# Metacognition as a Consequence of Competing Evolutionary Time Scales

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## Abstract

**:**

## 1. Introduction

## 2. Background

#### 2.1. Metacognition from an Evolutionary Perspective

#### 2.2. Computational Resources for Metaprocessing

#### 2.3. Interaction across a Markov Blanket

**$\mathcal{B}$**separating $S$ from $E$. Formally, the boundary

**$\mathcal{B}$**is a decompositional boundary in the state space of the joint system $SE$; it separates states of $S$ from states of $E$. The Hamiltonian ${H}_{SE}$ is, formally, a linear operator on this joint state space. The conservation of energy requires that the net energy flow between $S$ and $E$ is asymptotically zero, i.e., that the interaction is asymptotically adiabatic; we will assume for simplicity that it is adiabatic over whatever time scale is of interest.

**$\mathcal{B}$**, which $E$ then reads, and the cycle reverses [22,33,69]—see [70,71] for technical details. The information exchanged between $S$ and $E$—hence the information encoded on

**$\mathcal{B}$**in each interaction cycle—can be specified exactly: at each instant,

**$\mathcal{B}$**encodes the current eigenvalue of the operator ${H}_{SE}$.

**$\mathcal{B}$**functions as an MB between $S$ and $E$ [72]. An MB surrounding a system $S$ is, by definition, a set of states $m=\left(s,a\right)$ such that internal states $i$ of the blanketed system $S$ (if any) depend on external or environmental states e only via their effects on $m$ [25,26]. In practice, an MB exists whenever the state space of $S\left(E\right)$ includes states not on the boundary

**$\mathcal{B}$**. Friston [27,28] introduced the decomposition of the MB into sensory states $s$ and active states $a$, with the further constraint that sensory states are not influenced by internal states and active states are not influenced by external states; this distinction is illustrated in Figure 2b and, in two biological contexts, in Figure 3. Any MB is, clearly, completely symmetrical; the “sensory” states of $E$ are the “active” states of $S$ and vice versa. An MB generalizes the function of an API, in effect defining, and therefore limiting, the communication channel between any system $S$ and its environment $E$.

**$\mathcal{B}$**that implements the MB separating $S$ from $E$. The existence of the MB thus places significant restrictions on what $S$ can “know” about $E$ and vice-versa. Because $S$ has no direct access to the state $e$ of $E$, $S$ cannot measure the dimension of $e$, and hence cannot determine the number of degrees of freedom of $E$. The MB similarly prevents any direct access to the internal interaction ${H}_{E}$ of $E$. We can, therefore, construct any finite decomposition $E=FG$ with internal interaction ${H}_{E}={H}_{F}+{H}_{G}+{H}_{FG}$ without affecting what $S$ can detect, i.e., without affecting the information encoded on the MB by ${H}_{SE}$ in any way. Hence, $S$ cannot “know” about decompositions of $E$. Any “subsystems” of $E$ represented by $S$ are, effectively, sectors of the MB that are defined by computational processes implemented by $S$; see [22,33,69,73] for formal details and further discussion. As the MB is completely symmetrical, these considerations apply equally to $E$.

#### 2.4. Active Inference Framework

**Theorem**

**1.**

**Proof.**

**$\mathcal{B}$**within a fixed joint system $U$. This evolution corresponds to an exchange of degrees of freedom between $S$ and $E$; $S$ engulfing part of $E$ or vice versa would be an example. Any such evolution changes the Hamiltonian, ${H}_{SE}\to {H}_{{S}^{\prime}{E}^{\prime}}$. If by theoretical fiat we choose to regard $S$ and ${S}^{\prime}$ as “the same system”—e.g., we treat an organism as “the same thing” after it ingests a meal—we can conclude via Theorem 1 that the VFE experienced by that system has increased. Its VFE may, of course, thereafter decrease if the new interaction ${H}_{{S}^{\prime}{E}^{\prime}}$ proves more predictable than the previous ${H}_{SE}$.

