The determination of the value of Rado’s noncomputable function Σ(𝑘) for four-state Turing machines

Abstract

The well-defined but noncomputable functions Σ ( k ) \Sigma (k) and S ( k ) S(k) given by T. Rado as the "score" and "shift number" for the k-state Turing machine "Busy Beaver Game" were previously known only for k ⩽ 3 k \leqslant 3 . The largest known lower bounds yielding the relations Σ ( 4 ) ⩾ 13 \Sigma (4) \geqslant 13 and S ( 4 ) ⩾ 107 S(4) \geqslant 107 , reported by this author, supported the conjecture that these lower bounds are the actual particular values of the functions for k = 4 k = 4 . The four-state case has previously been reduced to solving the blank input tape halting problem of only 5,820 individual machines. In this final stage of the k = 4 k = 4 case, one appears to move into a heuristic level of higher order where it is necessary to treat each machine as representing a distinct theorem. The remaining set consists of two primary classes in which a machine and its tape are viewed as the representation of a growing string of cellular automata. The proof techniques, embodied in programs, are entirely heuristic, while the inductive proofs, once established by the computer, are completely rigorous and become the key to the proof of the new and original mathematical results: Σ ( 4 ) = 13 \Sigma (4) = 13 and S ( 4 ) = 107 S(4) = 107 .