calculator problem” [@RIES]. Another related question is, if such a reduced calculator is optimal for the task of constant recognition. Obviously, maximally reduced button set makes theoretical and statistical analysis convenient, therefore it is very useful as mathematical model. In practice, as we will show later, Kolmogorov complexity of formulas in maximally reduced language, which are considered as simple by humans, might become surprisingly large. On the contrary, formulas *simple* in reduced operation set, like nested power-towers, look inhuman, and are out of scope traditional mathematical aesthetics, despite small Kolmogorov complexity.

The simplest possible language we found has a length three ($n=3$). It is still able to perform all operations of the scientific calculator. It includes:

\[base-3\] $$\begin{aligned}
\label{e}
\text{either} & \qquad e \quad  \text{or}  \quad \pi. \end{aligned}$$ $$\begin{aligned}
\text{binary exponentiation} & \qquad x^y,\end{aligned}$$ $$\begin{aligned}
\text{arbitrary base (two-argument) logarithm} &\qquad  \log_x{y}.\end{aligned}$$

Another very simple language of length four is:

\[base-4\] $$\begin{aligned}
\text{any constant}& \qquad x.\end{aligned}$$ $$\begin{aligned}
\label{e1}
\text{natural exponential function}& \qquad \exp{x}  \equiv e^x,\end{aligned}$$ $$\begin{aligned}
\label{e2}
\text{natural logarithm}& \qquad \ln{x}  \equiv \log_e{x},\end{aligned}$$ $$\begin{aligned}
\text{subtraction}& \qquad x-y.\end{aligned}$$

Detailed proofs by exhaustion are presented in Appendices C,D in Supplemental Material. Above two examples clearly justify name $\exp-\log$ for class of explicit elementary 