## 3. Results

#### 3.1. Formal Investigation of Metacognition in Evolution

**$\mathcal{B}$**(i.e., sensation) and writing on

**$\mathcal{B}$**(i.e., action) on the part of both $S$ and $E$. Landauer’s principle [87,88] imposes a free-energy cost of at least $\epsilon ={k}_{B}Tln2$ per erased bit (and hence per rewritten bit) on (classical) information processing, where ${k}_{B}$ is Boltzmann’s constant and $T>0$ is the temperature of the system performing the computing. Note that this is a lower limit; a realistic system will have a per-bit processing cost of $\epsilon =\beta {k}_{B}T$, with $\beta >ln2$ a measure of thermodynamic efficiency. As this free energy must be sourced, and the waste heat dissipated, through

**$\mathcal{B}$**asymptotically, we assume for simplicity that it is sourced and dissipated through

**$\mathcal{B}$**on every cycle. We can, therefore, represent

**$\mathcal{B}$**as in Figure 2b. The bits allocated to free-energy acquisition and waste heat dissipation are uninformative to $S$, as they are burned as fuel for computation. The bits allocated to action by $S$ are similarly uninformative to $S$, although they are informative to $E$. The bits allocated to sensation by $S$ are informative to $S$ and serve as the inputs to $S$’s computations. This distinction between informative and uninformative bits is often not made explicitly (such as in Figure 3), where the flow of free energy as incoming fuel and outgoing waste is ignored; however, it is required whenever the joint system $SE$ is thermodynamically closed [33]. To emphasize this distinction, we will use $s$ for informative sensory states and $a$ for states encoding “interesting” actions, i.e., actions other than waste heat dissipation.

**$\mathcal{B}$**, from $S$’s perspective, τ is the response time of $E$ to actions by $S$. We assume for simplicity that $S$’s response time is also τ, i.e., that $S$ is capable of computing its next action in time τ from detecting $E$’s current action.

**$\mathcal{B}$**to sensation and action, i.e., $S$ computes a function $f:{\left\{0,1\right\}}^{n}\to {\left\{0,1\right\}}^{n}$, and that this function depends, in general, on the values of all n input bits (i.e., $S$ does not ignore informative bits). If computing this function requires m steps, including all relevant classical memory write operations, then $S$ has a free-energy cost of $nm\epsilon $ per sensation–action cycle.

**Theorem**

**2.**

**Proof.**

**Corollary**

**1.**

**Proof.**

#### 3.2. General Models of Two-System Interaction with Selection across Different Time Scales

#### 3.2.1. Multi-Agent Active Inference Networks

#### 3.2.2. Predator–Prey Models

#### 3.2.3. Coupled Genetic Algorithms

#### 3.2.4. Coupled Generative Adversarial Networks

#### 3.3. Spatio-Temporally Coarse-Grained Structures Emerge Naturally in Any Resource-Limited System with Sufficiently Complex Interaction Dynamics

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Generic architecture of a metaprocessor. A metaprocessor (red) regulates an object processor (blue) both internally and externally. The metaprocessor requires its own local memory of the object processor’s behavior.

**Figure 2.**General form of bipartite interaction dynamics in the absence of quantum entanglement. (

**a**) Finite systems

**S**and

**E**interact via a boundary

**$\mathcal{B}$**that serves as an MB. (

**b**) From

**S**’s perspective,

**$\mathcal{B}$**comprises sensory bits (in state s, red shading) that are inputs to

**S**’s computations and action bits (in state a, blue shading) that are

**S**’s outputs. When the free-energy costs of computation are taken into account, as in Section 3.1 below, some input bits must be allocated to fuel for computation and some output bits must be allocated to waste heat dissipation (hatched area); these bits are uninformative to

**S**.

**Figure 3.**Markov blanket state separation in active inference.

**Middle**: The internal (pink) and external states (white) of each agent are separated by a Markov blanket, comprised of sensory (grey) and active (green) states. This architecture can be applied to various forms of information processing, two of which are shown above and below. By associating the gradient flows of the Markov blanket partition with Bayesian belief updating, self-organization of internal states—in response to sensory fluctuations—can be thought of as perception, while active states couple internal states back to hidden external states vicariously, to provide a mathematical formulation of action and behavior.

