The Blog of Scott Aaronson If you take nothing else from this blog: quantum computers won't solve hard problems instantly by just trying all solutions in parallel.
Also, next pandemic, let's approve the vaccines faster!
Predictably, my last post attracted plenty of outrage (some of it too vile to let through), along with the odd commenter who actually agreed with what I consider my fairly middle-of-the-road, liberal Zionist stance. But since the outrage came from both sides of the issue, and the two sides were outraged about the opposite things, I guess I should feel OK about it.
Still, it’s hard not to smart from the burns of vituperation, so today I’d like to blog about a very different political issue: one where hopefully almost all Shtetl-Optimized readers will actually agree with me (!).
I’ve learned from colleagues that, over the past year, foreign-born scientists have been having enormously more trouble getting visas to enter the US than they used to. The problem, I’m told, is particularly severe for cryptographers: embassy clerks are now instructed to ask specifically whether computer scientists seeking to enter the US work in cryptography. If an applicant answers “yes,” it triggers a special process where the applicant hears nothing back for months, and very likely misses the workshop in the US that he or she had planned to attend. The root of the problem, it seems, is something called the Technology Alert List (TAL), which has been around for a while—the State Department beefed it up in response to the 9/11 attacks—but which, for some unknown reason, is only now being rigorously enforced. (Being marked as working in one of the sensitive fields on this list is apparently called “getting TAL’d.”)
The issue reached a comical extreme last October, when Adi Shamir, the “S” in RSA, Turing Award winner, and foreign member of the US National Academy of Sciences, was prevented from entering the US to speak at a “History of Cryptology” conference sponsored by the National Security Agency. According to Shamir’s open letter detailing the incident, not even his friends at the NSA, or the president of the NAS, were able to grease the bureaucracy at the State Department for him.
It should be obvious to everyone that a crackdown on academic cryptographers serves no national security purpose whatsoever, and if anything harms American security and economic competitiveness, by diverting scientific talent to other countries. (As Shamir delicately puts it, “the number of terrorists among the members of the US National Academy of Science is rather small.”) So:
Any readers who have more facts about what’s going on, or personal experiences, are strongly encouraged to share them in the comments section.
Any readers who might have any levers of influence to pull on this issue—a Congressperson to write to, a phone call to make, an Executive Order to issue (I’m talking to you, Barack), etc.—are strongly encouraged to pull them.
Hamas is trying to kill as many civilians as it can.
Israel is trying to kill as few civilians as it can.
Neither is succeeding very well.
Update (July 28): Please check out a superb essay by Sam Harris on the Israeli/Palestinian conflict. While, as Harris says, the essay contains “something to offend everyone”—even me—it also brilliantly articulates many of the points I’ve been trying to make in this comment thread.
See also a good HuffPost article by Ali A. Rizvi, a “Pakistani-Canadian writer, physician, and musician.”
So, the Templeton Foundation invited me to write a 1500-word essay on the above question. It’s like a blog post, except they pay me to do it! My essay is now live, here. I hope you enjoy my attempt at techno-futurist prose. You can comment on the essay either here or over at Templeton’s site. Thanks very much to Ansley Roan for commissioning the piece.
Seth Teller was a colleague of mine in CSAIL and the EECS department, and was one of my favorite people in all of MIT. He was a brilliant roboticist, who (among many other things) spearheaded MIT’s participation in the DARPA Grand Challenge for self-driving cars, and who just recently returned from a fact-finding trip to Fukushima, Japan, to see how robots could help in investigating the damaged reactor cores there. I saw Seth twice a week at lab and department lunches, and he often struck up conversations with me about quantum computing, cosmology, and other things. His curiosity was immense, wide-ranging, and almost childlike (in the best way). One small indication of his character is that, in the DARPA challenge, Seth opted not to preload MIT’s car with detailed data about the course, because he thought doing so made the challenge scientifically less interesting—even though DARPA’s rules allowed such preloading, the other teams did it, and it almost certainly would have improved MIT’s standing in the competition.
Seth was a phenomenal speaker, whose passion and clarity always won me over even though my research interests were far from his. I made it a point to show up for lab lunch whenever I knew he’d be speaking. Seth was also, from what I’ve heard, a superb mentor and teacher, who won an award earlier this year for his undergraduate advising.
