# Emergence at the Fundamental Systems Level: Existence Conditions for Iterative Specifications

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## Abstract

**:**

## 1. Introduction

- not defined at S8 because there is no trajectory emerging from it;
- non-deterministic at S1 because there are two distinct outbound transitions defined for it; and
- deterministic at S2 and S4 because there is only one outbound transition for each.

## 2. Formal Background

#### 2.1. Basic Discrete Event System Specification

- not defined at $S8$ because there is no transition pair with S8 as the left member in δ;
- non-deterministic at $S1$ because it is a left member of two transition pairs $(S1,S2)$ and $(S1,S3)$; and
- deterministic at $S2$ and $S4$ because there is only one transition pair involving each one as a left member.

#### 2.2. Coupled DEVS Models

## 3. Iterative System Specifications

**Theorem**

**1.**

- Existence of longest initial segments: $\omega \in {\Gamma}^{+}\Rightarrow max\left\{t\text{}\right|\text{}{\omega}_{t}\in \Gamma \}$ exists
- Closure under right segmentation: $\omega \in \Gamma \Rightarrow {\omega}_{<\tau}\in \Gamma $ for all $\tau \in dom(\omega )$

#### 3.1. DEVS Simulation of Iterative Specification

#### 3.2. Coupled Iterative Specification

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

**Theorem**

**2.**

**Theorem**

**3.**

**Theorem**

**4.**

**Proof.**

#### 3.3. Special Case: Memoryless Systems

$\rho 1=\beta 1(gr,\omega 1)\iff (\omega 1,\rho 1)\in R1$ |

$\rho 2=\beta 2(gr,\omega 2)\iff (\omega 2,\rho 2)\in R2$ |

$\omega 2=\rho 1$ |

$\omega 1=\rho 2$ |

$\rho 2=\beta 2(q2,\omega 2)\iff (\rho 1,\omega 1)\in R2\iff (\omega 1,\rho 1)\in R{2}^{-1}$ |

$f(\rho 2)=\beta 1(gr,\rho 2)$ |

$g(\rho 1)=\beta 2(gr,\rho 1)$ |

- no solutions if ${f}^{-1}$ does not exist, yielding no resultant;
- a unique solution for every input if ${f}^{-1}$ exists, yielding a deterministic resultant; and
- multiple solutions for a given segment $\rho \phantom{\rule{0.277778em}{0ex}}\mathrm{if}\phantom{\rule{0.277778em}{0ex}}{f}^{-1}(\rho )$ is multivalued, yielding a non-deterministic resultant.

$\beta (gr,\rho )=f(\rho )=\rho +1$ |

$i.e.,\forall t\in T,\beta (gr,\rho )(t)=\rho (t)+1$ |

- A NOT gate has an inverse $0\to 1,1\to 0$, so the composition has two solutions one for each of two complementary assignments, i.e., $F(gr,gr)=\{(1,0),(0,1)\}$ yielding a non-deterministic system.
- An AND gate with one of its input held to 0 always maps the other input into 0 so has a solution only for inputs of 0 to each component, i.e., $F(gr,gr)=(0,0)$, yielding a deterministic system.
- An AND gate with a stuck-at input of 1 is the identity mapping and has multiple solutions i.e., $F(gr,gr)=\{(0,0),(1,1)\}$, yielding a non-deterministic system.

#### 3.4. Active-Passive Systems

**Definition**

**6.**

## 4. Temporal Progress: Legitimacy, Zenoness

**Definition**

**7.**

**Definition**

**8.**

**Definition**

**9.**

**Definition**

**10.**

#### Fundamental Systems Existence: Probabilistic Characterization of Halting

## 5. Discussion: Future Research

## Appendix A. Turing Machine as a DEVS

#### Appendix A.1. Tape System

**;**in other words, the tape is an infinite sequence of bits, where $pos$ represents the position of the head, and $mv$ is a specification for moving left or right. An internal transition moves the head as specified; an external transition accepts a symbol, move pair, stores the symbol in the current position of the head and stores the move for subsequent execution. The slot for storing the move can also be null, which indicates that a move has taken place.

