On the Simplicity of Busy Beaver Sets

Abstract

l) 5 1. Introduction I n [28] RADO first described what he called the Busy Beaver n-game. Let F denote the class of all Turing Machines with alphabet {O, 1) which are always started on a blank tape and which always shift their tape head a t each step. The object of the Busy Beaver n-game is to discover a Turing Machine from the class F with n stsates which halts and which prints out, the maximum number of ones that can possibly be printed out by any halting n-state Turing Machine from the class .T (this maximum number is denoted by Z(n,)). RADO showed that 2 is not a computable funct,ion by showing that L' grows faster than every computable function. I n a similar manner RADO showed the non-comput'ability of t8he function S , where X ( n ) is the maximum number of shifts made by any halting n-state Turing Machine from the class F. Since the inception of the Busy Beaver n-games some atJtemptJs have been made to calculate , Z and X for small values of n,. RADO observed in [29] that Z(1) = S(1) = l and 4 2 ) = 4. LIX showed in [22] that, Z(3) = 6 and 8(3) = 21, and BRADY has recently shown in [6] that Z(4) = 13 and S(4) = 107.The original motivattion for RADO'S proposing tfhe Busy Beaver n-game was the construction of well defined non-comput'able functions, which indeed Z and X are.Paraphrasing his remarks in [as], tjhe main object,ive was the study of the phenomenon of non-comput,abilit>y in itms simplest, form, so as to gain insight into the limit,at#ions of comput,ability ( a view of computability from without, as it were). The purpose of this paper is to present a generalized version of tJhe Busy Beaver construction (more precisely, a generalization of the function S) which exhibits many properties of interest in recursive funct~ion theory as well as abstract computational complexit,y t'heory. It is often the case in these fields of st>udy that cliagonalization is required for the construction of sets with specific properties. RADO observed that the construct,ion of 2 and S involved no diagonalization. Likewise, it will be the c,ase below that diagona1izat)ion will not be needed. Moreover, all uses of the Recursion Theorem will be instances of what has been called hhe Inefficiency Lemma. The proof of t8he Inefficiency Lemma itself involves a fairly straight forward application of the Recursion Theorem, but more importantly the tJruth of the Inefficiency Lemma is intuitively obvious. This work, then, is presented in a spirit similar t o t'liat of R~n o ' s -the study of some of the non-computable recursive funct,ion theory and computat,ional complexitmy theory properties in their simplest form. For this reason many results whose proofs might just as easily be left t o citations of known results in t,he lit,erat,ure will be proved directly. In all cases the proofs are short and easy to understand. l)

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