Shtetl-Optimized

You might recall that last week I wrote a post criticizing Integrated Information Theory (IIT), and its apparent implication that a simple Reed-Solomon decoding circuit would, if scaled to a large enough size, bring into being a consciousness vastly exceeding our own.  On Wednesday Giulio Tononi, the creator of IIT, was kind enough to send me a fascinating 14-page rebuttal, and to give me permission to share it here:

Why Scott should stare at a blank wall and reconsider (or, the conscious grid)

If you’re interested in this subject at all, then I strongly recommend reading Giulio’s response before continuing further.   But for those who want the tl;dr: Giulio, not one to battle strawmen, first restates my own argument against IIT with crystal clarity.  And while he has some minor quibbles (e.g., apparently my calculations of Φ didn’t use the most recent, “3.0” version of IIT), he wisely sets those aside in order to focus on the core question: according to IIT, are all sorts of simple expander graphs conscious?

There, he doesn’t “bite the bullet” so much as devour a bullet hoagie with mustard.  He affirms that, yes, according to IIT, a large network of XOR gates arranged in a simple expander graph is conscious.  Indeed, he goes further, and says that the “expander” part is superfluous: even a network of XOR gates arranged in a 2D square grid is conscious.  In my language, Giulio is simply pointing out here that a √n×√n square grid has decent expansion: good enough to produce a Φ-value of about √n, if not the information-theoretic maximum of n (or n/2, etc.) that an expander graph could achieve.  And apparently, by Giulio’s lights, Φ=√n is sufficient for consciousness!

While Giulio never mentions this, it’s interesting to observe that logic gates arranged in a 1-dimensional line would produce a tiny Φ-value (Φ=O(1)).  So even by IIT standards, such a linear array would not be conscious.  Yet the jump from a line to a two-dimensional grid is enough to light the spark of Mind.

Personally, I give Giulio enormous credit for having the intellectual courage to follow his theory wherever it leads.  When the critics point out, “if your theory were true, then the Moon would be made of peanut butter,” he doesn’t try to wiggle out of the prediction, but proudly replies, “yes, chunky peanut butter—and you forgot to add that the Earth is made of Nutella!”

Yet even as we admire Giulio’s honesty and consistency, his stance might also prompt us, gently, to take another look at this peanut-butter-moon theory, and at what grounds we had for believing it in the first place.  In his response essay, Giulio offers four arguments (by my count) for accepting IIT despite, or even because of, its conscious-grid prediction: one “negative” argument and three “positive” ones.  Alas, while your Φ-lage may vary, I didn’t find any of the four arguments persuasive.  In the rest of this post, I’ll go through them one by one and explain why.

I. The Copernicus-of-Consciousness Argument

Like many commenters on my last post, Giulio heavily criticizes my appeal to “common sense” in rejecting IIT.  Sure, he says, I might find it “obvious” that a huge Vandermonde matrix, or its physical instantiation, isn’t conscious.  But didn’t people also find it “obvious” for millennia that the Sun orbits the Earth?  Isn’t the entire point of science to challenge common sense?  Clearly, then, the test of a theory of consciousness is not how well it upholds “common sense,” but how well it fits the facts.

The above position sounds pretty convincing: who could dispute that observable facts trump personal intuitions?  The trouble is, what are the observable facts when it comes to consciousness?  The anti-common-sense view gets all its force by pretending that we’re in a relatively late stage of research—namely, the stage of taking an agreed-upon scientific definition of consciousness, and applying it to test our intuitions—rather than in an extremely early stage, of agreeing on what the word “consciousness” is even supposed to mean.

Since I think this point is extremely important—and of general interest, beyond just IIT—I’ll expand on it with some analogies.

Suppose I told you that, in my opinion, the ε-δ definition of continuous functions—the one you learn in calculus class—failed to capture the true meaning of continuity.  Suppose I told you that I had a new, better definition of continuity—and amazingly, when I tried out my definition on some examples, it turned out that ⌊x⌋ (the floor function) was continuous, whereas x²  had discontinuities, though only at 17.5 and 42.

You would probably ask what I was smoking, and whether you could have some.  But why?  Why shouldn’t the study of continuity produce counterintuitive results?  After all, even the standard definition of continuity leads to some famously weird results, like that x sin(1/x) is a continuous function, even though sin(1/x) is discontinuous.  And it’s not as if the standard definition is God-given: people had been using words like “continuous” for centuries before Bolzano, Weierstrass, et al. formalized the ε-δ definition, a definition that millions of calculus students still find far from intuitive.  So why shouldn’t there be a different, better definition of “continuous,” and why shouldn’t it reveal that a step function is continuous while a parabola is not?

In my view, the way out of this conceptual jungle is to realize that, before any formal definitions, any ε’s and δ’s, we start with an intuition for we’re trying to capture by the word “continuous.”  And if we press hard enough on what that intuition involves, we’ll find that it largely consists of various “paradigm-cases.”  A continuous function, we’d say, is a function like 3x, or x², or sin(x), while a discontinuity is the kind of thing that the function 1/x has at x=0, or that ⌊x⌋ has at every integer point.  Crucially, we use the paradigm-cases to guide our choice of a formal definition—not vice versa!  It’s true that, once we have a formal definition, we can then apply it to “exotic” cases like x sin(1/x), and we might be surprised by the results.  But the paradigm-cases are different.  If, for example, our definition told us that x² was discontinuous, that wouldn’t be a “surprise”; it would just be evidence that we’d picked a bad definition.  The definition failed at the only task for which it could have succeeded: namely, that of capturing what we meant.

Some people might say that this is all well and good in pure math, but empirical science has no need for squishy intuitions and paradigm-cases.  Nothing could be further from the truth.  Suppose, again, that I told you that physicists since Kelvin had gotten the definition of temperature all wrong, and that I had a new, better definition.  And, when I built a Scott-thermometer that measures true temperatures, it delivered the shocking result that boiling water is actually colder than ice.  You’d probably tell me where to shove my Scott-thermometer.  But wait: how do you know that I’m not the Copernicus of heat, and that future generations won’t celebrate my breakthrough while scoffing at your small-mindedness?

I’d say there’s an excellent answer: because what we mean by heat is “whatever it is that boiling water has more of than ice” (along with dozens of other paradigm-cases).  And because, if you use a thermometer to check whether boiling water is hotter than ice, then the term for what you’re doing is calibrating your thermometer.  When the clock strikes 13, it’s time to fix the clock, and when the thermometer says boiling water’s colder than ice, it’s time to replace the thermometer—or if needed, even the entire theory on which the thermometer is based.

Ah, you say, but doesn’t modern physics define heat in a completely different, non-intuitive way, in terms of molecular motion?  Yes, and that turned out to be a superb definition—not only because it was precise, explanatory, and applicable to cases far beyond our everyday experience, but crucially, because it matched common sense on the paradigm-cases.  If it hadn’t given sensible results for boiling water and ice, then the only possible conclusion would be that, whatever new quantity physicists had defined, they shouldn’t call it “temperature,” or claim that their quantity measured the amount of “heat.”  They should call their new thing something else.

The implications for the consciousness debate are obvious.  When we consider whether to accept IIT’s equation of integrated information with consciousness, we don’t start with any agreed-upon, independent notion of consciousness against which the new notion can be compared.  The main things we start with, in my view, are certain paradigm-cases that gesture toward what we mean:

• You are conscious (though not when anesthetized).
• (Most) other people appear to be conscious, judging from their behavior.
• Many animals appear to be conscious, though probably to a lesser degree than humans (and the degree of consciousness in each particular species is far from obvious).
• A rock is not conscious.  A wall is not conscious.  A Reed-Solomon code is not conscious.  Microsoft Word is not conscious (though a Word macro that passed the Turing test conceivably would be).

