I am referring specifically to a very recent paper by Max Tegmark, http://arxiv.org/abs/1401.1219 In this paper he proposes 3 more criteria (independence, dynamics, and utility principle) in addition to Tononi's original criteria (information and integration).

I have seen the paper being discussed in Hacker News, reddit's r/neurophilosophy, and lesswrong, but I think they are mostly light jabs on the actual content of Tegmark's proposal.

  • It looks very interesting, but the content is too deep to admit an easy read (at least for me, even though it is clearly written).
    – Rex Kerr
    Jan 22, 2014 at 23:16
  • Yes, I agree. The paper is an order of magnitude harder to read than Tononi's original paper. The paper mentions "the integration paradox" which says that a consciousness as an integrated information can only be 37 bit, and attempts to address it.
    – pcr
    Jan 23, 2014 at 3:24
  • I caught that much, but that's based on a very dubious assumption which is that the entire brain is a Hopfield network. You can break it into 1000 27-bit-capacity Hopfield networks and then get 27k bits. Or into a million 17-bit-capacity networks and get 27M bits. That part of the paper is by far its weakest part (in my brief reading anyway).
    – Rex Kerr
    Jan 23, 2014 at 20:18
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    I have never heard of the need of breaking down into several Hopfield nets in brain simulation. Perhaps, you are implicitly claiming that the information content in the Hopfield net is not necessarily integrated (here is a calculation for its capacity caltechdemo.library.caltech.edu/3/1/ABUieeetit85.pdf)
    – pcr
    Jan 23, 2014 at 22:47
  • I'm claiming that Hopfield nets are more integrated than they need to be, and that relaxing this restriction yields a huge increase in the number of allowed states without losing meaningful integration.
    – Rex Kerr
    Jan 23, 2014 at 23:57

3 Answers 3


Although the paper is well-written and contains a lot of very interesting results, I think one of its premises is flawed, and hence it is not clear that Tononi's critera need to be amended.

First, Tegmark argues that a hypothetical state of matter with the minimum necessary conditions for self-awareness, hereafter called Perceptronium, must not merely have information and integration; it must also have autonomy, which arises from dynamics (future state is not the same as past state, though it is computed from it) and independence (insensitivity to external input).

But this is not to my mind very solid. It's an interesting idea to explore, and Tegmark does a good job of it in my opinion, but there is another equally plausible scenario: the system doesn't have much independence (it is highly sensitive to input) but it is oblivious to this fact. Without a strong case for independence, utility falters also.

One of the central claims in the paper is that there is an apparent conflict between the information and integration claims: you need a large number of states, but this information be integrated in a way that is not easily separable into independent components. The reason that this appears problematic is that our information is stored in our brains, and the best-known example of a neural network with fully integrated states is a Hopfield network, where it turns out that if you have N neurons you can maintain about N/7 distinct attractors--states around which the whole network converges.

Although a Hopfield network brilliantly delivers on the idea of integration (for each state is a global attractor; any neuron that starts going off on its own independent way will be pulled back into the state it's supposed to be in), and it has a measure of biological plausibility, the whole brain is certainly not actually a Hopfield network. The most obvious discrepancy is that in a Hopfield network every neuron provides input to every other, whereas in our brain one neuron provides input to only about 10,000 neurons out of >10,000,000 (a ten thousanth of one percent, or less).

But let's suppose that a Hopfield network is a good model. If so, we get on the order of 10 billion possible attractors, which we can encode in about 34 bits (Tegmark's math is a little different). So, consciousness only can flicker between 10 billion different states? That seems wrong somehow--surely there is more variety of experience than this, and so surely the criteria have to be altered somehow.

Tegmark acknowledges that there may be a way to encode more state, but doesn't really address the issue. But there is a very easy way to encode more state.

In computer science there is the idea of a hash code, which is a deterministic function that scrambles the input values so much that even a small change in input results in a very different output. Let's imagine a case where instead of one monolithic Hopfield network, we have a million Hopfield networks of size 10,000 (each holding about 10 bits of information). Now let's suppose that we compute from these networks a hash code that combines their input such that we disable half of each network's states. So we get 9 bits of information per Hopfield network, each of the million Hopfield networks depends rather strongly on all the others (because of this hash-code based selection function), and yet at any instant we have 9*1,000,000 bits of state represented. The state might be dynamic, as the hash-coding makes invalid certain states, which then causes a new hash code to be computed, which makes other state invalid, etc., and this dynamism may cause the network to settle into one of fewer than 2^(9M) states, but it seems exceedingly unlikely that it would go all the way down from 9M to 34 or whatever.

