Duplicating electronic consciousnesses

Assume naturalism and suppose that digital electronic systems can be significantly conscious. Suppose Alice is a deterministic significantly conscious digital electronic system. Imagine we duplicated Alice to make another such system, Bob, and fed them both the same inputs. Then there are two conscious beings with qualitatively the same stream of consciousness.

But now let’s add a twist. Suppose that we create a monitoring system that continually checks all of Alice and Bob’s components, and as soon as any corresponding components disagree—are in a different state—then the system pulls the plug on both, thereby resetting all components to state zero. In fact, however, everything works well, and the inputs are always the same, so there is never any deviation between Alice and Bob, and the monitoring system never does anything.

What happens to the consciousnesses? Intuitively, neither Alice nor Bob should be affected by a monitoring system that never actually does anything. But it is not clear that this is the conclusion that specific naturalist theories will yield.

First, consider functionalism. Once the monitoring system is in place, both Alice and Bob change with respect to their dispositional features. All the subsystems of Alice are now incapable of producing any result other than one synchronized to Bob’s subsystems, and vice versa. I think a strong case can be made that on functionalism, Alice and Bob’s subsystems lose their defining functions when the monitoring system is in place, and hence lose consciousness. Therefore, on functionalism, consciousness has an implausible extrinsicness to it. The duplication-plus-monitoring case is some evidence against functionalism.

Second, consider Integrated Information Theory. It is easy to see that the whole system, consisting of Alice, Bob and the monitoring system, has a very low Φ value. Its components can be thought of as just those of Alice and Bob, but with a transition function that sets everything to zero if there is a deviation. We can now split the system into two subsystems: Alice and Bob. Each subsystem’s behavior can be fully predicted from that subsystem’s state plus one additional bit of information that represents whether the other system agrees with it. Because of this, the Φ value of the system is at most 2 bits, and hence the system as a whole has very, very little consciousness.

Moreover, Alice remains significantly conscious: we can think of Alice as having just as much integrated information after the monitoring system is attached as before, but now having one new bit of environmental dependency, so the Φ measure does not change significantly from the monitoring being added. Moreover, because the joint system is not significantly conscious, Integrated Information Theory’s proviso that a system loses consciousness when it comes to be in a part-to-whole relationship with a more conscious system is irrelevant.

Likewise, Bob remains conscious. So far everything seems perfectly intuitive. Adding a monitoring system doesn’t create a new significantly conscious system, and doesn’t destroy the two existing conscious systems. However, here is the kicker. Let X be any subsystem of Alice’s components. Let S_X be the system consisting of the components in X together with all of Bob’s components that don’t correspond to the components in X. In other words, S_X is a mix of Alice’s and Bob’s components. It is easy to see the information theoretic behavior of S_X is exactly the same as the information theoretic behavior of Alice (or of Bob for that matter). Thus, the Φ value of S_X will be the same for all X.

Hence, on Integrated Information Theory, each of the S_X systems will be equally conscious. The number of these systems equals to 2ⁿ where n is the number of components in Alice. Of course, one of these 2ⁿ systems is Alice herself (that’s S_A where A is the set of Alice’s components) and another one is Bob himself (that’s S_∅). Conclusion: By adding a monitoring system to our Alice and Bob pair, we have created a vast number of new equally conscious systems: 2ⁿ − 2 of them!

The ethical consequences are very weird. Suppose that Alice has some large number of components, say 10¹¹ (that’s how many neurons we have). We duplicate Alice to create Bob. We’ve doubled the number of beings with whatever interests Alice had. And then we add a dumb monitoring that pulls the plug given a deviation between them. Suddenly we have created 2¹⁰¹¹ − 2 systems with the same level of consciousness. Suddenly, the moral consideration owed to to the Alice/Bob line of consciousness vastly outnumbers everything.

So both functionalism and Integrated Information Theory have trouble with our duplication story.

Another slit rectangle IIT system

One more observation on Integrated Information Theory (IIT), in Aaronson’s simplified formulation.

Let R_M, N be a wide rectangular grid of points (x,y) with x and y integers such that 0 ≤ x < M and 0 ≤ y < N. Suppose M ≫ 4N and M is divisible by four. Let R_M, N, t be the grid R_M, N with all points with coordinates (M/4,y) where y ≥ tN removed. This is a grid with a “bottleneck” at x = M/4.

