INDUCTIVE INFERENCE OF RECURSIVE CONCEPTS

Inductive Inference of Recursive Concepts Yasuhito Mukouchi Kyushu University December 1993 Inductive inference is a process of hypothesizing a general rule from examples. As a successful inference criterion for inductive inference of formal languages and models of logic programming, we have mainly used Gold's identification in the limit. An inference machine M is said to infer a concept L in the limit, if the sequence of guesses from M which is successively fed a sequence of examples of L converges to a correct expression of L, that is, all guesses from M become a unique expression in a finite time and that the expression is a correct one. A class, or a hypothesis space, is said to be inferable in the limit, if there is an inference machine M which infers every concept in the class. In the present thesis, we mainly investigate three criteria related to the identification in the limit. As we will see, they are necessary for practical applications of machine learning or machine discovery. The first criterion requires an inference machine to produce a unique guess. That is, we apply so-called finite identification to concept learning. As stated above, ordinary inductive inference is an infinite process. Thus we can not decide in general whether a sequence of guesses from an inference machine has converged or not at a certain time. To the contrary, in the criterion of finite identification, if an inference machine produces a guess, then it is a conclusive answer. The second criterion requires an inference machine to refute a hypothesis space in question, if a target concept is not in the hypothesis space. In the ordinary inductive inference, the behavior of an inference machine is not specified, when we feed examples of a target concept not belonging to the hypothesis space. That is, we implicitly assume that every target concept belongs to the hypothesis space. As far as data or facts are presented according to a concept that is unknown but guaranteed to be in the hypothesis space, the machine will eventually identify the hypothesis. However this assumption is not appropriate, if we want an inference machine to infer or to discover an unknown rule which explains examples or data obtained from scientific experiments. Thus we propose a successful inference criterion where, if there is no concept in the hypothesis space which coincides with a target concept, then an inference machine explicitly tells us this and stops in a finite time. The third criterion requires an inference machine to infer a minimal concept within the hypothesis space concerned. In actual applications of inductive inference, there are many cases where we want an inference machine to infer an approximate concept within the hypothesis space, even when there is no concept which exactly coincides with the target concept. Here we take a minimal concept as an approximate concept within the hypothesis space, and discuss inferability of a minimal concept of the target concept which may not belong to the hypothesis space. That is, we force an inference machine to converge to an expression of a minimal concept of the target concept, if there is a minimal concept of the target concept within the hypothesis space. In the present thesis, we discuss inferability of recursive concepts under the above three criteria, and show some necessary and sufficient conditions for inferability and some comparisons between inferable classes. Furthermore as practical and concrete hypothesis spaces, we take the classes definable by so-called lengt h-bounded elementary formal systems (EFS's, for short) and discuss their inferability in the above three criteria. In 1990, Shinohara showed that the classes definable by length-bounded EFS's with at most n axioms are inferable in the limit from positive data for any n 2: 1. In the present thesis, we show that the above classes are also refutably inferable from complete data, i.e. positive and negative data, as well as minimally inferable from positive data. This means that there are rich hypothesis spaces that are refutably inferable from complete data or minimally inferable from positive data.