Seth died ten days ago, on July 1st. (See here for MIT News’s detailed obituary, and here for an article in Cambridge Day.) While no cause of death was given at the time, according to an update yesterday in the MIT Tech, the death has been ruled a suicide. Seth is survived by his wife, Rachel, and by two daughters.
With his cheerful, can-do disposition, Seth is one of the last people on earth I’d imagine doing this: whatever he was going through, he did an unbelievable job of hiding it. I’m certain he wouldn’t abandon his family unless he was suffering unimaginable pain. If there’s a tiny atom of good to come out of this, I hope that at least one other person contemplating suicide will reflect on how much Seth had to live for, and that doing so will inspire that person to get the medical help they need.
Incidentally, outside of his research and teaching, Seth was also an activist for protecting the physical environment and open spaces of East Cambridge. At the “Wild and Crazy Ideas Session” of one CSAIL retreat, Seth floated a truly wild idea: to replace Memorial Drive, or at least the part of it that separates the MIT campus from the Charles River, by an underground tunnel, so that the land above the tunnel could be turned into a beautiful riverfront park. In his characteristic fashion, Seth had already done a pretty detailed engineering analysis, estimating the cost at “merely” a few hundred million dollars: a lot, but a worthy investment in MIT’s future. In any case, I can’t imagine a better way to memorialize Seth than to name some green space in East Cambridge after him, and I hope that happens.
Seth will be sorely missed. My thoughts go out to his family at this difficult time.
Foreword: Right now, I have a painfully-large stack of unwritten research papers. Many of these are “paperlets”: cool things I noticed that I want to tell people about, but that would require a lot more development before they became competitive for any major theoretical computer science conference. And what with the baby, I simply don’t have time anymore for the kind of obsessive, single-minded, all-nighter-filled effort needed to bulk my paperlets up. So starting today, I’m going to try turning some of my paperlets into blog posts. I don’t mean advertisements or sneak previews for papers, but replacements for papers: blog posts that constitute the entirety of what I have to say for now about some research topic. “Peer reviewing” (whether signed or anonymous) can take place in the comments section, and “citation” can be done by URL. The hope is that, much like with 17th-century scientists who communicated results by letter, this will make it easier to get my paperlets done: after all, I’m not writing Official Papers, just blogging for colleagues and friends.
Of course, I’ve often basically done this before—as have many other academic bloggers—but now I’m going to go about it more consciously. I’ve thought for years that the Internet would eventually change the norms of scientific publication much more radically than it so far has: that yes, instant-feedback tools like blogs and StackExchange and MathOverflow might have another decade or two at the periphery of progress, but their eventual destiny is at the center. And now that I have tenure, it hit me that I can do more than prognosticate about such things. I’ll start small: I won’t go direct-to-blog for big papers, papers that cry out for LaTeX formatting, or joint papers. I certainly won’t do it for papers with students who need official publications for their professional advancement. But for things like today’s post—on the power of a wooden mechanical computer now installed in the lobby of the building where I work—I hope you agree that the Science-by-Blog Plan fits well.
Oh, by the way, happy July 4th to American readers! I hope you find that a paperlet about the logspace-interreducibility of a few not-very-well-known computational models captures everything that the holiday is about.
The Power of the Digi-Comp II
by Scott Aaronson
Abstract
I study the Digi-Comp II, a wooden mechanical computer whose only moving parts are balls, switches, and toggles. I show that the problem of simulating (a natural abstraction of) the Digi-Comp, with a polynomial number of balls, is complete for CC (Comparator Circuit), a complexity class defined by Subramanian in 1990 that sits between NL and P. This explains why the Digi-Comp is capable of addition, multiplication, division, and other arithmetical tasks, and also implies new tasks of which the Digi-Comp is capable (and that indeed are complete for it), including the Stable Marriage Problem, finding a lexicographically-first perfect matching, and the simulation of other Digi-Comps. However, it also suggests that the Digi-Comp is not a universal computer (not even in the circuit sense), making it a very interesting way to fall short of Turing-universality. I observe that even with an exponential number of balls, simulating the Digi-Comp remains in P, but I leave open the problem of pinning down its complexity more precisely.
