More Is Not Always Better

In a previous post I discussed the status of Church’s Law in type theory, showing that it fails to hold internally to extensional type theory, even though one may see externally that the definable numeric functions in ETT are λ-definable, and hence Turing computable.  The distinction between internal and external is quite important in logic, mainly because a logical formalism may be unable to express precisely an externally meaningful concept.  The classical example is the Löwenheim-Skolem Theorem of first-order logic, which says that any theory with an infinite model has a countable model.  In particular the theory of sets has a countable model, which would seem to imply that the set of real numbers, for example, is countable.  But internally one can prove that the reals are uncountable (Cantor’s proof is readily expressed in the theory), which seems to be a paradox of some kind.  But no, all it says is that the function witnessing the countability of the term model cannot be expressed internally, and hence there is no contradiction at all.

A similar situation obtains with Church’s Law.  One may observe empirically, so to say, that Church’s Law holds externally of ETT, but this fact cannot be internalized.  There is a function given by Church’s Law that “decompiles” any (extensional) function of type N→N by providing the index for a Turing machine that computes it.  But this function cannot be definable internally to extensional type theory, because it may be used to obtain a decision procedure for halting of Turing machines, which is internally refutable by formalizing the standard undecidability proof.  In both of these examples it is the undefinability of a function that is important to the expressive power of a formalism, contrary to naïve analyses that would suggest that, when it comes to definability of functions, the more the merrier.  This is a general phenomenon in type theory.  The power of type theory arises from its strictures, not its affordances, in direct opposition to the ever-popular language design principle “first-class x” for all imaginable values of x.

Another perspective on the same issue is provided by Martin-Löf’s meaning explanation of type theory, which is closely related to the theory of realizability for constructive logic.  The high-level idea is that a justification for type theory may be obtained by starting with an untyped concept of computability (i.e., a programming language given by an operational semantics for closed terms), and then giving the meaning of the judgments of type theory in terms of such computations.  So, for example, the judgment A type, where A is a closed expression means that A evaluates to a canonical type, where the canonical types include, say, Nat, and all terms of the form A’→A”, where A’ and A” are types.  Similarly, if A is a type, the judgment a:A means that A evaluates to a canonical type A’ and that a evaluates to a canonical term a’ such that a’ is a canonical element of A’, where, say, any numeral for a natural number is a canonical member of Nat.  To give the canonical members of the function type A’→A” requires the further notion of equality of elements of a type, a=b:A, which all functions are required to respect.  A meaning explanation of this sort was suggested by Martin-Löf in his landmark paper Constructive Mathematics and Computer Programming, and is used as the basis for the NuPRL type theory, which extends that account in a number of interesting directions, including inductive and coinductive types, subset and quotient types, and partial types.

The relation to realizability emerges from applying the meaning explanation of types to the semantics of propositions given by the propositions-as-types principle (which, as I’ve previously argued, should not be called “the Curry-Howard isomorphism”).  According to this view a proposition P is identified with a type, the type of its proofs, and we say that P true iff P evaluates to a canonical proposition that has a canonical member.  In particular, for implication we say that P→Q true if and only if P true implies Q true (and, in addition, the proof respects equality, a condition that I will suppress here for the sake of simplicity).  More explicitly, the implication is true exactly when the truth of the antecedent implies the truth of the consequent, which is to say that there is a constructive transformation of proofs of P into proofs of Q.

In recursive realizability one accepts Church’s Law and demands that the constructive transformation be given by the index of a Turing machine (i.e., by a program written in a fixed programming language).  This means, in particular, that if P expresses, say, the decidability of the halting problem, for which there is no recursive realizer, then the implication P→Q is vacuously true!  By taking Q to be falsehood, we obtain a realizer for the statement that the halting problem is undecidable.  More generally, any statement that is not realized is automatically false  in the recursive realizability interpretation, precisely because the realizers are identified with Turing machine indices.  Pressing a bit further, there are statements, such as the statement that every Turing machine either halts or diverges on its own input, that are true in classical logic, yet have no recursive realizer, and hence are false in the realizability interpretation.

