Straight-Line Programs with One Input Variable

Let $\mathbb{C}$ be the set of all straight-line programs with one input variable, x, using the following instruction set: $y \leftarrow 0$, $y \leftarrow 1$, $y \leftarrow y + w$, $y \leftarrow y - w$, $y \leftarrow y * w$, and $y \leftarrow \lfloor y/w \rfloor $. We show that two programs in $\mathbb{C}$ are equivalent over integer inputs if and only if they are equivalent on all inputs x such that $|x| \leqq 2^{2^{\lambda r^2 } } $ ($\lambda $ is a fixed positive constant and r is the maximum of the lengths of the programs). In contrast, we prove that the zero-equivalence problem (deciding whether a program outputs 0 for all inputs) is undecidable for programs with two input variables. An interesting corollary is the following: Let $\mathbb{N}$ be the set of natural numbers and f be any total one-to-one function from $\mathbb{N}$ onto $\mathbb{N} \times \mathbb{N}$ (f is called a pair generator. Such functions are useful in recursive function theory and computability theory.) Then f cannot be computed ...