Let S_M, N, t be a system with a binary cell at each coordinate of R_M, N, t evolving according to the rule that at the next time step, each cell’s value changes to the xor of the up-to-four neighboring cells’ values (near the boundaries and the slit, the count will be less than four).

My intuitions about IIT say that the measure of integrated information Φ(S_M, N, t) will be equal to exactly 2N bits when t = 1, and will stay at 2N bits as we decrease t, until t is below or around 1/4, at which point it will jump to 2ceil(tN) bits. This shows two problems with this version of IIT. First, we as we cut a small slit in the rectangle, we should always be decreasing the amount of integrated information in the system—not just suddenly when the slit reaches around 3/4 of the width of the rectangle. Second, we would expect the amount of integrated information to vary much more continuously rather than suddenly jump from 2N to N/2 bits at around t = 1/4.

The problem here is the rather gerrymandered character in which IIT minimizes one quantity to generate an optimal decomposition of the system, and then defines the measure of integrated information using another quantity.

Specifically, we calculate Φ by finding a cut of the system into two subsystems A and B that minimizes Φ(A,B)/min (|A|,|B|), and intuitively, there are two types of candidates for an optimal cut if M ≫ 4N:

• cut the grid around x = M/2 into two equally sized portions using a cut that snips each horizontal line at exactly one cell (a vertical line is the most obvious option, but a diagonal cut will give the same Φ(A,B) value); the Φ(A,B) value is twice the vertical length of the cut, namely 2N bits

• cut the grid around the bottleneck into two portions, A and B, where A contains slightly more than a quarter of the cells in the system using a cut that follows the same rule as above: more precisely, we start the cut at the top of the bottleneck, and cut diagonally down and to the right (the result is that A is an (M/4) by N rectangle together with an isosceles triangle with equal sides of size tN; the triangle’s area is swamped by the rectangle’s area because M ≫ 4N); the Φ(A,B) value is (maybe modulo an off-by-one error) twice the vertical length of the cut, namely 2ceil(tN) bits.

Assuming these intuitions are right, when t is close to 1, the first type of cut results in a smaller Φ(A,B)/min (|A|,|B|) value, but when t is below or around 1/4, the second type of cut results in a smaller value.

Conjectures about a system in the context of Integrated Information Theory

I show really be done with Integrated Information Theory (IIT), in Aaronson’s simplified formulation, but I noticed a rather interesting difficult.

In my previous post on the subject, I noticed that a double grid system where there are two grids stacked on top of one another, with the bottom grid consisting of inputs and the upper grid of outputs, and each upper value being the logical OR of the (up to) five neighboring input values will be conscious according to IIT if all the values are zero and the grid is large enough.

In this post, I am going to give some conjectures about the mathematics rather than even a proof sketch. But I think the conjectures are pretty plausible and, if true, it shows something fishy about IIT’s measure of integrated information.

Consider our dual grid system, except now the grids are with some exceptions rectangular, with a length of M along the x-axis and a width of N along the y-axis (and the stacking along the z-axis). But there are the following exceptions to the rectangularity:

• at x-coordinates M/4 − 1 and M/4 the width instead of being N is N/8

• at x-coordinates M/2 − 1 and M/2 the width is N/10.

In other words, at two x-coordinate areas, the grids have bottlenecks, of slightly different sizes. We suppose M is significantly larger than N, and N is very, very large (say, 10¹⁵).

Let A_k be the components on the grids with x-coordinates less than k and let B_k be the remaining components. I suspect (with a lot of confidence) that the optimal choice for a partition {A, B} that minimizes the “modified Φ value” Φ(A,B)/min (|A|,|B|) will be pretty close to {A_k, B_k} where k is in one of the bottlenecks. Thus to estimate Φ, we need only look at the Φ and modified Φ values for {A_M/4, B_M/4} and {A_M/2, B_M/2}. Note that if k is M/4 or M/2, then min (|A|,|B|) is approximately 2MN/4 and 2MN/2, respectively, since there are two grids of components.

I suspect (again with a lot of confidence) that Φ(A_k,B_k) will be approximately proportional to the width of the grid around coordinate k. Thus, Φ(A_M/4,B_M/4)/min (A_M/4,B_M/4) will be approximately proportional to (N/8)/(2NM/4) = 0.25/M while Φ(A_M/2,B_M/2)/min (A_M/2,B_M/2) will be approximately proportional to (N/10)/(2NM/2) = 0.1/M.

