# A Bio-Inspired Model of Picture Array Generating P System with Restricted Insertion Rules

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Basic Notions

#### 2.2. Picture-Insertion System

**Definition**

**1.**

- (i)
- Σ is an alphabet;
- (ii)
- ${I}_{c}=\{{c}_{i}\mid 1\le i\le m\},$$(m\ge 1)$ where ${c}_{i},$ called a column insertion table, is a finite set of column insertion rules with alphabetic contexts of the form $(a,\alpha ,b),$$a,b\in {\Sigma}\cup \left\{\lambda \right\},\alpha \in {{\Sigma}}^{*}$ such that for any two rules $({a}_{1},\alpha ,{b}_{1}),({a}_{2},\beta ,{b}_{2})$ in ${c}_{i}$, we have $\left|\alpha \right|=\left|\beta \right|$ and either both the left contexts ${a}_{1}$ and ${a}_{2}$ are in Σ (likewise the right contexts ${b}_{1}$ and ${b}_{2}$ are in Σ) or both are λ;
- (iii)
- ${I}_{r}=\{{r}_{j}\mid 1\le j\le n\},$$(n\ge 1)$ where ${r}_{j},$ called a row insertion table, is a finite set of row insertion rules with alphabetic contexts of the form $(d,{}^{t}\gamma ,e),$$d,e\in {\Sigma}\cup \left\{\lambda \right\},\gamma \in {{\Sigma}}^{*}$ such that for any two rules $({d}_{1},{}^{t}\gamma ,{e}_{1}),({d}_{2},{}^{t}\delta ,{e}_{2})$ in ${r}_{j}$, we have $\left|\gamma \right|=\left|\delta \right|$ and either both the up contexts ${d}_{1}$ and ${d}_{2}$ are in Σ (likewise the down contexts ${e}_{1}$ and ${e}_{2}$ are in Σ) or both are λ;
- (iv)
- $A\subseteq {{\Sigma}}^{**}-\left\{\lambda \right\}$ is a finite set of axiom arrays.

**Example**

**1.**

## 3. Array P System with Restricted Picture Insertion Rules

**Definition**

**2.**

**Example**

**2.**

## 4. A Hierarchy between One Membrane and Two Membranes

**Theorem**

**1.**

**Proof.**

## 5. Comparison with Pure 2D Context-Free Grammars

**Theorem 2.**

- 1.
- $A{P}_{2}\left(RPIS\right)-P2DCFL\ne \varphi $
- 2.
- $A{P}_{2}\left(RPIS\right)-(l/u)P2DCFL\ne \varphi $
- 3.
- $A{P}_{2}\left(RPIS\right)-(r/d)P2DCFL\ne \varphi $
- 4.
- $A{P}_{2}\left(RPIS\right)-INPA\ne \varphi $

**Proof.**

## 6. Comparison with Certain Standard 2D Grammar Models

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

**Theorem**

**5.**

**Proof.**

## 7. An Application of the Model to “Kolam” Pattern Generation

## 8. Concluding Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 9.**Array representing the “kolam” in Figure 8.

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**MDPI and ACS Style**

Zhang, G.; Samdanielthompson, G.; David, N.G.; Nagar, A.K.; Subramanian, K.G.
A Bio-Inspired Model of Picture Array Generating *P* System with Restricted Insertion Rules. *Appl. Sci.* **2020**, *10*, 8306.
https://doi.org/10.3390/app10228306

**AMA Style**

Zhang G, Samdanielthompson G, David NG, Nagar AK, Subramanian KG.
A Bio-Inspired Model of Picture Array Generating *P* System with Restricted Insertion Rules. *Applied Sciences*. 2020; 10(22):8306.
https://doi.org/10.3390/app10228306

**Chicago/Turabian Style**

Zhang, Gexiang, G. Samdanielthompson, N. Gnanamalar David, Atulya K. Nagar, and K.G. Subramanian.
2020. "A Bio-Inspired Model of Picture Array Generating *P* System with Restricted Insertion Rules" *Applied Sciences* 10, no. 22: 8306.
https://doi.org/10.3390/app10228306