Orc in Haskell, now on Hackage

Orc is a concurrent scripting language, now available as an embedded DSL in Haskell. I like to think of Orc as the combination of three things: many-valued concurrency, external actions (effects), and managed resources, all packaged in a high-level set of abstractions that feel more like scripting rather than programming. It provides a very flexible way to manage multiple concurrent actions, like querying remote web sites, along with timeouts and default actions.Source code is available on Hackage; the easiest way to access it is with cabal (i.e. ‘cabal install orc’).Also available is a draft paper entitled Concurrent Orchestration in Haskell which explains how to use Orc, as well as describing the implementation in detail.Feedback welcome. Enjoy!

Read More

Galois Tech Talks, now on Vimeo!

For a number of years, Galois Inc. has been organizing technical seminars presented by visiting researchers, Galois engineers, and members of the vibrant Portland technical community. The seminars span a wide variety of topics, ranging from functional programming, formal methods, and compiler and language design, to cryptography, and operating system construction. The talks are free and open to the interested public. Announcements of upcoming talks are posted to this blog about a week in advance.Over the last few months we have received a number of requests to share videos of the talks with the wider community. As a result, we are very pleased to announce the Galois tech talk channel on Vimeo. Recent Galois talks should become available over the next few weeks, followed by future presentations.Enjoy!Galois tech talk channel: http://vimeo.com/channels/galois

Read More

Solving n-Queens in Cryptol

The eight queens puzzle asks how to place eight queens on a chess board such that none of them attacks any other. The problem easily generalizes to n queens, using an nxn board. In this post, we’ll see how to solve the n-Queens puzzle in Cryptol, without lifting a finger!

Representing the board

It is easy to see that any solution to the n-Queens puzzle will have precisely one queen on each row and column of the board. Therefore, we can represent the solution as a sequence of n row numbers, corresponding to subsequent columns. For instance, the board pictured below (taken from the wikipedia page), can be represented by the sequence3  7  0  2  5  1  6  4
recording the row of the queen appearing in each consecutive column, starting from the top-left position. (And yes, we always count from 0!)

