I have been working the last 8 years in try to convert from base n to base m without artihmetical operations.
My hypothesis is that by the same way that is possible convert bases without division (see the tables bellow.) based in a literal conversion.
Can you create an equivalence table that allow this direct translation?
***Partial answer: No (After 10 years), by Pumping Lemma. ***
Numeric systems are a subset of regular languajes, therefore they follow the Pumping Lemma. (Pending demonstration)
Working in the update for Direct translation where: $$ m = k·n | k ∈ N $$
$$ log_{(log_{n}(k))}n ∈ I $$
- Is possible numeric systems which their tokens represent real values instead of natural values?
-Working on the formal explanation-
$$ 00011111_{2} $$ For converting from base 2 up to base 4 you have to get pumping length[1]. $$ (Source Base)^x = (Destination Base) $$
$$ x = log_{sourceBase}(Destination Base) | x ∈ n $$
$$ 2^x = 4 $$
$$ x = 2 $$
And re create the new value representation based in the finite alphabet transition table[2] between the source and destination alphabets.
[01][11][11]| You don't need a math operation for this operation because
The follow symbol '3'(4) -> '11'(2)
[1][3][3]|
[0011][111]| You don't need a math operation for this operation because
The follow symbol
Transition Table from base 2 to base 8
$$ 0_{8} = 000_{2} $$
$$ 1_{8} = 001_{2} $$
$$ 2_{8} = 010_{2} $$ $$ 3_{8} = 011_{2} $$
$$ 4_{8} = 100_{2} $$
$$ 5_{8} = 101_{2} $$
$$ 6_{8} = 110_{2} $$
$$ 7_{8} = 111_{2} $$
$$ 37_{8} = 133_{4} = 011111_{2} $$
As it is very know the number of groups that you have to use is:
$$log_{n}=m $$ where ( n can be represented as m^c , and c is and Integer) (m)
This operation:
'7'(8) -> '111'(2) (Symbol(7)->111)
In my case, is a not math operations, because I have an state machine that is able to understand and reflect (bad joke) that symbol 7 means 111 in the default output (or default queue).
As you know when c is not an integer, we have a very complex problem therefore I was creating random table-states based on random rules (it means jumps of states using genetic algorithms) but It has been a real waste of time/energy.
Now I share my Idea, I believe that all bases must be represented as a sub-languages for other bases and they creates a cycle , It I couldn't demostrate it as a formal theorem. But my heart, my soul and my migth believes that it could be possible.
For example:
(3) 0 1 2 / 10 11 12 20 /
_______
3^1 (grp) 3^2(grp)
(2) 0 1 / 10 11 /
_______
2^1 (grp) 2^2(grp)
(4) 0 1 2 3 / 10 11 12 13 20 21 22 23 30 31 32 33 /
_______
4^1 (grp) 4^2(grp)
Conversion table:
GroupSize = Log(2)^4
Rule: (0/0 1/1 2/10 3/11 )
_______________________________________________________
Do you have any formalism to define a base as sub-language of other base for cases like this?
Post In Edition.
[1]: Pumping Lemme https://en.wikipedia.org/wiki/Pumping_lemma_for_regular_languages
[2]: Transition Table https://en.wikipedia.org/wiki/State-transition_table
