Phenomenology of the Busy Beaver Problem as Dynamical Systems

Abstract

We reconsider the busy beaver problem, which is a modification of the Turing machine’s halting problem, in the light of a dynamical systems approach. The Turing machine has finite internal states, a tape, and a transition table. In the busy beaver problem, the initial tape is filled with blank symbols and the transition table are made at random. When starting from such conditions, some machines eventually halt, or never halt. The busy beaver is a machine with a given state size that halts with the maximum number of non‐blank symbols left on the tape [2]. In this work, we regard the Turing machine as dynamical systems and plot the spatio‐temporal patterns of machines motion on the tape. In [1], we showed that the spatio‐temporal patterns of 5‐state machines can be roughly classified into four types. In particular, well‐regulated and self‐similar patterns are found in most longer‐lived machines, which include known best busy beaver machines. We are now scanning higher (10 or more) states machines using Monte‐Carlo methods and conjecture such self‐similar patterns will hold the majority. The detailed analysis is yet to be understand. We shall, however, present a relevant analysis in analogy with random walks.