Read More

POPL 2010, Day 1

Here are some talks that caught my attention on the opening day of POPL 2010.Reconfigurable Asynchronous Logic Automata, Neil GershenfeldIn this invited talk, Neil proposed that we throw out the sequential model of computation from Turing and Von Neumann, and re-imagine computing from the ground up, more along the line of physics as it is manifested in the world. This might be the way forward given the huge power demands that will arise from traditional architectures as they become more and more powerful. Rather than imposing a top-down structure on the machine, Neil propose we consider systems built up from cellular automata. In ALA each cell is able to do a basic logic operation or, in RALA, reconfigure other cells. Built around 2D grids, and computing in parallel without any global clock, the automata are able to compute many basic functions in linear time, including multiplication and sorting (of course, they also take a linear number of cells to do so).What I liked about this talk is that it called on me to think more broadly about the nature of computation. I don’t think I buy the line that how we do computing today is wrong. But I do believe that computing and information theory is going to become more and more important in the development of many other subjects, including physics and biology. I also believe that we will need to harness many more structures to do computing rather than simply relying on gates burned in silicon. Coincidentally, at lunch today Luca Cardelli was telling me about his explorations in computing with DNA. Fascinating stuff!At the lowest levels, the primitive components of computation reflect the computational substrate, so they are very different from one instantiation to another. As primitive computational elements become combined, however, the different descriptions become more abstract, and start to look more and more alike. The same mathematical operations may arise in each form, for example. That makes me wonder what the space of truly computationally-neutral specifications looks like: how might we program in such a way that we really don’t have a preference for what substrate we map the computation onto. We already consider FPGAs, GPUs and CPUs as standard targets for languages like Cryptol. Let’s conceptually add DNA and RALA to the list, and see if that would have us change anything.A Simple, Verified Validator for Software Pipelining, Jean-Baptiste Tristan & Xavier LeroySoftware pipelining is a compiler transformation that reorders loop code to make more efficient use of the CPU and memory accesses. Code from many different iterations of the loop might be drawn together and overlapped to execute out of order, and perhaps in parallel. The question is how to verify that the pipelining has been done correctly.The paper proposes using symbolic evaluation. It describes an equivalence principle which requires proving two separate commutativity properties. These may both be shown by symbolic evaluation of the instruction code, being careful to handle the small differences (such as different temporary variables in the two paths). Overall the process is simple enough that it could be a standard component of any compiler, even if the compiler is non-verifying overall.A Verified Compiler for an Impure Functional Language, Adam ChlipalaHandling object-level variables in proofs is a real pain, one that is hard to eliminate. This paper has an interesting way of dealing with them. It describes a Coq verification of a compiler for Untyped Mini-ML with references and exceptions, compiling to an idealized assembly code.Now, using explicit object variables leads to lots of lemmas about substitution. On the other hand, full higher-order abstract syntax (HOAS) would lead quickly to non-termination, and hence unsoundness. So instead, the paper introduces a closure semantics in which variables are just numbers pointing into a heap. Many operations that would reorder substitutions (and hence require big proofs) don’t affect the closure heap, so the proofs go through easily.The paper also uses a high degree of proof automation: it has a generic proof script that looks at the hypotheses and goals, and decides what approaches to try. The proof script is a bit slow because the evaluation mechanism of Coq wasn’t designed for this particularly (but see my blog post from PADL about accelerating normalization). However, because the proof is highly automatic, it is not hard to evolve the compiler and proof simultaneously. In fact, the compiler and proof were developed incrementally in exactly this way.Verified Just-in-Time Compiler on x86, Magnus MyreenJIT compiling has become standard for accelerating web pages, amongst other things. Good JIT compilers can take into account the input to the program as well as the program itself. But it is hard to get them right, as witnessed apparently by a recent Firefox bug.An aspect that makes verification tricky is that code and data are mixed: a JIT compiler essentially produces self-modifying code. The paper provides a programming logic for self-modifying code by adapting Hoare triples so that there is both “before” and “after” code. Some practical issues also need to be handled: for example, the frame-property has subtleties regarding the instruction cache, and code-updates can sometime happen too recently to show up in the current instruction stream. The x86 documentation is a little vague about how soon code-modifications can be relied on to be present.The end result is a verified JIT compiler from a toy input byte-code language, but mapping to a realistic semantics for x86. The compiler produces comparable code quality to C for a GCD example.

