Richardson's theorem

In mathematics, Richardson's theorem establishes the undecidability of the equality of real numbers defined by expressions involving integers, π, ${\displaystyle \ln 2,}$ and exponential and sine functions. It was proved in 1968 by the mathematician and computer scientist Daniel Richardson of the University of Bath.

Specifically, the class of expressions for which the theorem holds is that generated by rational numbers, the number π, the number ln 2, the variable x, the operations of addition, subtraction, multiplication, composition, and the sin, exp, and abs functions.

For some classes of expressions generated by other primitives than in Richardson's theorem, there exist algorithms that can determine whether an expression is zero.^[1]