Moreover, I conjecture that the optimal partition will be close to {A_k, B_k} for some k in one of the bottlenecks. If so, then our best choice will be close to {A_M/2, B_M/2}, and it will yield a Φ value approximately proportional to N/10.

Now modify the system by taking each output component at an x-coordinate less than M/4 and putting four more output components besides the original output component, and with the very same value as the original output component. 

I strongly suspect that the optimal partition will again be obtained by cutting the system at one of the two bottlenecks. The Φ values of at the M/4 and M/2 bottlenecks will be unchanged—mere duplication of outputs does not affect information content—but the modified Φ values (obtained by dividing Φ(A,B) by min (|A|,|B|)) will be (N/8)/(6NM/4) = 0.083/M and (N/10)/(2NM/2) = 0.1/M. Thus the optimal choice will be to partition the system at the M/4 bottleneck. This will yield a Φ value approximately proportional to N/8. Which is bigger than N/10.

For concreteness, let’s now imagine that each output is an LED. We now see that if we replace some of the LEDs by five LEDs (namely, the ones in the left-hand quarter of the system), we increase the amount of integrated information from N/10 to N/8. This has got to be wrong. Simply by duplicating LEDs we don’t add anything to the information content. And we certainly don’t make a system more conscious just by lighting up a portion of it with additional LEDs.

Notice, too, that IIT has a special proviso: if one system is a part of another with a higher degree of consciousness, the part system has no consciousness. So now imagine that a Φ value proportional to N/10 is sufficiently large for significant consciousness, so our original system, without extra output LEDs, is conscious. Now, besides the left quarter of the LEDs, add the quadruples of new LEDs that simply duplicate the original LED values (they might not even be electrically connected to the original system: they might sense whether the original LED is on, and light up if so). According to IIT, then, the new system is more conscious than the old—and the old system has had its consciousness destroyed, simply by adding enough duplicates of its LEDs. This seems wrong.

Of course, my conjectures and back-of-the-evelope calculations could be false. 

A bit more fun with Integrated Information Theory

I hope this is my last post for a while on Integrated Information Theory (IIT), in Aaronson’s simplified formulation.

One of the fun and well-known facts is that if you have an impractically large square two-dimensional grid of interconnected logic gates (presumably with some constant time-delay in each gate between inputs and outputs to prevent race conditions) in a fixed point (i.e., nothing is changing), the result can still have a degree of integrated information proportional to the square root of the number of gates. A particular known case is where you have a very large grid of XOR gates, with each gate’s output being connected to the inputs of its neighbors, all of them at 0.

That said, that kind of a grid does give off the “every part affects the rest of the system” vibe that IIT says consciousness consists in, so objecting that this grid isn’t conscious doesn’t impress IIT afficionados. Moreover, such a grid is more complex than it seems at first sight, because to avoid race conditions while maintaining the ostensible state transitions a practical implementation would require some kind of a synchronization between the gates.

Today I want to note that there seems to be an even less intuitive conscious system according to IIT. Imagine a large N by N grid of binary data, “the inputs”, and then another large N by N grid of binary data, “the outputs”, aligned above the first grid. Each value on the output grid is then the logical OR of the input value under it with the four (or three for edge points and two for corner points) neighbors of that input value. And all of the input grid is at zero.

This does not give off any “every part affects the rest of the system” vibe. And getting consciousness out of zeroed OR gates with no feedback system seems really absurd.

To see the alleged consciousness, recall the IIT measure Φ of integration information, which is supposed to be be proportional to the amount of consciousness. For any partition of the components into two nonempty subsets A and B, we compute the “effective information” EI(A→B) that A provides for B. This is the entropy of the new values of the components in B given the old values of the components in A while randomizing over all possible values of the components in A. Let Φ(A,B) = EI(A→B) + EI(B→A) be the two-way effective information in the partition. Then choose A and B to minimize Φ(A,B)/min (|A|,|B|), and let the system’s Φ be Φ(A,B) for that choice of A and B. Aaronson says it’s not clear what to do if the minimum is not unique. To be conservative (i.e., count fewer systems as conscious), if there are multiple pairs that minimize Φ(A,B)/min (|A|,|B|), I’ll assume we choose one that also minimizes Φ(A,B).

Let’s now do a handwavy proof that Φ for our pair of grids when everything is at zero is at least proportional to N, and hence for large N we have consciousness according to IIT. Let’s say A and B minimize Φ(A,B)/min (|A|,|B|). Let C be the set of all inputs x such that x is connected to both an output in A and an output in B.