Read More

PADL/PEPM 2010, Day 2

Again, here are a few talks that caught my attention and I thought I would share my impressions. Enjoy!An Ode to Arrows, Hai Liu & Paul Hudak (PADL 2010)Representing the infinite tower of derivatives of a function as a lazy list has become popular in the Haskell world, the most recent example being Conal Elliott’s paper on Beautiful Differentiation at ICFP 2009. This work explores taking the same ideas and using them to express systems of Ordinary Differential Equations.A natural translation of these ideas into Haskell requires recasting the ODEs in an integral form, rather than a purely differential form, leading to a shallow embedding of the equations as recursive equations in Haskell. Unfortunately, this simple translation leads to quadratic computational complexity, in both time and space. Memoization can address this moderately well, except that one is left with the problems of managing (and freeing) the memo-tables.Moving to arrows leads to a better solution. The recursion is expressed using Paterson’s ‘loop’ construct, which puts it in the control of the programmer. When combined with Causal Commutative Arrows (http://lambda-the-ultimate.org/node/3659) the circuits can be optimized into tight iterative loops, with dramatic improvements in performance.A DSL Approach to Protocol Stack Implementation, Yan Wang & Veronica Gaspes (PADL 2010)Implementations of network code are very hard to connect with the specifications, leading to major challenges for maintenance. This is significant for embedded devices which need implementations that appropriately trade off power, speed, and memory usage. To that end, IPS is a DSL for generating network code. It allows packet formats to be defined using dependencies between fields, handles the interaction of protocol and packet, and provides a framework for constructing the protocol stacks.In a comparison exercise, IPS was used to generate code for the Rime protocol stack. Compared with the published hand-written version in C, the IPS code was 1/4 of the size of the C, but produced an executable 25% larger. The performance on sending packets was 10% worse in the IPS version, but both versions were equivalent for receiving packets.No measurements were given for ease of maintenance or design exploration between the two versions, but I know which one I would prefer to have to work with…I/O Guided Detection of List Catamorphisms, Martin Hofmann & Emanuel Kitzelmann (PEPM 2010)Having the computer write its own programs has always been appealing. You provide some input/output pairs and the machine comes up with a generalization expressed as a program that has the same behavior. Broadly, there are two ways to do this: analytical (where the structure of the data is examined to build the program), and generate & test (where many programs are produced, and their behavior improved, perhaps by evolutionary combinations).This work uses the idea of catamorphisms (also known an fold or reduce functions) as a template to explore, when the i/o pairs comes from a recursive data type (just lists at the moment, I think). Using the fact that if a function g satisfies ‘g [] = v’ and ‘g (x:xs) = f x (g xs)’ for some ‘f’ and ‘v’, then ‘g = fold f v’, the system figures out what the i/o pairs for the sub-functions would be, and then tries to deduce the definitions. It includes specific smarts that notice when the form of the function corresponds to a ‘map’ or ‘filter’.Bridging the Gap Between Symbolic and Efficient AES Implementation, Andrew Moss & Dan Page (PEPM 2010)Different implementations of AES, the standard for block encryption, run at very different speeds. The original reference implementation takes about 840 cycles per round, the Open SSL implementation takes 30, and a top performing assembly code version takes 21.6 cycles per round. The CAO DSL and compiler described in this paper takes a high-level spec and compiles it, producing code that takes 23 cycles per round. Impressive.CAO is a sequential language, designed to be easily related to the pseudo code that crypto specifications often use. A significant amount of effort in the compiler then goes into deducing what parallelism is possible and how best to balance the work. CAO has matrices as first class objects, to allow the compiler opportunity to deduce good representations. There are also optimizations to convert references from the matrices in their entirety to references to the appropriate sub-portions, so as to eliminate unnecessary additional sequential dependencies. The compiler also has some optimizations that are specific to AES.Clone Detection and Elimination for Haskell, Christopher Brown & Simon Thompson (PEPM 2010)I liked this talk partly because it reminded me to go and play with HaRe (https://www.cs.kent.ac.uk/projects/refactor-fp/hare.html). The Haskell Refactorer is an Emacs and Vim tool for manipulating Haskell 98 programs, providing structural transformation (like fold/unfold), altering data types (like adding or eliminating arguments), slicing, and now, clone elimination (cue Star Wars music).The clone eliminator finds repeated code forms above a threshold size, and deduces the least-general generalization of the code to invent new functions if necessary. Clone detection is entirely automatic, but it is up to the programmer to decide which ones to act upon — sometimes adding abstraction helps the structure of a program, and sometimes it obscures it. It works within monadic code too.HaRe works mostly at the AST level, though it does use the raw token stream for some of its discovery. It also works across modules.Making “Stricterness” More Relevant, Stefan Holdermans & Jurriaan Hage (PEPM 2010)The old approaches to strictness were all denotational, but to actually use strictness within the compiler it is better to have a syntax-driven formulation, i.e. in logic. There is a nice approach which uses the concept of ‘relevance’, where ‘x’ is relevant to ‘e’ if every evaluation of ‘e’ demands ‘x’. Clearly, relevance is equivalent to strictness.But actually it doesn’t work so well in the presence of ‘seq’. The original formulation of relevance assumed that the only time a function would be evaluated was at its point of application. Then there is no way to distinguish ‘undefined’ from ‘x->undefined’. ‘Seq’ changes all that. The paper shows how to fix relevance by tracking which lambdas are used applicatively (i.e. applied somewhere in the program in a place that will be evaluated). The definition is circular, but the least solution is the one intended.

