An algebraic approach to a calculus of functional differences - Core

NPS52-87-041

NAVAL POSTGRADUATE SCHOOL Monterey, California

AN ALGEBRAIC APPROACH TO A CALCULUS OF FUNCTIONAL DIFFERENCES: FIXED DIFFERENCES AND INTEGRALS

Bruce J. MacLennan

September 1987 Approved for public release; distribution is unlimited, Prepared for: Chief of Naval Research Arlington, VA 22217

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AN ALGEBRAIC APPROACH TO A CALCULUS OF FUNCTIONAL DIFFERENCES' FIXED DIFFERENCES AND INTEGRALS PERSONAL AUTHOR(S)

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We introduce a notion of functional differences in which the difference of a function f with respect to a function h is that function g that describes how the value of f changes when its argument is altered by h: f[h(x)] = gff(x)] We

also introduce the inverse operation of functional integration and derive useful properties of both operations The result is a calculus that facilitates derivation and reasoning about recursive programs inis is illustrated in a number of simple examples. The present report uses algebraic methods to establish preliminary results pertaining to fixed differences that is functional differences that do not depend on the value of the argument x. .

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t

An A

Algebraic Approach to

Calculus of Functional Differences: Fixed Differences and Integrals

B. J. MacLennan Computer Science Department

Naval Postgraduate School Monterey, CA 93943

Abstract

We

introduce a notion of functional differences in which the difference of a function / with respect is that function g that describes how the value of / changes when its argument is altered by h: fh{x)\ = g[f(x)\. We also introduce the inverse operation of functional integrato a function h

tion

and derive

examples.

The present report

to fixed differences, that

ment

1.

The

useful properties of both operations.

derivation and reasoning about recursive programs.

is,

uses algebraic

This

methods

result is

is

a calculus that facilitates

illustrated in a

number

of simple

to establish preliminary results pertaining

functional differences that do not depend on the value of the argu-

x.

Motivation

Simple recursive definitions often take the following form:

=

y

f(hx)

=

fao

The assumption here cations of h.

That

is

is,

that an arbitrary

g{fx), for

domain value

for all acceptable x there is

z*z

(1)

x can be reached by finitely

an n such that x = h

terns of recursive definition will be considered in Section 6

n

x

.

More

and more generally

in

many

appli-

general pat-

[MacLennan

'88].

Many

of the results in this report are included in

methods from the calculus of

more easier

*

easily by use of the

relations [Carnap '58].

methods

[MacLennan

The

of abstract algebra.

'86j,

however, that report used

present report obtains the

The reason

same

results

that the algebraic approach

is

seems to be that the algebraic notion of a function, which incorporates the domain and

Author's current address: 37996-1301.

Department

of

Computer

Science, Ayres Hall, University of Tennessee, Knoxville,

TN

codomain as part of dled explicitly.

In

[MacLennan

about

domains

the

and the present report

'86]

is

of

a

in

differences.

and

algebraic approach

our

restrict

attention

a recursive definition for a particular

and

g, k, x

y

.

Since h and x

are usually determined by the

determined from the definition of

To

see

to

fixed

unknowns that must be

there are four

/,

domain

in

question

they are zero and the successor function for the domain of natural numbers), and easily

is

their associated integrals that will be

now we

for

'88];

The

regard.

in

1

In deriving

found,

[MacLennan

report

later

in this

functions are

all

Comparing the derivations

functions.

illuminating

essential for the investigation of variable differences

presented

assumption that

particular, the algebraic notation (and its

reasoning

simplifies

total)

automatically takes care of issues that must otherwise be han-

its definition,

how

this

main problem

the

/,

can be done consider the second

is

j/

usually

is

determining the function

(e.g.,

g.

line in Eq. 1:

f(hx) = g(fx) The

tells

us

how much

That

is,

if

function g

changed by equation

is

h.

/'s

analogous to the

the value of the function / changes

argument

changed by

is

finite difference

difference

is

that in the

first

then

value

its

=

Note that

A

1:

of

/:

become important

different notion

problem

Suppose

S

for fized differences there

distinction will 2.

g.

This

This

is

because

we want

to be

(e.g., lists, sets, relations).

2

analogy we introduce

Definition

1.

changed by

is

equation the "amounts of change" are expressed as functions

able to deal with functions whose domains and ranges are nonnumeric this

argument

g+(fx)

rather than numbers, as they are in the finite difference equation.

Based on

is

its

equation

f(h+x) The

k,

when

of functional

in

-»

is

T,

g:

T

-*

T

and

h:

S

-»

S.

