Integrated Information Theory and Isomorphic Feed-Forward Philosophical Zombies

Jake R. Hanson; Sara I. Walker

REVIEW 3 major objections 4 minor 28 references

Integrated Information Theory and Isomorphic Feed-Forward Philosophical Zombies

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Two circuits with identical state-transition diagrams can have opposite values of the integrated-information measure $\Phi$, differing only by a permutation of binary labels, so the paper argues any theory of consciousness must be…

desk verdict A clean mathematical construction that gives IIT critics a sharper zombie argument, but the paper's own conclusion overstates its case by sliding from automata isomorphism to functional equivalence. read the letter →

arxiv 1908.09621 v2 pith:OFPCCBN2 submitted 2019-08-03 cs.IT math.IT

Jake R. Hanson , Sara I. Walker This is my paper · ORCID

classification cs.ITmath.IT MSC 68Q7068Q45

keywords integratedinformationtheoryphilosophicalzombiesfeed-forwarddecompositioncascadepreservedpartitionsfinite-stateautomatastate-transitionisomorphismconsciousnessmeasurement

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper argues that any quantitative theory of consciousness should assign the same value to two systems whose state-transition diagrams are identical up to relabeling. The authors focus on the integrated-information measure $\Phi$, the central quantity of a leading phenomenological theory of consciousness, and build explicit pairs of small circuits: one with feedback and $\Phi > 0$, the other strictly feed-forward with $\Phi = 0$, yet with exactly the same state-transition topology. The only mathematical difference between the two is which binary strings label the internal states. If state-transition topology is what a computation is, then $\Phi$ rewards an arbitrary labeling convention rather than the computation itself, and no phenomenological difference can justify the large difference in $\Phi$. The paper therefore proposes invariance under isomorphism as a minimal test any theory of consciousness must pass to avoid philosophical zombies.

What carries the argument

The central object is the preserved partition of a finite-state automaton's state space: a grouping of states into blocks such that every state in a block transitions to a single block. A nested sequence of preserved partitions, each splitting the previous blocks evenly in half, supplies the coordinates of a new automaton whose logic gates are read off from block-to-block transitions. For the paper's examples, this sequence yields a cascade, or strictly feed-forward, architecture with the same global state-transition diagram as the original feedback architecture. The cascade form guarantees $\Phi = 0$ because any unidirectional partition of it leaves the dynamics unchanged, while the original circuit has $\Phi > 0$. This machinery isolates the effect of internal labeling: the isomorphism is a dictionary between representations, and the only mathematical difference between the two systems is the permutation of binary labels used to instantiate the same computation.

What would settle it

Take a concrete finite-state automaton with feedback and enumerate every bijective relabeling of its internal states, constructing each as a distinct logical architecture. The paper's claim predicts that at least one relabeling yields a strictly feed-forward architecture with $\Phi = 0$ while the original has $\Phi > 0$ under the standard $\Phi$ algorithm; if no such relabeling exists for any finite-state automaton, or if every relabeling of a given transition diagram yields the same $\Phi$ value, the central claim is falsified.

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Extended reading notes

Core claim

The central claim is that a system satisfying the integrated-information criteria for consciousness ($\Phi > 0$) can have an isomorphic counterpart that is a philosophical zombie by the same theory's standards ($\Phi = 0$), where the isomorphism is a bijective relabeling of internal binary states that preserves every state transition. The authors demonstrate this by explicit construction: an AND/OR feedback pair and a three-bit XNOR/XOR/XOR system are each decomposed into cascade form through a nested sequence of preserved partitions. In the decomposed system information flows strictly forward, so any unidirectional partition leaves the transition probability matrix unchanged, forcing $\Phi = 0$ for all states; yet the global state-transition diagram is exactly the original diagram with states renamed. The paper concludes that $\Phi$ depends on the internal representation of a computation, not on the computation itself, and that any quantitative measure of consciousness should be invariant under such isomorphisms: measurable differences in consciousness must correspond to measurable differences in the state-transition function.

Load-bearing premise

The argument depends on treating a bijective relabeling of internal binary states as producing the same computation with no physical or semantic difference; if the actual physical wiring or the environment can privilege one labeling over another, then the isomorphic zombie is not genuinely equivalent and the challenge to $\Phi$ collapses.

