Some Decidability Results on Grammatical Inference and Complexity

The grammatical inference problem is easy to state: One must choose, on the basis of a finite set of symbol strings, a grammar for the language of which the given set is a sample. Precisely the same problem arises in trying to choose a model (theory, function, etc.) to explain a collection of sample data. This is one of the most important information processing problems known and it is surprising that there has been so little work on its formalization. The papers that have been written (e.g., Solomonoff, 1964) all discuss the importance of the problem so we will forego that discussion here. Any attempt to formalize the grammatical inference problem must include precise formulations of several concepts left vague in the statement above. The four central notions are: the hypothesis space, the measure of adequacy, the rules by which samples are drawn, and the criterion for success in the limit of the inference process. For this paper, the hypothesis spaces will be subsets of the general rewriting systems expressed in a particular notation. A grammar will adequately represent a sample when it generates all of the known strings and none of the non-strings of the sample; in Section 3 we introduce an additional requirement of minimum complexity. For the presentation of sample strings, we require only that each string ultimately be included, (Horning, 1969) makes use of stronger requirements. The criterion for success in the limit of the grammatical inference process is one of the main subjects of this paper. In the remainder of this section we