method would o n l y b e a v a i l a b l e i f
t h e r e c u r s i v e f u n c t i o n i s equal t o some v e r y simple
a l g e b r a i c f u n c t i o n , such a s i s the case w i t h the
Tower of Hanoi f u n c t i o n .
For most o t h e r r e c u r s i v e
f u n c t i o n s such as the F i b o n a c c i f u n c t i o n s we have
suggested t h a t the b e l i e f s needed b y the g e n e r a l i z ­
a t i o n method would not be a v a i l a b l e .
In s e c t i o n 3 we d e s c r i b e how the b e l i e f s need­
ed by the method of successive r e f i n e m e n t might be
a u t o m a t i c a l l y produced.
2.4 The Supplementary Method of E x i s t e n t i a l
Functions
We have one f i n a l p o i n t to make about the
F i b o n a c c i example:
If we had a s l i g h t l y more
s o p h i s t i c a t e d b e l i e f a s t o what the c l o s e d - f o r m
s o l u t i o n might b e , then our d e d u c t i v e system would
b e a b l e t o o b t a i n the s o l u t i o n i n a v e r y d i r e c t
manner.
L e t us suppose t h a t we have the b e l i e f
t h a t t h e c l o s e d - f o r m s o l u t i o n o f the F i b o n a c c i i n ­
v o l v e s a l i n e a r combination of two non-zero f u n c ­
tions :
( B e l i e f Fn i n v o l v e s α a n + βb n )
Then the h y p o t h e s i s i s :
=
αa n
+βb n
A s o l u t i o n is o b t a i n e d by i n d u c t i n g t w i c e on n:
3. Temporal I n d u c t i v e Methods
We now d e s c r i b e methods of i n d u c t i v e reasoning
based on i n f o r m a t i o n o b t a i n e d from t h e s o l u t i o n s of
p r e v i o u s l y solved problems.
Such methods are c a l l ­
ed temporal i n d u c t i v e methods.
There are two
temporal i n d u c t i v e methods, d i s t i n g u i s h e d by t h e
t y p e of i n f o r m a t i o n on which they are based.
The
f i r s t temporal method i s based s o l e l y o n i n f o r m a ­
t i o n o b t a i n e d from the theorem which expresses a
p r e v i o u s l y solved problem, whereas t h e second
temporal method is based on i n f o r m a t i o n o b t a i n e d
from the p r o o f of a p r e v i o u s l y solved problem.
R e f e r r i n g back to s e c t i o n 2.5 we s h a l l see in
s e c t i o n 3.2 how t h e second temporal method may be
a p p l i e d t o p r o d u c i n g the k i n d o f b e l i e f s needed b y
t h e method of successive r e f i n e m e n t .
3 . 1 The Temporal Method based on Theorems
The b a s i c idea of the temporal method based on
theorems i s f o r any g i v e n problem F t o t r y t o f i n d
a s i m i l a r problem H which has a l r e a d y been s o l v e d ,
and then t o use something s i m i l a r t o t h e s o l u t i o n
f o r H as a h y p o t h e s i s f o r t h e s o l u t i o n of F.
For example l e t H be the Tower of Hanoi p r o b ­
lem whose s o l u t i o n we d i s c o v e r e d in S e c t i o n 2, and
l e t F be t h e F i b o n a c c i f u n c t i o n whose s o l u t i o n we
are t r y i n g t o f i n d .
The theorem which expresses
t h e s o l u t i o n t o t h e Hanoi problem i s t h e n :
(Vn H ( n + l ) = 2 ( H n ) + l A H O 0 ) -> Hn=2 n -1
and t h e (as y e t incomplete) theorem which expresses
S p e c i a l i z e d S y s t e m s - 4 : Brown
848
We see t h a t t h e r e is no reasonable analogy t h a t we
can make which w i l l h e l p us to s o l v e F.
For e x ample, if we form the analogy t h a t m is n - 1 from
(Hn+1 Fm+2), then we might c o n j e c t u r e t h a t :
Fm=2m-1 -1
which is f a l s e .
In any case, so much syntax has
been l e f t unexplained b y t h i s analogy ( f o r example
the 2. , + 1 , HO = 0, +Fm, and Fl = 1) t h a t we can
have no confidence in t h i s c o n j e c t u r e anyway.
Furthermore even i f our g e n e r a l i z a t i o n system
were s o p h i s t i c a t e d enough to r e w r i t e the H theorem
as the e q u i v a l e n t theorem:
(VnH(n+2)=2H(n+l)+l ' H l « l ^ H O = 0 ) + Hn=2 n -1
(VroF(m+2)-F(ra+l)+Fm / F l - 1 > FO=0) + Fm=?
The c l o s e s t analogy would s t i l l not e x p l a i n the 2* ,
+ 1 , and +Fm.
The problem here is t h a t H and F are so d i s s i m i l a r t h a t no immediate analogy can be made.