${\delta}_{int}(tape,pos,mv)=(tape,move(pos,mv),null)$ |

${\delta}_{ext}((tape,pos,null),e,(sym,mv))=(store(tape,pos,sym),pos,mv)$ |

$ta(tape,pos,mv)=1$ |

$ta(tape,pos,null)=\infty $ |

$\lambda (tape,pos,mv)=getSymbol(tape,pos)$ |

$move(pos.mv)=pos+mv$ |

$store(tape,pos,sym)=tap{e}^{\prime}\phantom{\rule{0.277778em}{0ex}}where\phantom{\rule{0.277778em}{0ex}}tap{e}^{\prime}(pos)=sym,tap{e}^{\prime}(i)=tape(i)$ |

$getSymbol(tape,pos)=tape(pos)$ |

#### Appendix A.2. TM Control

${\delta}_{int}(st,sym)=(TMState(st,sym),null)$ |

${\delta}_{ext}((st,null),e,sym)=(st,sym)$ |

$ta(st,sym)=1$ |

$ta(st,null)=\infty $ |

$\lambda (st,sym)=TMOutput(st,sym)$ |

$TMState(st,sym)=s{t}^{\prime}$ |

$TMOutput(st,sym)=sy{m}^{\prime}$ |

## Appendix B. Iterative Specification of Systems

#### Appendix B.1. Generator Segments

**Theorem**

**B1.**

- Existence of longest initial segments: $\omega \in {\Gamma}^{+}\Rightarrow max\left\{t\text{}\right|\text{}{\omega}_{t}\in \Gamma \}$ exists
- Closure under right segmentation: $\omega \in \Gamma \Rightarrow {\omega}_{<t}\in \Gamma $ for all $\tau \in dom(\omega )$

#### Appendix B.2. Generator State Transition Systems

**Theorem**

**B2.**

- Existence of longest prefix segments: $\omega \in {\mathsf{\Omega}}_{G}^{+}\Rightarrow max\left\{t\text{}\right|\text{}omeg{a}_{t}\in {\mathsf{\Omega}}_{G}\}$ exists
- Closure under right segmentation: $\omega \in {\mathsf{\Omega}}_{G}\Rightarrow {\omega}_{<t}\in {\mathsf{\Omega}}_{G}$ for $t\in dom(\omega )$
- Closure under left segmentation: $\omega \in {\mathsf{\Omega}}_{G}\Rightarrow {\omega}_{t>}\in {\mathsf{\Omega}}_{G}$ for $t\in dom(\omega )$
- Consistency of composition: ${\delta}_{G}^{+}(q,{\omega}_{1}\u2022{\omega}_{2}\u2022\cdots \u2022{\omega}_{n})={\delta}_{G}({\delta}_{G}(\cdots {\delta}_{G}({\delta}_{G}(q,{\omega}_{1}),{\omega}_{2}),\cdots ),{\omega}_{n})$

## Appendix C. DEVS Atomic Model Simulation of an Iterative Specification

**Theorem**

**C1.**

**Proof.**

$g:{\mathsf{\Omega}}_{G}\to {\mathsf{\Omega}}_{DEVS}$ using mls |

$g(\omega )={\left[\omega \right]}_{l(\omega )>}=\left[\omega \right]{\varphi}_{l(\omega )>}$ |

$h:Q\to Q\times {\mathsf{\Omega}}_{G}$ |

$h(q)=(q,dummy)$ |

$S=Q\times {\mathsf{\Omega}}_{G}$ |

${\delta}_{int}(q,\omega )=({\delta}_{G}(q,\omega ),dummy)$ |

$ta(q,\omega )=l(\omega )$ |

${\delta}_{ext}((q,\omega ),e,{\omega}^{\prime})=({\delta}_{G}(q,{\omega}_{e>}),{\omega}^{\prime})$ |

