A way to identify a unique number '$a_1$' has performed operations on '$x$'

Question

The problem I have is bit vague, so I will try to explain as best as I can.

Suppose we have a base number '$x$' and a set of unique number $'a_1', 'a_2', 'a_3', 'a_4'$ and so on

Now $a_1$ perform some operation on $x$ and results in $y_1$ From $y_1$, I should be able to identify that $a_1$ has performed operations on this number

Now on $y_1, a_2$ performed similar operations and results in $y_2$. From $y_2$, I should be able to infer that either $a_1$ or $a_2$ has performed operations on it and none other numbers had ($a_3, a_4$ etc)

So I am looking for some logic, that can help me create this scenario.

Answer 1

The operations need to be specified, otherwise there is no solution. Let me define a family of operations $f_z(x,y)=z$ (one for each real $z$, independent of $x, y$). Clearly you cannot recover $x$ from the results of these operations.

Answer 2

The logic should be similar to Gödel Numbering and the more sophisticated Gödel Numbering for Sequences.

Answer 3

Assuming the numbers $a_i$ are to be natural numbers, then a simple approach is to define the operation $\otimes:\mathbb{N}\times\mathbb{N}\rightarrow\mathbb{N}$ such that $a \otimes b=a p(a)^{b}$, where $p(a)$ is the smallest prime that does not divide $a$. Starting with $x=1$ produces $$ (((x\otimes a_1)\otimes a_2)\otimes a_3)...=2^{a_1} 3^{a_2} 5^{a_3} ..., $$ from which the sequence $(a_1,a_2,a_3,...)$ is easily retrieved.