Introduction
To celebrate his 60th birthday, my colleague Charles Leiserson (who some of you might know as the “L” in the CLRS algorithms textbook) had a striking contraption installed in the lobby of the MIT Stata Center. That contraption, pictured below, is a custom-built, supersized version of a wooden mechanical computer from the 1960s called the Digi-Comp II, now manufactured and sold by a company called Evil Mad Scientist.
Click here for a short video showing the Digi-Comp’s operation (and here for the user’s manual). Basically, the way it works is this: a bunch of balls (little steel balls in the original version, pool balls in the supersized version) start at the top and roll to the bottom, one by one. On their way down, the balls may encounter black toggles, which route each incoming ball either left or right. Whenever this happens, the weight of the ball flips the toggle to the opposite setting: so for example, if a ball goes left, then the next ball to encounter the same toggle will go right, and the ball after that will go left, and so on. The toggles thus maintain a “state” for the computer, with each toggle storing one bit.
Besides the toggles, there are also “switches,” which the user can set at the beginning to route every incoming ball either left or right, and whose settings aren’t changed by the balls. And then there are various wooden tunnels and ledges, whose function is simply to direct the balls in a desired way as they roll down. A ball could reach different locations, or even the same location in different ways, depending on the settings of the toggles and switches above that location. On the other hand, once we fix the toggles and switches, a ball’s motion is completely determined: there’s no random or chaotic element.
“Programming” is done by configuring the toggles and switches in some desired way, then loading a desired number of balls at the top and letting them go. “Reading the output” can be done by looking at the final configuration of some subset of the toggles.
Whenever a ball reaches the bottom, it hits a lever that causes the next ball to be released from the top. This ensures that the balls go through the device one at a time, rather than all at once. As we’ll see, however, this is mainly for aesthetic reasons, and maybe also for the mechanical reason that the toggles wouldn’t work properly if two or more balls hit them at once. The actual logic of the machine doesn’t care about the timing of the balls; the sheer number of balls that go through is all that matters.
The Digi-Comp II, as sold, contains a few other features: most notably, toggles that can be controlled by other toggles (or switches). But I’ll defer discussion of that feature to later. As we’ll see, we already get a quite interesting model of computation without it.
One final note: of course the machine that’s sold has a fixed size and a fixed geometry. But for theoretical purposes, it’s much more interesting to consider an arbitrary network of toggles and switches (not necessarily even planar!), with arbitrary size, and with an arbitrary number of balls fed into it. (I’ll give a more formal definition in the next section.)
The Power of the Digi-Comp
So, what exactly can the Digi-Comp do? As a first exercise, you should convince yourself that, by simply putting a bunch of toggles in a line and initializing them all to “L” (that is, Left), it’s easy to set up a binary counter. In other words, starting from the configuration, say, LLL (in which three toggles all point left), as successive balls pass through we can enter the configurations RLL, LRL, RRL, etc. If we interpret L as 0 and R as 1, and treat the first bit as the least significant, then we’re simply counting from 0 to 7 in binary. With 20 toggles, we could instead count to 1,048,575.
But counting is not the most interesting thing we can do. As Charles eagerly demonstrated to me, we can also set up the Digi-Comp to perform binary addition, binary multiplication, sorting, and even long division. (Excruciatinglyslowly, of course: the Digi-Comp might need even more work to multiply 3×5, than existing quantum computers need to factor the result!)
To me, these demonstrations served only as proof that, while Charles might call himself a theoretical computer scientist, he’s really a practical person at heart. Why? Because a theorist would know that the real question is not what the Digi-Comp can do, but rather what it can’t do! In particular, do we have a universal computer on our hands here, or not?
If the answer is yes, then it’s amazing that such a simple contraption of balls and toggles could already take us over the threshold of universality. Universality would immediately explain why the Digi-Comp is capable of multiplication, division, sorting, and so on. If, on the other hand, we don’t have universality, that too is extremely interesting—for we’d then face the challenge of explaining how the Digi-Comp can do so many things without being universal.
