Classes of bounded functions that are semantically equivalent to Turing-machine are PAC learnable

Kevin Xu

Keywords: Learnability, computability, database, bounded function, lambda calculus

TL;DR: A symbolic but bottom-up approach toward the PAC learnability for semantically Turing-machine equivalent functions

Abstract: A contemporary programming language is a top-down approach in the sense that we know exactly what a function is to be constructed and we construct the exact function. Learning is a bottom-up approach in the sense that we don't know how to exactly program a targeted function but we can program an algorithm that automatically constructs another function that converges to the target by accepting sample data. Learning can be done not only through statistical methods but also through symbolic computing. While statistical learning methods have their unique positions in many applications including pattern recognition, symbolic approaches toward learning persist in keeping the realizability assumption for convergence. In addition to many known symbolic Probably Approximately Correct (PAC) learnables such as conjunctions of Boolean literals and rectangle learning games, there are other symbolic computing systems that are PAC learnable as well. In this paper, we show a class of functions that is semantically equivalent to Turing machine is PAC learnable. This learnability is realized through the Enterprise-Participant (EP) data model, a database language representing such a class of functions, called bounded functions as they have a finite co-domain while an infinite domain. An EP database is mathematically capable of inventorying all the properties of partial recursive functions with the hypothesis of infinite space and time.

Submission Number: 3