We

define g to be a (fixed) func-

no distinction between forward differences and backward

differences.

This

the study of variable differences.

differences

updating data structures such as

is

in Paige & Koenig an imperative context.

described

sets in

-2-

|

'82j.

Their's

is

addressed to the

tional difference of / with respect to h

the functional difference

is

unique,

if

and only

we write

=

g

if it

A

h

satisfies /

/,

and

»

call g

=

h

g

°

f.

the "h difference of /."

also call g ''the change in / with respect to /i" or "the change in / given the

when

the difference exists and

is

Furthermore,

if

We

change k." Thus,

unique we have

/

h

.

=

(fc

A

/)

•

/ (2)

2.

Existence and Uniqueness of Fixed Differences

Equation solve

A

for h

it

an explicit definition we need to

2 defines the functional difference implicitly; to get

-1

This can be accomplished by composing with the inverse of

/.

/,

/

,

on both

sides of the equation to yield:

h

a

f = /

•

h

r

.

1

This seems to yield an explicit formula for the functional difference, but careful, since

with /

l °

assumes the existence of the (right) inverse f~ 1

it

=

f~

Theorem

I r , exists

1:

Let

/:

satisfying the equation /

We

Proof:

g-

f

g

f

?

.

=

.

/

h

->

h

•

T =

/

is

surjective.

be a surjection, g

»

/.

Then

right with f~

x

g

(We

let

=

necessary to be more

Recall that a right inverse f~

write 1 T for the identity I:»T

S

h:

f

°

h

->•

5,

and

°

/r

let

T — T

g:

_1

_1 ,

where / r

is

-»

l ,

T.)

be any function

a right inverse of/.

on both sides of the equation and simplify:

h

= f.

f-l

•

S

if

compose on the

/•

=

only

.

it is

h

.

f-l

1 °

f;

The preceding theorem assumes a sufficient for its existence.

difference exists.

The

following theorem establishes conditions

Theorem h

/

g

I'roof:

2:

/:

S

T

-»

f has solutions /

"

«

be an h

'.

f,

and

injection

each

for

let

h:

inverse

left

S /,

-»

*

The functional equation

5.

of

/.

For convenience we represent composition by juxtaposition when no ambiguity

Since /

result.

Let

is

injective

has a

it

left

inverse /f

1 ,

with f[ x f = I s

.

Hence, letting g =

-1 /A/,

will

,

we

have

Hence, the difference equation

We

observe

surjective.

in

wr

=

0/

1

)/

satisfied,

is

=

1

/)

fhi s =

^

passing that none of the preceding results depend on h being either injective or

S

h:

-» S.

3

2 establishes sufficient conditions for the existence of solutions to the difference

equation, namely that /

is

an injection. The following theorems establish necessary and sufficient

conditions for the existence and uniqueness of solutions. cept of an equivalence kernel

Definition

S x

=

o

Thus, they apply to any function

Theorem

/M/r

[MacLane

The equivalence

2:

&;

Birkhoff

However, to state them we need the con-

'67].

kernel E, of a function

/:

S

-»

T

is

the following relation on

5:

E}

=

{ (z,

x)

E

5 x S\

fx = fx'

}

(3)

Thus

(x, x'

)

f E,

Theorem Proof:

3:

and only

if

A

if

fx

=

fx'

3.

Now we

can state our existence theorem:

solution g to the difference equation fh = gf exists

To prove

the "if" part,

assume that E, C

ten as a composition / = 4>F in which

tion.

.

4>:

S! Ej

— T

£"«.

is

if

and only

Every function

f:

an injection and F: S

S -»

->

if

T

E, C E*.

can be writ-

5/E/

is

a surjec-

Also, since £/ C £"«, /A can be written as a composition

In iMacLennan '86] we introduced at this point the idea of an "isomorphic image" from the calculus of relations. This was used to prove a number of existence theorems for functional differences. Algebraic methods obviate the need for this approach. However, Hasse diagrams are still useful for visualizing functional differences.

-4-

ttPF

=

fh

(4)

where

S/E^

n:

x 9 = 4>K1

is

-*

T

an injection and P: S/E,

is

a solution to the difference equation.

=

gf

Now Ei C by

E

/.

to

is

,

show the

Theorem if

if

a surjection.

is

Now we

claim that

checked by substitution:

is

=

7rF

fh

E, C E^.

if part assume that

4:

We

not surjective.

E^ = E

fh = gf,

and only

Proof:

=

E^

there

a g such that fh =

is

Observe that

gf.

always true, since g cannot separate values that have already been mapped together

But since

unique

"'only

This

(4>n^)(F)

a solution exists

S/

-»

Suppose S if

/

is

^=

,,

and therefore Ei C

and the equation

1

E^

Q

.