Suppose first that C is nonempty. Then |C| is approximately proportional to the size the boundary of the set of outputs in A, or of the set of outputs in B, which will be at least proportional to the square root of the smaller of |A| and |B|. Moreover, EI(A→B) + EI(B→A) will be at least proportional to |C| given how the dependencies are arranged and given that all the values are at zero so that if you have any unknowns amount the values that an output value depends on, then the output value is also unknown. So, Φ(A,B)/min(|A|,|B|) will be at least proportional to (min(|A|,|B|))^−1/2. Thus if A and B minimize Φ(A,B)/min (|A|,|B|), then A and B will have to be both of the same order of magnitude, namely N², since the greater the disparity, the bigger (min(|A|,|B|))^−1/2 will be. In that case, Φ(A,B)/min (|A|,|B|) will be at least proportional to 1/N, and so Φ(A,B) will be at least proportional to (1/N) ⋅ N² = N.

Now suppose C is empty. Then one of A and B contains all the outputs. Let’s say it’s A. Then B consists solely of inputs, so EI(A→B) = 0, and Φ(A,B) = EI(B→A) will be at least proportional to the size of B. Then for large N, Φ(A,B)/min (|A|,|B|) will be bigger if instead A and B contain respectively the left and right halves of the grids as then Φ(A,B) would be at most proportional to the size of their boundaries, i.e., to N, and hence Φ(A,B)/min (|A|,|B|) would be at most proportional to 1/N. So the case where C is empty cannot be a case where A and B are optimal, at least if N is large.

Integrated Information Theory doesn't seem to get integrated information right

I’m still thinking about Integrated Information Theory (IIT), in Aaronson’s simplified formulation. Aaronson’s famous criticisms show pretty convincingly that IIT fails to correctly characterize consciousness: simple but large systems of unchanging logic gates end up having human-level consciousness on IIT.

However, IIT attempts to do two things: (a) provide an account of what it is for a system to have integrated information in terms of a measure Φ, and (b) equate conscious systems with ones that have integrated information.

In this post, I want to offer some evidence that IIT fails at (a). If IIT fails at (a), then it opens up the option that notwithstanding the counterexamples, IIT gets (b) right. I am dubious of this option. For one, the family of examples in this post suggests that IIT’s account of integrated information is too restrictive, and making it less restrictive will only make it more subject to Aaronson-style counterexample. For another, I have a conclusive reason to think that IIT is false: God is conscious but has no parts, whereas IIT requires all conscious systems to have parts.

On to my argument against (a). IIT implies that a system lacks integrated information provided that it can be subdivided into two subsystems of roughly equal size such that each subsystem’s evolution over the next time step is predictable on the basis of that subsystem alone, as measured by information-theoretic entropy, i.e., only a relatively small number of additional bits of information need to be added to perfectly predict the subsystem’s evolution.

The family of systems of interest to me are what I will call “low dependency input-output (ldio) systems”. In these systems, the components can be partitioned into input components and output components. Input component values do not change. Output component values depend deterministically on the input components values. Moreover, each output component value depends only on a small number of input components. It is a little surprising that any ldio systems counts as having integrated information in light of the fact that the input components do not depend on output components, but there appear to be examples, even if details of proof have not yet been given. Aaronson is confident that low density parity check codes are an example. Another example is two large grids of equal size where the second (output) grid’s values consist of applying a step of an appropriate cellular automaton to the first (input) grid. For instance, one could put a one at the output grid provided that the neighboring points on the input grid have an odd number of ones, and otherwise put a zero.

Now suppose we have an ldio system with a high degree of integrated information as measured by IIT’s Φ measure. Then we can easily turn it into a system with a much, much lower Φ using a trick. Instead of having the system update all its outputs at once, have the system update the outputs one-by-one. To do this, add to the system a small number of binary components that hold hold an “address” for the “current” output component—say, an encoding of a pair of coordinates if the system is a grid. Then at each time step have the system update only the specific output indicated by the address, and also have the system advance the address to the address of the next output component, wrapping around to the first output component once done with all of them. We could imagine that these steps are performed really, really fast, so in the blink of an eye we have updated all the outputs—but not all at once.