In contrast in the meaning explanation for NuPRL Church’s Law is not assumed.  Although one may show that there is no Turing machine to decide halting for Turing machines, it is impossible to show that there is no constructive transformation that may do so.  For example, an oracle machine would be able to make the required decision.  This is entirely compatible with intuitionistic principles, because although intuitionism does not affirm LEM, neither does it deny it.  This point is often missed in some accounts, leading to endless confusions.  Intuitionistic logic, properly conceived, is compatible with classical logic in that classical logic may be seen as an idealization of intuitionistic logic in which we heuristically postulate that all propositions are decidable (all instances of LEM hold).

The crucial point distinguishing the meaning explanation from recursive realizability is precisely the refusal to accept Church’s Law, a kind of comprehension principle for functions as discussed earlier.  This refusal is often called computational open-endedness because it amounts to avoiding a commitment to the blasphemy of limiting God’s programming language to Turing machines (using an apt metaphor of Andrej Bauer’s).  Rather, we piously accept that richer notions of computation are possible, and avoid commitment to a ”final theory” of computation in which Church’s Law is postulated outright.  By avoiding the witnessing function provided by Church’s Law we gain expressive power, rather than losing it, resulting in an elegant theory of constructive mathematics that enriches, rather than diminishes, classical mathematics.    In short, contrary to “common sense” (i.e., uninformed supposition), more is not always better.

Update: corrected minor technical error and some typographical errors.

Update: clarified point about incompatibility of recursive realizability with classical logic.

Exceptions are shared secrets

It’s quite obvious to me that the treatment of exceptions in Haskell is wrong. Setting aside the example I gave before of an outright unsoundness, exceptions in Haskell are nevertheless done improperly, even if they happen to be sound. One reason is that the current formulation is not stable under seemingly mild extensions to Haskell that one might well want to consider, notably any form of parameterized module or any form of shadowing of exception declarations. For me this is enough to declare the whole thing wrong, but as it happens Haskell is too feeble to allow full counterexamples to be formulated, so one may still claim that what is there now is ok … for now.

But I wish to argue that Haskell exceptions are nevertheless poorly designed, because it misses the crucial point that exception values are shared secrets. Let us distinguish two parties, the raiser of the exception, and the handler of it. The fundamental idea of exceptions is to transfer a value from the raiser to the handler without the possibility of interception by another party. While the language of secrecy seems appropriately evocative, I hasten to add that I am not here concerned with “attackers” or suchlike, but merely with the difficulties of ensuring modular composition of programs from components. In such a setting the “attacker” is yourself, who is not malicious, but who is fallible.

By raising an exception the raiser is “contacting” a handler with a message. The raiser wishes to limit which components of a program may intercept that message. More precisely, the raiser wishes to ensure that only certain previously agreed-upon components may handle that exception, perhaps only one. This property should remain stable under extension to the program or composition with any other component. It should not be possible for an innocent third party to accidentally intercept a message that was not intended for it.

Achieving this requires a secrecy mechanism that allows the raiser and the handler(s) to agree upon their cooperation. This is accomplished by dynamic classification, exactly as it is done properly in Standard ML (but not O’Caml). The idea is that the raiser has access to a dynamically generated constructor for exception values, and any handler has access to the corresponding dynamically generated matcher for exception values. This means that the handler, and only the handler, can decode the message sent by the raiser; no other party can do anything with it other than pass it along unexamined. It is “perfectly encrypted” and cannot be deciphered by any unintended component.

The usual exception mechanisms, as distinct from exception values, allow for “wild-card handlers”, which means that an exception can be intercepted by a third party. This means that the raiser cannot ensure that the handler actually receives the message, but it can ensure, using dynamic classification, that only a legitimate handler may decipher it. Decades of experience with Standard ML shows that this is a very useful thing indeed, and has application far beyond just the simple example considered here. For full details, see my forthcoming book, for a full discussion of dynamic classification and its role for ensuring integrity and confidentiality in a program. Dynamic classification is not just for “security”, but is rather a good tool for everyday programming.