Fetuses, coma patients, fish, and hypothetical AIs are the x sin(1/x)’s of consciousness: they’re the tougher cases, the ones where we might actually need a formal definition to adjudicate the truth.

Now, given a proposed formal definition for an intuitive concept, how can we check whether the definition is talking about same thing we were trying to get at before?  Well, we can check whether the definition at least agrees that parabolas are continuous while step functions are not, that boiling water is hot while ice is cold, and that we’re conscious while Reed-Solomon decoders are not.  If so, then the definition might be picking out the same thing that we meant, or were trying to mean, pre-theoretically (though we still can’t be certain).  If not, then the definition is certainly talking about something else.

What else can we do?

II. The Axiom Argument

According to Giulio, there is something else we can do, besides relying on paradigm-cases.  That something else, in his words, is to lay down “postulates about how the physical world should be organized to support the essential properties of experience,” then use those postulates to derive a consciousness-measuring quantity.

OK, so what are IIT’s postulates?  Here’s how Giulio states the five postulates leading to Φ in his response essay (he “derives” these from earlier “phenomenological axioms,” which you can find in the essay):

1. A system of mechanisms exists intrinsically if it can make a difference to itself, by affecting the probability of its past and future states, i.e. it has causal power (existence).
2. It is composed of submechanisms each with their own causal power (composition).
3. It generates a conceptual structure that is the specific way it is, as specified by each mechanism’s concept — this is how each mechanism affects the probability of the system’s past and future states (information).
4. The conceptual structure is unified — it cannot be decomposed into independent components (integration).
5. The conceptual structure is singular — there can be no superposition of multiple conceptual structures over the same mechanisms and intervals of time.

From my standpoint, these postulates have three problems.  First, I don’t really understand them.  Second, insofar as I do understand them, I don’t necessarily accept their truth.  And third, insofar as I do accept their truth, I don’t see how they lead to Φ.

To elaborate a bit:

I don’t really understand the postulates.  I realize that the postulates are explicated further in the many papers on IIT.  Unfortunately, while it’s possible that I missed something, in all of the papers that I read, the definitions never seemed to “bottom out” in mathematical notions that I understood, like functions mapping finite sets to other finite sets.  What, for example, is a “mechanism”?  What’s a “system of mechanisms”?  What’s “causal power”?  What’s a “conceptual structure,” and what does it mean for it to be “unified”?  Alas, it doesn’t help to define these notions in terms of other notions that I also don’t understand.  And yes, I agree that all these notions can be given fully rigorous definitions, but there could be many different ways to do so, and the devil could lie in the details.  In any case, because (as I said) it’s entirely possible that the failure is mine, I place much less weight on this point than I do on the two points to follow.

I don’t necessarily accept the postulates’ truth.  Is consciousness a “unified conceptual structure”?  Is it “singular”?  Maybe.  I don’t know.  It sounds plausible.  But at any rate, I’m far less confident about any these postulates—whatever one means by them!—than I am about my own “postulate,” which is that you and I are conscious while my toaster is not.  Note that my postulate, though not phenomenological, does have the merit of constraining candidate theories of consciousness in an unambiguous way.

I don’t see how the postulates lead to Φ.  Even if one accepts the postulates, how does one deduce that the “amount of consciousness” should be measured by Φ, rather than by some other quantity?  None of the papers I read—including the ones Giulio linked to in his response essay—contained anything that looked to me like a derivation of Φ.  Instead, there was general discussion of the postulates, and then Φ just sort of appeared at some point.  Furthermore, given the many idiosyncrasies of Φ—the minimization over all bipartite (why just bipartite? why not tripartite?) decompositions of the system, the need for normalization (or something else in version 3.0) to deal with highly-unbalanced partitions—it would be quite a surprise were it possible to derive its specific form from postulates of such generality.

I was going to argue for that conclusion in more detail, when I realized that Giulio had kindly done the work for me already.  Recall that Giulio chided me for not using the “latest, 2014, version 3.0” edition of Φ in my previous post.  Well, if the postulates uniquely determined the form of Φ, then what’s with all these upgrades?  Or has Φ’s definition been changing from year to year because the postulates themselves have been changing?  If the latter, then maybe one should wait for the situation to stabilize before trying to form an opinion of the postulates’ meaningfulness, truth, and completeness?

III. The Ironic Empirical Argument

Or maybe not.  Despite all the problems noted above with the IIT postulates, Giulio argues in his essay that there’s a good a reason to accept them: namely, they explain various empirical facts from neuroscience, and lead to confirmed predictions.  In his words:

[A] theory’s postulates must be able to explain, in a principled and parsimonious way, at least those many facts about consciousness and the brain that are reasonably established and non-controversial.  For example, we know that our own consciousness depends on certain brain structures (the cortex) and not others (the cerebellum), that it vanishes during certain periods of sleep (dreamless sleep) and reappears during others (dreams), that it vanishes during certain epileptic seizures, and so on.  Clearly, a theory of consciousness must be able to provide an adequate account for such seemingly disparate but largely uncontroversial facts.  Such empirical facts, and not intuitions, should be its primary test…

[I]n some cases we already have some suggestive evidence [of the truth of the IIT postulates’ predictions].  One example is the cerebellum, which has 69 billion neurons or so — more than four times the 16 billion neurons of the cerebral cortex — and is as complicated a piece of biological machinery as any.  Though we do not understand exactly how it works (perhaps even less than we understand the cerebral cortex), its connectivity definitely suggests that the cerebellum is ill suited to information integration, since it lacks lateral connections among its basic modules.  And indeed, though the cerebellum is heavily connected to the cerebral cortex, removing it hardly affects our consciousness, whereas removing the cortex eliminates it.

I hope I’m not alone in noticing the irony of this move.  But just in case, let me spell it out: Giulio has stated, as “largely uncontroversial facts,” that certain brain regions (the cerebellum) and certain states (dreamless sleep) are not associated with our consciousness.  He then views it as a victory for IIT, if those regions and states turn out to have lower information integration than the regions and states that he does take to be associated with our consciousness.

But how does Giulio know that the cerebellum isn’t conscious?  Even if it doesn’t produce “our” consciousness, maybe the cerebellum has its own consciousness, just as rich as the cortex’s but separate from it.  Maybe removing the cerebellum destroys that other consciousness, unbeknownst to “us.”  Likewise, maybe “dreamless” sleep brings about its own form of consciousness, one that (unlike dreams) we never, ever remember in the morning.

Giulio might take the implausibility of those ideas as obvious, or at least as “largely uncontroversial” among neuroscientists.  But here’s the problem with that: he just told us that a 2D square grid is conscious!  He told us that we must not rely on “commonsense intuition,” or on any popular consensus, to say that if a square mesh of wires is just sitting there XORing some input bits, doing nothing at all that we’d want to call intelligent, then it’s probably safe to conclude that the mesh isn’t conscious.  So then why shouldn’t he say the same for the cerebellum, or for the brain in dreamless sleep?  By Giulio’s own rules (the ones he used for the mesh), we have no a-priori clue whether those systems are conscious or not—so even if IIT predicts that they’re not conscious, that can’t be counted as any sort of success for IIT.