This sequentialized version of the ldio is still an ldio: each output value depends on a small number of input values, plus the relatively small number of bits needed to specify the address (log₂N where N is the number of outputs). But the Φ value is apt to be immensely reduced compared to the original system. For divide up the sequentialized version into any two subsystems of roughly equal size. The outputs (if any) in each subsystem can be determined by specifying the current address (log₂N bits) plus a small number of bits for the values of the inputs that the currently addressed output depends on. Thus each subsystem has low number of bits of entropy when we randomize the values of the other subsystem, and hence the Φ measure will be low. While, say, the original system’s Φ measure is of the order N^p, the new system’s Φ measure will be at most of the order log₂N plus the maximum number of inputs that an output depends on.

But the sequentialized system will have the same time evolution as the original simultaneous-processing system as long as we look at the output of the sequentialized system after N steps, where N is the number of outputs. Intuitively, the sequentialized system has a high degree of integrated information if and only if the original system does (and is conscious if and only if the original system is).

I conclude that IIT has failed to correctly characterize integrated information.

There is a simple fix. Given a system S, there is a system S^k with the same components but each of whose steps consists in k steps of the system S. We could say that a system S has integrated information provided that there is some k such that Φ(S^k) is high. (We might even define the measure of integrated information as sup_kΦ(S^k).) I worry that this move will make it too easy to have a high degree of integrated information. Many physical systems are highly predictable over a short period of time but become highly unpredictable over a long period of time, with results being highly sensitive to small variation in most of the initial values: think of weather systems. I am fairly confident that if we fix IIT as suggested, then planetary weather systems will end up having super-human levels of consciousness.

A very simple counterexample to Integrated Information Theory?

I’ve been thinking a bit about Integerated Information Theory (IIT) as a physicalist-friendly alternative to functionalism as an account of consciousness.

The basic idea of IIT is that we measure the amount of consciousness in a system by subdividing the system into pairs of subsystems and calculating how well one can predict the next state of each of the two subsystems without knowing the state of the other. If there is a partition which lets you make the predictions well, then the system is considered reducible, with low integrated information, and hence low consciousness. So you look for the best-case subdivision—one where you can make the best predictions as measured by Shannon entropy with a certain normalization—and say that the amount Φ of “integrated information” in the system varies in reverse order with the quality of these best predictions. And then the amount of consciousness Φ in the system corresponds to the amount of integrated information.

Aaronson gives a simple mathematical framework and what sure look like counterexamples: systems that intuitively don’t appear to be mind-like and yet have a high Φ value. Surprisingly, though, Tononi (the main person behind IIT) has responded by embracing these counterexamples as cases of consciousness.

In this post, I want to offer a counterexample with a rather different structure. My counterexample has an advantage and a disadvantage with respect to Aaronson’s. The advantage is that it is a lot harder to embrace my counterexample as an example of consciousness. The disadvantage is that my example can be avoided by an easy tweak to the definition of Φ.

It is even possible that my tweak is already incorporated in the official IIT 4.0. I am right now only working with Aaronson’s perhaps simplified framework (for one, his framework depends on a deterministic transition function), because the official one is difficult for me to follow. And it is also possible that I am just missing something obvious and making some mistake. Maybe a reader will point that out to me.

The idea of my example is very simple. Imagine a system consisting of two components each of which has N possible states. At each time step, the two components swap states. There is now only one decomposition of the system into two subsystems, which makes things much simpler. And note that each subsystem’s state at time n has no predictive power for its own state at n + 1, since it inherits the other subsystem’s state at n + 1. The Shannon entropies corresponding to the best predictions are going to be log₂N, and so Φ of the system is 2log₂N. By making N arbitrarily large, we can make Φ arbitrarily large. In fact, if we have an analog system with infinitely many states, then Φ is infinite.

Advantage over Aaronson’s counterexamples: There is nothing the least consciousness-like in this setup. We are just endlessly swapping states between two components. That’s not consciousness. Imagine the components are hard drives and we just endlessly swap the data between them. To make it even more vivid, suppose the two hard drives have the same data, so nothing actually changes in the swaps!

Disadvantage: IIT can escape the problem by modifying the measure Φ of integrated information in some way in the special case where the components are non-binary. Aaronson’s counterexamples use binary components, so they are unaffected. Here are three such tweaks. (i) Just to divide by the logarithm of the maximum number of states in a component (seems ad hoc). (ii) Restrict the system to one with binary components, and therefore require that any component with more that two possible states be reinterpreted as a collection of binary components encoding the non-binary state (but which binarization should one choose?). (iii) Define Φ of a non-binary system as a minimum of the Φ values over all possible binarizations. Either (i) or (iii) kills my counterexample.