A MODULAR SYSTEM $M$ is a finite set of "rules" of the form $ax+b\to cx+d$, with $a,b,c,d\in\mathbb{Z}$. If $u,v\in\mathbb{Z}$, then $u$ is "derivable" from $v$ in $M$ if one can get from $u$ to $v$ by applying rules in $M$. For example, the well-known Collatz problem asks whether for all positive integers $u$, 1 is derivable from $u$ in the modular system with the two rules $2x\to x, 2x+1\to 6x+4$.

The general problem of whether $u$ is derivable from $v$ in a given modular system $M$ is undecidable. (Proved in Borger, "Computability, Complexity and Logic").