The Extended BG-Simulation

The BG-simulation is currently the main tool, short of arguing directly about topology, to show the impossibility of r/w protocol to solve a task T (res) on n processors, t-resiliently. It is done by reducing T (res) from a counterpart task T (wf) on t + 1 processors and showing that T (wf) is not wait-free solvable. The task T (wf) derived from T (res), is driven by the capability of the BG-simulation. Yet, the task T (wf) proved meaningful only when derived from a task T (res) which was “colorless” essentially a decision task in which any processor may adopt the output of any other processor as it own. It lacked, when a terminating simulator cannot easily adopt any simulated output for itself. In this paper we extend the capability of the BG-simulation to this case. As a result, the definition of T (wf) as derived from T (res) becomes tighter, i.e. the new task is a subtask of the previous one. This makes for a T (wf) that is of higher-level and therefore a better tool. We demonstrate the convenience that the Extended BG-simulation provides, in two ways. First, we prove a new impossibility result: we show that n processes cannot solve t-resilient weak renaming with n+ (t+ 1)− 2 names, where n > 1 and 0 < t < n. Second, we show that the solvability n-processors tasks, t-resiliently, for t > 1 and n > 2, is undecidable by a simple reduction to the undecidability of the solvability of 3-processors tasks, wait-free.