Read More

PADL/PEPM 2010, Day 1

Here are some papers which caught my imagination at this year’s PADL and PEPM conferences in Madrid.O Partial Evaluator, Where art thou? Lennart Augustsson (PEPM 2010)In this invited talk, Lennart walked us through his personal engagement with PE-like code transformations that he has confronted directly in his career, which has consisted of Haskell programming in a wide variety of commercial and industrial settings. The examples included optimizing airline scheduling code for Lufthansa, eliminating unnecessary hardware elements in Bluespec compilation, and generating efficient Monte-Carlo computations in investment banking.In Bluespec, Lennart needed to introduce a ‘tabulate’ function which produces a table-driven implementation of its function argument. It is a powerful idea, but it requires a sufficiently powerful set of transformations for its boilerplate to get simplified away. In this case, Lennart needed to introduce a term ‘_’ which means “don’t care”, i.e. any convenient result can be returned. During unfolding and partial evaluation, ‘_’ could be replaced with any term that would help the partial evaluator make good progress on simplification. Semantically, this is a form of refinement of non-determinism performed during code transformation (in fact, I’m still not sure what the correct denotational approach to its semantics would be). Lennart then explored using ‘tabulate’ directly in Haskell, but found it next to impossible to get the GHC simplifier to do enough transformation.The Paradise ESDL is like Bluespec in that it uses the Haskell framework to bolt together primitive elements from a lower level language. In this case, the efficiency opportunities arise from, say, 18-argument functions being called repeatedly with 15 known arguments. Lennart introduced a ‘compile’ function which is semantically the identity function, but which invokes a run-time compilation. Yet again, Lennart found himself writing essentially the same kind of code he had previously written in a different setting. Each time it is fairly simple partial evaluation.Where oh where is the partial evaluator for Haskell? And don’t worry about self-application or anything fancy like that. Just something that is solid and reliable.Explicitly Typed Exceptions for Haskell, Jose Iborra (PADL 2010)Java gives you safe exceptions, in that if you don’t handle an exception that could be raised, the compiler will complain. Currently Haskell does not. This paper builds on Simon Marlow’s work of providing an exception library with existentially-quantified exceptions, that provide access to the exception through class methods. Marlow’s exceptions work very nicely with the type system, allowing exception and non-exception code to be intermingled. If only there was a way still to know which particular exceptions could be raised …Iborra to the rescue! He adapts Marlow’s exceptions with additional class constraints that track exactly which exception could be raised, fulfilling those constraints when a handler for the exception is provided. Unhandled exceptions show up as missing instances (of handlers), pinpointing the code that is at fault. It all works very beautifully. The downside? He needs to use overlapping instances to be able to ‘subtract’ that constraints. Hmmm. Normally I’m very down on overlapping instances: it seems really fragile to rely on details of the type-inference engine to select what code should run. But this is a particularly benign case: here the overlapping instances are over phantom type variables, so the same code will run in either case. Maybe Haskell should allow that in general; overlapping instances are fine if the same run-time code is produced however the instances are resolved.Conversion by Evaluation, Mathieu Boespflug (PADL 2010)Proof tactics in theorem provers often require large lambda terms to be normalized. For example some Coq tactics perform reflective computation over other Coq terms. This reduction can be quite inefficient when done explicitly, because of the interpretive overhead, yet the result of evaluation needs to be in a form where the final lambda term can be examined.This paper shows the how to interpret explicit lambda terms efficiently by mapping them into a underlying evaluation mechanism of the host language (e.g. Haskell). The approach is deceptively simple and consists of a set of straightforward optimizations (higher-order abstract syntax, currying, de Bruijn indices etc). Astonishingly, the result is that the evaluation incurs only a 10-30% overhead compared with using the underlying evaluator directly! A must-read for anyone interested in reflective evaluation.Optimizing Generics Is Easy! Jose Pedro Magalhaes et al (PEPM 2010)Generic programming is a way to provide a programmable version of the deriving concept that arises in Haskell. The problem is that the current libraries for implementing generics introduce performance overheads from 50% to 16x compared with writing the original by hand. This paper presents a framework for doing generics using type families to represent the components of algebraic datatypes, together with overloaded operators for mapping between the generic datatype and any user datatype.GHC does a good job of rewriting much of the overhead away, but needs to be encouraged to do the best it can. There are GHC flags to control decision thresholds for inlining etc that turn out to enable GHC to completely remove the overhead of the generic code, producing in some cases exactly the code that would get written by hand. The open problem is that increasing the threshold over the whole program might have negative effects elsewhere in the compilation, so a mechanism for localizing the effect of increased thresholds would be valuable.