So why does Haskell not do it this way? Well, I’m not the one to answer that question, but my guess is that doing so conflicts with the monadic separation of effects. To do exceptions properly requires dynamic allocation, and this would force code that is otherwise functional into the IO monad. Alternatively, one would have to use unsafePerformIO—as in ezyang’s implementation—to “hide” the effects of exception allocation. But this would then be further evidence that the strict monadic separation of effects is untenable.

Update: Reworked last paragraph to clarify the point I am making; the previous formulation appears to have invited misinterpretation.

Update: This account of exceptions also makes clear why the perennial suggestion to put exception-raising information into types makes no sense to me. I will write more about this in a future post, but meanwhile contemplate that a computation may raise an exception that is not even in principle nameable in the type. That is, it is not conservativity that’s at issue, it’s the very idea.

PFPL is out!

Practical Foundations for Programming Languages, published by Cambridge University Press, is now available in print! It can be ordered from the usual sources, and maybe some unusual ones as well. If you order directly from Cambridge using this link, you will get a 20% discount on the cover price (pass it on).

Since going to press I have, inevitably, been informed of some (so far minor) errors that are corrected in the online edition. These corrections will make their way into the second printing. If you see something fishy-looking, compare it with the online edition first to see whether I may have already corrected the mistake. Otherwise, send your comments to me.

By the way, the cover artwork is by Scott Draves, a former student in my group, who is now a professional artist as well as a researcher at Google in NYC. Thanks, Scott!

Update: The very first author’s copy hit my desk today!

Univalent Foundations at IAS

As many of you may know, the Institute for Advanced Study is sponsoring a year-long program, called “Univalent Foundations for Mathematics” (UF), which is developing the theory and applications of Homotopy Type Theory (HTT).  The UF program is organized by Steve Awodey (CMU), Thierry Coquand (Chalmers), and Vladimir Voevodsky (IAS).  About two dozen people are in residence at the Institute to participate in the program, including Peter Aczel, Andrej Bauer, Peter Dybjer, Dan Licata, Per Martin-Löf, Peter Lumsdaine, Mike Shulman, and many others.  I have been shuttling back and forth between the Institute and Carnegie Mellon, and will continue to do so next semester.

The excitement surrounding the program is palpable.  We all have the sense that we are doing something important that will change the world.  A typical day consists of one or two lectures of one or two hours, with the rest of the day typically spent in smaller groups or individuals working at the blackboard.  There are many strands of work going on simultaneously, including fundamental type theory, developing proof assistants, and formulating a body of informal type theory.  As visitors come and go we have lectures on many topics related to HTT and UF, and there is constant discussion going on over lunch, tea, and dinner each day.  While there I work each day to the point of exhaustion, eager to pursue the many ideas that are floating around.

So, why is homotopy type theory so exciting?  For me, and I think for many of us, it is the most exciting development in type theory since its inception.  It brings together two seemingly disparate topics, algebraic topology and type theory, and provides a gorgeous framework in which to develop both mathematics and computer science.  Many people have asked me why it’s so important.  My best answer is that it’s too beautiful to be ignored, and such a beautiful concept bmust be good for something!  We’ll be at this for years, but it’s too soon to say yet where the best applications of HTT will arise.  But I am sure in my bones that it’s as important as type theory itself.

Homotopy type theory is based on two closely related concepts:

1. Constructivity.  Proofs of propositions are mathematical objects classified by their types.
2. Homotopy.  Paths between objects of a type are proofs of their interchangeability in all contexts.  Paths in a type form a type whose paths are homotopies (deformations of paths).

Homotopy type theory is organized so that maps and families respect homotopy, which, under the identification of paths with equality proofs, means that they respect equality.  The force of this organization arises from axioms that specify what are the paths within a type.   There are two major sources of non-trivial paths within a type, the univalence axiom, and higher inductive types.

The univalence axiom specifies that there is an equivalence between equivalences and equalities of the objects of a universe.  Unravelling a bit, this means that for any two types inhabiting a universe, evidence for their equivalence (a pair of maps that are inverse up to higher homotopy, called weak equivalence) is evidence for their equality.  Put another way, weak equivalences are paths in the universe.  So, for example, a bijection between two elements of the universe of sets constitutes a proof of the equality (universal interchangeability) of the two sets.

