Content Or Predicate Addressable Sequences

[Jon Awbrey]

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COPAS.  Note 2

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sequiturs,

the boundary conditions always give me fits,
and i thought that i had thought about that
whole issue more carefully somwehere before.
i think that it went a bit like this:

let !N! = {1, 2, 3, ...}, the multiplicative naturals.

let (!N! -> !N!) be the set of functions from !N! to !N!.

let [!N! -> !N!] be the set of finite partial functions from !N! to !N!.

let p_k = k^th prime, p_1 = 2, p_2 = 3, etc.

decode element j of !N! as the finite partial function f_j : !N! -> !N!,
via prime decomposition.

1   =  {}               =  empty function      =  . . . . . ... .

2   =  (p_1)^1          =  {<1, 1>}            =  1 . . . . ... .

3   =  (p_2)^1          =  {<2, 1>}            =  . 1 . . . ... .

4   =  (p_1)^2          =  {<1, 2>}            =  2 . . . . ... .

5   =  (p_3)^1          =  {<3, 1>}            =  . . 1 . . ... .

6   =  (p_1)^1 (p_2)^1  =  {<1, 1>, <2, 1>}    =  1 1 . . . ... .

7   =  (p_4)^1          =  {<4, 1>}            =  . . . 1 . ... .

8   =  (p_1)^3          =  {<1, 3>}            =  3 . . . . ... .

9   =  (p_2)^2          =  {<2, 2>}            =  . 2 . . . ... .

10  =  (p_1)^1 (p_3)^1  =  {<1, 1>, <3, 1>}    =  1 . 1 . . ... .

11  =  (p_5)^1          =  {<5, 1>}            =  . . . . 1 ... .

12  =  (p_1)^2 (p_2)^1  =  {<1, 2>, <2, 1>}    =  2 1 . . . ... .

then code the probe P_j on the sequences f in (!N! -> !N!)
as a functional or a "partial information predicate" (pip)
such that P_j (f) = 1 iff f fits the data given.

or something along those lines ...

i remember thinking quite a bit about the extension to
the non-negative integer codomain N = {0, 1, 2, 3, ...},
but can't recall right at the moment what came of it.

let me know if you see any problems with this,
as my recollection is very fuzzy at present,

jon awbrey

archived at:  http://stderr.org/pipermail/inquiry/

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