# 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

## References

- Rees, G.; Kreiman, G.; Koch, C. Neural correlates of consciousness in humans. Nat. Rev. Neurosci.
**2002**, 3, 261. [Google Scholar] [CrossRef] [PubMed] - Chalmers, D.J. Facing Up to the Problem of Consciousness. J. Conscious. Stud.
**1995**, 2, 200–219. [Google Scholar] - Searle, J.R. Minds, brains, and programs. Behav. Brain Sci.
**1980**, 3, 417–424. [Google Scholar] [CrossRef][Green Version] - Tononi, G. Consciousness as integrated information: A provisional manifesto. Biol. Bull.
**2008**, 215, 216–242. [Google Scholar] [CrossRef] - Oizumi, M.; Albantakis, L.; Tononi, G. From the phenomenology to the mechanisms of consciousness: Integrated information theory 3.0. PLoS Comput. Biol.
**2014**, 10, e1003588. [Google Scholar] [CrossRef] - Tononi, G.; Boly, M.; Massimini, M.; Koch, C. Integrated information theory: From consciousness to its physical substrate. Nat. Rev. Neurosci.
**2016**, 17, 450. [Google Scholar] [CrossRef] - Shannon, C.E. A mathematical theory of communication. Bell Syst. Tech. J.
**1948**, 27, 379–423. [Google Scholar] [CrossRef] - Tononi, G. An information integration theory of consciousness. BMC Neurosci.
**2004**, 5, 42. [Google Scholar] [CrossRef] - Balduzzi, D.; Tononi, G. Integrated information in discrete dynamical systems: Motivation and theoretical framework. PLoS Comput. Biol.
**2008**, 4, e1000091. [Google Scholar] [CrossRef] - Godfrey-Smith, P. Theory and Reality: An Introduction to the Philosophy of Science; University of Chicago Press: Chicago, IL, USA, 2009. [Google Scholar]
- Marcus, E. Why Zombies Are Inconceivable. Australas. J. Philos.
**2004**, 82, 477–490. [Google Scholar] [CrossRef] - Kirk, R. Zombies 2003. Available online: https://plato.stanford.edu/entries/zombies/ (accessed on 24 October 2019).
- Harnad, S. Why and how we are not zombies. J. Conscious. Stud.
**1995**, 1, 164–167. [Google Scholar] - Turing, A. Computing Machinery and Intelligence. Mind
**1950**, 59, 433. [Google Scholar] [CrossRef] - Doerig, A.; Schurger, A.; Hess, K.; Herzog, M.H. The unfolding argument: Why IIT and other causal structure theories cannot explain consciousness. Conscious. Cognit.
**2019**, 72, 49–59. [Google Scholar] [CrossRef] [PubMed] - Krohn, K.; Rhodes, J. Algebraic theory of machines. I. Prime decomposition theorem for finite semigroups and machines. Trans. Am. Math. Soc.
**1965**, 116, 450–464. [Google Scholar] [CrossRef] - Zeiger, H.P. Cascade synthesis of finite-state machines. Inf. Control
**1967**, 10, 419–433. [Google Scholar] [CrossRef][Green Version] - Hopcroft, J.E.; Motwani, R.; Ullman, J.D. Automata theory, languages, and computation. Int. Ed.
**2006**, 24, 19. [Google Scholar] - Ginzburg, A. Algebraic Theory of Automata; Academic Press: Cambridge, MA, USA, 2014. [Google Scholar]
- Egri-Nagy, A.; Nehaniv, C.L. Computational holonomy decomposition of transformation semigroups. arXiv
**2015**, arXiv:1508.06345. [Google Scholar] - Zeiger, P. Yet another proof of the cascade decomposition theorem for finite automata. Theory Comput. Syst.
**1967**, 1, 225–228. [Google Scholar] [CrossRef] - Arbib, M.; Krohn, K.; Rhodes, J. Algebraic Theory of Machines, Languages, and Semi-Groups; Academic Press: Cambridge, MA, USA, 1968. [Google Scholar]
- Shannon, C.E.; McCarthy, J. Automata Studies (AM-34); Princeton University Press: Princeton, NJ, USA, 2016; Volume 34. [Google Scholar]
- DeDeo, S. Effective Theories for Circuits and Automata. Chaos (Woodbury, N.Y.)
**2011**, 21, 037106. [Google Scholar] [CrossRef] - Maler, O. A decomposition theorem for probabilistic transition systems. Theor. Comput. Sci.
**1995**, 145, 391–396. [Google Scholar] [CrossRef][Green Version] - Tegmark, M. Improved measures of integrated information. PLoS Comput. Biol.
**2016**, 12, e1005123. [Google Scholar] [CrossRef] [PubMed] - Albantakis, L.; Hintze, A.; Koch, C.; Adami, C.; Tononi, G. Evolution of Integrated Causal Structures in Animats Exposed to Environments of Increasing Complexity. PLoS Comput. Biol.
**2014**, 10, e1003966. [Google Scholar] [CrossRef] [PubMed] - Hartmanis, J. Algebraic Structure Theory of Sequential Machines; Prentice-Hall International Series in Applied Mathematics; Prentice-Hall, Inc.: Upper Saddle River, NJ, USA, 1966. [Google Scholar]
- Egri-Nagy, A.; Nehaniv, C.L. Hierarchical coordinate systems for understanding complexity and its evolution, with applications to genetic regulatory networks. Artif. Life
**2008**, 14, 299–312. [Google Scholar] [CrossRef] [PubMed] - Mayner, W.G.; Marshall, W.; Albantakis, L.; Findlay, G.; Marchman, R.; Tononi, G. PyPhi: A toolbox for integrated information theory. PLoS Comput. Biol.
**2018**, 14, e1006343. [Google Scholar] [CrossRef] [PubMed] - Karnaugh, M. The map method for synthesis of combinational logic circuits. Trans. Am. Inst. Electr. Eng. Part I Commun. Electron.
**1953**, 72, 593–599. [Google Scholar] [CrossRef] - Ellis, G. How Can Physics Underlie the Mind; Springer: Berlin, Germany, 2016. [Google Scholar]
- Kim, J. Concepts of supervenience. In Supervenience; Routledge: Abingdon, UK, 2017; pp. 37–62. [Google Scholar]
- Auletta, G.; Ellis, G.F.; Jaeger, L. Top-down causation by information control: From a philosophical problem to a scientific research programme. J. R. Soc. Interface
**2008**, 5, 1159–1172. [Google Scholar] [CrossRef]

**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.
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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