Higher inductive types allow one to define types by specifying their elements, any paths between their elements, any paths between those paths, and so on to any level, or dimension.  For example, the interval, , has as elements the endpoints , and a path between and within .  The circle, has an element and a path from to itself within .

Respect for homotopy means that, for example, a family of types indexed by the type must be such that if and are isomorphic sets, then there must be an equivalence between the types and allowing us to transport objects from one “fiber” to the other.  And any function with domain must respect bijection—it could be the cardinality function, for example, but it cannot be a function that would distinguish from .

Univalence allows us to formalize the informal convention of identifying things “up to isomorphism”.  In the presence of univalence equivalence types (spaces) are, in fact, equal.  So rather than rely on convention, we have a formal account of such identifications.

Higher inductives generalize ordinary inductive definitions to higher dimensions.  This means that we can now define maps (computable functions!) between, say, the 4-dimensional sphere and the 3-dimensional sphere, or between the interval and the torus.  HTT makes absolutely clear what this even means, thanks to higher inductive types.  For example, a map out of is given by two pieces of data:

1. What to do with the base point.  It must be mapped to a point in the target space.
2. What to do with the loop.  It must be mapped to a loop in the target space based at the target point.

A map out of is given similarly by specifying

1. What to do with the endpoints.  These must be specified points in the target space.
2. What to do with the segment.  It must be a path between the specified points in the target space.

It’s all just good old functional programming!  Or, rather, it would be, if we were to have a good computational semantics for HTT, a topic of intense interest at the IAS this year.  It’s all sort-of-obvious, yet also sort-of-non-obvious, for reasons that are difficult to explain briefly.  (If I could, they would probably be considered obvious, and not in need of much explanation!)

A game-changing aspect of all of this is that HTT provides a very nice foundation for mathematics in which types (-groupoids) play a fundamental role as classifying all mathematical objects, including proofs of propositions, which are just types.  Types may be classified according to the structure of their paths—and hence propositions may be classified according to the structure of their proofs.  For example, any two proofs of equivalence of two natural numbers are themselves equivalent; there’s only one way to say that , for example.  Of course there is no path between and .  And these two situations exhaust the possibilities: any two paths between natural numbers are equal (but there may not even be one).  This unicity of paths property lifts to function spaces by extensionality, paths between functions being just paths between the range elements for each choice of domain element.  But the universe of Sets is not like this: there are many paths between sets (one for each bijection), and these are by no means equivalent.  However, there is at most one way to show that two bijections between sets are equivalent, so the structure “peters out” after dimension 2.

The idea to apply this kind of analysis to proofs of propositions is a distinctive feature of HTT, arising from the combination of constructivity, which gives proofs status as mathematical objects, and homotopy, which provides a powerful theory of the equivalence of proofs.  Conventional mathematics ignores proofs as objects of study, and is thus able to express certain ideas only indirectly.  HTT brings out the latent structure of proofs, and provides an elegant framework for expressing these concepts directly.

Update: edited clumsy prose and added concluding paragraph.

Introductory FP Course Materials

The course materials for our first-semester introductory programming course (which I’ve discussed elsewhere on this blog) are now available here.

The course materials for our second-semester data structures and algorithms course are available here.

Thanks to Dan Licata and Guy Blelloch for helping make these available.  Comments are most welcome.

Update: I will be giving a talk about this at Google Pittsburgh on Thursday the 20th of September.

Yet Another Reason Not To Be Lazy Or Imperative

In an earlier post I argued that, contrary to much of the literature in the area, parallelism is all about efficiency, and has little or nothing to do with concurrency.  Concurrency is concerned with controlling non-determinism, which can arise in all sorts of situations having nothing to do with parallelism.  Process calculi, for example, are best viewed as expressing substructural composition of programs, and have very little to do with parallel computing.  (See my PFPL and Rob Simmons’ forthcoming Ph.D. dissertation for more on this perspective.)  Parallelism, on the other hand, is captured by analyzing the cost of a computation whose meaning is independent of its parallel execution.  A cost semantics specifies the abstract cost of a program that is validated by a provable implementation that transfers the abstract cost to a precise concrete cost on a particular platform.  The cost of parallel execution is neatly captured by the concept of a cost graph that captures the dynamic data dependencies among subcomputations.  Details such as the number of processors or the nature of the interconnect are factored into the provable implementation, which predicts the asymptotic behavior of a program on a hardware platform based on its cost graph.  One advantage of cost semantics for parallelism is that it is easy to teach freshmen how to write parallel programs; we’ve been doing this successfully for two years now, with little fuss or bother.

