ISSN:
1432-0770
Source:
Springer Online Journal Archives 1860-2000
Topics:
Biology
,
Computer Science
,
Physics
Description / Table of Contents:
Résumé Le tabeleau 1 résume les propriétés topologiques sur l'espace d'apprentissage. Celles sur l'espace caractéristique s'en obtiendra aisément en utilisant l'aspect héréditaire des propriétés d'un espace à son sousespace.
Notes:
Abstract There have been two contrasting doctrines concerning learning, more generally about acquisition of knowledge: empiricism and rationalism. The theory of learning in such a field as artificial intelligence seems to fall within the empiricist framework. On the hand, N. Chomsky and his followers have discussed, during the last decade, concerning learning, especially about language learning, from the rationalist point of view (Chomsky, 1965). The main feature in the rationalist approach toward a theory of learning lies in the speculation that in order to acquire knowledge it is indispensable for a learner to be endowed with “innate ideas”, and that “experience” in the external world are merely subsidiary types of information for the learner. If this is acceptable, we can inquire: Under what kind of innate ideas can the learner understand the structure of the external world? In our previous paper (Uesaka, Aizawa, Ebara, and Ozeki, 1973), we formalized this by introducing the mathematical notion of “learnability”, and gave a partial answer to the above inquiry. In this formalization we assumed that the set F of objects to be learned consists of mappings of N to itself, where N is the set of positive integers. Then, constructing a topological space (F, $$\mathcal{O}$$ ) by an appropriate family $$\mathcal{O}$$ of open sets, we observed that the notion of learnability can be well described in terms of topological properties of the learning space (F, $$\mathcal{O}$$ ). Many problems must be solved, however, before we raise the theory to a complete model of the rationalist theory of learning. The topological study of the space (F, $$\mathcal{O}$$ ) is, we believe, the first step toward this approach. In this context, we discuss the topological aspects of this space. Now we define $$\mathcal{O}$$ as follows: By N 2 we mean the direct product of two N's. Let s be a subset of N 2. If, for any (x, y), (x′, y′) in s, x=x′ implies y=y′, then we say that s is single-valued. Let f ∈ F, If, for any (x, y) in s, y=f(x), then f is said to be on s, denoted as $$f\underline \supseteq s$$ . Let $$\pi \left( s \right) = \left\{ {g;g \in F,g\underline \supseteq s} \right\}$$ . A single-valued finite subset of N 2 is called datum. Let D denote the family of all data. Let $$\mathcal{O}* = \left\{ \phi \right\} \cup \left\{ {\pi \left( d \right);d \in D} \right\}$$ , and $$\mathcal{O}$$ denote the family of all subsets of F, each of which is written as $$\mathop \cup \limits_\alpha W_{\alpha }$$ , where W α is in $$\mathcal{O}*$$ . Then, it is easily seen that $$\mathcal{O}$$ satisfies the axiom of the open system of a topological space. It is shown that the learning space (F, $$\mathcal{O}$$ ) has the following properties: (1) It satisfies the first and the second countability axioms. (2) It is separable and is totally disconnected. (3) It is a Hausdorff space and, further, is regular and normal. (4) It is neither compact nor locally compact. (5) It is metrizable, or more precisely there exists a complete but not totally bounded metric space which is homeomorphic to learning space. (6) Any of its subspace can be embedded into its special subspace.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF00288915
Permalink