Read More

GHC Nominated for Programming Language Award

ACM SIGPLAN recently announced a new award:

The SIGPLAN Programming Languages Software Award is awarded to an institution or individual(s) to recognize the development a software system that has had a significant impact on programming language research, implementations, and tools. The impact may be reflected in the wide-spread adoption of the system or its underlying concepts by the wider programming language community either in research projects, in the open-source community, or commercially. The award includes a prize of $2,500.

I think that GHC (Glasgow Haskell Compiler) and the two Simons (Peyton Jones and Marlow) are prime candidates for this. So, being careful to stay within the 500 word limit, I submitted a nomination statement for GHC as follows:

For the last decade, Haskell has been at the center of the most exciting innovations within programming languages, with work on software transactional memory, generalized algebraic data types, rank-N polymorphism, monads, multi-parameter type classes, embedded domain-specific languages, property-based testing, data parallelism, thread-profiling, and so on, generating hundreds of research papers from many diverse research groups.GHC, the Glasgow Haskell Compiler, is the vehicle that made this research possible.It is hard to explore radical ideas on real systems, yet the GHC team created a flexible platform that allows other researchers to explore the implications of their ideas and to test whether they really work in the large. From the first beta release in 1991, GHC emphasized collaboration and open “bazaar” style development, as opposed to the “cathedral” development of most of its contemporaries. GHC was open source even before Linux made open source cool. GHC has continued in the same vein, now listing over 60 contributors to the codebase.In those early days, efficient compilation of a higher-order, allocation-rich, lazy functional language seemed to be a pipe dream. Yet GHC has risen to be a top-flight performer in the online language performance shootout (shootout.alioth.debian.org), comparable with Java Server-6, and approaching native C in performance overall. This is a tribute to the incredible amount of profound optimization built into the compiler, with techniques like cross-module code migration, unboxed data types, and automated removal of intermediate data structures, all done through correctness-preserving transformations that exploit the algebraic simplicity of Haskell terms, even in the presence of monadic effects.The impact GHC has had on programming language research would be sufficient to merit an award by itself, but GHC is having a corresponding influence in industry. By showing the feasibility of purely functional, statically-typed programming in the large, GHC Haskell has also had clear influence on many of the newest generation of languages, such as C#, F#, Java Generics, LINQ, Perl 6, Python, and Visual Basic 9.0. As Soma Somasegar, Microsoft Developer Division Chief, said in 2007, “One of the important themes in programming languages over recent years has been a move to embrace ideas from functional programming, [which] are helping us address some of the biggest challenges facing the industry today, from the impedance mismatch between data and objects to the challenges of the multi-core and parallel computing space.”GHC now supports a burgeoning professional Haskell world. The O’Reilly book Real World Haskell, targeted to professional programmers and oriented to GHC, was published in 2008. It went on to win the Jolt Award for best technical book of the year. In 2009 there were 3500+ Haskell package updates, with more than 100,000 package downloads in November alone. GHC is now used across the financial sector in institutions like Credit Suisse and Standard Chartered Bank, and for high assurance software in companies like Amgen, Eaton, and Galois. Some of these companies came together in 2009 to create the Industrial Haskell Group, whose purpose is to ensure the health and longevity of GHC.

499 words. Whew! There is so much that could be said, but let’s hope this is enough. I think the case is very strong, and both Simon’s deserve honor and accolade for their work. Thank you both so much!

Read More

Shuffling a deck of cards, Cryptol style

I can never shuffle cards properly. They seem to go every which way when I try, and a perfect random shuffle seems nigh-impossible to achieve, even though the experts claim it takes a mere 7 moves. (The mathematical argument is surprisingly quite technical.) Luckily, we shall concern ourselves with a much simpler problem today: How many perfect out-shuffles does it take to restore a deck back to its original order? We’ll throw in a couple of variants of the problem for fun, but rest assured that we’ll let computers do the work. And, of course, we’ll use Cryptol to help us along the way.