= gf has a solution

fh

g.

Then

this solution

is

T

is

surjective.

prove the contrapositive of the "only Therefore there

y

$

R

is

a y 6

T

Therefore suppose that

/:

S

-»

such that

Im / =

=

if" part.

y

{

y = fz for

|

some

x €

5

}

(5)

We

construct a g' different from g that also solves the equation.

sible so long as

S ^

1.

Note that since

For the "if part, suppose that both

y $

g

R, g' f =

and

g'f

gf,

and hence 9'/ =

g' are solutions

=

fh

-

Let g'y = a

and that /

^

yj/,

which

is

pos-

fh.

is

a surjection.

Hence,

gf (6)

Since /

is

surjective

it

has a right inverse; compose with both sides of Eq. 6 to yield:

9

Hence,

g

=

g'

and the so the solution

l

=

g'ff7

is

unique.

=

-5-

1

gffr'

=

g

Obviously

a critical issue whether /

is

it

relationships between the differences of surjective

Definition

The

3:

nonhomogeneous

/

if

is

investigate the

and nonsurjective functions.

difference equation fh = gf

homogeneous

called

is

if

/

surjective,

is

and

not surjective.

Homogeneous

Corollary 4-1:

we

Therefore

surjective or not.

is

difference equations have at

ous difference equations do not have unique solutions

(i.e.,

most one solution; nonhomogene-

they have no solutions or

more than

one solution).

Proof: Follows from

Theorem

4

and the definition of homogeneous.

Next we characterize the many solutions of a nonhomogeneous equation as a family of functions derived from an associated homogeneous equation.

S C

we

Definition

4:

If

Definition

5:

Suppose that

be written uniquely

in

T,

use js _ T for the trivial injection of

S

/:

the form / =

-*

7

1 ,

g:

T

->

T, h:

S

-> 5,

S

and

(f

-

g): (S,

6:

If

+ S2 ) -

/:

S1

-»

T

and

g:

S2

->

is

Let

a solution to the

/:

/.

then

T,

we

= Im call

fh

Note that / can = gf the homo-

gf.

the

define

sum,

direct

i*«eS,

ifx

5:

R

7\ by

=

('^)*

Theorem

js ^ T x = x.

Jn^jf where f':S->R. Then we

geneous equation associated with the nonhomogeneous equation fh =

Definition

into T:

5 -

7\

g:

T -

T,

h:

\g X

S -

nonhomogeneous equation

9

=

S,

R

fh = gf

j{g

+

lf

X

eS

= Im if

/,

(7)

2

Q

and only

-

T - R and

if it

j = jR ^ T

can be written

.

Then

in the

g

form

c) (8)

where

gQ

:

R — R

is

the solution of the associated homogeneous equation, and

c:

Q

->

R.

Proof: For the "only if part assume that fh = gf has a solution.

Im Hence, Im y C

show that 9

= j(9o

+

Im gf =

is

Im /

C

fh

a solution to the homogeneous equation

=

/?

f'h=

g

sum

f

.

in

Eq. 10.

It

remains to

Since fh = gf, f =

jf and

we know:

c),

=

jf'h

Since j

Im

This means that y can be written as the direct

fi.

is

g

C

g

Therefore,

an injection, we can compose

its

]{9o

+

c

)Jf

inverse on the

left of

both sides of this equation to

yield:

/'*

Now

+

notice that [g

In fact

it is

fh

fh

= g

+ c)jf

f

1

part

we assume

fh = gf.

show

that there

is

Noting again that (g

and we have proved that

is

a solution.

homogeneous.

is

a g such that

+

gQ

c)j = g

,

fh

=

gQ

f

.

Let g = j(g

+

c);

derive:

= jf'h = = =

Hence

the "if

to

is

therefore

;

(

S, j = jR

_

R

+

q and

c:

Q —

R.

This

is call

the general

form

nonhomogeneous equation.

of the solution of the

Proof: Follows from the preceding proof.

Examples of Differences

3.

we

In this section

most

tions, since they are

We

cessor function.

-

6]

x

=

x

-

a:

We

(power a

call

power a

f) n

=

power a /

Let IN be the natural numbers and

familiar.

Also,

We

7:

begin with numerical funclet a:

use the presection and postsection notations [Wile

6, etc.

Definition

We

give several examples of functional differences.

we

let 't

1

represent exponentiation, a

write '(power ffl

n

f

a.

