On the Error Correcting Power of Pluralism in BC-Type Inductive Inference

In this paper we show that it is always possibk to reduce errors for some forms of inductive Inference by increasing the numller of machines involved in the inference process. Moreover, we obtain precise bounds for the number of machines required to reduce any given number of errors. The type of inference we consider here was originally defined by Barzdin [l] (called GN” inference 1, and later independently by Case [2] (called BC inference) who expanded th,definition to inciude the inference of programs which are allowed a finite numhcr of errors (called BC’” inference). This latter type of inference has been studied extensively by Case and Smith [2,4]. We use here the definitions and notations from Case and Smith. We say that an inductive inference machine M BC”’ idcrztific~ ;1 total function f (written f~ BC”‘(MH if and only if when M is successively fed the graph of f’ as input it outputs over time a sequence of programs po, pI, . . . , such that NXi ) M,l, = “’ f], where g =“I /z means that g and Cz disagree at at most HZ places. Thus, when M is presented a function f it is permitted to change its mind infinitely often so long as eventually each 0;’ its conjectures contain at most oz errors. Moreover, if for some integer k s m at most finitely many of the programs p, in the above sequence have more than k ‘errors of commission’ (i.e., an error at a location .Y such that &,(x) is defined and unequal to f(_x )I, then we write f E BC”*‘k (M 1. We will assume without loss of generality that the machine M :Is always presented with fmite initial segments of the graph of f. We define the following classes,