For me, the point is even stronger: I, personally, would be a million times more inclined to ascribe consciousness to the human cerebellum, or to dreamless sleep, than I would to the mesh of XOR gates.  For it’s not hard to imagine neuroscientists of the future discovering “hidden forms of intelligence” in the cerebellum, and all but impossible to imagine them doing the same for the mesh.  But even if you put those examples on the same footing, still the take-home message seems clear: you can’t count it as a “success” for IIT if it predicts that the cerebellum in unconscious, while at the same time denying that it’s a “failure” for IIT if it predicts that a square mesh of XOR gates is conscious.  If the unconsciousness of the cerebellum can be considered an “empirical fact,” safe enough for theories of consciousness to be judged against it, then surely the unconsciousness of the mesh can also be considered such a fact.

IV. The Phenomenology Argument

I now come to, for me, the strangest and most surprising part of Giulio’s response.  Despite his earlier claim that IIT need not dovetail with “commonsense intuition” about which systems are conscious—that it can defy intuition—at some point, Giulio valiantly tries to reprogram our intuition, to make us feel why a 2D grid could be conscious.  As best I can understand, the argument seems to be that, when we stare at a blank 2D screen, we form a rich experience in our heads, and that richness must be mirrored by a corresponding “intrinsic” richness in 2D space itself:

[I]f one thinks a bit about it, the experience of empty 2D visual space is not at all empty, but contains a remarkable amount of structure.  In fact, when we stare at the blank screen, quite a lot is immediately available to us without any effort whatsoever.  Thus, we are aware of all the possible locations in space (“points”): the various locations are right “there”, in front of us.  We are aware of their relative positions: a point may be left or right of another, above or below, and so on, for every position, without us having to order them.  And we are aware of the relative distances among points: quite clearly, two points may be close or far, and this is the case for every position.  Because we are aware of all of this immediately, without any need to calculate anything, and quite regularly, since 2D space pervades most of our experiences, we tend to take for granted the vast set of relationship[s] that make up 2D space.

And yet, says IIT, given that our experience of the blank screen definitely exists, and it is precisely the way it is — it is 2D visual space, with all its relational properties — there must be physical mechanisms that specify such phenomenological relationships through their causal power … One may also see that the causal relationships that make up 2D space obtain whether the elements are on or off.  And finally, one may see that such a 2D grid is necessary not so much to represent space from the extrinsic perspective of an observer, but to create it, from its own intrinsic perspective.

Now, it would be child’s-play to criticize the above line of argument for conflating our consciousness of the screen with the alleged consciousness of the screen itself.  To wit:  Just because it feels like something to see a wall, doesn’t mean it feels like something to be a wall.  You can smell a rose, and the rose can smell good, but that doesn’t mean the rose can smell you.

However, I actually prefer a different tack in criticizing Giulio’s “wall argument.”  Suppose I accepted that my mental image of the relationships between certain entities was relevant to assessing whether those entities had their own mental life, independent of me or any other observer.  For example, suppose I believed that, if my experience of 2D space is rich and structured, then that’s evidence that 2D space is rich and structured enough to be conscious.

Then my question is this: why shouldn’t the same be true of 1D space?  After all, my experience of staring at a rope is also rich and structured, no less than my experience of staring at a wall.  I perceive some points on the rope as being toward the left, others as being toward the right, and some points as being between two other points.  In fact, the rope even has a structure—namely, a natural total ordering on its points—that the wall lacks.  So why does IIT cruelly deny subjective experience to a row of logic gates strung along a rope, reserving it only for a mesh of logic gates pasted to a wall?

And yes, I know the answer: because the logic gates on the rope aren’t “integrated” enough.  But who’s to say that the gates in the 2D mesh are integrated enough?  As I mentioned before, their Φ-value grows only as the square root of the number of gates, so that the ratio of integrated information to total information tends to 0 as the number of gates increases.  And besides, aren’t what Giulio calls “the facts of phenomenology” the real arbiters here, and isn’t my perception of the rope’s structure a phenomenological fact?  When you cut a rope, does it not split?  When you prick it, does it not fray?

Conclusion

At this point, I fear we’re at a philosophical impasse.  Having learned that, according to IIT,

1. a square grid of XOR gates is conscious, and your experience of staring at a blank wall provides evidence for that,
2. by contrast, a linear array of XOR gates is not conscious, your experience of staring at a rope notwithstanding,
3. the human cerebellum is also not conscious (even though a grid of XOR gates is), and
4. unlike with the XOR gates, we don’t need a theory to tell us the cerebellum is unconscious, but can simply accept it as “reasonably established” and “largely uncontroversial,”

I personally feel completely safe in saying that this is not the theory of consciousness for me.  But I’ve also learned that other people, even after understanding the above, still don’t reject IIT.  And you know what?  Bully for them.  On reflection, I firmly believe that a two-state solution is possible, in which we simply adopt different words for the different things that we mean by “consciousness”—like, say, consciousness_Real for my kind and consciousness_WTF for the IIT kind.  OK, OK, just kidding!  How about “paradigm-case consciousness” for the one and “IIT consciousness” for the other.

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Completely unrelated announcement: Some of you might enjoy this Nature News piece by Amanda Gefter, about black holes and computational complexity.

Update (June 3): A few days after we posted this paper online, Brent Werness, a postdoc in probability theory at the University of Washington, discovered a serious error in the “experimental” part of the paper.  Happily, Brent is now collaborating with us on producing a new version of the paper that fixes the error, which we hope to have available within a few months (and which will replace the version currently on the arXiv).

To make a long story short: while the overall idea, of measuring “apparent complexity” by the compressed file size of a coarse-grained image, is fine, the “interacting coffee automaton” that we study in the paper is not an example where the apparent complexity becomes large at intermediate times.  That fact can be deduced as a corollary of a result of Liggett from 2009 about the “symmetric exclusion process,” and can be seen as a far-reaching generalization of a result that we prove in our paper’s appendix: namely, that in the non-interacting coffee automaton (our “control case”), the apparent complexity after t time steps is upper-bounded by O(log(nt)).  As it turns out, we were more right than we knew to worry about large-deviation bounds giving complete mathematical control over what happens when the cream spills into the coffee, thereby preventing the apparent complexity from ever becoming large!

But what about our numerical results, which showed a small but unmistakable complexity bump for the interacting automaton (figure 10(a) in the paper)?  It now appears that the complexity bump we saw in our data is likely to be explainable by an incomplete removal of what we called “border pixel artifacts”: that is, “spurious” complexity that arises merely from the fact that, at the border between cream and coffee, we need to round the fraction of cream up or down to the nearest integer to produce a grayscale.  In the paper, we devoted a whole section (Section 6) to border pixel artifacts and the need to deal with them: something sufficiently non-obvious that in the comments of this post, you can find people arguing with me that it’s a non-issue.  Well, it now appears that we erred by underestimating the severity of border pixel artifacts, and that a better procedure to get rid of them would also eliminate the complexity bump for the interacting automaton.

Once again, this error has no effect on either the general idea of complexity rising and then falling in closed thermodynamic systems, or our proposal for how to quantify that rise and fall—the two aspects of the paper that have generated the most interest.  But we made a bad choice of model system with which to illustrate those ideas.  Had I looked more carefully at the data, I could’ve noticed the problem before we posted, and I take responsibility for my failure to do so.