It might be said that the Digi-Comp is certainly not a universal computer, since if nothing else, it’s incapable of infinite loops. Indeed, the number of steps that a given Digi-Comp can execute is bounded by the number of balls, while the number of bits it can store is bounded by the number of toggles: clearly we don’t have a Turing machine. This is true, but doesn’t really tell us what we want to know. For, as discussed in my last post, we can consider not only Turing-machine universality, but also the weaker (but still interesting) notion of circuit-universality. The latter means the ability to simulate, with reasonable efficiency, any Boolean circuit of AND, OR, and NOT gates—and hence, in particular, to compute any Boolean function on any fixed number of input bits (given enough resources), or to simulate any polynomial-time Turing machine (given polynomial resources).
The formal way to ask whether something is circuit-universal, is to ask whether the problem of simulating the thing is P-complete. Here P-complete (not to be confused with NP-complete!) basically means the following:
There exists a polynomial p such that any S-step Turing machine computation—or equivalently, any Boolean circuit with at most S gates—can be embedded into our system if we allow the use of poly(S) computing elements (in our case, balls, toggles, and switches).
Of course, I need to tell you what I mean by the weasel phrase “can be embedded into.” After all, it wouldn’t be too impressive if the Digi-Comp could “solve” linear programming, primality testing, or other highly-nontrivial problems, but only via “embeddings” in which we had to do essentially all the work, just to decide how to configure the toggles and switches! The standard way to handle this issue is to demand that the embedding be “computationally simple”: that is, we should be able to carry out the embedding in L (logarithmic space), or some other complexity class believed to be much smaller than the class (P, in this case) for which we’re trying to prove completeness. That way, we’ll be able to say that our device really was “doing something essential”—i.e., something that our embedding procedure couldn’t efficiently do for itself—unless the larger complexity class collapses with the smaller one (i.e., unless L=P).
So then, our question is whether simulating the Digi-Comp II is a P-complete problem under L-reductions, or alternatively, whether the problem is in some complexity class believed to be smaller than P. The one last thing we need is a formal definition of “the problem of simulating the Digi-Comp II.” Thus, let DIGICOMP be the following problem:
We’re given as inputs:
A directed acyclic graph G, with n vertices. There is a designated vertex with indegree 0 and outdegree 1 called the “source,” and a designated vertex with indegree 1 and outdegree 0 called the “sink.” Every internal vertex v (that is, every vertex with both incoming and outgoing edges) has exactly two outgoing edges, labeled “L” (left) and “R” (right), as well as one bit of internal state s(v)∈{L,R}.
For each vertex v, an “initial” value for its internal state s(v).
A positive integer T (encoded in unary notation), representing the number of balls dropped successively from the source vertex.
Computation proceeds as follows: each time a ball appears at the source vertex, it traverses the path induced by the L and R states of the vertices that it encounters, until it reaches a terminal vertex, which might or might not be the sink. As the ball traverses the path, it flips s(v) for each vertex v that it encounters: L goes to R and R goes to L. Then the next ball is dropped in.
The problem is to decide whether any balls reach the sink.
Here the internal vertices represent toggles, and the source represents the chute at the top from which the balls drop. Switches aren’t included, since (by definition) the reduction can simply fix their values to “L” to “R” and thereby simplify the graph.
Of course we could consider other problems: for example, the problem of deciding whether an odd number of balls reach the sink, or of counting how many balls reach the sink, or of computing the final value of every state-variable s(v). However, it’s not hard to show that all of these problems are interreducible with the DIGICOMP problem as defined above.
The Class CC
My main result, in this paperlet, is to pin down the complexity of the DIGICOMP problem in terms of a complexity class called CC (Comparator Circuit): a class that’s obscure enough not to be in the Complexity Zoo (!), but that’s been studied in several papers. CC was defined by Subramanian in his 1990 Stanford PhD thesis; around the same time Mayr and Subramanian showed the inclusion NL ⊆ CC (the inclusion CC ⊆ P is immediate). Recently Cook, Filmus, and Lê revived interest in CC with their paper The Complexity of the Comparator Circuit Value Problem, which is probably the best current source of information about this class.
