Determining Smallest Path Size of Multiplication Transducers Without a Restricted Digit Set

TL;DR

This paper investigates the smallest closed loop size in directed multiplication transducers across various bases, revealing recursive patterns and ranges of multipliers, with potential extensions to digit set reductions.

Contribution

It introduces a recursive method to determine the smallest loop size in multiplication transducers and identifies related patterns and ranges for different bases and multipliers.

Findings

Recursive pattern for smallest closed loop length

Range of multipliers with specific loop lengths

General recurrence relation for transducer loops

Abstract

Directed multiplication transducers are a tool for performing non-decimal base multiplication without an additional conversion to base 10. This allows for faster computation and provides easier visualization depending on the problem at hand. By building these multiplication transducers computationally, new patterns can be identified as these transducers can be built with much larger bases and multipliers. Through a recursive approach, we created artificial multiplication transducers, allowing for the formation of several unique conjectures specifically focused on the smallest closed loop around a multiplication transducer starting and ending at zero. We show a general recursive pattern for this loop; through this recurrence relation, the length of the smallest closed loop for a particular transducer base b along with the range of multipliers having this particular length for multiplier m was also identified. This research is expected to be explored further by testing reductions of the digit set and determining whether similar properties will hold.