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

Benedek Nagy

TL;DR

This paper considers inputs which are necklaces, i.e., they represent circular DNA molecules, and presents some hierarchy results based on restricted variants of the WK automata, such as stateless or all-final variants.

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.

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

TL;DR

This paper considers inputs which are necklaces, i.e., they represent circular DNA molecules, and presents some hierarchy results based on restricted variants of the WK automata, such as stateless or all-final variants.

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.
Paper Structure (5 sections, 19 theorems, 11 equations, 5 figures)

This paper contains 5 sections, 19 theorems, 11 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. On weakly accepted necklace language classes
  4. On strongly accepted necklace language classes
  5. Conclusions

Key Result

Proposition 1

The cyclic closure $cycl(\mathcal{L}_{LIN})$ is weakly accepted by sensing $5'\rightarrow 3'$ WK finite automata, that is, for each linear language $L$, its cyclic closure $cycl(L)$ is in $\mathcal{W}_*$ and for each language $L' \in \mathcal{W}_*$ there is a linear language $L"$ such that $L' = cyc i.e., the class of weakly accepted necklace languages by a restricted class of sensing $5'\rightarr

Figures (5)

  • Figure 1: A sensing $5'\rightarrow 3'$ WK automaton in the initial configuration.
  • Figure 2: A sensing $5'\rightarrow 3'$ WK automaton in a configuration during a computation and in an accepting configuration with a final state $q_f$ (bottom).
  • Figure 3: A sensing F1$5' \rightarrow 3'$ WK automaton that is accepting a non linear context free language of necklaces in the strong mode.
  • Figure 4: Hierarchy of necklace languages weakly accepted by sensing $5' \rightarrow 3'$ WK finite automata in a Hasse diagram. Each of the shown inclusions is proper.
  • Figure 5: Hierarchy of necklace languages strongly accepted by sensing $5' \rightarrow 3'$ WK finite automata in a Hasse diagram. Arrows represent proper inclusions, while lines represent inclusions where the properness is left open.

Theorems & Definitions (21)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • ...and 11 more