3.2 Temporal Method based on Proofs
The b a s i c idea of the temporal method based
on p r o o f s is as f o l l o w s :
1) F i r s t u s i n g the temporal method based on t h e o r ems f i n d a problem H which is s i m i l a r to the p r o b lem F you are t r y i n g to s o l v e and c o n j e c t u r e a
solution for F.
2) Using the d e d u c t i v e system produce t h e p r o o f of
H and a p r o t o c o l of F w i t h t h a t h y p o t h e s i s .
3 ) F i n a l l y compare t h e p r o o f w i t h the p r o t o c o l t o
f i n d o u t why no s o l u t i o n was o b t a i n e d , and form a
new h y p o t h e s i s .
Suppose, now t h a t we are t r y i n g to f i n d a s o l u t i o n
to the F i b o n a c c i f u n c t i o n F, g i v e n t h a t we a l r e a d y
know t h e s o l u t i o n to t h e Tower of Hanoi f u n c t i o n H.
We s h a l l use the p r o o f of the s o l u t i o n of H which
is given as described in section 2 . 2 , s t a r t i n g w i t h :
3a Vn Hn = a n - 1
The h y p o t h e s i s f o r t h e s o l u t i o n o f F i s
3 a Vn Fn - a n - l
(We c o u l d throw in a few e x i s t e n t i a l c o n s t a n t s , f o r
example:
Fn = a.a n - 3 , b u t it w o n ' t make any d i f ference a t t h i s s t a g e . )
Using our d e d u c t i v e system we o b t a i n the f o l lowing p r o t o c o l :
We now compare t h i s p r o t o c o l w i t h the p r o o f of Hn.
We see t h a t in b o t h cases t h e i n d u c t i o n p r i n c i p l e
was f i r s t a p p l i e d .
We then see t h a t t h e steps in
the base case of p r o o f and p r o t o c o l were t h e same,
and t h a t the f i r s t few steps on the i n d u c t i o n s t e p
branch of the p r o o f and p r o t o c o l were the same. In
f a c t we f i n d t h a t t h e y d i f f e r e x a c t l y where H(n+1)
is replaced by 2(Hn)+l
H(n+l)=a(Hn)+a-l
T h i s then is t h e problem.
On F the theorem p r o v e r
i n d u c t s because i t c a n ' t r e c u r s e , because F recurses
on n+2 n o t n + 1 , whereas H immediately recurses to
obtain a solution.
So g r a n t e d t h a t F recurses on n+2 and t h a t the
theorem proven i n d u c t s t w i c e the q u e s t i o n , i s : why
i s not a s o l u t i o n o b t a i n e d a f t e r the second i n d u c tion?
Let us compare the steps b e g i n n i n g w i t h the
second i n d u c t i o n i n F t o t h e steps i n Hn:
3a Vn Hn = an - 1
3a Vn. F(n+1) = aFn+a - 1
We see t h a t Hn i n v o l v e s some term " a " e x p o n e n t i a t e d
by t h e i n d u c t i o n v a r i a b l e n.
Comparing Hn to Fn+1
t h i s leads u s t o suspect t h a t Fn+1 might a l s o i n v o l v e some term e x p o n e n t i a t e d by the i n d u c t i o n term
n+1.
Thus we would hypothesize t h a t Fn+1 i n v o l v e s
some term b
or r a t h e r than Fn i n v o l v e s some
term b .
The reader should bear in mind t h a t the
hypothesis is not a
f o r t h i s a would c l a s h w i t h
the bound v a r i a b l e a a l r e a d y o c c u r i n g in the above
Fibonacci expression.
The p o i n t i s t h a t the " a "
i n the Hanoi e x p r e s s i o n and " a " i n the F i b o n a c c i
expression are two d i s t i n c t bound v a r i a b l e s , w h i c h
do n o t n e c e s s a r i l y r e f e r to t h e same number.
There are two o t h e r i n d i c a t i o n s t h a t F n i n v o l v e s a term of the form b .
I n d u c t i n g on n in
each ca^se we get r e s p e c t i v e l y t h e base cases:
Note t h a t in the p r o o f of H a skolem f u n c t i o n d i s appears by having a r e p l a c e d by 1 whereas in the
p r o t o c o l of F no skolem f u n c t i o n disappears in
t h i s manner.
Here then is a second i n d i c a t i o n t h a t
a term of the form b or r a t h e r b is needed in t h e
Fibonacci p r o t o c o l .
F i n a l l y , b y comparing the i n d u c t i o n steps o f
H and F we note t h a t in H t h e e q u a l i t y i t e m r e p l a c e d a by an e x p r e s s i o n , whereas in F it merely
r e p l a c e d " a " by an e x p r e s s i o n .
Here then is a
t h i r d i n d i c a t i o n t h a t F n i n v o l v e s a n expression o f
the form b
(where b is n o t n e c e s s a r i l y equal to a ) .
Thus, f o r t h r e e reasons we are lead to the b e l i e f
t h a t Fn i n v o l v e d some term b where b is n o t necess a r i l y a.