Editorial extensions

If this is right

Any measure of consciousness that satisfies the proposed invariance criterion will assign identical values to isomorphic circuits, so it cannot single out physical feedback as a necessary condition for consciousness in these examples.
The integrated-information measure $\Phi$, as currently defined, fails the criterion: isomorphic systems exist with $\Phi > 0$ and $\Phi = 0$.
The two members of each pair cannot be told apart by an outside observer tracking state transitions; if one is conscious and the other is not, the difference is causally silent in the state dynamics.
The invariance criterion is necessary but not sufficient for a theory of consciousness to be free of philosophical zombies; further constraints are still required.
Consciousness measures should be evaluated on equivalence classes of physical implementations that realize the same state-transition topology, rather than on a single wiring diagram.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editor's inference: the same invariance test could be applied to other causal-structure theories of consciousness or to measures of causal emergence; any such measure that changes under a bijective relabeling of internal states would face the same isomorphic-zombie problem.
Editor's inference: the examples suggest a sharper, computable claim: enumerating all bijective relabelings of a given transition diagram and computing $\Phi$ for each isomorphic circuit would quantify how much of $\Phi$ is labeling artifact rather than graph structure.
Editor's inference: if the invariance criterion is accepted, the integrated-information framework would need to define $\Phi$ on the quotient of circuits under isomorphism, effectively turning $\Phi$ into a graph-theoretic quantity on the state-transition diagram.
Editor's inference: because $\Phi$ is independent of computation on the paper's account, any future theory making $\Phi$ central must explain why organisms would evolve high-$\Phi$ wirings rather than their equivalent low-$\Phi$ isomorphic counterparts.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The construction is real and worth knowing. The paper gives explicit, worked examples of feedback systems with Phi > 0 that are isomorphic at the level of the global state-transition diagram to feed-forward systems with Phi = 0, preserving system size and state count. That is genuinely sharper than the earlier Krohn-Rhodes or Doerig et al. zombie arguments, which only gave homomorphic (many-to-one) emulation. The nested preserved partition method is clearly explained, and the two examples check out as internally consistent. I verified the logic of the AND/OR and XNOR/XOR/XOR rewrites; the cascade forms do have unidirectional partitions that leave the TPM unchanged, so Phi = 0 follows from IIT's own rules. The authors also honestly cite prior work and do not oversell the novelty relative to that literature.

The soft spot is the interpretive leap, and it is load-bearing. The mathematical fact is that two automata with the same unlabeled transition diagram can have different Phi. The paper's stronger claim, that the only difference is a 'functionally arbitrary' label permutation, only holds if you grant that the unlabeled state-transition topology is the right object for a theory of consciousness. IIT's axioms explicitly target the physical mechanism decomposition, not the abstract automaton. The two systems also differ in wiring and element decomposition; the bijection maps global states, not mechanisms. The paper acknowledges this in Section 3.1 and then dismisses the implementation-based reading as untestable. That is not a refutation, it is a statement of the hard problem. So the paper does not establish that IIT is wrong; it establishes that IIT is non-invariant under a specific abstraction choice, and that the burden is on IIT to justify why its abstraction is the correct one.

The invariance criterion at the end is a reasonable proposal, but it is a philosophical premise, not a derivation. The paper is honest about this in the final paragraph, which mitigates the overreach in the abstract. I would not call the central argument unsound; I would call it conditional on functionalism. Within that frame, the mathematics is solid and the examples are reproducible.

Who should read this? Consciousness researchers, especially anyone working on IIT or causal structure theories, and automata theorists interested in decomposition. It deserves a serious referee: the formal result is novel and the philosophical challenge is well posed, even if the conclusion needs to be softened. I would send it to peer review and ask the authors to clearly separate the mathematical theorem from the functionalist interpretation.

