Transducer Descriptions of DNA Code Properties and Undecidability of Antimorphic Problems

TL;DR

This paper introduces a transducer-based framework for describing DNA code properties, demonstrating increased expressiveness and undecidability results for certain problems, including maximality and antimorphic variants.

Contribution

It defines DNA code properties via transducers, proving this approach is more expressive than trajectory descriptions and establishing undecidability of key problems.

Findings

Transducer descriptions are strictly more expressive than trajectory descriptions.

Decidability of satisfaction questions remains efficient under the new framework.

Maximality and antimorphic Post Correspondence problems are undecidable.

Abstract

This work concerns formal descriptions of DNA code properties, and builds on previous work on transducer descriptions of classic code properties and on trajectory descriptions of DNA code properties. This line of research allows us to give a property as input to an algorithm, in addition to any regular language, which can then answer questions about the language and the property. Here we define DNA code properties via transducers and show that this method is strictly more expressive than that of trajectories, without sacrificing the efficiency of deciding the satisfaction question. We also show that the maximality question can be undecidable. Our undecidability results hold not only for the fixed DNA involution but also for any fixed antimorphic permutation. Moreover, we also show the undecidability of the antimorphic version of the Post Corresponding Problem, for any fixed antimorphic permutation.