Scott Aaronson and the Expander Graph Objection to Integrated Information Theory
Scott Aaronson, a computational complexity theorist at the University of Texas at Austin, published Why I Am Not An Integrated Information Theorist in May 2014. The argument is short and it has never been answered in a way most computer scientists accept. Aaronson constructed simple systems, grids and expander graphs performing elementary operations such as parity, and showed that integrated information theory assigns them values of phi far exceeding those of a human brain. Giulio Tononi replied that the grid is therefore conscious. That reply is the reason the exchange still matters, because it converted a technical objection into a question about what a theory of consciousness is allowed to conclude.
| Step | Aaronson | Tononi’s reply |
|---|---|---|
| Construct a simple system with enormous phi | An expander graph or 2D grid computing a parity function | Accepts the construction is correct |
| Note it is obviously not conscious | Treats this as a reductio of the theory | Denies the premise. The grid is conscious |
| Ask what phi is measuring | Concludes phi tracks a network property, not experience | Concludes intuition about what is conscious is unreliable |
The Construction
Phi measures how much a system’s causal structure exceeds the sum of its parts, formalised as the extent to which the whole cannot be reduced to independent components. Systems that are richly interconnected, with no clean way to cut them into pieces, score highly.
Aaronson’s point is that this is a property of connectivity, and connectivity is cheap. Expander graphs are sparse graphs with strong connectivity properties, well understood in computer science, and easy to build. A grid of simple logic elements arranged as an expander, computing something as trivial as the parity of its inputs, is hard to decompose in exactly the way phi rewards.
Run the numbers and the grid scores higher than a brain. It does nothing. It has no behaviour, no sensory input, no responses, and no capacity to do anything with the information it holds. Its only property is that its wiring resists partition.
Aaronson’s conclusion is that a measure which ranks a parity grid above a human is not measuring consciousness. It is measuring a graph theoretic property that happens to be present in brains alongside all the other things brains do.
Why Tononi’s Reply Made It Worse
Tononi responded with a post titled “Why Scott should stare at a blank wall and reconsider”, and he did not dispute the mathematics. He accepted that the grid has high phi and drew the consequence the theory requires. The grid is conscious. Our confidence that it is not is an intuition, intuitions about which systems are conscious have been wrong before, and a theory that follows its axioms to an uncomfortable place is doing its job.
This is a serious position and it is internally consistent. It is also the point at which the theory becomes very hard to test, because it has explicitly detached its predictions from any independent check. If every counterexample is answered by saying the counterexample is conscious, no observation can count against the theory.
Aaronson’s follow up made this the crux. His objection was never that IIT gives a strange answer. It was that IIT gives an answer nobody can verify, on a quantity nobody can compute for real systems, and then treats disagreement as a failure of imagination.
The Peer Reviewed Version
The blog exchange was informal and the substance reached the literature. Michael Cerullo published The Problem with Phi. A Critique of Integrated Information Theory in PLOS Computational Biology in 2015, developing the same line with additional cases.
The technical objections cluster into three.
Phi is computationally intractable. Calculating it exactly requires evaluating every possible partition of the system, which grows super-exponentially. Every phi value ever reported for a real system is an approximation using a proxy measure, and the proxies do not agree with each other.
Phi is defined over a specified system with specified boundaries and a specified grain. Change where you draw the edges of the system, or the timescale, and phi changes. The theory does not supply a principled way to choose, so the answer depends on a modelling decision made by the researcher.
Phi tracks integration, and integration is not obviously the thing. Aaronson’s grid is the sharpest illustration, and the general worry is that a system can be maximally irreducible while doing nothing that resembles perception, memory, or control.
What This Means for AI
The objection cuts in a specific direction for artificial systems, and it is not the direction usually assumed.
IIT is normally cited as the theory that rules out current AI, because feedforward transformer architectures have near zero phi. Aaronson’s argument undermines the reassurance rather than supporting it. If phi can be enormous in a parity grid, a low phi score for a language model tells you about its wiring topology and not about whether anything is happening inside it. The measure is not tracking the thing people want it to track in either direction.
It also predicts something buildable. A system deliberately architected as an expander, doing nothing useful, would score higher than any deployed model. If that system is conscious, the field’s practical concern is aimed at entirely the wrong hardware. If it is not, phi is not the measure.
The adversarial testing programme has tried to settle related questions empirically, with results neither camp accepted as decisive, documented in the adversarial test of IIT against global neuronal workspace. Where IIT sits relative to its rivals is set out in the index of consciousness theories and what each predicts about AI.
What Survives
IIT’s defenders have a real answer to part of this, which is that the theory starts from axioms about experience and derives the physical requirements, so it was never going to be constrained by intuitions about grids. Working from phenomenology outward is a legitimate method and it is the theory’s distinctive contribution.
What Aaronson establishes is narrower and it holds. Phi is not usable as a consciousness detector. It cannot be computed for real systems, its value depends on modelling choices the theory does not fix, and its ranking of systems conflicts with every other criterion anyone has proposed. A theory can be true and its central quantity still be useless as an instrument, and that is the position IIT is in.
That distinction matters for the search for testable indicators described in the current scientific consensus on AI consciousness. Frameworks that include IIT derived indicators are importing a quantity nobody can measure, and they should say so.
What Follows
The exchange is unusual because both parties were right about different things. Aaronson was right that phi fails as an instrument. Tononi was right that a theory should not be abandoned because its conclusions feel wrong.
The unresolved question is what to do when those two are the only options. Nagel’s argument, examined in Thomas Nagel and what is it like to be a bat applied to AI, says no objective measurement settles a subjective fact. If that is right, every candidate measure will eventually face a version of Aaronson’s grid, and the response will always be the one Tononi gave.