OK, so what isCC? Informally, it’s the class of problems that you can solve using a comparator circuit, which is a circuit that maps n bits of input to n bits of output, and whose only allowed operation is to sort any desired pair of bits. That is, a comparator circuit can repeatedly apply the transformation (x,y)→(x∧y,x∨y), in which 00, 01, and 11 all get mapped to themselves, while 10 gets mapped to 01. Note that there’s no facility in the circuit for copying bits (i.e., for fanout), so sorting could irreversibly destroy information about the input. In the comparator circuit value problem (or CCV), we’re given as input a description of a comparator circuit C, along with an input x∈{0,1}n and an index i∈[n]; then the problem is to determine the final value of the ith bit when C is applied to x. Then CC is simply the class of all languages that are L-reducible to CCV.
As Cook et al. discuss, there are various other characterizations of CC: for example, rather than using a complete problem, we can define CC directly as the class of languages computed by uniform families of comparator circuits. More strikingly, Mayr and Subramanian showed that CC has natural complete problems, which include (decision versions of) the famous Stable Marriage Problem, as well as finding the lexicographically first perfect matching in a bipartite graph. So perhaps the most appealing definition of CC is that it’s “the class of problems that can be easily mapped to the Stable Marriage Problem.”
It’s a wide-open problem whether CC=NL or CC=P: as usual, one can give oracle separations, but as far as anyone knows, either equality could hold without any dramatic implications for “standard” complexity classes. (Of course, the conjunction of these equalities would have a dramatic implication.) What got Cook et al. interested was that CC isn’t even known to contain (or be contained in) the class NC of parallelizable problems. In particular, linear-algebra problems in NC, like determinant, matrix inversion, and iterated matrix multiplication—not to mention other problems in P, like linear programming and greatest common divisor—might all be examples of problems that are efficiently solvable by Boolean circuits, but not by comparator circuits.
One final note about CC. Cook et al. showed the existence of a universal comparator circuit: that is, a single comparator circuit C able to simulate any other comparator circuit C’ of some fixed size, given a description of C’ as part of its input.
DIGICOMP is CC-Complete
I can now proceed to my result: that, rather surprisingly, the Digi-Comp II can solve exactly the problems in CC, giving us another characterization of that class.
I’ll prove this using yet another model of computation, which I call the pebbles model. In the pebbles model, you start out with a pile of x pebbles; the positive integer x is the “input” to your computation. Then you’re allowed to apply a straight-line program that consists entirely of the following two operations:
Given any pile of y pebbles, you can split it into two piles consisting of ⌈y/2⌉ and ⌊y/2⌋ pebbles respectively.
Given any two piles, consisting of y and z pebbles respectively, you can combine them into a single pile consisting of y+z pebbles.
Your program “accepts” if and only if some designated output pile contains at least one pebble (or, in a variant that can be shown to be equivalent, if it contains an odd number of pebbles).
As suggested by the imagery, you don’t get to make “backup copies” of the piles before splitting or combining them: if, for example, you merge y with z to create y+z, then y isn’t also available to be split into ⌈y/2⌉ and ⌊y/2⌋.
Note that the ceiling and floor functions are the only “nonlinear” elements of the pebbles model: if not for them, we’d simply be applying a sequence of linear transformations.
I can now divide my CC-completeness proof into two parts: first, that DIGICOMP (i.e., the problem of simulating the Digi-Comp II) is equivalent to the pebbles model, and second, that the pebbles model is equivalent to comparator circuits.
Let’s first show the equivalence between DIGICOMP and pebbles. The reduction is simply this: in a given Digi-Comp, each edge will be associated to a pile, with the number of pebbles in the pile equal to the total number of balls that ever traverse that edge. Thus, we have T balls dropped in to the edge incident to the source vertex, corresponding to an initial pile with T pebbles. Multiple edges pointing to the same vertex (i.e., fan-in) can be modeled by combining the associated piles into a single pile. Meanwhile, a toggle has the effect of splitting a pile: if y balls enter the toggle in total, then ⌈y/2⌉ balls will ultimately exit in whichever direction the toggle was pointing initially (whether left or right), and ⌊y/2⌋ balls will ultimately exit in the other direction. It’s clear that this equivalence works in both directions: not only does it let us simulate any given Digi-Comp by a pebble program, it also lets us simulate any pebble program by a suitably-designed Digi-Comp.
