Regarding "Informational derivation of quantum theory", "PHYSICAL REVIEW A 84, 012311 (2011)", Chiribella
I was provoked to cast Information Compression as an Iterated Function
So given a string S, of length L, find an iterated function F, that produces a new string S’ of length L’ where L’ < L.
One might also require an iterated function G = F-1 that takes S’ and produces S.
If such a G exists, then F would be called a lossless or invertible iterated compression function.
S = "The rain in Spain stays mainly in the plain."
L = 45
F1 = { s/ain/1/g }
S1 = "The r1 in Sp1 stays m1ly in the pl1."
L1 = 37
F2 = { s/he/2/g }
S2 = "T2 r1 in Sp1 stays m1ly in t2 pl1."
L2 = 35
F() = F1(F2())
S' = F(S)
L' < L since 35 < 45
G1 = { s/1/ain/g }
G2 = { s/2/he/g }
G = G1(G2())
S = G(S')
F is invertible, iterated and compresses.
Sidebar 1: F can be packaged with S:
Sp = F + S = /ain/1/he/2/"The rain in Spain stays mainly in the plain."
Next Step: Find an Iterated Function that performs Information Expansion
So given a string S, of length L, find an iterated function F, that produces a new string S’ of length L’ where L’ > L.
One might also require an iterated function G = F-1 that takes S’ and produces S.
If such a G exists, then F would be called a lossless or invertible iterated expansion function.
S = "The rain in Spain stays mainly in the plain."
L = 45
F = { s/[a-zA-Z]/[%3d]/ }
S1 = "084he rain in Spain stays mainly in the plain."
L1 = 47
S2 = "084104e rain in Spain stays mainly in the plain.
L2 = 49
S' = "0841041010321140971051100321051100320831120971051100
3211511609712111503210909710511010812103210511003211
6104101032112108097105110046010"
L' > L since 135 > 45
G = { s/[%3d]/[a-zA-Z]/ }
S = G(S')
F is invertible, iterated and expands.
Sidebar 1: F can be packaged with S:
Sp = F + S = /[a-zA-Z]/[%3d]/"The rain in Spain stays mainly in the plain."
In this context we call F, the "transcription apparatus".
Next Step: Show that the cost of mutagenesis is higher in the compressed case.
This is trivial and will be left as an exercise to the reader.
Source for F in Expansion Section:
#include
/* convert any number to itself and any letter to a number */
/* hint add a comma to each phrase and watch intronic explosion */
/* hint remove the commas and watch loss of reading frame */
/* hint add triplets and watch need for reading frame */
int isNumber(int c)
{
return ( ('0' <= c) && (c <= '9') );
}
void echo(int c)
{
printf("%c", c);
}
void number(int c)
{
printf("%03d", c);
}
main()
{
int i;
int c;
for(i = 0; (c = getchar()) != EOF; i++ )
{
if( isNumber(c) )
{
echo(c);
}
else
{
number(c);
break;
}
}
for(; (c = getchar()) != EOF;)
{
echo(c);
}
}
