Recently, I've read about the ANS, where each symbol get described on a basis with size that is relative to its frequency.
I was wondering if it is possible to obtain meaningful arithmetic operations on such systems.
In particular, assuming that the symbols have an numerical meaning, then, in the r-ANS formulation:
Letting $S = (s_1,s_2,s_3,\ldots, s_n)$ be the input string of numeral symbols from the alphabet set $\mathcal{A} = \{a_1,a_2,\ldots,a_k \}$ of size $k$. Further, let's assume the data has the frequency counts $\mathcal{F} = \{ F_{a_1}, F_{a_2},\ldots, F_{a_k} \}$, which are integers proportional to the probability mass distribution $\{p_1,p_2,\ldots,p_k\}$ of the symbols, such that $M = \sum_{i=1}^k F_i$, (and hence, $p_i = \frac{F_{a_i}}{M}$) and the cumulative frequency counts is denoted by $C_{a_i} = \sum_{j=1}^{i-1} F_{a_j}$.
Then, in this system, a symbol of $t+1$ digits is described as:
$$X_t = \left\lfloor \frac{X_{t-1}}{F_{s_t}} \right\rfloor * M + C_{s_t} + mod(X_{t-1}, F_{s_t})$$ My questions are:
- is it possible to obtain the sum of two numbers in the symmetric system from a sum in the asymmetric system? (and vice-versa?)
- what is the meaning (in the symmetrical sense), if any, of the sum of two numbers in this asymmetrical system?
- is it reasonable to look for mappings of arithmetical operations between the symmetric and asymmetric numeral systems?
Any clues and ideas are much appreciated