This summer Guy Blelloch and I began thinking about other characterizations of the complexity of programs besides the familiar abstractions of execution time and space requirements of a computation.  One important measure, introduced by Jeff Vitter, is called I/O Complexity.  It measures the efficiency of algorithms with respect to memory traffic, a very significant determiner of performance of programs.  The model is sufficiently abstract as to encompass several different interpretations of I/O complexity.  Basically, the model assumes an unbounded main memory in which all data is ultimately stored, and considers a cache of blocked into chunks of size that provides quick access to main memory.  The complexity of algorithms is analyzed in terms of these parameters, under the assumption that in-cache accesses are cost-free, so that the only significant costs are those incurred by loading and flushing the cache.  You may interpret the abstract concepts of main memory and cache in the standard way as a two-level hierarchy representing, say, on- and off-chip memory access, or instead as representing a disk (or other storage medium) loaded into memory for processing.  The point is that the relative costs of processing cached versus uncached data is huge, and worth considering as a measure of the efficiency of an algorithm.

As usual in the algorithms world Vitter makes use of a low-level machine model in which to express and evaluate algorithms.  Using this model Vitter obtains a lower-bound for sorting in the I/O model, and a matching upper bound using a -way merge sort, where is chosen as a function of and (that is, it is not cache oblivious in the sense of Leiserson, et al.)  Although such models provide a concrete, and well-understood, basis for analyzing algorithms, we all know full well that programming at such a low-level is at best a tedious exercise.  Worse, machine models provide no foundation for composition of programs, the single most important characteristic of higher-level language models.  (Indeed, the purpose of types is to mediate composition of components; without types, you’re toast.)

The point of Guy’s and my work this summer is to adapt the I/O model to functional programs, avoiding the mess, bother, and futility of trying to work at the machine level.  You might think that it would be impossible to reason about the cache complexity of a functional program (especially if you’re under the impression that functional programming necessarily has something to do with Haskell, which it does not, though you may perhaps say that Haskell has something to do with functional programming).  Traditional algorithms work, particularly as concerns cache complexity, is extremely finicky about memory management in order to ensure that reasonable bounds are met, and you might reasonably suspect that it will ever be thus.  The point of our paper, however, is to show that the same asymptotic bounds obtained by Vitter in the I/O model may be met using purely functional programming, provided that the functional language is (a) non-lazy (of course), and (b) implemented properly (as we describe).

Specifically, we give a cost semantics for functional programs (in the paper, a fragment of ML) that takes account of the memory traffic engendered by evaluation, and a provable implementation that validates the cost semantics by describing how to implement it on a machine-like model.  The crux of the matter is to account for the cache effects that arise from maintaining a control stack during evaluation, even though the abstract semantics has no concept of a stack (it’s part of the implementation, and cannot be avoided).  The cost semantics makes explicit the reading and allocation of values in the store (using Felleisen, Morrisett, and H’s “Abstract Models of Memory Management”), and imposes enough structure on the store to capture the critical concept of locality that is required to ensure good cache (or I/O) behavior.  The abstract model is parameterized by and described above, but interpreted as representing the number of objects in the cache and the neighborhood of an object in memory (the objects that are allocated near it, and that are therefore fetched along with the object whenever the cache is loaded).