$\lambda (q,dummy)={\lambda}_{G}(q)$ |

${\delta}_{G}^{+}(q,\omega {\omega}^{\prime})={\delta}_{G}({\delta}_{G}(q,\omega ),{\omega}^{\prime})$ |

${\delta}_{DEVS}(h(q),g(\omega {\omega}^{\prime}))$ |

$={\delta}_{DEVS}((q,dummy),g(\omega )g({\omega}^{\prime}))$ |

$={\delta}_{DEVS}((q,dummy),{\left[\omega \right]}_{l(\omega )>},{\left[{\omega}^{\prime}\right]}_{l({\omega}^{\prime})>})$ |

$={\delta}_{DEVS}({\delta}_{ext}((q,dummy),\left[\omega \right]),{\varphi}_{l(\omega )>}),{\omega}_{l({\omega}^{\prime})>}^{\prime})$ |

$={\delta}_{DEVS}((q,\left[\omega \right]),{\varphi}_{l(\omega )>}),{\omega}_{l({\omega}^{\prime})>}^{\prime})$ |

$={\delta}_{DEVS}({\delta}_{int}(q,\left[\omega \right]),{\varphi}_{l(\omega )>}),{\omega}_{l({\omega}^{\prime})>}^{\prime})$ |

$={\delta}_{DEVS}(({\delta}_{G}(q,\omega ),dummy),{\omega}_{l({\omega}^{\prime})>}^{\prime})$ |

$=({\delta}_{G}({\delta}_{G}(q,\omega ),{\omega}^{\prime}),dummy)$ |

$h({\delta}_{G}^{+}(q,\omega {\omega}^{\prime}))$ |

$=h({\delta}_{G}({\delta}_{G}(q,\omega ),{\omega}^{\prime}))$ |

$=({\delta}_{G}({\delta}_{G}(q,\omega ),{\omega}^{\prime}),dummy)$ |

$={\delta}_{DEVS}(h(q),g(\omega {\omega}^{\prime}))$ |

## Appendix D. Coupled Iterative Specification at the I/O System level

**Theorem**

**D1.**

**Proof.**

**Theorem**

**D2.**

**Proof.**

## Appendix E. A Statistical Experiment Sampling from 2-Symbol, 3-State TMs

State | Symbol | Next State {A, B, C} | Move {1, −1} | Print Symbol {0, 1} |
---|---|---|---|---|

A | 0 | x | x | x |

A | 1 | x | x | x |

B | 0 | x | x | x |

B | 1 | x | x | x |

C | 0 | x | x | x |

C | 1 | x | x | x |

Halts at N | Frequency | Geometric Model |
---|---|---|

1 | 0.161 | 0.161 |

2 | 0.1 | 0.09 |

3 | 0.042 | 0.039 |

4 | 0.023 | 0.022 |

5 | 0.016 | 0.015 |

6 | 0.005 | 0.00488 |

7 | 0.004 | 0.00390 |

11 | 0.001 | 0.000990 |

16 | 0.001 | 0.000985 |

10000 | 0.647 |

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Zeigler, B.P.; Muzy, A. Emergence at the Fundamental Systems Level: Existence Conditions for Iterative Specifications. *Systems* **2016**, *4*, 34.
https://doi.org/10.3390/systems4040034

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Zeigler BP, Muzy A. Emergence at the Fundamental Systems Level: Existence Conditions for Iterative Specifications. *Systems*. 2016; 4(4):34.
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**Chicago/Turabian Style**

Zeigler, Bernard P., and Alexandre Muzy. 2016. "Emergence at the Fundamental Systems Level: Existence Conditions for Iterative Specifications" *Systems* 4, no. 4: 34.
https://doi.org/10.3390/systems4040034