Bounded Counter Languages

TL;DR

This paper explores the computational equivalences of finite automata with multiple heads and counter machines on bounded languages, and investigates the limits of linear speed-up in such computational models.

Contribution

It establishes the equivalence of certain automata and counter machines on bounded languages and introduces a technique for speeding up counter machine computations.

Findings

Deterministic finite automata with k two-way heads are equivalent to machines with one two-way head and k-1 counters on bounded languages.

Linear speed-up generally does not hold for counter machines.

A method is developed to accelerate computations by any constant factor using additional counters.

Abstract

We show that deterministic finite automata equipped with $k$ two-way heads are equivalent to deterministic machines with a single two-way input head and $k-1$ linearly bounded counters if the accepted language is strictly bounded, i.e., a subset of $a_1^*a_2^*... a_m^*$ for a fixed sequence of symbols $a_1, a_2,..., a_m$. Then we investigate linear speed-up for counter machines. Lower and upper time bounds for concrete recognition problems are shown, implying that in general linear speed-up does not hold for counter machines. For bounded languages we develop a technique for speeding up computations by any constant factor at the expense of adding a fixed number of counters.