Regarding "Informational Derivation of Quantum Theory" by Chiribella, et.al.

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);
    }
}