apply predicate to a list

From: Mikito Harakiri (mikharakiri_at_iahu.com)
Date: 07/12/04 

Date: Mon, 12 Jul 2004 13:30:35 -0700

Is the CS concept of map -- predicate applied to a list -- well-defined from
math perspective?

For comparison, let's focus on sets, first. Any set can be represented by
its characteristic function, so that applying predicate to a set is just
multiplying 2 characteristic functions (and, perhaps, applying sum/integral
operator on top of it).

Lists/sequences, however, are different. Any list is a formal univariable
polynomial. In this representation, however, I can't figure out what is the
operator that applies predicate.

An example:

Assuming domain of reals, the set

{4,5,-1,3}

is represented by

Dirac(x-4)+Dirac(x-5)+Dirac(x+1)+Dirac(x-3)

If we want to apply predicate "x>0", then we just multiply it by
Heaviside(y) and get

Dirac(x-4)+Dirac(x-5)+Dirac(x-3)

For equality predicate, say x=5, I multiply by Dirac(y-5) and integrate.
(This begs a question why do I integrate in one case, but not in the other).

List, however,

[4,5,-1,3]

is represented as a polynomial

sum(a_n*z^n)=4+5*z-z^2+3*z^3

Then, is there a way to apply predicate a_n>0 and get

4+5*z+3*z^3

?

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