Referee Report

3 major / 4 minor

Summary. This manuscript challenges Integrated Information Theory (IIT) by constructing feedback systems with Φ>0 and isomorphic feed-forward systems with Φ=0. The authors work with finite-state automata and use preserved partitions to produce cascade decompositions that preserve the global state-transition diagram up to a bijective relabeling. The two worked examples are an AND/OR system mapped to COPY/OR and a three-element system with two XNOR gates and one XOR mapped to a NOT/COPY/COPY cascade. The paper argues that because the only apparent difference between the two realizations is a permutation of binary labels, Φ is sensitive to functionally arbitrary internal representation, and it proposes that any quantitative theory of consciousness should be invariant under isomorphisms of the state-transition diagram.

Significance. If the central claim were fully established, this would be a significant challenge to IIT: it would exhibit equal-size, equal-state-count systems with identical global state-transition graphs but different IIT verdicts. The examples are explicit and reproducible, and the paper is commendably clear that the invariance criterion is a proposal rather than a theorem. The main weakness is that the conclusion depends on identifying 'computation' with the unlabeled global state-transition topology, which is precisely the functionalist premise that IIT's constitutive decomposition into physical mechanisms rejects. The manuscript acknowledges this alternative in Section 3.1 but does not refute it, so the paper is best read as a conditional critique that sharpens the philosophical disagreement rather than as a definitive refutation of IIT.

major comments (3)

[Abstract and Section 3 (Figures 6–8)] The claim that the only mathematical difference between Y and Y′ is a permutation of binary labels is inaccurate with respect to IIT's formal objects. IIT defines Φ on a transition probability matrix together with a specified decomposition into physical elements (Oizumi et al. 2014). The two systems differ in element-level wiring—two XNOR gates and one XOR with cyclic coupling versus NOT/COPY/COPY in cascade—and the bijection h maps global states but does not map the element decomposition. Therefore Y and Y′ are not isomorphic in the mathematical category in which Φ is defined, and the conclusion that Φ is sensitive to 'functionally arbitrary' labeling requires the additional premise that the unlabeled global transition topology is the correct notion of function. The paper should state this premise explicitly and defend it, or the conclusion should be weakened to a conditional one.
[Section 3.1] The proposed invariance criterion ('any quantitative theory of consciousness ... should be invariant under isomorphisms') is a normative assertion, not a consequence of IIT's axioms or of the preceding mathematical construction. The manuscript acknowledges that IIT defines behavior in terms of the specific logical implementation and dismisses this as untestable from outside. But the untestability of a constitutive assumption from a third-person perspective does not refute it; the hard problem is precisely that such assumptions are not directly observable. The paper therefore does not so much refute IIT as locate the disagreement in the choice between functionalist and implementation-based notions of computation. This is a legitimate contribution, but the abstract and title overstate the result as demonstrating the existence of isomorphic zombies rather than zombies under a contested functionalist criterion.
[Section 2.3.1 (Figure 4)] The first example begins by 'leaving off the binary labels' of the physical AND/OR system and treating the unlabeled diagram as the computation. This move is not neutral: the labels of a physical circuit are determined by the outputs of its gates, and an isomorphic relabeling forces a different gate implementation (COPY/OR instead of AND/OR). The statement that both systems 'operate on the same support' conflates the abstract state set with the physical states of the circuits. The 'isomorphism' is between the abstract directed graph and the relabeled graph, not between the two physical circuits, so the phrase 'only a permutation' should be reserved for the global state-transition diagram and explicitly accompanied by the caveat that the physical architectures differ.

minor comments (4)

[Section 3] There is a typo: 'Karnuagh maps' should be 'Karnaugh maps'.
[Section 2.2] The definition of a homomorphism refers to a map on states, stimuli, and transitions, but the commuting diagram in Figure 2 and the subsequent examples use only a single input symbol (the passage of time). The paper would be clearer if it explicitly stated that the input alphabet is trivial in the examples and that the isomorphism is of the underlying deterministic state-transition graph.
[Figure 8 caption] The caption says the two diagrams 'differ only by a permutation of labels'; given the difference in gate wiring, this should be qualified as 'at the level of the global state-transition diagram' to avoid contradicting the paper's own Section 3.1 discussion of the alternative interpretation.
[Acknowledgements] The word 'feed-back' should be 'feedback' for consistency with the rest of the text.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the isomorphic zombie is constructed directly from automata theory, the Phi values are computed from the transition probability matrices, and the invariance criterion is an explicitly stated philosophical premise rather than a disguised input.