What is a riffle shuffle?

According to wikipedia, a riffle (or dovetail) shuffle goes like this:

… half of the deck is held in each hand with the thumbs inward, then cards are released by the thumbs so that they fall to the table interleaved. Many also lift the cards up after a riffle, forming what is called a bridge which puts the cards back into place…

Well, I read that a couple times, and watched a couple of videos on the internet showing how to do it, but no luck so far. Luckily, this sort of shuffling is quite easy to express programmatically, and Cryptol has the right abstractions to code this in a couple of lines.

Bisecting the deck

The first step in the shuffle is bisecting the deck into two equal halves:

bisect : {a b} (fin a) => [2*a]b -> ([a]b, [a]b);bisect xs = (take (w, xs), drop (w, xs))where w = width xs / 2;

We simply compute the mid-point, and divide the given sequence xs into two, by take‘ing and drop‘ping the correct amounts from the input sequence. In fact, the type of bisect is more interesting than its definition: It succinctly captures the following four facts:

  1. The input has to be of even length (2*a),
  2. The input has to be finite (fin a),
  3. The output has two components, each of which has precisely a elements, that is, half the input,
  4. The actual contents of the sequence can be of any type (b), i.e., the function bisect is shape-polymorphic in the contents.

The ability to express precise size/shape-polymorphic properties using types is one of the strengths of Cryptol.

Out-shuffle vs in-shuffle

Once the deck is split into two, we proceed by picking the cards alternatingly from each half. We have two choices: We can either start with the first half, or the second. If you start with the first half, that’s called an out-shuffle. If you start with the second half, then it’s an in-shuffle. These two functions are actually quite easy to code in Cryptol:

 outShuffle : {a b} (fin a) => [2*a]b -> [2*a]b;outShuffle xs = join [| [x y] || x <- fh || y <- sh |]where (fh, sh) = bisect xs;inShuffle : {a b} (fin a) => [2*a]b -> [2*a]b;inShuffle xs = join [| [y x] || x <- fh || y <- sh |]where (fh, sh) = bisect xs;

The definitions are almost identical, except for the order in which we put the cards from the halves (fh and sh) together. In the outShuffle, the first card in each pair comes from the first-half. In the inShuffle, it comes from the second half. Easier done than said! Let’s make sure they behave as we expect:

 Cryptol> bisect [1..8]([1 2 3 4], [5 6 7 8])Cryptol> outShuffle [1..8][1 5 2 6 3 7 4 8]Cryptol> inShuffle [1..8][5 1 6 2 7 3 8 4]

Good! It’s interesting to see what happens when we apply bisect to an odd-length sequence:

 Crytpol> bisect [1..9]In a top-level expression: with inferred type:{a} ([a][4],[a][4])encountered the following unresolved constraints:fin a2*a == 9

Cryptol is basically telling us that there is no a such that 2*a is 9, resulting in a type-error. Note that this is a static-check before you run your program! In other words, if your program type-checks, then you can rest assured that whenever you call bisect, it is guaranteed to get an even-length sequence as its argument. Strong static typing and size-polymorphism of Cryptol really pays off in this case!

Sequences of shuffles

Before proceeding to the properties of shuffles, we need one last notion: The application of a shuffle repeatedly to a given input, yielding an infinite sequence of transformations:

 iterate : {a} (a -> a, a) -> [inf]a;iterate (f, x) = [x] # iterate (f, f x);outShuffles, inShuffles :{a b} (fin a) => [2*a]b -> [inf][2*a]b;outShuffles xs = iterate(outShuffle, xs);inShuffles xs = iterate(inShuffle, xs);

The high-order function iterate gives us the infinite sequence of results of applying a function to a value over and over. We simply use this helper to define outShuffles and inShuffles to apply the corresponding functions indefinitely to their input. Note that the resulting type shows that we get an infinite sequence as output, as indicated by the size inf.