The good news is that ultimately, I think the truth only makes our story more interesting.  For it turns out that apparent complexity, as we define it, is not something that’s trivial to achieve by just setting loose a bunch of randomly-walking particles, which bump into each other but are otherwise completely independent.  If you want “complexity” along the approach to thermal equilibrium, you need to work a bit harder for it.  One promising idea, which we’re now exploring, is to consider a cream tendril whose tip takes a random walk through the coffee, leaving a trail of cream in its wake.  Using results in probability theory—closely related, or so I’m told, to the results for which Wendelin Werner won his Fields Medal!—it may even be possible to prove analytically that the apparent complexity becomes large in thermodynamic systems with this sort of behavior, much as one can prove that the complexity doesn’t become large in our original coffee automaton.

So, if you’re interested in this topic, stay tuned for the updated version of our paper.  In the meantime, I wish to express our deepest imaginable gratitude to Brent Werness for telling us all this.

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Good news!  After nearly three years of procrastination, fellow blogger Sean Carroll, former MIT undergraduate Lauren Ouellette, and yours truly finally finished a paper with the above title (coming soon to an arXiv near you).  PowerPoint slides are also available (as usual, you’re on your own if you can’t open them—sorry!).

For the background and context of this paper, please see my old post “The First Law of Complexodynamics,” which discussed Sean’s problem of defining a “complextropy” measure that first increases and then decreases in closed thermodynamic systems, in contrast to entropy (which increases monotonically).  In this exploratory paper, we basically do five things:

1. We survey several candidate “complextropy” measures: their strengths, weaknesses, and relations to one another.
2. We propose a model system for studying such measures: a probabilistic cellular automaton that models a cup of coffee into which cream has just been poured.
3. We report the results of numerical experiments with one of the measures, which we call “apparent complexity” (basically, the gzip file size of a smeared-out image of the coffee cup).  The results confirm that the apparent complexity does indeed increase, reach a maximum, then turn around and decrease as the coffee and cream mix.
4. We discuss a technical issue that one needs to overcome (the so-called “border pixels” problem) before one can do meaningful experiments in this area, and offer a solution.
5. We raise the open problem of proving analytically that the apparent complexity ever becomes large for the coffee automaton.  To underscore this problem’s difficulty, we prove that the apparent complexity doesn’t become large in a simplified version of the coffee automaton.

Anyway, here’s the abstract:

In contrast to entropy, which increases monotonically, the “complexity” or “interestingness” of closed systems seems intuitively to increase at first and then decrease as equilibrium is approached. For example, our universe lacked complex structures at the Big Bang and will also lack them after black holes evaporate and particles are dispersed. This paper makes an initial attempt to quantify this pattern. As a model system, we use a simple, two-dimensional cellular automaton that simulates the mixing of two liquids (“coffee” and “cream”). A plausible complexity measure is then the Kolmogorov complexity of a coarse-grained approximation of the automaton’s state, which we dub the “apparent complexity.” We study this complexity measure, and show analytically that it never becomes large when the liquid particles are non-interacting. By contrast, when the particles do interact, we give numerical evidence that the complexity reaches a maximum comparable to the “coffee cup’s” horizontal dimension. We raise the problem of proving this behavior analytically.

Questions and comments more than welcome.

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In unrelated news, Shafi Goldwasser has asked me to announce that the Call for Papers for the 2015 Innovations in Theoretical Computer Science (ITCS) conference is now available.

Happy birthday to me!

Recently, lots of people have been asking me what I think about IIT—no, not the Indian Institutes of Technology, but Integrated Information Theory, a widely-discussed “mathematical theory of consciousness” developed over the past decade by the neuroscientist Giulio Tononi.  One of the askers was Max Tegmark, who’s enthusiastically adopted IIT as a plank in his radical mathematizing platform (see his paper “Consciousness as a State of Matter”).  When, in the comment thread about Max’s Mathematical Universe Hypothesis, I expressed doubts about IIT, Max challenged me to back up my doubts with a quantitative calculation.

So, this is the post that I promised to Max and all the others, about why I don’t believe IIT.  And yes, it will contain that quantitative calculation.

But first, what is IIT?  The central ideas of IIT, as I understand them, are:

(1) to propose a quantitative measure, called Φ, of the amount of “integrated information” in a physical system (i.e. information that can’t be localized in the system’s individual parts), and then

(2) to hypothesize that a physical system is “conscious” if and only if it has a large value of Φ—and indeed, that a system is more conscious the larger its Φ value.

I’ll return later to the precise definition of Φ—but basically, it’s obtained by minimizing, over all subdivisions of your physical system into two parts A and B, some measure of the mutual information between A’s outputs and B’s inputs and vice versa.  Now, one immediate consequence of any definition like this is that all sorts of simple physical systems (a thermostat, a photodiode, etc.) will turn out to have small but nonzero Φ values.  To his credit, Tononi cheerfully accepts the panpsychist implication: yes, he says, it really does mean that thermostats and photodiodes have small but nonzero levels of consciousness.  On the other hand, for the theory to work, it had better be the case that Φ is small for “intuitively unconscious” systems, and only large for “intuitively conscious” systems.  As I’ll explain later, this strikes me as a crucial point on which IIT fails.

The literature on IIT is too big to do it justice in a blog post.  Strikingly, in addition to the “primary” literature, there’s now even a “secondary” literature, which treats IIT as a sort of established base on which to build further speculations about consciousness.  Besides the Tegmark paper linked to above, see for example this paper by Maguire et al., and associated popular article.  (Ironically, Maguire et al. use IIT to argue for the Penrose-like view that consciousness might have uncomputable aspects—a use diametrically opposed to Tegmark’s.)

Anyway, if you want to read a popular article about IIT, there are loads of them: see here for the New York Times’s, here for Scientific American‘s, here for IEEE Spectrum‘s, and here for the New Yorker‘s.  Unfortunately, none of those articles will tell you the meat (i.e., the definition of integrated information); for that you need technical papers, like this or this by Tononi, or this by Seth et al.  IIT is also described in Christof Koch’s memoir Consciousness: Confessions of a Romantic Reductionist, which I read and enjoyed; as well as Tononi’s Phi: A Voyage from the Brain to the Soul, which I haven’t yet read.  (Koch, one of the world’s best-known thinkers and writers about consciousness, has also become an evangelist for IIT.)

So, I want to explain why I don’t think IIT solves even the problem that it “plausibly could have” solved.  But before I can do that, I need to do some philosophical ground-clearing.  Broadly speaking, what is it that a “mathematical theory of consciousness” is supposed to do?  What questions should it answer, and how should we judge whether it’s succeeded?

The most obvious thing a consciousness theory could do is to explain why consciousness exists: that is, to solve what David Chalmers calls the “Hard Problem,” by telling us how a clump of neurons is able to give rise to the taste of strawberries, the redness of red … you know, all that ineffable first-persony stuff.  Alas, there’s a strong argument—one that I, personally, find completely convincing—why that’s too much to ask of any scientific theory.  Namely, no matter what the third-person facts were, one could always imagine a universe consistent with those facts in which no one “really” experienced anything.  So for example, if someone claims that integrated information “explains” why consciousness exists—nope, sorry!  I’ve just conjured into my imagination beings whose Φ-values are a thousand, nay a trillion times larger than humans’, yet who are also philosophical zombies: entities that there’s nothing that it’s like to be.  Granted, maybe such zombies can’t exist in the actual world: maybe, if you tried to create one, God would notice its large Φ-value and generously bequeath it a soul.  But if so, then that’s a further fact about our world, a fact that manifestly couldn’t be deduced from the properties of Φ alone.  Notice that the details of Φ are completely irrelevant to the argument.

