Unpredictable strings are collectives

Abstract

In [Z] we defined the notion of an “inference method” which given a finite binary string guesses its continuation. We showed that a special class of these inference methods could be used to construct a very general theory of confirmation. In this paper we investigate the class of strings which are unpredictable with respect to these inference method-that is, infinite strings which are such that given any initial segment a given inference method will guess its continuation incorrectly. We show that such unpredictable strings are also what Von Mises [6] called “collectives” (where the ambiguity in Von Mises’ definition is appropriately resolved). Thus strings which are unpredictable from initial segments are also statistically random. This can be regarded as a combinatorial analogue of a result of MartinLof [S] on computable inference methods. Martin-Liif showed that if the shortest algorithmic descriptions of longer and longer initial segments of an infinite string are of length rougly equal that of the initial segments they describe, then that infinite string is statistically random. We discuss the application of minimal algorithmic descriptions to inductive inference in [4]. We will use the following notation. S is a (finite or infinite) binary string. (S)j is the jth bit of S. [Sli is the j-length initial segment of S. ISI is the length of S. ISI 1 is the number of l’s in the string S. For a finite S, the density of S, D(S)= ISI,/ISI. For an infinite S, D(S)=lim,,, D([S],).