OK, next let’s show the equivalence between pebbles and comparator circuits. As a first step, given any comparator circuit, I claim that we can simulate it by a pebble program. The way to do it is simply to use a pile of 0 pebbles to represent each “0” bit, and a pile of 1 pebble to represent each “1” bit. Then, any time we want to sort two bits, we simply merge their corresponding piles, then split the result back into two piles. The result? 00 gets mapped to 00, 11 gets mapped to 11, and 01 and 10 both get mapped to one pebble in the ⌈y/2⌉ pile and zero pebbles in the ⌊y/2⌋ pile. At the end, a given pile will have a pebble in it if and only if the corresponding output bit in the comparator circuit is 1.
One might worry that the input to a comparator circuit is a sequence of bits, whereas I said before that the input to a pebble program is just a single pile. However, it’s not hard to see that we can deal with this, without leaving the world of logspace reductions, by breaking up an initial pile of n pebbles into n piles each of zero pebbles or one pebble, corresponding to any desired n-bit string (along with some extra pebbles, which we subsequently ignore). Alternatively, we could generalize the pebbles model so that the input can consist of multiple piles. One can show, by a similar “breaking-up” trick, that this wouldn’t affect the pebbles model’s equivalence to the DIGICOMP problem.
Finally, given a pebble program, I need to show how to simulate it by a comparator circuit. The reduction works as follows: let T be the number of pebbles we’re dealing with (or even just an upper bound on that number). Then each pile will be represented by its own group of T wires in the comparator circuit. The Hamming weight of those T wires—i.e., the number of them that contain a ‘1’ bit—will equal the number of pebbles in the corresponding pile.
To merge two piles, we first merge the corresponding groups of T wires. We then use comparator gates to sort the bits in those 2T wires, until all the ‘1’ bits have been moved into the first T wires. Finally, we ignore the remaining T wires for the remainder of the computation.
To split a pile, we first use comparator gates to sort the bits in the T wires, until all the ‘1’ bits have been moved to the left. We then route all the odd-numbered wires into “Pile A” (the one that’s supposed to get ⌈y/2⌉ pebbles), and route all the even-numbered wires into “Pile B” (the one that’s supposed to get ⌊y/2⌋ pebbles). Finally, we introduce T additional wires with 0’s in them, so that piles A and B have T wires each.
At the end, by examining the leftmost wire in the group of wires corresponding to the output pile, we can decide whether that pile ends up with any pebbles in it.
Since it’s clear that all of the above transformations can be carried out in logspace (or even smaller complexity classes), this completes the proof that DIGICOMP is CC-complete under L-reductions. As corollaries, the Stable Marriage and lexicographically-first perfect matching problems are L-reducible to DIGICOMP—or informally, are solvable by easily-described, polynomial-size Digi-Comp machines (and indeed, characterize the power of such machines). Combining my result with the universality result of Cook et al., a second corollary is that there exists a “universal Digi-Comp”: that is, a single Digi-Comp D that can simulate any other Digi-Comp D’ of some polynomially-smaller size, so long as we initialize some subset of the toggles in D to encode a description of D’.
How Does the Digi-Comp Avoid Universality?
Let’s now step back and ask: given that the Digi-Comp is able to do so many things—division, Stable Marriage, bipartite matching—how does it fail to be a universal computer, at least a circuit-universal one? Is the Digi-Comp a counterexample to the oft-repeated claims of people like Stephen Wolfram, about the ubiquity of universal computation and the difficulty of avoiding it in any sufficiently complex system? What would need to be added to the Digi-Comp to make it circuit-universal? Of course, we can ask the same questions about pebble programs and comparator circuits, now that we know that they’re all computationally equivalent.
