5' -> 3' Watson-Crick Automata accepting Necklaces

TL;DR

This paper explores 5'->3' Watson-Crick automata processing circular DNA-like structures called necklaces, analyzing their acceptance modes and hierarchical relations, extending automata theory to cyclic DNA models.

Contribution

It introduces and studies the acceptance modes of 5'->3' Watson-Crick automata on necklaces, revealing their relation to linear and cyclic language classes and establishing hierarchy results.

Findings

Weak acceptance languages are cyclic shifts of linear context-free languages.

Strong acceptance languages are maximal cyclic shift-closed subsets of linear languages.

Hierarchy results for restricted automata variants are established.

Abstract

Watson-Crick (WK) finite automata work on a Watson-Crick tape representing a DNA molecule. They have two reading heads. In 5'->3' WK automata, the heads move and read the input in opposite physical directions. In this paper, we consider such inputs which are necklaces, i.e., they represent circular DNA molecules. In sensing 5'->3' WK automata, the computation on the input is finished when the heads meet. As the original model is capable of accepting the linear context-free languages, the necklace languages we are investigating here have strong relations to that class. Here, we use these automata in two different acceptance modes. On the one hand, in weak acceptance mode the heads are starting nondeterministically at any point of the input, like the necklace is cut at a nondeterministically chosen point), and if the input is accepted, it is in the accepted necklace language. These languages can be seen as the languages obtained from the linear context-free languages by taking their closure under cyclic shift operation. On the other hand, in strong acceptance mode, it is required that the input is accepted starting the heads in the computation from every point of the cycle. These languages can be seen as the maximal cyclic shift closed languages included in a linear language. On the other hand, as it will be shown, they have a kind of locally testable property. We present some hierarchy results based on restricted variants of the WK automata, such as stateless or all-final variants.