We now go back and modify our o r i g i n a l
S p e c i a l i z e d S y s t e m s - 4 : Brown
849
belief:
t h a t Fn i n v o l v e s a :
( B e l i e f Fn i n v o l v e s a )
to the new b e l i e f t h a t Fn i n v o l v e s b o t h a and b :
( B e l i e f Fn i n v o l v e s a + b )
And t h i s new b e l i e f , as we have a l r e a d y seen in
s e c t i o n 2 . 3 , s u f f i c i e s t o f i n d the c l o s e d - f o r m s o l u ­
t i o n t o the F i b o n a c c i f u n c t i o n .
3.3 Summary
In summary we have d e s c r i b e d two temporal i n ­
d u c t i v e methods, one based on comparing the s t a t e ­
ment of a problem w i t h the statement of a s i m i l a r
p r e v i o u s l y solved problem, and the o t h e r based on
comparing the unsuccessful attempts to solve a
problem w i t h the p r o o f of an analogous problem. In
e i t h e r case we have found t h a t i n d u c t i v e reasoning
depends on the a b i l i t y to make use of p r e v i o u s s o l u ­
tions.
We have a l s o seen how t h e type of b e l i e f need­
ed to apply t h e method of succession r e f i n e m e n t to
the F i b o n a c c i problem might be generated.
4 . Conclusion
We have i n v e s t i g a t e d and d e s c r i b e d s e v e r a l
f e a s i b l e methods of i n d u c t i v e reasoning which are
u s e f u l i n the domain o f s o l v i n g d i f f e r e n c e equations.
We s t r e s s t h a t these methods use no mathematical
knowledge in terms of lemmas, s t r a t e g i e s or r u l e s
o t h e r than t h a t which would be a v a i l a b l e to a simple
a l g e b r a theorem prover which has o n l y the f o l l o w i n g
rules*.
1) Rules to s i m p l i f y a l g e b r a i c e x p r e s s i o n s .
2) The l o g i c a l r u l e s needed to p u t formulas in
c o n j u n c t i v e and d i s j u n c t i v e normal f o r m .
3) The two e q u a l i t y r u l e s :
4) A mathematical i n d u c t i o n p r i n c i p l e .
5) Rules to equate t h e c o e f f i c i e n t s of v a r i o u s types
of expressions.
(This a b i l i t y i s needed b y t h e
method o f e x i s t e n t i a l f u n c t i o n s )
Of c o u r s e , if we were to a l l o w the use of s o p h i s ­
t i c a t e d mathematical techniques we c o u l d e a s i l y
solve t h e problems we have considered in t h i s paper.
In p a r t i c u l a r , we c o u l d implement a theorem p r o v e r
based e i t h e r o n the r u l e s o f the f i n i t e d i f f e r e n c e
c a l c u l u s (Spiegel [ 1 9 7 1 ] , Jordan [ 1 9 6 5 ] ) , or on
the method o f g e n e r a t i n g f u n c t i o n s ( L i n [ 1 9 6 8 ] ,
Bekenbach [ 1 9 6 4 ] , Knuth [ 1 9 6 9 ] , Jordan [ 1 9 6 5 ] ) .
But t h i s is j u s t what we do n o t want to d o , f o r
t h i s would n o t e x p l a i n how t h e mathematical
knowledge embedded in these techniques was a c t u a l l y
created.
In our view of mathematics, we see t h a t
t h i s knowledge i s b a s i c a l l y c r e a t e d b y t r y i n g t o
solve a number of problems using o n l y o n e ' s g e n e r a l
i n d u c t i v e a b i l i t i e s and the c u r r e n t s t a t u s o f o n e ' s
deductive a b i l i t i e s .
Then one g e n e r a l i z e s and
organizes the methods t h a t one has found u s e f u l in
s o l v i n g these problems i n t o some s o r t of deductive
c a l c u l u s , such a s f o r example the t h e o r y o f f i n i t e
d i f f e r e n c e equations or t h e method of g e n e r a t i n g
functions.
One then adds t h i s c a l c u l u s to o n e ' s
deductive a b i l i t i e s , h e n c e f o r t h making what were
d i f f i c u l t problems q u i t e easy, thus a l l o w i n g one
t o t r y t o s o l v e problems i n more d i f f i c u l t domains
by repeating t h i s process.
Acknowledgements
We thank P r o f e s s o r B. M e l t z e r f o r c r i t i c i z i n g a
d r a f t „ o f t h i s paper and f o r a r r a n g i n g f o r
Dr. Tarnlund t o v i s i t the Department o f A r t i f i c i a l
I n t e l l i g e n c e so t h a t we could c o l l a b o r a t e on t h i s
research.
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* These r u l e s are embedded in an algebra theorem
prover which one of t h e a u t h o r s (Brown) implemented
about a year ago.
S n p c i a l i z e d S v s t e m s - 4 : Brown
850