This

is

/) n' for the n

n

=

'73]:

a

IN be the suc-

->

=

[a +] x

a

+

x,

n .

power of function / applied

to initial value

defined recursively:

(power a

/)

=

(power a

/) (n

+ 1) = /

a

power from a of

'the

(

]

IN

[(power a /)

/'.

If

S

/:

->

n],

for

S and

n^O

a G 5,

(10)

then

it

is

clear

that

/: IN -> S.

Theorem

6:

The

difference of the

power (from

a

A

power tt

a) of / with respect to successor

/ =

is /,

/,

(11)

provided that power a /

is

surjective.

Proof: This follows directly from the definition of 'power':

(power a

Therefore, (power a /)

Theorem

a = /

•

/) (a n)

(power a

=

f [(power a /)

/).

7:



Thus,

/?.

power a / (13)

Since

7r

is

surjective,

Im

Therefore,

4>:

R

->

R

if

/'s

A

o~

Im [/(power a /)]

=

//'

Observe that

exists.

7r

domain

=

restricted to

is

Im [(power a /)

(14)

The

definition of

power a /

is

equivalent

(power a /)n.

to:

left

fl

/)

=

fjx

=

j

+

jc

=

IJ

+

jc

+

c)

5-1,

Now

observe that

power a a power a a power --

a

.

power

[a +]

powerj

[a x]

Notice that the last two of these are not surjective. The differences of these functions follow from the theorem

o

A

[a

+

o

A

[a

-

a

A

[- a



S with

respect to any function

S

->•

I is surjective.

S with

c

respect to itself

is itself.

Further,

if

h

h.

Proof: Observe g = h satisfies hh = gh.

Corollary 9-1:

S

h.

Proof: Observe that g = h satisfies Ik = gl and that

Theorem

I:

difference of a

a

power of a function with respect

10-

to that function

is

that

Also,

function.

if

h

n

surjective then h

is

n n Proof: Let g = h in h h = gh

Theorem

A

10:

A

h

n

=

h.

.

inverse of a difference of a function with respect to a surjection

(left)

difference of the function with respect to a (right) inverse of the surjection.

That

is,

if

is

a

fh = gf

then

fK

l

=

l

sr f (16)

If

/

is

surjective, then (h

Proof: Suppose h inverse of

surjective

—

It

1

gf

f.

Theorem

11:

difference of the

xp

h

is

A



fg = {h

Proof:

is

a

also

A

drop the

A

A

.

a

is

g satisfying fh

sides of fh = gf.

/

and

r

difference

A

A

sides of / =

subscripts and write (h

f.

x

g[ fh

1

is

a

left

Since h

yields:

is

fh~

l

.

_1

A

=

/)

h'

1

two functions with respect

A

/.

to a third

is

a

of

fg

with

respect

xp is

to

h.

,

then

surjective,

f.

If

ii>{fg).

g)

Suppose that g^

gf.

This yields g[ l fh =

Composing with h~ on both

a difference of g with respect to h and

g)

=

function with respect to a difference of the second with respect to the third.

first

Corollary 11-1: fg = {h

/.

difference of a composition of

The derivation

Hence, (fg)h =

A

A

a functional difference of / with respect to h~ l

is

(fg)h

h

h~ l

=

has a right inverse h~

Hence gf 1

is, if

*

a surjection and there

is

will usually be clear to

That

/)f

Then compose g[ l with both

g.

it

A

If

f.

is

-

direct.

f(gh)

Suppose gh =

=

-

f{H)

(14>)g

ff)g

Then

=

*{&).

/ and g are surjective then the differences are unique.

g

is

surjective

and /

is

any function such that the differences

exist,

then

Proof:

Suppose that / =

]

+

c]

]d>

third line above follows from the preceding theorem.

Theorem

The

12:

difference of a bijection with respect to a composition of functions

the

is

composition of the differences of the bijection with respect to each of the functions:

ghAf

= (gA f)(h

A

f)

(17)

Proof: Since /

gkAf=

is

bijective

fghf

l

=

(fgr'Kfhr g

)

Af)(hAf)

the following theorem

is

A

we extend

it

of chain rule for evaluating differences of bijections.

then

That xp

is

is, if

is

In

to any functions with differences.

functional difference of a function with respect to a composition of functions

the composition of the difference of the

tions.

A,

13:

1:

1

The preceding theorem provides a kind

Theorem

hence by Theorem

x

fg{r f)hr

(

,

1

-

=

has the two-sided inverse f~

it

first

function with respect to each of the other func-

a difference of / with respect to

a difference of / with respect to gh.