Properties of shuffles

It turns out that if one applies 8 out-shuffles to a deck, a remarkable thing happens: Nothing! The deck goes back to its original order! This is a bit hard to believe, and harder to realize using a real deck of cards. (A friend of mine says he saw it done at college once by hand, but I’m yet to meet anyone who can do this successfully so far!)Well, the good thing about programming is that we can manipulate the sequences at will, without fear of messing up the cards. Even better, we can assert the above claim as a theorem in Cryptol:

 type Deck = [52][6];outShuffle8 : Deck -> Bit;theorem outShuffle8: {deck}.outShuffles(deck) @ 8 == deck;

We have defined a Deck to be a sequence of 52 things, each of which is 6-bits wide, which is more than enough to cover all the 52-unique elements that appear in an ordinary deck. (6-bits can encode 64 values, so we have 12 unused elements.) The theorem simply states that the 8’th element of the infinite sequence of outShuffle applied to an arbitrary deck gives us back the original deck.Let’s ask Cryptol to prove this theorem: (Cryptol’s symbolic and sbv backends can perform these proofs, so we first set our mode accordingly below.)

 Cryptol> :set sbvCryptol> :prove outShuffle8Q.E.D.

Voila! The proof completes instantaneously, with almost no time elapsed. (This might be surprising at first, since the input space to the theorem has 52*6 = 312 bits, which is quite large. However, we note that the theorem is actually fairly easy to prove since all shuffling does is a mere re-ordering of things with no specific computation; which is easy to manipulate symbolically for Cryptol’s proof engine.)

Reversing the deck

Can we reverse a deck of cards using outShuffle‘s? Turns out that this cannot be done. In particular, an outShuffle never moves the first element of the deck anywhere:

 outShuffleFirstCard : Deck -> Bit;theorem outShuffleFirstCard: {deck}.outShuffle deck @ 0 == deck @ 0;

We have:

 Cryptol> :prove outShuffleFirstCardQ.E.D.

Since the first card remains stationary, there is no way to reverse a deck of cards by just using outShuffles.How about with inShuffle? Turns out the magic number is 26 for reversing a deck of cards in this particular case:

inShuffle26Rev : Deck -> Bit;theorem inShuffle26Rev : {deck}.inShuffles(deck) @ 26 == reverse deck;

Again, the proof is immediate:

 Cryptol> :prove inShuffle26RevQ.E.D.

If 26 in-shuffle’s reverse the deck, then 52 of them will restore it back. Here’s the corresponding theorem:

inShuffle52 : Deck -> Bit;theorem inShuffle52: {deck}.inShuffles(deck) @ 52 == deck;

Again, the proof is immediate.

The Mongean Shuffle

There is one more variation on the shuffle that we w
ill consider. The mongean shuffle picks the even and odd numbered elements, reverses the odds and adds the evens at the back. For instance, given the sequence: 0 1 2 3 4 5 6 7 8 9, we first construct two sub-sequences: The even index elements: 0 2 4 6 8, and the odd ones 1 3 5 7 9. We then reverse the latter to get 9 7 5 3 1, and append the former, obtaining: 9 7 5 3 1 0 2 4 6 8. Luckily, the Cryptol definition is much easier to read:

monge xs = reverse odds # evenswhere { w = width xs;evens = xs @@ [0 2 .. (w-1)];odds = xs @@ [1 3 .. (w-1)]};monges xs = iterate(monge, xs);

With a monge shuffle, it takes 12 rounds to restore a deck:

monge12 : Deck -> Bit;theorem monge12: {deck}. monges(deck) @ 12 == deck;

We will leave it to the reader to construct the argument that no sequence of monge shuffles would reverse a deck. (In particular, one can prove that the 18th element from top will never move in a deck of 52. Proving this theorem in Cryptol is again quite trivial.)

