Gathering Semi-Synchronously Scheduled Two-State Robots

We study the problem \emph{Gathering} for $n$ autonomous mobile robots in synchronous settings with a persistent memory called \emph{light}. It is well known that Gathering is impossible in the basic model ($OBLOT$) where robots have no lights, even if the system is semi-synchronous (called SSYNCH). Gathering becomes possible, however, if each robot has a light of some type that can be set to a constant number of colors. In the $FCOM$ model, the robots can only see the lights of other robots. In the $FSTA$ model, each robot can only observe its own light. In the $LUMI$ model, all robots can see all lights. This paper focuses on $FSTA$ robots with 2-colored lights in synchronous settings. We show that 2-color $FSTA$ and $FCOM$ robots cannot solve Gathering in SSYNCH without additional conditions, even with rigid movement and agreement of chirality and the minimum moving distance. We also improve the condition of the previous gathering algorithm for $FSTA$ robots with 2-color working in SSYNCH.