-12-

g

and

is

a difference of / with respect to

Proof: Since fg = rJ>)f,

Corollary 13-1: exist.

Note that

if

and

/

is

=

(fg)h

derive:

=

(0f)h

A

any function, gh

is

{rPf)

=

(4>v)f

a functional difference of / with respect to gh.



be written

h

c),

A

/ =

j(hA

+

c')

observe the product:

=

j{gA4> + c)j{hAc/> + c')

=

j(gA)(hA

=

][{gA4>){h&) +{g±)c'\

=

;[( ? A0)(/iA^)

=

j(ghA

+

c')

+c"\

+ c")

by the preceding theorem;

A

it is

the general form of gh

difference of a function with respect to the nth

n nth power of a difference: h

We

may

the differences exist, then they

A

=

/

(h

A

/)".

an inductive application of the previous theorems.

use a product notation for compositions:

]1

F = {

F,

.

F2

F._,

Using this notation we can express

13-

Fn

A

/.

power of a function

is

the

A

Corollary 13-3:

difference of a function with respect to a product of functions

is

the pro-

duct of differences with respect to each of those functions:

x

n

bx Q = g bx Q

a solution provided the g

-21-

b

g

' .

n

35,

'

bx Q

Hence,



is

single valued.

ignore generation.

Lemma

17-3:

the g

If

k

ignore the generation of 5, then

b

=

hbg

9 hbg

(36)

For an arbitrary x G S we show hx

We

n*

1

=

h

=

g

=

gg bz

=

g hb

the g

b

->

is

S -

T,

T

g.

->

T, and suppose

Suppose

generated from

D

by

h.

and

17-3.

the unique solution of the difference equation.

S,

b:

= fh n jx

S

is

the g b ignore the generation of 5.

17-1

for x Q G -D.

h:

S

k

if

D

T,

->

g:

T

-* 5,

By Cor.

=

b:

D

n

T,

S -*

T.

If

the g

k

b

ignore the

.

A6ff

13-4,

=

g fjx

-»

T, and /:

-»

a solution to fh = gf, then / =

fx

18:

D

and only

h:

n Proof: Suppose x = h x

Theorem

b:

Lemmas

Proof: This follows from

Our next

S,

5:

ignore this generation, then

22-

T

g

n

-»

bx Q

=

T.

If

cf>

5

hbg x

is

generated from Z? by A, but

*.

*

f

-

*„, (37)

Proof: This follows from

7.

Lemmas

17-3

and

18-1.

Properties of Integrals

The

following corollaries follow easily from the preceding theorems and from the properties of

differences in section 4.

Corollary 18-1:

[h$ b

{h

g\

n

x

)

=

g

n

bx

,

torx Q

eD (38)

Proof: Follows from

Theorem

Corollary 18-2: h

Proof: Follows from

Corollary 18-3:

18-4:

=

Thm.

8.

h

Thm.

Lemma

17-2.

I.

=

h

h
,

6

by hypothesis

\ Cor.

13-4

by definition of

the h integral from b of

b

g:

h

=

f.

Combining with

g

= h

A

/ yields the desired

result.

Theorem

20:

Suppose

h:

S

->

S,

b:

D

-*

T,

Then,

-24-

g:

T

-»

7\ and

S

is

generated from

Z)

by

/i.

hA\h$ if

and only

T

if

and be unique

/

if

and Ej C

surjective

is

we compute a formula

g

(4i)

integral / = h

E^. To

image of

for the

=

g.

Under the stated conditions the

Proof:

jective

generated from b[D] by

is

b g\

b

The

g exists.

difference will exist

determine the conditions under which /

is

sur-

/:

Im / = rng * Wff = rng{ {k n x = =

Now

n

g

bx

|J

Im

{

observe that

n

g

,

x

\

n

g

e

bx

)

x

\

D, n ^

eD,n>0} }

b

T =

(J

Im

n

g b

if

and only

if

T

is

generated from b[D] by

Next we must show E, C E^. Therefore assume fx = fx' 17-2, fh

fhx

mx

= =

= g m bx and fh n x

fh

m-

g(g

l

m bx

= g* + Hx =

=

x

)

'

= g bx

'

n

g(g bx Q

Hence, g

=

n

g

bx Q

.

is,

fh

Now

mx

= fh n x Q

'.

By Lem.

derive:

= fh n + l x

')

'

fhx'

Hence E, C E* and the difference

The

'.

m bx

that

m + 1 bx

g

=

n

;

g.

is

unique,

n

following corollary shows that the difference of a nonsurjective integral

a remainder term.

Corollary 20-1:

If

T

is

not generated from b[D] by

25-