A note on theorems

The attentive reader might worry that our Deck type does not quite correspond to a deck-of-cards. This is indeed the case. There are two discrepancies. First, as we mentioned before, our decks can represent 64 elements, while a deck of cards has only 52 distinct cards. On the plus side, this just makes our theorems “stronger,” since we allow for more cards then possible. More importantly, the properties are intended for decks that have no-repeating cards in them. (That is, every card occurs precisely once.) But our theorems apply to arbitrary deck‘s, even those that have repeating elements in them. Again, this just makes our theorems stronger, as the unique-sequence cases directly follow from them. We can rest assured that our proofs are conclusive, even though our model of playing-cards is not perfect.


Free evaluation licenses of Cryptol are available at www.cryptol.net. The Cryptol source code for shuffling cards is also available as well.

Read More

Domain Specific Languages for Domain Specific Problems

We have a new position paper on the use of EDSLs and Haskell for tackling the “programmability gap” of emerging high performance computing architectures — such as GPGPUs. It will be presented tomorrow at LACSS in Santa Fe. (Download) :: PDFSlides for the talk, including a 10 minute guide to EDSLs in Haskell, and a 10 minute guide to multicore programming in Haskell, can be found here :: PDF.

Domain Specific Languages for Domain Specific ProblemsDon Stewart, Galois.Workshop on Non-Traditional Programming Models for High-Performance Computing, LACSS 2009.

As the complexity of large-scale computing architecture increases, the effort needed to program these machines efficiently has grown dramatically. The challenge is how to bridge this “programmability gap”, making the hardware more accessible to domain experts. We argue for an approach based onexecutable embedded domain specific languages (EDSLs)—small languages with focused expressive power hosted directly in existing high-level programming languages such as Haskell. We provide examples of EDSLs in use in industry today, and describe the advantages EDSLs have over general purpose languages in productivity, performance, correctness and cost.Thanks to Magnus Carlsson, Dylan McNamee, Wouter Swiestra, Derek Elkins and Alex Mason for feedback on drafts.

Read More

Substitution ciphers in Cryptol

Substitution ciphers are one of the oldest encryption methods, dating back to at least the 15th century. In a substitution cipher, each character in the plain-text is simply “substituted” according to a predefined map. Decryption is simply the substitution in the reverse direction. Wikipedia has a nice description of these ciphers. Obviously, you wouldn’t want your bank to use such a cipher when executing your web-based transactions! But they are fun to play around, especially when entertaining kids in hot summer days. In this post, we’ll see how to code simple substitution ciphers in Cryptol, and go a step further and actually prove that our implementation is correct.


The simplest form of substitution ciphers use a permutation of the input alphabet. That is, each letter in the input alphabet gets mapped to another in the same  alphabet. (Strictly speaking, input and output alphabets need not be the same, but nothing essential changes by making that assumption.) For instance, you might decide that your substitution will map ‘a’ to ‘q’, and ‘b’ to ‘d’, …, etc., making sure no two letters are mapped to the same target. Once this mapping is agreed on, all you have to do to encrypt a given message is to map each character to the corresponding element according to your predefined mapping rules.Here’s our Cryptol encoding of these ciphers. First, some preliminary declarations:

type Char = [8];type String(l) = [l]Char;type Table(n) = [n](Char, Char);

We’ll simply assume that the input consist of “characters,” each of which will be 8-bit quantities (i.e., numbers from 0 to 255). We will simply use ASCII encoding for normal English characters. This is captured by the Char type declaration above, which simply gives a convenient name for 8-bit wide words. The second type declaration captures sized-strings: For any given size l, the type String(l) represents a sequence of length l, containing 8-bit words. For instance, String(16) is the type of all sequences of length 16, containing numbers from 0 to 255 as elements. Finally a Table of size n is simply n-pairings of characters that form a substitution. Here’s the example table we will use:

cipherTable : Table(28);cipherTable = [| (x, y) || x <- plain || y <- cipher |]where { plain = "abcdefghijklmnopqrstuvwxyz .";cipher = "oebxa.cdf hijklmnzpqtuvwrygs"};

Note that our table has 28 entries (the lower-case English alphabet, plus space and the dot). A simple Cryptol sequence-comprehension succinctly zips the sequences up, forming our example table.

Read More