Is it possible to write a program that takes a natural number n as input, and finds the shortest arithmetic formula which computes n?

Question

Is it possible to write a program that takes a natural number $n$ as input, and finds the shortest arithmetic formula which computes $n$?
A formula is a sequence consisting of digits, $+$, $\times$, ^ operator that raises to a power, and parentheses.
The length of formula is the number of characters in the formula.
Each digit, operator, or parentheses is counted as 1 character.
How to approach this problem?
I knew that it is always possible to construct an arithmetic formula since each natural number $n$ can be treated as a sequence of digits.
However, is it possible that we can enumerate the set of all possible arithmetic formulas and find out the minimum length formula for each natural number $n$?