full rationale

The paper's central result—that a feedback system with Phi > 0 can be accompanied by an isomorphic feed-forward system with Phi = 0—is established by explicit construction, not assumed. Section 2.3.1 builds the cascade counterpart X' from a nested sequence of preserved partitions, and Section 3 does the same for Y, with the global state-transition diagrams shown to differ only by a relabeling (Figures 5, 7, 8). The claim that the cascade system has Phi = 0 follows from IIT's own unidirectional partition rule applied to the transition probability matrix (Section 2.2), and the positive Phi values for the feedback systems are checked against the independent PyPhi toolbox [25]. No load-bearing step reduces to a fitted parameter or to the paper's own prior results: the automata-theoretic facts cited (Krohn-Rhodes, preserved partitions from Arbib/Hartmanis) are external and used to motivate the construction, and there are no self-citations. The paper explicitly acknowledges in Section 3.1 that an alternative, implementation-level interpretation exists, 'for example as IIT adopts,' and concedes the hard problem cannot adjudicate it. The final recommendation that consciousness measures be invariant under isomorphisms is a normative criterion built on the premise that behavior is defined by state-transition topology; that is a philosophical assumption, not a circular derivation from or reduction to the paper's own definitions.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted; Phi values are computed from the published IIT 3.0 definition via PyPhi. No new physical entities are introduced; the paper proposes a normative criterion (isomorphism invariance), which is a formal principle rather than an entity per the ledger definition.

assumptions (4)

standard math Krohn-Rhodes theorem
Every finite automaton can be decomposed into cascade form (proved in [14,15]); used to argue feed-forward emulation is always possible, though size may increase.
standard math Preserved partitions yield isomorphic cascade decompositions
The construction in Section 2.3 relies on the standard automata theory result that a nested sequence of evenly splitting preserved partitions induces a cascade decomposition; used to build Y'.
domain assumption IIT's partition rule for feed-forward systems
Under IIT 3.0, a unidirectional partition leaves a cascade TPM unchanged, so Phi=0 (Section 2.2, following Oizumi et al. [5]); the authors use this to assert Phi(Y')=0.
domain assumption Functionalist equivalence of isomorphic automata
Two automata with the same state-transition diagram, differing only by a permutation of labels, are regarded as performing the same computation (Section 2.2 and 3.1); this is the premise that makes the zombie claim meaningful.

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Cite this review

Pith. "Pith review of Integrated Information Theory and Isomorphic Feed-Forward Philosophical Zombies." pith.science (2026). https://pith.science/paper/OFPCCBN2

@misc{pith2026190809621,
  author       = {Pith},
  title        = {Pith review of: Integrated Information Theory and Isomorphic Feed-Forward Philosophical Zombies},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OFPCCBN2}},
  note         = {Machine review of arXiv:1908.09621}
}

abstract

Any theory amenable to scientific inquiry must have testable consequences. This minimal criterion is uniquely challenging for the study of consciousness, as we do not know if it is possible to confirm via observation from the outside whether or not a physical system knows what it feels like to have an inside - a challenge referred to as the "hard problem" of consciousness. To arrive at a theory of consciousness, the hard problem has motivated the development of phenomenological approaches that adopt assumptions of what properties consciousness has based on first-hand experience and, from these, derive the physical processes that give rise to these properties. A leading theory adopting this approach is Integrated Information Theory (IIT), which assumes our subjective experience is a "unified whole", subsequently yielding a requirement for physical feedback as a necessary condition for consciousness. Here, we develop a mathematical framework to assess the validity of this assumption by testing it in the context of isomorphic physical systems with and without feedback. The isomorphism allows us to isolate changes in $\Phi$ without affecting the size or functionality of the original system. Indeed, we show that the only mathematical difference between a "conscious" system with $\Phi>0$ and an isomorphic "philosophical zombies" with $\Phi=0$ is a permutation of the binary labels used to internally represent functional states. This implies $\Phi$ is sensitive to functionally arbitrary aspects of a particular labeling scheme, with no clear justification in terms of phenomenological differences. In light of this, we argue any quantitative theory of consciousness, including IIT, should be invariant under isomorphisms if it is to avoid the existence of isomorphic philosophical zombies and the epistemological problems they pose.