The reason for the failure of universality is perhaps easiest to see in the case of comparator circuits. As Steve Cook pointed out in a talk, comparator circuits are “1-Lipschitz“: that is, if you have a comparator circuit acting on n input bits, and you change one of the input bits, your change can affect at most one output bit. Why? Well, trace through the circuit and use induction. So in particular, there’s no amplification of small effects in comparator circuits, no chaos, no sensitive dependence on initial conditions, no whatever you want to call it. Now, while chaos doesn’t suffice for computational universality, at least naïvely it’s a necessary condition, since there exist computations that are chaotic. Of course, this simpleminded argument can’t be all there is to it, since otherwise we would’ve proved CC≠P. What the argument does show is that, if CC=P, then the encoding of a Boolean circuit into a comparator circuit (or maybe into a collection of such circuits) would need to be subtle and non-obvious: it would need to take computations with the potential for chaos, and reduce them to computations without that potential.
Once we understand this 1-Lipschitz business, we can also see it at work in the pebbles model. Given a pebble program, suppose someone surreptitiously removed a single pebble from one of the initial piles. For want of that pebble, could the whole kingdom be lost? Not really. Indeed, you can convince yourself that the output will be exactly the same as before, except that one output pile will have one fewer pebble than it would have otherwise. The reason is again an induction: if you change x by 1, that affects at most one of ⌈x/2⌉ and ⌊x/2⌋ (and likewise, merging two piles affects at most one pile).
We now see the importance of the point I made earlier, about there being no facility in the piles model for “copying” a pile. If we could copy piles, then the 1-Lipschitz property would fail. And indeed, it’s not hard to show that in that case, we could implement AND, OR, and NOT gates with arbitrary fanout, and would therefore have a circuit-universal computer. Likewise, if we could copy bits, then comparator circuits—which, recall, map (x,y) to (x∧y,x∨y)—would implement AND, OR, and NOT with arbitrary fanout, and would be circuit-universal. (If you’re wondering how to implement NOT: one way to do it is to use what’s known in quantum computing as the “dual-rail representation,” where each bit b is encoded by two bits, one for b and the other for ¬b. Then a NOT can be accomplished simply by swapping those bits. And it’s not hard to check that comparator gates in a comparator circuit, and combining and splitting two piles in a pebble program, can achieve the desired updates to both the b rails and the ¬b rails when an AND or OR gate is applied. However, we could also just omit NOT gates entirely, and use the fact that computing the output of even a monotone Boolean circuit is a P-complete problem under L-reductions.)
In summary, then, the inability to amplify small effects seems like an excellent candidate for the central reason why the power of comparator circuits and pebble programs hits a ceiling at CC, and doesn’t go all the way up to P. It’s interesting, in this connection, that while transistors (and before them, vacuum tubes) can be used to construct logic gates, the original purpose of both of them was simply to amplify information: to transform a small signal into a large one. Thus, we might say, comparator circuits and pebble programs fail to be computationally universal because they lack transistors or other amplifiers.
I’d like to apply exactly the same analysis to the Digi-Comp itself: that is, I’d like to say that the reason the Digi-Comp fails to be universal (unless CC=P) is that it, too, lacks the ability to amplify small effects (wherein, for example, the drop of a single ball would unleash a cascade of other balls). In correspondence, however, David Deutsch pointed out a problem: namely, if we just watch a Digi-Comp in action, then it certainly looks like it has an amplification capability! Consider, for example, the binary counter discussed earlier. Suppose a column of ten toggles is in the configuration RRRRRRRRRR, representing the integer 1023. Then the next ball to fall down will hit all ten toggles in sequence, resetting them to LLLLLLLLLL (and thus, resetting the counter to 0). Why isn’t this precisely the amplification of a small effect that we were looking for?
Well, maybe it’s amplification, but it’s not of a kind that does what we want computationally. One way to see the difficulty is that we can’t take all those “L” settings we’ve produced as output, and feed them as inputs to further gates in an arbitrary way. We could do it if the toggles were arranged in parallel, but they’re arranged serially, so that flipping any one toggle inevitably has the potential also to flip the toggles below it. Deutsch describes this as a “failure of composition”: in some sense, we do have a fan-out or copying operation, but the design of the Digi-Comp prevents us from composing the fan-out operation with other operations in arbitrary ways, and in particular, in the ways that would be needed to simulate any Boolean circuit.