Faced with this point, many scientifically-minded people start yelling and throwing things.  They say that “zombies” and so forth are empty metaphysics, and that our only hope of learning about consciousness is to engage with actual facts about the brain.  And that’s a perfectly reasonable position!  As far as I’m concerned, you absolutely have the option of dismissing Chalmers’ Hard Problem as a navel-gazing distraction from the real work of neuroscience.  The one thing you can’t do is have it both ways: that is, you can’t say both that the Hard Problem is meaningless, and that progress in neuroscience will soon solve the problem if it hasn’t already.  You can’t maintain simultaneously that

(a) once you account for someone’s observed behavior and the details of their brain organization, there’s nothing further about consciousness to be explained, and

(b) remarkably, the XYZ theory of consciousness can explain the “nothing further” (e.g., by reducing it to integrated information processing), or might be on the verge of doing so.

As obvious as this sounds, it seems to me that large swaths of consciousness-theorizing can just be summarily rejected for trying to have their brain and eat it in precisely the above way.

Fortunately, I think IIT survives the above observations.  For we can easily interpret IIT as trying to do something more “modest” than solve the Hard Problem, although still staggeringly audacious.  Namely, we can say that IIT “merely” aims to tell us which physical systems are associated with consciousness and which aren’t, purely in terms of the systems’ physical organization.  The test of such a theory is whether it can produce results agreeing with “commonsense intuition”: for example, whether it can affirm, from first principles, that (most) humans are conscious; that dogs and horses are also conscious but less so; that rocks, livers, bacteria colonies, and existing digital computers are not conscious (or are hardly conscious); and that a room full of people has no “mega-consciousness” over and above the consciousnesses of the individuals.

The reason it’s so important that the theory uphold “common sense” on these test cases is that, given the experimental inaccessibility of consciousness, this is basically the only test available to us.  If the theory gets the test cases “wrong” (i.e., gives results diverging from common sense), it’s not clear that there’s anything else for the theory to get “right.”  Of course, supposing we had a theory that got the test cases right, we could then have a field day with the less-obvious cases, programming our computers to tell us exactly how much consciousness is present in octopi, fetuses, brain-damaged patients, and hypothetical AI bots.

In my opinion, how to construct a theory that tells us which physical systems are conscious and which aren’t—giving answers that agree with “common sense” whenever the latter renders a verdict—is one of the deepest, most fascinating problems in all of science.  Since I don’t know a standard name for the problem, I hereby call it the Pretty-Hard Problem of Consciousness.  Unlike with the Hard Hard Problem, I don’t know of any philosophical reason why the Pretty-Hard Problem should be inherently unsolvable; but on the other hand, humans seem nowhere close to solving it (if we had solved it, then we could reduce the abortion, animal rights, and strong AI debates to “gentlemen, let us calculate!”).

Now, I regard IIT as a serious, honorable attempt to grapple with the Pretty-Hard Problem of Consciousness: something concrete enough to move the discussion forward.  But I also regard IIT as a failed attempt on the problem.  And I wish people would recognize its failure, learn from it, and move on.

In my view, IIT fails to solve the Pretty-Hard Problem because it unavoidably predicts vast amounts of consciousness in physical systems that no sane person would regard as particularly “conscious” at all: indeed, systems that do nothing but apply a low-density parity-check code, or other simple transformations of their input data.  Moreover, IIT predicts not merely that these systems are “slightly” conscious (which would be fine), but that they can be unboundedly more conscious than humans are.

To justify that claim, I first need to define Φ.  Strikingly, despite the large literature about Φ, I had a hard time finding a clear mathematical definition of it—one that not only listed formulas but fully defined the structures that the formulas were talking about.  Complicating matters further, there are several competing definitions of Φ in the literature, including Φ_DM (discrete memoryless), Φ_E (empirical), and Φ_AR (autoregressive), which apply in different contexts (e.g., some take time evolution into account and others don’t).  Nevertheless, I think I can define Φ in a way that will make sense to theoretical computer scientists.  And crucially, the broad point I want to make about Φ won’t depend much on the details of its formalization anyway.

We consider a discrete system in a state x=(x₁,…,x_n)∈Sⁿ, where S is a finite alphabet (the simplest case is S={0,1}).  We imagine that the system evolves via an “updating function” f:Sⁿ→Sⁿ. Then the question that interests us is whether the x_i‘s can be partitioned into two sets A and B, of roughly comparable size, such that the updates to the variables in A don’t depend very much on the variables in B and vice versa.  If such a partition exists, then we say that the computation of f does not involve “global integration of information,” which on Tononi’s theory is a defining aspect of consciousness.

More formally, given a partition (A,B) of {1,…,n}, let us write an input y=(y₁,…,y_n)∈Sⁿ to f in the form (y_A,y_B), where y_A consists of the y variables in A and y_B consists of the y variables in B.  Then we can think of f as mapping an input pair (y_A,y_B) to an output pair (z_A,z_B).  Now, we define the “effective information” EI(A→B) as H(z_B | A random, y_B=x_B).  Or in words, EI(A→B) is the Shannon entropy of the output variables in B, if the input variables in A are drawn uniformly at random, while the input variables in B are fixed to their values in x.  It’s a measure of the dependence of B on A in the computation of f(x).  Similarly, we define

EI(B→A) := H(z_A | B random, y_A=x_A).

We then consider the sum

Φ(A,B) := EI(A→B) + EI(B→A).

Intuitively, we’d like the integrated information Φ=Φ(f,x) be the minimum of Φ(A,B), over all 2ⁿ-2 possible partitions of {1,…,n} into nonempty sets A and B.  The idea is that Φ should be large, if and only if it’s not possible to partition the variables into two sets A and B, in such a way that not much information flows from A to B or vice versa when f(x) is computed.

However, no sooner do we propose this than we notice a technical problem.  What if A is much larger than B, or vice versa?  As an extreme case, what if A={1,…,n-1} and B={n}?  In that case, we’ll have Φ(A,B)≤2log₂|S|, but only for the boring reason that there’s hardly any entropy in B as a whole, to either influence A or be influenced by it.  For this reason, Tononi proposes a fix where we normalize each Φ(A,B) by dividing it by min{|A|,|B|}.  He then defines the integrated information Φ to be Φ(A,B), for whichever partition (A,B) minimizes the ratio Φ(A,B) / min{|A|,|B|}.  (Unless I missed it, Tononi never specifies what we should do if there are multiple (A,B)’s that all achieve the same minimum of Φ(A,B) / min{|A|,|B|}.  I’ll return to that point later, along with other idiosyncrasies of the normalization procedure.)

Tononi gives some simple examples of the computation of Φ, showing that it is indeed larger for systems that are more “richly interconnected” in an intuitive sense.  He speculates, plausibly, that Φ is quite large for (some reasonable model of) the interconnection network of the human brain—and probably larger for the brain than for typical electronic devices (which tend to be highly modular in design, thereby decreasing their Φ), or, let’s say, than for other organs like the pancreas.  Ambitiously, he even speculates at length about how a large value of Φ might be connected to the phenomenology of consciousness.

To be sure, empirical work in integrated information theory has been hampered by three difficulties.  The first difficulty is that we don’t know the detailed interconnection network of the human brain.  The second difficulty is that it’s not even clear what we should define that network to be: for example, as a crude first attempt, should we assign a Boolean variable to each neuron, which equals 1 if the neuron is currently firing and 0 if it’s not firing, and let f be the function that updates those variables over a timescale of, say, a millisecond?  What other variables do we need—firing rates, internal states of the neurons, neurotransmitter levels?  Is choosing many of these variables uniformly at random (for the purpose of calculating Φ) really a reasonable way to “randomize” the variables, and if not, what other prescription should we use?