Figures

Figures reproduced from arXiv: 1908.09621 by the authors.

**Figure 1.** The "right-shift automaton" A in terms of its state-transition diagram (1a), transition function δ (1b), and logical architecture (1c). It is important to note that not all automata require multiple input symbols and it is common to find examples of automata with a single-letter input alphabet. In fact, any deterministic state-transition diagram can be represented in this way, with a single input letter signaling th… view at source ↗

**Figure 2.** For the map h to be a homomorphism from A 0 onto A, updating the dynamics then applying h (top) must yield the same state of A as applying h then updating the dynamics (bottom). Because we are interested in isolating the role of feedback, the specific type of decomposition we seek is a feed-forward or cascade decomposition of the logical architecture of a given system. Cascade decomposition takes the automaton A and… view at source ↗

**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 information flows unidirectionally. Pertinently, the Krohn-Rhodes theorem proves that every automaton can be decomposed into cascade form [14,15], which implies every system for which we can measure non-zero Φ allows a feed-forward decomposition with Φ = 0. The… view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: The goal of an isomorphic cascade decomposition is to decompose the integrated logical architecture of the system X (4a) so that it is in cascade form X 0 (4b) without affecting the state-transition topology of the original system (4c). (a) (b) (c) [PITH_FULL_IMAGE:fi…

**Figure 5.** Figure 5: The nested sequence of preserved partitions in 5a yields the isomorphism 5b between X and X 0 which can be translated into the strictly feed-forward logical architecture with Φ = 0 shown in 5c. At this point, the isomorphic cascade decomposition is complete. We have co…

**Figure 6.** Figure 6: The transition probability matrix (6a), logical architecture (6b), and all available Φ values (6c) for the example system Y (n/a implies Φ is not defined for a given state because it is unreachable). We first evenly partition the state space of Y into two blocks B0 = {…

**Figure 7.** Figure 7: Nested sequence of preserved partitions used to decompose Y into cascade form. (a) (b) (c) (d) [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]

**Figure 8.** Figure 8: Side-by-side comparison of the feedback system Y with Φ > 0 (8a) and its isomorphic feed-forward counterpart Y 0 with Φ = 0 (8c). The global state-transition diagrams (8b and 8d, respectively) differ only by a permutation of labels. labeling scheme. And, indeed, it is …

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Neural correlates of consciousness in humans

Rees, G.; Kreiman, G.; Koch, C. Neural correlates of consciousness in humans. Nature Reviews Neuroscience 2002, 3, 261

work page 2002

[2] [2]

Facing Up to the Problem of Consciousness

Chalmers, D.J. Facing Up to the Problem of Consciousness. Journal of Consciousness Studies 1995, 2, 200–19

work page 1995

[3] [3]

Minds, brains, and programs

Searle, J.R. Minds, brains, and programs. Behavioral and brain sciences 1980, 3, 417–424

work page 1980

[4] [4]

Consciousness as integrated information: a provisional manifesto

Tononi, G. Consciousness as integrated information: a provisional manifesto. The Biological Bulletin 2008, 215, 216–242

work page 2008

[5] [5]

From the phenomenology to the mechanisms of consciousness: integrated information theory 3.0

Oizumi, M.; Albantakis, L.; Tononi, G. From the phenomenology to the mechanisms of consciousness: integrated information theory 3.0. PLoS computational biology 2014, 10, e1003588

work page 2014

[6] [6]

Integrated information theory: from consciousness to its physical substrate

Tononi, G.; Boly, M.; Massimini, M.; Koch, C. Integrated information theory: from consciousness to its physical substrate. Nature Reviews Neuroscience 2016, 17, 450

work page 2016

[7] [7]

A mathematical theory of communication

Shannon, C.E. A mathematical theory of communication. Bell system technical journal 1948, 27, 379–423

work page 1948

[8] [8]

An information integration theory of consciousness

Tononi, G. An information integration theory of consciousness. BMC neuroscience 2004, 5, 42

work page 2004