  Recursive self-improving (RSI) systems have been dreamed of since the early
days of computer science and artificial intelligence. However, many existing
studies on RSI systems remain philosophical, and lacks clear formulation and
results. In this paper, we provide a formal definition for one class of RSI
systems, and then demonstrate the existence of computable and efficient RSI
systems on a restricted version. We use simulation to empirically show that we
achieve logarithmic runtime complexity with respect to the size of the search
space, and these results suggest it is possible to achieve an efficient
recursive self-improvement.
