# A Formal Model for Adaptive Free Choice in Complex Systems

## Abstract

**:**

## 1. Introduction

## 2. Adaptive Free Choice

## 3. Process Ontology

**Definition**

**1**(State).

**Definition**

**2**(Process)

**.**

## 4. The Model

**Definition**

**3**(Choice)

**.**

**Axiom 1**.The most fundamental systems are irreducible to other systems, i.e., they contain no interactions and cannot be partitioned.**Axiom 2**. All possible configurations, i.e., microstates, of fundamental systems are equally likely in the long run.**Axiom 3**. A system’s macrostates are formed via interacting (micro)processes.**Axiom 4**. The probability that a choice will lead to its macrostate is arbitrarily high if the choice is free, i.e., a free choice inevitably leads to a nearly deterministic outcome.**Axiom 5**. The number N of possible choices that a system has must be small enough to be read into the system’s memory in a finite time.**Axiom 6**. The choices do not change at any time during the agent’s processing of the choices nor during the agent’s enactment of the choice.**Axiom 7**. The choices must be distinct.

**Definition**

**4**(Free choice)

**.**

**Definition**

**5**(Free will)

**.**

## 5. Weighting the Choices

## 6. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**A process ${\pi}_{ij}$ takes a system $\sigma $ from some fixed state ${\mathit{\psi}}_{\mathit{i}}\left({\lambda}_{i}\right)$ to one (and only one) of n possible states ${\mathit{\psi}}_{\mathit{j}}\left({\lambda}_{j}\right)$.

**Figure 2.**The macroprocess ${\mathsf{\Pi}}_{1J}$ has four possible outcomes determined by the possible outcomes of each of the independent, non-interacting two-state subsystems. The probability of each possible outcome is the same. Recall that, once enacted, a process ultimately only leads to a single outcome, e.g., while the act of flipping a coin can lead to two possible states, once the flip has concluded, the coin ultimately ends up in only one state.

**Figure 3.**The normalized probability distribution for a two-state system is shown for (

**a**) $N=10$, (

**b**) $N=100$, and (

**c**) $N=1000$ subsystems as a function of ${N}_{2}/N$. As can be seen, for larger values of N, a smaller number of macrostates tend to dominate over the rest. As $N\to \infty $, the distribution tends toward a delta function.

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Durham, I. A Formal Model for Adaptive Free Choice in Complex Systems. *Entropy* **2020**, *22*, 568.
https://doi.org/10.3390/e22050568

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Durham I. A Formal Model for Adaptive Free Choice in Complex Systems. *Entropy*. 2020; 22(5):568.
https://doi.org/10.3390/e22050568

**Chicago/Turabian Style**

Durham, Ian. 2020. "A Formal Model for Adaptive Free Choice in Complex Systems" *Entropy* 22, no. 5: 568.
https://doi.org/10.3390/e22050568