The provable implementation is given in two steps.  First, we show how to transfer the abstract cost assigned to a computation into the amount of memory traffic incurred on an abstract machine with an explicit control stack.  The key idea here is an amortization argument that allows us to obtain tight bounds on the overhead required to maintain the stack.  Second, we show how to implement the crucial read and allocate operations that underpin the abstract semantics and the abstract machine.  Here we rely on a competitive analysis, given by Sleator, et al., of the ideal cache model, and on an amortization of the cost of garbage collection in the style of Appel.  We also make use of an original (as far as I know) technique for implementing the control stack so as to avoid unnecessary interference with the data objects in cache.  The net result is that the cost semantics provides an accurate asymptotic analysis of the I/O complexity of a functional algorithm, provided that it is implemented in the manner we describe in the paper (which, in fact, is not far from standard implementation techniques, the only trick being how to manage the control stack properly).  We then use the model to derive bounds for several algorithms that are comparable to those obtained by Vitter using a low-level machine model.

The upshot of all of this is that we can reason about the I/O or cache complexity of functional algorithms, much as we can reason about the parallel complexity of functional algorithms, namely by using a cost semantics.  There is no need to drop down to a low-level machine model to get a handle on this important performance metric for your programs, provided, of course, that you’re not stuck with a lazy language (for those poor souls, there is no hope).

Believing in Computer Science

It’s not every day that I can say that I agree with Bertrand Meyer, but today is an exception. Meyer has written an opinion piece in the current issue of C.ACM about science funding that I think is worth amplifying. His main point is that funding agencies, principally the NSF and the ERC, are constantly pushing for “revolutionary” research, at the expense of “evolutionary” research. Yet we all (including the funding agencies) know full well that, in almost every case, real progress is made by making seemingly small advances on what is already known, and that whether a body of research is revolutionary or not can only be assessed with considerable hindsight. Meyer cites the example of Hoare’s formulation of his logic of programs, which was at the time a relatively small increment on Floyd’s method for proving properties of programs. For all his brilliance, Hoare didn’t just invent this stuff out of thin air, he built on and improved upon the work that had gone before, as of course have hundreds of others built on his in turn. This all goes without saying, or ought to, but as Meyer points out, we computer scientists are constantly bombarded with direct and indirect exhortations to abandon all that has gone before, and to make promises that no one can honestly keep.

Meyer’s rallying cry is for incrementalism. It’s a tough row to hoe. Who could possibly argue against fostering earth-shattering research that breaks new ground and summarily refutes all that has gone before? And who could possibly defend work that is obviously just another slice of the same salami, perhaps with a bit of mustard this time? And yet what he says is obviously true. Funding agencies routinely beg the very question under consideration by stipulating a priori that there is something wrong with a field, and that an entirely new approach is required. With all due respect to the people involved, I would say that calls such as these are both ill-informed and outrageously arrogant.

But where does this attitude come from? Meyer cites “market envy” as one particularly powerful influence. Funding agencies wish to see themselves as analogous to venture capitalists investing in the next big thing, losing track of the fundamental differences between basic research and product development. (We see this sort of nonsense all the time in national politics; there is always a constituency for the absurd proposition that a government should be run like a business, as if there were any similarity at all between the two. What they really mean is, turn the government’s money over to business, but that can’t be said too loudly for fear that people will catch on.)

Another influence, which Meyer doesn’t mention, seems to be a long-standing problem in computer science. It seems to me that many researchers who move into political and administrative roles are either bored with, or do not believe in, computer science as an academic discipline. Their own research area is, or maybe always was, boring, or has been obviated by technological developments or scientific advances. So they move into politics, perhaps carrying a sense that there is something wrong with the field that can only be corrected by radical surgery. They then demand that researchers do what they themselves never did in their own careers, kicking the ladder out from behind them.

From this somewhat contentious premise, much follows. The constant implication, stated and unstated, that CS research is not worth doing for its own sake. The emphasis on interdisciplinarity as an end in itself, rather than a means to an end. The emphasis on applications to other disciplines being more important than CS itself. The sense that “broader impacts” are far more important than the “innovative claims” (the actual work) in a research proposal.

Seen from this perspective, Meyer’s position is much more easily defensible. When funding agencies are talking about “breakthroughs” and “paradigm shifts”, what they really mean is “anything but computer science”. When Meyer talks about incrementalism, what he really means is “computer science is worth doing for its own sake”.

And I, for once, agree with him.