# Integrated Information Theory and Isomorphic Feed-Forward Philosophical Zombies

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## Abstract

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## 1. Introduction

## 2. Methods

#### 2.1. Finite-State Automata

#### 2.2. Cascade Decomposition

- Use the stimulus of ${A}^{\prime}$ to update the state of ${A}^{\prime}$ then map the resulting state onto A.
- Map the stimulus of ${A}^{\prime}$ and the state of ${A}^{\prime}$ to the corresponding stimulus/state in A then update the state of A using the stimulus of A.

#### 2.3. Feed-Forward Isomorphisms via Preserved Partitions

#### 2.3.1. Example: `AND`/`OR` ≅ `COPY`/`OR`

`AND`gate and an

`OR`gate, as shown in Figure 4a. As it stands, X is not in cascade form because information flows bidirectionally between the components ${Q}_{1}$ and ${Q}_{2}$. While this feedback alone is insufficient to guarantee $\mathsf{\Phi}>0$, one can readily check that X does indeed have $\mathsf{\Phi}>0$ for all possible states (e.g., [30]). The global state-transition diagram for the system X is shown in Figure 4c. Note that we have purposefully left off the binary labels that X uses to instantiate these computational states, as the goal is to relabel them in a way that results in a different (feed-forward) instantiation of the same underlying computation. In general, one typically starts from the computation and derives a single logical architecture but, here, we must start and end with fixed (isomorphic) logical architectures—passing through the underlying computation in between. The general form of the feed-forward logical architecture ${X}^{\prime}$ that we seek is shown in Figure 4b.

`COPY`gate receiving its previous state as input.

`OR`gate receiving input from both ${Q}_{1}^{\prime}$ and ${Q}_{2}^{\prime}$.

## 3. Results

`XNOR`gates and one

`XOR`gate, clearly contains feedback between components and has $\mathsf{\Phi}>0$ for all states for which $\mathsf{\Phi}$ can be calculated (Figure 6c). As in Section 2.3, the goal of the decomposition is an isomorphic relabeling of the finite-state machine representing the global behavior of the system, such that the induced logical architecture is strictly feed-forward.

`NOT`gate. Note that this choice is not unique, as we could just as easily have chosen a different preserved partition such as ${B}_{0}=\{A,D,E,H\}$ and ${B}_{1}=\{B,C,F,G\}$, in which case the first coordinate would be a

`COPY`gate; as long as the partition is preserved, the choice here is arbitrary and amounts to selecting one of several different feed-forward logical architectures—all in cascade form. For the second preserved partition, we let ${P}_{2}=\{\{{B}_{00},{B}_{01}\},\{{B}_{10},{B}_{11}\}\}$ with ${B}_{00}=\{C,G\}$, ${B}_{01}=\{A,H\}$, ${B}_{10}=\{B,F\}$. and ${B}_{11}=\{D,E\}$. The transition function for the automaton representing the second coordinate, given by the movement of these blocks, is: ${\delta}_{Q{2}^{\prime}}=\{00\to 0;01\to 1;10\to 0;11\to 1\}$, which is again a

`COPY`gate receiving input from itself. The third and final partition ${P}_{3}$ assigns each state to its own unique block. As is always the case, this last partition is trivially preserved because individual states are guaranteed to transition to a single block. The transition function for this coordinate, read off the bottom row of Figure 7, is given by:

`COPY`gate receiving input from ${Q}_{2}^{\prime}$. With the specification of the logic for the third coordinate, the cascade decomposition is complete and the new labeling scheme is shown in Figure 7. A side-by-side comparison of the original system Y and the feed-forward system ${Y}^{\prime}$ is shown in Figure 8. As required, the feed-forward system has $\mathsf{\Phi}=0$ but executes the same sequence of state transitions as the original system, modulo a permutation of the labels used to instantiate the states of the global state-transition diagram.

## 4. Discussion

## Author Contributions

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The “right-shift automaton” A in terms of its: state-transition diagram (

**a**); transition function $\delta $ (

**b**); and logical architecture (

**c**).

**Figure 2.**For the map h to be a homomorphism from ${A}^{\prime}$ onto A, updating the dynamics then applying h (

**top**) must yield the same state of A as applying h then updating the dynamics (

**bottom**).

**Figure 3.**An example of a fully-connected three-component system in cascade form. Any subset of the connections drawn above meets the criteria for cascade form because all information flows unidirectionally.

**Figure 4.**The goal of an isomorphic cascade decomposition is to decompose the integrated logical architecture of the system X (

**a**) so that it is in cascade form ${X}^{\prime}$ (

**b**) without affecting the state-transition topology of the original system (

**c**).

**Figure 5.**The nested sequence of preserved partitions in (

**a**) yields the isomorphism (

**b**) between X and ${X}^{\prime}$ which can be translated into the strictly feed-forward logical architecture with $\mathsf{\Phi}$ = 0 shown in (

**c**).

**Figure 6.**The transition probability matrix (

**a**); logical architecture (

**b**); and all available $\mathsf{\Phi}$ values (

**c**) for the example system Y (n/a implies $\mathsf{\Phi}$ is not defined for a given state because it is unreachable).

**Figure 7.**Nested sequence of preserved partitions used to isomorphically decompose Y into cascade form.

**Figure 8.**Side-by-side comparison of the feedback system Y with $\mathsf{\Phi}$ > 0 (a) and its isomorphic feed-forward counterpart Y’ with $\mathsf{\Phi}$ = 0 (

**c**). The global state-transition diagrams (

**b**,

**d**, respectively) differ only by a permutation of labels.

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Hanson, J.R.; Walker, S.I.
Integrated Information Theory and Isomorphic Feed-Forward Philosophical Zombies. *Entropy* **2019**, *21*, 1073.
https://doi.org/10.3390/e21111073

**AMA Style**

Hanson JR, Walker SI.
Integrated Information Theory and Isomorphic Feed-Forward Philosophical Zombies. *Entropy*. 2019; 21(11):1073.
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**Chicago/Turabian Style**

Hanson, Jake R., and Sara I. Walker.
2019. "Integrated Information Theory and Isomorphic Feed-Forward Philosophical Zombies" *Entropy* 21, no. 11: 1073.
https://doi.org/10.3390/e21111073