The third and final difficulty is that, even if we knew exactly what we meant by “the f and x corresponding to the human brain,” and even if we had complete knowledge of that f and x, computing Φ(f,x) could still be computationally intractable.  For recall that the definition of Φ involved minimizing a quantity over all the exponentially-many possible bipartitions of {1,…,n}.  While it’s not directly relevant to my arguments in this post, I leave it as a challenge for interested readers to pin down the computational complexity of approximating Φ to some reasonable precision, assuming that f is specified by a polynomial-size Boolean circuit, or alternatively, by an NC⁰ function (i.e., a function each of whose outputs depends on only a constant number of the inputs).  (Presumably Φ will be #P-hard to calculate exactly, but only because calculating entropy exactly is a #P-hard problem—that’s not interesting.)

I conjecture that approximating Φ is an NP-hard problem, even for restricted families of f’s like NC⁰ circuits—which invites the amusing thought that God, or Nature, would need to solve an NP-hard problem just to decide whether or not to imbue a given physical system with consciousness!  (Alas, if you wanted to exploit this as a practical approach for solving NP-complete problems such as 3SAT, you’d need to do a rather drastic experiment on your own brain—an experiment whose result would be to render you unconscious if your 3SAT instance was satisfiable, or conscious if it was unsatisfiable!  In neither case would you be able to communicate the outcome of the experiment to anyone else, nor would you have any recollection of the outcome after the experiment was finished.)  In the other direction, it would also be interesting to upper-bound the complexity of approximating Φ.  Because of the need to estimate the entropies of distributions (even given a bipartition (A,B)), I don’t know that this problem is in NP—the best I can observe is that it’s in AM.

In any case, my own reason for rejecting IIT has nothing to do with any of the “merely practical” issues above: neither the difficulty of defining f and x, nor the difficulty of learning them, nor the difficulty of calculating Φ(f,x).  My reason is much more basic, striking directly at the hypothesized link between “integrated information” and consciousness.  Specifically, I claim the following:

Yes, it might be a decent rule of thumb that, if you want to know which brain regions (for example) are associated with consciousness, you should start by looking for regions with lots of information integration.  And yes, it’s even possible, for all I know, that having a large Φ-value is one necessary condition among many for a physical system to be conscious.  However, having a large Φ-value is certainly not a sufficient condition for consciousness, or even for the appearance of consciousness.  As a consequence, Φ can’t possibly capture the essence of what makes a physical system conscious, or even of what makes a system look conscious to external observers.

The demonstration of this claim is embarrassingly simple.  Let S=F_p, where p is some prime sufficiently larger than n, and let V be an n×n Vandermonde matrix over F_p—that is, a matrix whose (i,j) entry equals i^j-1 (mod p).  Then let f:Sⁿ→Sⁿ be the update function defined by f(x)=Vx.  Now, for p large enough, the Vandermonde matrix is well-known to have the property that every submatrix is full-rank (i.e., “every submatrix preserves all the information that it’s possible to preserve about the part of x that it acts on”).  And this implies that, regardless of which bipartition (A,B) of {1,…,n} we choose, we’ll get

EI(A→B) = EI(B→A) = min{|A|,|B|} log₂p,

and hence

Φ(A,B) = EI(A→B) + EI(B→A) = 2 min{|A|,|B|} log₂p,

or after normalizing,

Φ(A,B) / min{|A|,|B|} = 2 log₂p.

Or in words: the normalized information integration has the same value—namely, the maximum value!—for every possible bipartition.  Now, I’d like to proceed from here to a determination of Φ itself, but I’m prevented from doing so by the ambiguity in the definition of Φ that I noted earlier.  Namely, since every bipartition (A,B) minimizes the normalized value Φ(A,B) / min{|A|,|B|}, in theory I ought to be able to pick any of them for the purpose of calculating Φ.  But the unnormalized value Φ(A,B), which gives the final Φ, can vary greatly, across bipartitions: from 2 log₂p (if min{|A|,|B|}=1) all the way up to n log₂p (if min{|A|,|B|}=n/2).  So at this point, Φ is simply undefined.

On the other hand, I can solve this problem, and make Φ well-defined, by an ironic little hack.  The hack is to replace the Vandermonde matrix V by an n×n matrix W, which consists of the first n/2 rows of the Vandermonde matrix each repeated twice (assume for simplicity that n is a multiple of 4).  As before, we let f(x)=Wx.  Then if we set A={1,…,n/2} and B={n/2+1,…,n}, we can achieve

EI(A→B) = EI(B→A) = (n/4) log₂p,

Φ(A,B) = EI(A→B) + EI(B→A) = (n/2) log₂p,

and hence

Φ(A,B) / min{|A|,|B|} = log₂p.

In this case, I claim that the above is the unique bipartition that minimizes the normalized integrated information Φ(A,B) / min{|A|,|B|}, up to trivial reorderings of the rows.  To prove this claim: if |A|=|B|=n/2, then clearly we minimize Φ(A,B) by maximizing the number of repeated rows in A and the number of repeated rows in B, exactly as we did above.  Thus, assume |A|≤|B| (the case |B|≤|A| is analogous).  Then clearly

EI(B→A) ≥ |A|/2,

while

EI(A→B) ≥ min{|A|, |B|/2}.

So if we let |A|=cn and |B|=(1-c)n for some c∈(0,1/2], then

Φ(A,B) ≥ [c/2 + min{c, (1-c)/2}] n,

and

Φ(A,B) / min{|A|,|B|} = Φ(A,B) / |A| = 1/2 + min{1, 1/(2c) – 1/2}.

But the above expression is uniquely minimized when c=1/2.  Hence the normalized integrated information is minimized essentially uniquely by setting A={1,…,n/2} and B={n/2+1,…,n}, and we get

Φ = Φ(A,B) = (n/2) log₂p,

which is quite a large value (only a factor of 2 less than the trivial upper bound of n log₂p).

Now, why did I call the switch from V to W an “ironic little hack”?  Because, in order to ensure a large value of Φ, I decreased—by a factor of 2, in fact—the amount of “information integration” that was intuitively happening in my system!  I did that in order to decrease the normalized value Φ(A,B) / min{|A|,|B|} for the particular bipartition (A,B) that I cared about, thereby ensuring that that (A,B) would be chosen over all the other bipartitions, thereby increasing the final, unnormalized value Φ(A,B) that Tononi’s prescription tells me to return.  I hope I’m not alone in fearing that this illustrates a disturbing non-robustness in the definition of Φ.

But let’s leave that issue aside; maybe it can be ameliorated by fiddling with the definition.  The broader point is this: I’ve shown that my system—the system that simply applies the matrix W to an input vector x—has an enormous amount of integrated information Φ.  Indeed, this system’s Φ equals half of its entire information content.  So for example, if n were 10¹⁴ or so—something that wouldn’t be hard to arrange with existing computers—then this system’s Φ would exceed any plausible upper bound on the integrated information content of the human brain.