**Top**: Visual processing. Internal states are made up of the brain, which directs movement of the eye through abductor muscles as active states. Sensory states as the photoreceptors in the eye perceive the external states in the visual field of view.

**Bottom**: Transcriptional machinery. Internal states are given by the gene expression levels plus epigenetic modifications. Intracellular components such as ribosomes, smooth and rough endoplasmic reticulum, and Golgi apparatus implement protein translation and delivery as the active states. Sensory states correspond to the surface states of the cell membrane, such as ligand receptors, ion channel states, and gap junctions between cells. External states are associated with extracellular concentrations and the states of other cells. Image credits: The cell schematic has been adapted from an image by Judith Stoffer supplied at the National Institute of General Medical Sciences, NIH (CC BY-NC 2.0). The brain schematic has been adapted from an image of a head cross-section on openclipart.org by Kevin David Pointon under a public domain license, and an image of an eye with abductor muscles from Wikimedia by Patrick J. Lynch (CC BY-NC 2.5).

**Figure 4.**Schematic of adaptive learning in predator–prey models.

**Left**: A typical predator–prey model simulation based on Equation (5), characterized by persistent population size oscillations of both predator and prey, where peaks in prey precede peaks in predator populations.

**Right**: Predator–prey model simulations with learned adaptive coevolution of both predator and prey based on Equation (6) and the work done by Park et al. [106], characterized by initially dampened oscillations followed by only marginally oscillating, stable population sizes of both predator and prey.

**Figure 5.**Coupled Genetic Algorithm Schematic.

**Top**: Schematic of a standard genetic algorithm. An initial population (purple) of individuals or chromosomes (orange), consisting of individual bits or genes (black), reproduce with crossover of random sets of genes and mutations. Fitness according to a fixed fitness function is calculated for each chromosome, after which individuals with highest fitness are selected and cloned to form a new population. After multiple iterations of this process, the algorithm stops when no higher fitness can be generated in any new individual.

**Bottom**: Schematic of a coupled genetic algorithm. Coevolving populations undergo reproduction as usual but are then coupled in the computation of each other’s fitness function, which in turn are variable and can incorporate integration of past fitness functions as a form of memory, at which point learning methods can be implemented.

**Figure 6.**Schematic of Coupled Generative Adversarial Networks.

**Top**: Schematic of a standard generative adversarial network applied to the problem of camouflage. A generator (blue) neural network generates data x

_{g}with the goal of emulating the potential distributions of real data x

_{r}as best as possible. In this example, inspired by Talas et al. [139], the generator generates stripe patterns that, when emerged in the habitat of a Siberian tiger, are aimed at fooling the discriminator in not being able to discern generated tiger patterns from tree patterns in the background. A second neural network called discriminator receives unlabeled data from both real and generated distributions and aims to label them correctly by origin.

**Bottom**: Schematic of a coupled generative adversarial network. When fitness landscapes of two populations are coupled, each generative adversarial network type agent tries to internally generate and discern trait patterns that correspond to the observed traits of the other population, thereby learning and predicting the traits of the other population. Because in a coupled generative adversarial network weights between the first few generator layers, and last few discriminator layers are shared, this configuration allows the coupled generative adversarial network to learn a joint fitness distribution underlying traits without correspondence supervision [140]. Tiger and birch tree images are adapted from images taken from flickr.com (last accessed 02/09/2022) under the CC BY-NC-SA 2.0 license.

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Kuchling, F.; Fields, C.; Levin, M. Metacognition as a Consequence of Competing Evolutionary Time Scales. *Entropy* **2022**, *24*, 601.
https://doi.org/10.3390/e24050601

**AMA Style**

Kuchling F, Fields C, Levin M. Metacognition as a Consequence of Competing Evolutionary Time Scales. *Entropy*. 2022; 24(5):601.
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**Chicago/Turabian Style**

Kuchling, Franz, Chris Fields, and Michael Levin. 2022. "Metacognition as a Consequence of Competing Evolutionary Time Scales" *Entropy* 24, no. 5: 601.
https://doi.org/10.3390/e24050601