Distributed computing with alien technology

Introduction

Suppose we are given a function of boolean arguments that returns a boolean result. Alice has bits, and Bob has another bits . Alice and Bob are widely separated and don't know each other's bits. What is the total number of bits that Alice has to send to Bob and that Bob has to send to Alice so that between them they can compute ? Think about how complex might get. The and might each describe half of a "voxelised" region of space and might answer a question about a computational fluid dynamics (CFD) simulation running in that space. CFD simulations can be chaotic and so we might expect that in the worst case many bits have to be transferred back and forth between Alice and Bob. In the worst case we might expect that Alice has to send Bob all of her bits, or vice versa.

But in fact Alice needs to send Bob just one bit.

A loophole

To get the communication requirements down to one bit we need to use a loophole. But I hope to (1) justify the cheat to some extent and (2) justify that it's even worthwhile to think about cheats.

Alice and Bob have access to some Ancient technology. They each have one of a pair of boxes. At prearranged times, Alice puts a bit into her box, and Bob puts a bit into his box. A bit pops back out of Alice's box and a bit pops back out of Bob's box. Whatever the input, both Alice and Box have a 0.5 chance of seeing a one or zero pop out of their respective boxes. But when the two outputs are XORed together the result is the logical AND of the two inputs. With such boxes, Alice can compute after Bob sends a single bit down a conventional communication channel.

"But this is a total cheat!" you complain before I even start to explain their technique. It seems Alice receives a bit that depends on what Bob input, and so Bob is communicating with Alice. But look closely and you'll see that the boxes don't allow any communication. No matter what Bob inputs, Alice has a 0.5 chance of getting zero or one. There is no way Bob can use this to communicate anything. It's like intercepting a message encrypted with a one time pad. Without the pad, the message is basically a sequence of random bits. Nonetheless, it is true that the outputs that Alice and Bob see are correlated.

I hope I've convinced you that Alice and Bob can't send any bits with these boxes. Despite this, it is pretty clear that the behaviour of the boxes is non-local. We'll call any kind of boxes that allow instantaneous long range correlations that can't be explained by purely local behaviour non-local boxes. Boxes that can't be used for message sending are called non-signalling local boxes. And the particular non-local box I describe above is called a PR box (eg. see here).

(BTW As an aside note that as the box results in widely separated outputs that are correlated, but doesn't allow communication, it's an example of how non-locality doesn't imply communication. Usually when people want to give examples of such a thing they talk about quantum mechanics. But there's no need to mention quantum mechanics to explain the behaviour of these particular non-local boxes.)

The method

Any single bit boolean function of a finite sequence of bits can be written as a polynomial modulo 2. Each monomial in the polynomial can be written as a product of terms involing just the and terms involving just the , ie.

where depends only on the , depends only on the and is drawn from some finite set. Alice can compute the and Bob can compute the . Now Alice and Bob, in parallel, feed and respectively into their PR boxes. We know that we could evaluate each term in the sum we want by adding Alice's output to Bob's output. But that would require sending one one-bit message for each . But we don't need each term one by one; we just want the sum. So Alice and Bob can individually sum their separate outputs knowing that adding Alice's output and Bob's output modulo 2 will be the correct sum. So Bob sends his sum to Alice. Alice adds that number to her own (modulo 2) and that's the value we want. Only one one-bit message was sent.

But what about reality?

Non-local boxes don't exist, do they? So why are we talking about them?

Actually, non-local boxes exist both theoretically and in the lab. Non-local correlations in quantum mechanics allow them to be constructed. But for this article I wanted to abstract from quantum mechanics and talk about the behaviour of a non-local box without getting my hands dirty with the details of quantum mechanics. Having said that, although non-local boxes do exist, the special case of the PR box can't in fact be constructed with quantum mechanics. In some sense it allows correlations that are "too strong". An article I wrote a while back describes the closest you can get to building a PR box with quantum correlations. Curiously, if you restrict yourself to the kind of non-local box quantum mechanics allows you to build you find that some functions can still be computed with less communication than you'd need if non-local correlations are disallowed. Nonetheless, the worst case scenario with QM still requires the sending of bits.

Going further there's an interesting conjecture. It says that any non-local box that is even marginally better (in some sense) than what quantum mechanics allows is powerful enough to allow the computation of any with only a single bit of communication. It suggests that quantum mechanics is right at the edge of the space of possible physics that make life difficult for us. If quantum mechanics were to be tweaked the tiniest amount to make correlations any stronger, large numbers of difficult distributed computing problems would suddenly collapse to become trivial. If the conjecture is true it means that nature looks a bit like a conspiracy to keep computer scientists in work. (It's possible the conjecture has been decided one way or the other by now.)

Final words

There are a couple of papers about universes where PR boxes can be built; so called boxworlds. There is a lot of interesting theoretical work in characterising quantum mechanics. In particular there are a number of theorems and conjectures that describe QM in the form "the most X theory that doesn't allow Y" where X is an interesting property and Y is something you'd like to do.

References

I learnt all of this from the paper Implausible Consequences of Superstrong Nonlocality by Wim van Dam.

What stops us defining Truth?