And if we look at the architecture of the cortex, we find that the architecture supports this kind of model better than a universal Hopfield network.

So I don't think Tegmark provides a very compelling case that there's even a problem to solve. Without a problem to solve, the dynamics principle isn't needed. That said, it seems intuitively that Tegmark is right that dynamics are needed, or at least that we in practice are dynamic and therefore our kind of Perceptronium is one which contains dynamics, so I'm not arguing that the paper is wrong, either. Just that there is plenty of wiggle room if you want to stick to Tononi's original criteria.

  • what about this assertion: I don't see how a group of conscious individuals like a society having a collective consciousness by sharing information and integrating it into commonality like works of governance, science and art are differentiated from a conscious individual by either Tononi's or Tegmark's current level of specificity.
    – user5375
    Feb 3, 2014 at 3:14
  • @caseyr547 - I would be delighted by a theory of consciousness that could identify a system with embedded consciousness, whether it was one or many. This alone is hard enough; after finding out if there are more than zero, one can then wonder about how many.
    – Rex Kerr
    Feb 3, 2014 at 5:26
  • @caseyr547 - I didn't intend to validate your assertion. I didn't evaluate it since at this point the hard part is discriminating between consciousness and not.
    – Rex Kerr
    Feb 3, 2014 at 15:30
  • Thats fine. If I understand you I think we agree that in its current state both papers do not differentiate collective consciousness from a single consciousness. right? Can we also agree that it doesn't identify the type of consciousness whether it be primary or of higher order?
    – user5375
    Feb 3, 2014 at 16:00
  • @caseyr547 - I don't know whether they would differentiate it or not. I haven't thought about it deeply, they don't say, and it seems a lot less interesting of a question than being able to distinguish some consciousness vs no consciousness (or to quantify the amount of consciousness).
    – Rex Kerr
    Feb 3, 2014 at 20:34

I think we are a long way from understanding the kind of complexity which gives rise to cognition. For example one could argue "feedback" loops are key and some in fact have. But, deeper questions are unanswered by IIT or Tegmark's "upgrade". The most important is the relationship between cognitive complexity and time. Another is the nature of transformations which leave cognition invariant. I have addressed these issues in my paper, Causal Event Network:


IIT is a statistical measure of complexity between two sets of random variables. Even if this measure is important to consciousness, the complexity comes about because of a transformation, a set of events in space and time. One can and I think should address the question of complexity directly on those event structures, on the process itself and not on the statistics which result from it. Cognitive processes, and hence our subjective experience can be viewed as transformation of information in space and time. Exploring the relationship between matter-energy and spacetime led us from the development of Classical Mechanics and Electro-Magnetism to the Standard Model of Physics and General Relativity. It is highly unlikely that without directly understanding how event structures relate to space and time we are able to understand the problem as they are more fundamental regardless of upgrades. In the above paper I present a possible point of view.

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    Is there any chance you could explore this a little further? What arguments do you find most persuasive here, etc.?
    – Joseph Weissman
    Aug 15, 2014 at 0:30

http://arxiv.org/pdf/1004.1999.pdf Towards a mathematical theory of meaningful communication by Bernat Corominas-Murtra et al approaches communication of meaning in much the same way that Tononi approaches consciousness. Both arrive at entropy-like concepts which appear relate to Kullback-Leibler divergence. The Tegmark articles (including his just released book (Our Mathematical Universe: My Quest for the Ultimate Nature of Reality) seem to unnecessarily complicate the simplicity of Tononi's ideas, adding a lot of ad hoc conditions on his requirements for "Perceptronium." One aspect of Tegmark's ideas that is not a part of Tononi's is their broad applicability to non-human subjects. I would welcome extending this conversation toward the goal of building on Tononi's ideas, attempting to reach clarity and rigor.

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