And yet this Vandermonde system doesn’t even come close to doing anything that we’d want to call intelligent, let alone conscious!  When you apply the Vandermonde matrix to a vector, all you’re really doing is mapping the list of coefficients of a degree-(n-1) polynomial over F_p, to the values of the polynomial on the n points 0,1,…,n-1.  Now, evaluating a polynomial on a set of points turns out to be an excellent way to achieve “integrated information,” with every subset of outputs as correlated with every subset of inputs as it could possibly be.  In fact, that’s precisely why polynomials are used so heavily in error-correcting codes, such as the Reed-Solomon code, employed (among many other places) in CD’s and DVD’s.  But that doesn’t imply that every time you start up your DVD player you’re lighting the fire of consciousness.  It doesn’t even hint at such a thing.  All it tells us is that you can have integrated information without consciousness (or even intelligence)—just like you can have computation without consciousness, and unpredictability without consciousness, and electricity without consciousness.

It might be objected that, in defining my “Vandermonde system,” I was too abstract and mathematical.  I said that the system maps the input vector x to the output vector Wx, but I didn’t say anything about how it did so.  To perform a computation—even a computation as simple as a matrix-vector multiply—won’t we need a physical network of wires, logic gates, and so forth?  And in any realistic such network, won’t each logic gate be directly connected to at most a few other gates, rather than to billions of them?  And if we define the integrated information Φ, not directly in terms of the inputs and outputs of the function f(x)=Wx, but in terms of all the actual logic gates involved in computing f, isn’t it possible or even likely that Φ will go back down?

This is a good objection, but I don’t think it can rescue IIT.  For we can achieve the same qualitative effect that I illustrated with the Vandermonde matrix—the same “global information integration,” in which every large set of outputs depends heavily on every large set of inputs—even using much “sparser” computations, ones where each individual output depends on only a few of the inputs.  This is precisely the idea behind low-density parity check (LDPC) codes, which have had a major impact on coding theory over the past two decades.  Of course, one would need to muck around a bit to construct a physical system based on LDPC codes whose integrated information Φ was provably large, and for which there were no wildly-unbalanced bipartitions that achieved lower Φ(A,B)/min{|A|,|B|} values than the balanced bipartitions one cared about.  But I feel safe in asserting that this could be done, similarly to how I did it with the Vandermonde matrix.

More generally, we can achieve pretty good information integration by hooking together logic gates according to any bipartite expander graph: that is, any graph with n vertices on each side, such that every k vertices on the left side are connected to at least min{(1+ε)k,n} vertices on the right side, for some constant ε>0.  And it’s well-known how to create expander graphs whose degree (i.e., the number of edges incident to each vertex, or the number of wires coming out of each logic gate) is a constant, such as 3.  One can do so either by plunking down edges at random, or (less trivially) by explicit constructions from algebra or combinatorics.  And as indicated in the title of this post, I feel 100% confident in saying that the so-constructed expander graphs are not conscious!  The brain might be an expander, but not every expander is a brain.

Before winding down this post, I can’t resist telling you that the concept of integrated information (though it wasn’t called that) played an interesting role in computational complexity in the 1970s.  As I understand the history, Leslie Valiant conjectured that Boolean functions f:{0,1}ⁿ→{0,1}ⁿ with a high degree of “information integration” (such as discrete analogues of the Fourier transform) might be good candidates for proving circuit lower bounds, which in turn might be baby steps toward P≠NP.  More strongly, Valiant conjectured that the property of information integration, all by itself, implied that such functions had to be at least somewhat computationally complex—i.e., that they couldn’t be computed by circuits of size O(n), or even required circuits of size Ω(n log n).  Alas, that hope was refuted by Valiant’s later discovery of linear-size superconcentrators.  Just as information integration doesn’t suffice for intelligence or consciousness, so Valiant learned that information integration doesn’t suffice for circuit lower bounds either.

As humans, we seem to have the intuition that global integration of information is such a powerful property that no “simple” or “mundane” computational process could possibly achieve it.  But our intuition is wrong.  If it were right, then we wouldn’t have linear-size superconcentrators or LDPC codes.

I should mention that I had the privilege of briefly speaking with Giulio Tononi (as well as his collaborator, Christof Koch) this winter at an FQXi conference in Puerto Rico.  At that time, I challenged Tononi with a much cruder, handwavier version of some of the same points that I made above.  Tononi’s response, as best as I can reconstruct it, was that it’s wrong to approach IIT like a mathematician; instead one needs to start “from the inside,” with the phenomenology of consciousness, and only then try to build general theories that can be tested against counterexamples.  This response perplexed me: of course you can start from phenomenology, or from anything else you like, when constructing your theory of consciousness.  However, once your theory has been constructed, surely it’s then fair game for others to try to refute it with counterexamples?  And surely the theory should be judged, like anything else in science or philosophy, by how well it withstands such attacks?

But let me end on a positive note.  In my opinion, the fact that Integrated Information Theory is wrong—demonstrably wrong, for reasons that go to its core—puts it in something like the top 2% of all mathematical theories of consciousness ever proposed.  Almost all competing theories of consciousness, it seems to me, have been so vague, fluffy, and malleable that they can only aspire to wrongness.

[Endnote: See also this related post, by the philosopher Eric Schwetzgebel: Why Tononi Should Think That the United States Is Conscious.  While the discussion is much more informal, and the proposed counterexample more debatable, the basic objection to IIT is the same.]

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Update (5/22): Here are a few clarifications of this post that might be helpful.

(1) The stuff about zombies and the Hard Problem was simply meant as motivation and background for what I called the “Pretty-Hard Problem of Consciousness”—the problem that I take IIT to be addressing.  You can disagree with the zombie stuff without it having any effect on my arguments about IIT.

(2) I wasn’t arguing in this post that dualism is true, or that consciousness is irreducibly mysterious, or that there could never be any convincing theory that told us how much consciousness was present in a physical system.  All I was arguing was that, at any rate, IIT is not such a theory.

(3) Yes, it’s true that my demonstration of IIT’s falsehood assumes—as an axiom, if you like—that while we might not know exactly what we mean by “consciousness,” at any rate we’re talking about something that humans have to a greater extent than DVD players.  If you reject that axiom, then I’d simply want to define a new word for a certain quality that non-anesthetized humans seem to have and that DVD players seem not to, and clarify that that other quality is the one I’m interested in.

(4) For my counterexample, the reason I chose the Vandermonde matrix is not merely that it’s invertible, but that all of its submatrices are full-rank.  This is the property that’s relevant for producing a large value of the integrated information Φ; by contrast, note that the identity matrix is invertible, but produces a system with Φ=0.  (As another note, if we work over a large enough field, then a random matrix will have this same property with high probability—but I wanted an explicit example, and while the Vandermonde is far from the only one, it’s one of the simplest.)

(5) The n×n Vandermonde matrix only does what I want if we work over (say) a prime field F_p with p>>n elements.  Thus, it’s natural to wonder whether similar examples exist where the basic system variables are bits, rather than elements of F_p.  The answer is yes. One way to get such examples is using the low-density parity check codes that I mention in the post.  Another common way to get Boolean examples, and which is also used in practice in error-correcting codes, is to start with the Vandermonde matrix (a.k.a. the Reed-Solomon code), and then combine it with an additional component that encodes the elements of F_p as strings of bits in some way.  Of course, you then need to check that doing this doesn’t harm the properties of the original Vandermonde matrix that you cared about (e.g., the “information integration”) too much, which causes some additional complication.

(6) Finally, it might be objected that my counterexamples ignored the issue of dynamics and “feedback loops”: they all consisted of unidirectional processes, which map inputs to outputs and then halt.  However, this can be fixed by the simple expedient of iterating the process over and over!  I.e., first map x to Wx, then map Wx to W²x, and so on.  The integrated information should then be the same as in the unidirectional case.

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Update (5/24): See a very interesting comment by David Chalmers.