Introduction
Recall the standard cartoon sketch of the proof of Gödel's first incompleteness theorem. We start by defining a predicate, , that is true if and only if its argument is provable. (Or more accurately, is true if is the Gödel number of a provable proposition.) With some quining we can use this to construct the proposition which says . The proposition asserts its own unprovability.
Suppose instead we define a predicate which holds if its argument is true. We can use this to construct the proposition which says . Then if is true it must also be false and if it's false then it must be true. We seem to have a paradox. The loophole is that we assumed the existence of the predicate . So this argument demonstrates that there is actually no such predicate. This is Tarski's undefinability theorem.
But what exactly stops us defining ? What goes wrong if we attempt to define a predicate that analyses the parts of a proposition to tell us whether or not it is true?

Note
This article is written in English. But as is standard in much of mathematics, unless I state otherwise, I'm using English largely as shorthand for an argument that could, in principle, be written in the formal language of Set Theory. So I will allow myself to use all of the usual reasoning methods that are available in ZF, even when talking about other formal systems such as Peano Arithmetic.

Defining Truth for Propositional Calculus
Suppose we're given a proposition from propositional calculus like . We can use a syntactic approach to determining whether or not it is true. We determine whether or not is true, then whether or not is true, and then the whole proposition is true if both and are true. Similarly is true if either or is true. Of course and might themselves be compound propositions using , and . But that's fine, that simply means that to define truth for such propositions we need to employ recursion. In fact, we can straightforwardly turn such a definition into a recursive computer program.
(Ultimately with propositional calculus we hit the leaves which are atomic propositions like . Typically when we ask about the truth of a proposition in propositional calculus we've already made an assignment of truth values to the atomic propositions. So the base case for the recursion is straightforward.)
We can illustrate the process with a diagram:

The truth value of a node in the tree is determined by the truth of the propositions hanging underneath it. We have a parent-child relation between a proposition and its subexpressions. Recursion allows us to make a definition by defining what happens on the leaves of such a tree, and by saying how the definition at a node is built from that of its children.

Defining truth for Peano Arithmetic
We can go further and attempt this approach with Peano Arithmetic (PA). The catch is that we need to consider quantifiers. For example, consider this proposition from Peano arithmetic: . This proposition is true if and only if is true whatever number we substitute for in the expression.

The proposition at the top of the tree above is true if all of the immediate children are true and their truth is in turn determined by the truth of the propositions immediately below them. With some work this eventually leads to a perfectly good definition of truth for propositions in PA. Because we have nodes with infinitely many children we don't get an algorithm guaranteed to terminate, but that's not a problem for a definition in ZF. Note that we don't literally prove the infinitely many child propositions one at a time. Instead what happens is that to define the truth of we define it in terms of the truth of some infinite family of propositions all based on . ZF is perfectly good at dealing with such definitions without us having to list every element of our family explicitly.
Note how in this case the tree isn't the parse tree of the proposition. It's much bigger with nodes that have infinite branching. But that's fine, there's nothing about infinite branching that prevents us making a recursive definition. So we can ultimately extend the idea for defining truth in propositional calculus to include quanifiers and then all of Peano arithmetic.

Defining truth for ZF
But the approach used for PA looks like it might work perfectly well for ZF as well. For example, our definition of truth would say that is true if is true whatever set we substitute for . In ZF there is no difficulty in defining a predicate that uses quantification over all sets. So it seems we can define for ZF in ZF, contradicting Tarski's theorem.

What went wrong?
Recursive definitions typically rely on the parent-child relation I mentioned above. To recursively define something we (1) define it for all leaves and then (2) specify how the definition at a parent is given in terms of the value for all of its children. We then invoke a recursion theorem of some sort to show how this uniquely defines our object for everything in our universe. For example, one form of recursion in Peano arithmetic has as its leaf and the only child of is . The induction axiom for PA can be used to show that definitions using this parent-child relation are valid.
Similarly in ZF we have the empty set as leaf and the children of a set are simply its elements. But now we need to look closely at the recursion principle we need. For ZF we need to invoke the Transfinite Recursion Theorem. Transfinite recursion is very powerful. It's not just limited to induction over sets. It can also be used for induction over classes. For example if you need to recursively define a function on the class of all sets it can allow this. (Strictly speaking it'll be a function class rather than a function.) But now comes the catch. If you take a look at the Wikipedia article it mentions that the parent-child relation, , needs to be set-like (though as the article is currently written it's almost an afterthought). For this theorem to apply we need the collection of children of a proposition to form a set. But to prove the truth of a proposition with a quantifier at the front we need to prove something is true for all children where there is one child for each set. This means the children don't form a set. So we can't use transfinite recursion. And this means the informal definition of truth I gave above can't be turned into a rigorous definition.

Conclusion
I think this issue is quite subtle. It's really easy to say in English "this thing is true if that thing is true for all sets". Such a sentence in isolation can often be turned into a rigorous proposition in ZF. But if that sentence is part of a collection of sentences that refer to each other forming an attempt at a mutually recursive definition, you need to check precisely what parent-child relation you're using.

Acknowledgement
Thanks to Sridar Ramesh for making clear to me why the attempted definition of truth in ZF doesn't work. But I've probably made some mistakes above and they have nothing to do with Sridar.