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Characterizations of Controlled Generation of Right Linear Grammars with Unknown Behaviors

Daihei Ise, Satoshi Kobayashi

TL;DR

This work develops a rigorous theoretical framework for the controlled generation of right linear grammars with unknown behaviors (RLUBs) under a partial order on the control alphabet. By introducing Condition (C) and its relaxed form (C'), it provides necessary and sufficient conditions for realizing a given finite class of languages $\mathcal{L}$ via synchronous (and partially non-synchronous) erasing-mode generation using RLUBs and their control systems. The main contribution is a constructive method that yields an RLUB $H_*=(G,M)$ and a control system $C_*=(\Gamma,\phi,T)$ whenever (C) holds, together with a converse showing that any such generation implies the existence of an injection $\theta$ and Boolean mappings $\delta_{\mathcal{L}}$ satisfying (C). This advances the theoretical understanding of regulation and hierarchy in RLUBs, with potential applications to modeling control devices in DNA-based nano-scale systems and other regulated rewriting frameworks.

Abstract

This paper deals with the control generation of right linear grammars with unknown behaviors (RLUBs, for short) in which derivation behavior is not determined completely. In particular, we consider a physical property of control devices used in control systems and formulate it as a partial order over control alphabet of the control system. We give necessary and sufficient conditions for given finite language classes to be generated by RLUBs and their control systems using a given partial order over control alphabet.

Characterizations of Controlled Generation of Right Linear Grammars with Unknown Behaviors

TL;DR

This work develops a rigorous theoretical framework for the controlled generation of right linear grammars with unknown behaviors (RLUBs) under a partial order on the control alphabet. By introducing Condition (C) and its relaxed form (C'), it provides necessary and sufficient conditions for realizing a given finite class of languages via synchronous (and partially non-synchronous) erasing-mode generation using RLUBs and their control systems. The main contribution is a constructive method that yields an RLUB and a control system whenever (C) holds, together with a converse showing that any such generation implies the existence of an injection and Boolean mappings satisfying (C). This advances the theoretical understanding of regulation and hierarchy in RLUBs, with potential applications to modeling control devices in DNA-based nano-scale systems and other regulated rewriting frameworks.

Abstract

This paper deals with the control generation of right linear grammars with unknown behaviors (RLUBs, for short) in which derivation behavior is not determined completely. In particular, we consider a physical property of control devices used in control systems and formulate it as a partial order over control alphabet of the control system. We give necessary and sufficient conditions for given finite language classes to be generated by RLUBs and their control systems using a given partial order over control alphabet.
Paper Structure (9 sections, 10 theorems, 38 equations, 1 figure)

This paper contains 9 sections, 10 theorems, 38 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. RLUB and its Control
  4. Important Definitions and Lemma
  5. Characterization of Controlled Generation of RLUBs
  6. Condition (C)
  7. Construction of $H_*$ and $C_*$
  8. Necessary and sufficient conditions for the controlled generation of RLUBs
  9. Conclusions and Future Works

Key Result

Theorem 1

(Theorem 2 in Kimoto for RLUBs) Let $M=(\mu_t,\mu_b)$ be a GC, $G=(V,\Sigma,S,P)$ be a right linear grammar, $L$ be a language over $\Sigma$, and $C=(\Gamma,\phi,T)$ be a control system. Consider RLUBs $H=(G,M)$, $H_1=(G,(\mu_t,\mu_t))$, $H_2=(G,(\mu_b,\mu_b))$, and any control sequence $\tau \in T$

Figures (1)

  • Figure 1: Set inclusion relations of $S(i,j)$

Theorems & Definitions (41)

  • Remark 1
  • Remark 2
  • Example 1
  • Theorem 1
  • Proposition 1
  • Definition 1
  • Example 2
  • Proposition 2
  • proof
  • Proposition 3
  • ...and 31 more