Learning in a Computable Setting. Applications of Gold’s Inductive Inference Model

Abstract

The Theory of Recursive Functions is here employed for a rigorous formalization of the learning process. We adopt the model proposed by Gold (1967) in the context of natural language acquisition. Among the merits of this literature is that of providing a fairly articulated theoretical grammar to tackle a complex phenomenon such as learning.

The question we ask is whether a boundedly rational agent can learn how to form rational expectations. Our stylized learner is described as a Turing Machine in the attempt to coherently incorporate in the analysis the notion of procedural rationality. The results we present cast some serious doubts on the relevance of the Rational Expectations Hypothesis even as the reference point for an as if assumption.

Concentrating our attention on a “two step learning procedure” suggested by Spear (1989), on the one hand we show how hysteresis can greatly reduce the positive learning results obtained so far in the literature; on the other hand, we suggest that institutions may play an important role in the learning process thanks to the pre-processing of information they perform, the uncertainty reduction they foster, and the collective memory resources they make available.

As a by-product we also present a version of strong nativism in line with the concept introduced by Chomsky (1981). In particular, imposing some realistic features on the learner and assuming that the learning process converges, we obtain — following Osherson et al. (1989) — that the cardinality of the class of regimes that can be successfully learned is finite. In other words, the spectrum of economic organizations going from perfect decentralization to absolute central planification should contain only a finite number of intermediate shades.

A previous version of this paper appeared as OFCE Document de travail No. 94-05. We would like to thank Professors Kumaraswamy Velupillai and Axel Leijonhufvud for their precious comments and encouragement. Any remaining errors are, as usual, the author’s. A Human Capital and Mobility Grant is gratefully acknowledged.