Compatibility of convergence algorithms for autonomous mobile robots

We consider several convergence problems for autonomous mobile robots under the $\cal SSYNC$ model. Let $\Phi$ and $\Pi $ be a set of target functions and a problem, respectively. If the robots whose target functions are chosen from $\Phi$ always solve $\Pi$, we say that $\Phi$ is compatible with respect to $\Pi$. If $\Phi$ is compatible with respect to $\Pi$, every target function $\phi \in \Phi$ is an algorithm for $\Pi$. Note that even if both $\phi$ and $\phi'$ are algorithms for $\Pi$, $\{ \phi, \phi' \}$ may not be compatible with respect to $\Pi$. We investigate, the convergence, the fault tolerant ($n,f$)-convergence (FC($f$)), the fault tolerant ($n,f$)-convergence to $f$ points (FC($f$)-PO), the fault tolerant ($n,f$)-convergence to a convex $f$-gon (FC($f$)-CP), and the gathering problem, assuming crash failures. We classify these problems from the viewpoint of compatibility; the group of the convergence, FC(1), FC(1)-PO and FC($f$)-CP, and the group of the gathering and FC($f$)-PO for $f \geq 2$ have completely opposite properties. FC($f$) for $f \geq 2$ is placed in between.