# Insertion Operations on Deterministic Reversal-Bounded Counter Machines

## TL;DR

This paper investigates how insertion operations based on various language relations affect the acceptance capabilities of deterministic reversal-bounded multicounter machines, focusing on whether the resulting languages remain within the same class.

## Contribution

It introduces and analyzes insertion operations on languages accepted by deterministic reversal-bounded multicounter machines, exploring their closure properties and acceptance power.

## Key findings

- Certain insertion operations preserve acceptance by the same class of machines.
- Some operations lead to languages that require more counters or input turns to accept.
- The study clarifies the boundaries of deterministic reversal-bounded multicounter machine capabilities.

## Abstract

Several insertion operations are studied applied to languages accepted by one-way and two-way deterministic reversal-bounded multicounter machines. These operations are defined by the ideals obtained from relations such as the prefix, infix, suffix, and outfix relations, as well as operations defined from inverses of a type of deterministic transducer with reversal-bounded counters attached. The question of whether the resulting languages can always be accepted by deterministic machines with the same number (or larger number) of input-turns (resp., counters, counter-reversals, etc.) is investigated.