Eight years ago, I put up a post entitled The Ten Most Annoying Questions in Quantum Computing.  One of the ten wasn’t a real question—it was simply a request for readers to submit questions—so let’s call it nine.  I’m delighted to say that, of the nine questions, six have by now been completely settled—most recently, my question about the parallel-repeated value of the CHSH game, which Andris Ambainis pointed out to me last week can be answered using a 2008 result of Barak et al. combined with a 2013 result of Dinur and Steurer.

To be clear, the demise of so many problems is exactly the outcome I wanted. In picking problems, my goal wasn’t to shock and awe with difficulty—as if to say “this is how smart I am, that whatever stumps me will also stump everyone else for decades.” Nor was it to showcase my bottomless profundity, by proffering questions so vague, multipartite, and open-ended that no matter what progress was made, I could always reply “ah, but you still haven’t addressed the real question!” Nor, finally, was my goal to list the biggest research directions for the entire field, the stuff everyone already knows about (“is there a polynomial-time quantum algorithm for graph isomorphism?”). My interest was exclusively in “little” questions, in weird puzzles that looked (at least at the time) like there was no deep obstruction to just killing them one by one, whichever way their answers turned out. What made them annoying was that they hadn’t succumbed already.

So, now that two-thirds of my problems have met the fate they deserved, at Andris’s suggestion I’m presenting a new list of Ten Most Annoying Questions in Quantum Computing—a list that starts with the three still-unanswered questions from the old list, and then adds seven more.

But we’ll get to that shortly. First, let’s review the six questions that have been answered.

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CLOSED, NO-LONGER ANNOYING QUESTIONS IN QUANTUM COMPUTING

1. Given an n-qubit pure state, is there always a way to apply Hadamard gates to some subset of the qubits, so as to make all 2ⁿ computational basis states have nonzero amplitudes?  Positive answer by Ashley Montanaro and Dan Shepherd, posted to this blog in 2006.

3. Can any QMA(2) (QMA with two unentangled yes-provers) protocol be amplified to exponentially small error probability?  Positive answer by Aram Harrow and Ashley Montanaro, from a FOCS’2010 paper.

4. If a unitary operation U can be applied in polynomial time, then can some square root of U also be applied in polynomial time?  Positive answer by Lana Sheridan, Dmitri Maslov, and Michele Mosca, from a 2008 paper.

5. Suppose Alice and Bob are playing n parallel CHSH games, with no communication or entanglement. Is the probability that they’ll win all n games at most pⁿ, for some p bounded below 0.853?

OK, let me relay what Andris Ambainis told me about this question, with Andris’s kind permission. First of all, we’ve known for a while that the optimal success probability is not the (3/4)ⁿ that Alice and Bob could trivially achieve by just playing all n games separately. I observed in 2006 that, by correlating their strategies between pairs of games in a clever way, Alice and Bob can win with probability (√10 / 4)ⁿ ~ 0.79ⁿ. And Barak et al. showed in 2008 that they can win with probability ((1+√5)/4)ⁿ ~ 0.81ⁿ. (Unfortunately, I don’t know the actual strategy that achieves the latter bound!  Barak et al. say they’ll describe it in the full version of their paper, but the full version hasn’t yet appeared.)

Anyway, Dinur-Steurer 2013 gave a general recipe to prove that the value of a repeated projection game is at most αⁿ, where α is some constant that depends on the game in question. When Andris followed their recipe for the CHSH game, he obtained the result α=(1+√5)/4—thereby showing that Barak et al.’s strategy, whatever it is, is precisely optimal! Andris also observes that, for any two-prover game G, the Dinur-Steurer bound α(G) is always strictly less than the entangled value ω^*(G), unless the classical and entangled values are the same for one copy of the game (i.e., unless ω(G)=ω^*(G)). This implies that parallel repetition can never completely eliminate a quantum advantage.

6. Forget about an oracle relative to which BQP is not in PH (the Polynomial Hierarchy). Forget about an oracle relative to which BQP is not in AM (Arthur-Merlin). Is there an oracle relative to which BQP is not in SZK (Statistical Zero-Knowledge)?  Positive answer by me, posted to this blog in 2006.  See also my BQP vs. PH paper for a different proof.

9. Is there an n-qubit pure state that can be prepared by a circuit of size n³, and that can’t be distinguished from the maximally mixed state by any circuit of size n²?  A positive answer follows from this 2009 paper by Richard Low—thanks very much to Fernando Brandao for bringing that to my attention a few months ago.

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OK, now on to:

THE NEW TEN MOST ANNOYING QUESTIONS IN QUANTUM COMPUTING

1. Can we get any upper bound whatsoever on the complexity class QMIP—i.e., quantum multi-prover interactive proofs with unlimited prior entanglement? (Since I asked this question in 2006, Ito and Vidick achieved the breakthrough lower bound NEXP⊆QMIP, but there’s been basically no progress on the upper bound side.)

2. Given any n-qubit unitary operation U, does there exist an oracle relative to which U can be (approximately) applied in polynomial time? (Since 2006, my interest in this question has only increased. See this paper by me and Greg Kuperberg for background and related results.)

3. How many mutually unbiased bases are there in non-prime-power dimensions?

4. Since Chris Fuchs was so thrilled by my including one of his favorite questions on my earlier list (question #3 above), let me add another of his favorites: do SIC-POVMs exist in arbitrary finite dimensions?

5. Is there a Boolean function f:{0,1}ⁿ→{0,1} whose bounded-error quantum query complexity is strictly greater than n/2?  (Thanks to Shelby Kimmel for this question!  Note that this paper by van Dam shows that the bounded-error quantum query complexity never exceeds n/2+O(√n), while this paper by Ambainis et al. shows that it’s at least n/2-O(√n) for almost all Boolean functions f.)

6. Is there a “universal disentangler”: that is, a superoperator S that takes n^O(1) qubits as input; that produces a 2n-qubit bipartite state (with n qubits on each side) as output; whose output S(ρ) is always close in variation distance to a separable state; and that given an appropriate input state, can produce as output an approximation to any desired separable state?  (See here for background about this problem, originally posed by John Watrous. Note that if such an S existed and were computationally efficient, it would imply QMA=QMA(2).)

7. Suppose we have explicit descriptions of n two-outcome POVM measurements—say, as d×d Hermitian matrices E₁,…,E_n—and are also given k=(log(nd))^O(1) copies of an unknown quantum state ρ in d dimensions.  Is there a way to measure the copies so as to estimate the n expectation values Tr(E₁ρ),…,Tr(E_nρ), each to constant additive error?  (A forthcoming paper of mine on private-key quantum money will contain some background and related results.)

8. Is there a collection of 1- and 2-qubit gates that generates a group of unitary matrices that is (a) not universal for quantum computation, (b) not just conjugate to permuted diagonal matrices or one-qubit gates plus swaps, and (c) not conjugate to a subgroup of the Clifford group?

9. Given a partial Boolean function f:S→{0,1} with S⊆{0,1}ⁿ, is the bounded-error quantum query complexity of f always polynomially related to the smallest degree of any polynomial p:{0,1}ⁿ→R such that (a) p(x)∈[0,1] for all x∈{0,1}ⁿ, and (b) |p(x)-f(x)|≤1/3 for all x∈S?

10. Is there a quantum finite automaton that reads in an infinite sequence of i.i.d. coin flips, and whose limiting probability of being found in an “accept” state is at least 2/3 if the coin is fair and at most 1/3 if the coin is unfair?  (See this paper by me and Andy Drucker for background and related results.)