Automatic functions, linear time and learning

TL;DR

This paper characterizes linear time computable automatic functions and explores their learnability, demonstrating how additional resources like work tapes, queues, or stacks enhance learning capabilities within linear time constraints.

Contribution

It precisely characterizes automatic functions via linear time Turing machines and shows how adding resources like work tapes, queues, or stacks enables full learnability in linear time.

Findings

Automatic functions are characterized by linear time Turing machines with a single tape.

Adding one work tape extends learning power beyond automatic learners.

Two work tapes, one queue, or two stacks suffice for full learning power.

Abstract

The present work determines the exact nature of {\em linear time computable} notions which characterise automatic functions (those whose graphs are recognised by a finite automaton). The paper also determines which type of linear time notions permit full learnability for learning in the limit of automatic classes (families of languages which are uniformly recognised by a finite automaton). In particular it is shown that a function is automatic iff there is a one-tape Turing machine with a left end which computes the function in linear time where the input before the computation and the output after the computation both start at the left end. It is known that learners realised as automatic update functions are restrictive for learning. In the present work it is shown that one can overcome the problem by providing work tapes additional to a resource-bounded base tape while keeping the update-time to be linear in the length of the largest datum seen so far. In this model, one additional such work tape provides additional learning power over the automatic learner model and two additional work tapes give full learning power. Furthermore, one can also consider additional queues or additional stacks in place of additional work tapes and for these devices, one queue or two stacks are sufficient for full learning power while one stack is insufficient.