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Determinism in Multi-Soliton Automata

Henning Bordihn, Helena Schulz

Abstract

Soliton automata are mathematical models of soliton switching in chemical molecules. Several concepts of determinism for soliton automata have been defined. The concept of strong determinism has been investigated for the case in which only a single soliton can be present in a molecule. In the present paper, several different concepts of determinism are explored for the multi-soliton case. It is shown that the degree of non-determinism is a connected measure of descriptional complexity for multi-soliton automata. A characterization of the class of strongly deterministic multi-soliton automata is presented. Finally, the concept of perfect determinism, forming a natural extension of strong determinism, is introduced and considered for multi-soliton automata.

Determinism in Multi-Soliton Automata

Abstract

Soliton automata are mathematical models of soliton switching in chemical molecules. Several concepts of determinism for soliton automata have been defined. The concept of strong determinism has been investigated for the case in which only a single soliton can be present in a molecule. In the present paper, several different concepts of determinism are explored for the multi-soliton case. It is shown that the degree of non-determinism is a connected measure of descriptional complexity for multi-soliton automata. A characterization of the class of strongly deterministic multi-soliton automata is presented. Finally, the concept of perfect determinism, forming a natural extension of strong determinism, is introduced and considered for multi-soliton automata.
Paper Structure (4 sections, 5 theorems, 20 equations, 6 figures)

This paper contains 4 sections, 5 theorems, 20 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Soliton Automata
  3. Graph Properties and Determinism
  4. Concluding Remarks

Key Result

Theorem 1

The degree of non-determinism is a connected measure of descriptional complexity, that is, for every positive integer $g$, there is a soliton automaton $A_g$ such that its degree of non-determinism is $g$.

Figures (6)

  • Figure 1: A soliton graph with two external nodes.
  • Figure 2: Part of a configuration trail for the burst $(1,1)\|_1(1,1)\bot$. The first soliton is depicted as a black pebble, while the second one is depicted as a white pebble.
  • Figure 3: Soliton graphs $G_1$, $G_2$ and $G_3$
  • Figure 4: A cycle with even length and two edges with weight 2 leading into it.
  • Figure 5: A node of degree $3$ as entry point of a cycle.
  • ...and 1 more figures

Theorems & Definitions (26)

  • Definition 1: DassowJurg:Soliton
  • Definition 2: Bursts of Inputs BorJur:Multiwave
  • Definition 3: Position Map BorJur:Multiwave
  • Definition 4: Initial Position Map for a Burst BorJur:Multiwave
  • Definition 5: Final Position Map BorJur:Multiwave
  • Definition 6: Potential Successor Map BorJur:Multiwave
  • Definition 7: Configuration and Configuration Trail BorJur:Multiwave
  • Definition 8: Soliton Path
  • Definition 9: Result of a Burst BorJur:Multiwave
  • Definition 10: Multi-Soliton Automaton BorJur:Multiwave
  • ...and 16 more