So, what features could we add to the Digi-Comp to make it universal? Here’s the simplest possibility I was able to come up with: suppose that, scattered throughout the device, there were balls precariously perched on ledges, in such a way that whenever one was hit by another ball, it would get dislodged, and both balls would continue downward. We could, of course, chain several of these together, so that the two balls would in turn dislodge four balls, the four would dislodge eight, and so on. I invite you to check that this would provide the desired fan-out gate, which, when combined with AND, OR, and NOT gates that we know how to implement (e.g., in the dual-rail representation described previously), would allow us to simulate arbitrary Boolean circuits. In effect, the precariously perched balls would function as “transistors” (of course, painfully slow transistors, and ones that have to be laboriously reloaded with a ball after every use).
As a second possibility, Charles Leiserson points out to me that the Digi-Comp, as sold, has a few switches and toggles that can be controlled by other toggles. Depending on exactly how one modeled this feature, it’s possible that it, too, could let us implement arbitrary fan-out gates, and thereby boost the Digi-Comp up to circuit-universality.
Open Problems
My personal favorite open problem is this:
What is the complexity of simulating a Digi-Comp II if the total number of balls dropped in is exponential, rather than polynomial? (In other words, if the positive integer T, representing the number of balls, is encoded in binary rather than in unary?)
From the equivalence between the Digi-Comp and pebble programs, we can already derive a conclusion about the above problem that’s not intuitively obvious: namely, that it’s in P. Or to say it another way: it’s possible to predict the exact state of a Digi-Comp with n toggles, after T balls have passed through it, using poly(n, log T) computation steps. The reason is simply that, if there are T balls, then the total number of balls that pass through any given edge (the only variable we need to track) can be specified using log2T bits. This, incidentally, gives us a second sense in which the Digi-Comp is not a universal computer: namely, even if we let the machine “run for exponential time” (that is, drop exponentially many balls into it), unlike a conventional digital computer it can’t possibly solve all problems in PSPACE, unless P=PSPACE.
However, this situation also presents us with a puzzle: if we let DIGICOMPEXP be the problem of simulating a Digi-Comp with an exponential number of balls, then it’s clear that DIGICOMPEXP is hard for CC and contained in P, but we lack any information about its difficulty more precise than that. At present, I regard both extremes—that DIGICOMPEXP is in CC (and hence, no harder than ordinary DIGICOMP), and that it’s P-complete—as within the realm of possibility (along with the possibility that DIGICOMPEXP is intermediate between the two).
By analogy, one can also consider comparator circuits where the entities that get compared are integers from 1 to T rather than bits—and one can then consider the power of such circuits, when T is allowed to grow exponentially. In email correspondence, however, Steve Cook sent me a proof that such circuits have the same power as standard, Boolean comparator circuits. It’s not clear whether this tells us anything about the power of a Digi-Comp with exponentially many balls.
A second open problem is to formalize the feature of Digi-Comp that Charles mentioned—namely, toggles and switches controlled by other toggles—and see whether, under some reasonable formalization, that feature bumps us up to P-completeness (i.e., to circuit-universality). Personally, though, I confess I’d be even more interested if there were some feature we could add to the machine that gave us a larger class than CC, but that still wasn’t all of P.
A third problem is to pin down the power of Digi-Comps (or pebble programs, or comparator circuits) that are required to be planar. While my experience with woodcarving is limited, I imagine that planar or near-planar graphs are a lot easier to carve than arbitrary graphs (even if the latter present no problems of principle).
A fourth problem has to do with the class CC in general, rather than the Digi-Comp in particular, but I can’t resist mentioning it. Let CCEXP be the complexity class that’s just like CC, but where the comparator circuit (or pebble program, or Digi-Comp) is exponentially large and specified only implicitly (that is, by a Boolean circuit that, given as input a binary encoding of an integer i, tells you the ith bit of the comparator circuit’s description). Then it’s easy to see that PSPACE ⊆ CCEXP ⊆ EXP. Do we have CCEXP = PSPACE or CCEXP = EXP? If not, then CCEXP would be the first example I’ve ever seen of a natural complexity class intermediate between PSPACE and EXP.
Acknowledgments
I thank Charles Leiserson for bringing the Digi-Comp II to MIT, and thereby inspiring this “research.” I also thank Steve Cook, both for giving a talk that first brought the complexity class CC to my attention, and for helpful correspondence. Finally I thank David Deutsch for the point about composition.
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