Learning formulae from elementary facts

Abstract

Since the seminal paper by E.M. Gold [Gol67] the computational learning theory community has been presuming that the main problem in the learning theory on the recursion-theoretical level is to restore a grammar from samples of language or a program from its sample computations. However scientists in physics and biology have become accustomed to looking for interesting assertions rather than for a universal theory explaining everything.

The language for the formulation of the interesting statements is, of course, most important. We use first order predicate logic. Three types of the formulae learning machines are considered. Machines of the first type produce each result in a finite number of steps. This learning type is very restricted. Machines of the second type produce the output in the limit. The third type machine supplies the result with what we call assurance levels. For success, we require that the assurance level grows indefinitley.

We have proved that there is the best finite formulae learner and there is the best nondeterministic assurance formulae learner while for the other types of learning there is no best learner. No ∀x(P(x)) type formula is learnable in a finite mode. All ∃x∀y(P(x, y)) type formulae are assurance learnable but not vice versa. Formulae involving only monadic predicates are both learnable in the limit and assurance learnable. Nondeterministic assurance learners can learn all the predicate formulae while probabilistic assurance learners can learn only ∃x∀y∃z(P(x, y,z)) type formulae. Nondeterministic and probabilistic limit learners can be simulated by deterministic ones.

This project was supported by an International Agreement under NSF Grant 9421640.

The first author was supported by Latvian Science Council Grant No. 93.593.

The second author was supported by Latvian Science Council Grant No. 93.599.

On leave from the Department of Computer Science, University of Maryland, College Park, MD 20742 USA, The third author was supported in part by NSF Grant 9301339 and a grant from the Netherlands Organization for Scientific Research.