Existentials and Zero-Knowledge Proofs

When I use "∃ x . P(x)" in typed programming (using intuitionistic logic), I must have in hand an x and a proof of P(x). This is not the case in the classical logic, in which I might have a proof of ∃ x . P(x) without knowing any details of x at all.

This idea that I know such an x exists though I have no knowledge of it reminds me of zero-knowledge proofs. Of course, the zero-knowledge proofs referenced in the Wikipedia article aren't so much proofs as they are assurances with high probability. I looked around for zero-knowledge proofs that were more proof-like, but I didn't find much. I wonder if there is any deeper connection between classical existentials and zero-knowledge proofs?

Conor's Rule?

Edwin Brady writes in "How to write programs in two easy steps", that

I think it's also an area where dependently typed languages like Epigram will shine, because it will one day be possible to write verified interpreters for domain specific languages …

I immediately thought "I can write dependently typed programs right now with GADTs, and I can do so in GHC Haskell, Scala, or the upcoming C#". This is, of course, only mostly true, since none of these does termination checking, and the syntax is awkward.

The same blog post later references Greenspun's Tenth Rule:

Any sufficiently complicated C or Fortran program contains an ad hoc, informally-specified, bug-ridden, slow implementation of half of Common Lisp

and I thought that my programs follow a similar pattern with dependent types: I'm always simulating them informally. Based on Section 6.1: "The Trouble with Faking It" in Conor McBride's Faking It (Simulating Dependent Types in Haskell) (mirror), I wonder if there's any truth to something we might call

Conor's Rule

Any sufficiently complicated Haskell or ML program contains an ad hoc, informally-specified, bug-ridden, half-completed simulation of dependent types.

Definability and Continuity

In How Many Functions are There of Type Bool -> Bool?, I stated "The trick is to remember that if f is continuous, then f _|_ =/= _|_ implies that for all x, f x = f _|_". That is false. It would have been true had I said "… remember that if f is definable in Haskell, then …". The correct statement for continuous functions is: for all x, f x ≥ f _|_. The "≥" here is that used in domain theory to talk about CPOs.

In addition to only considering functions definable in Haskell, I was also only considering flat CPOs, in which each element is comparable only to itself and _|_. An example of a CPO that is not flat:

data Wrap = Wrap Unit
data Unit = Unit

In the CPO Wrap_{_|_}, _|_ < Wrap _|_ < Wrap Unit. Though Wrap_{_|_} has three values, just like Bool_{_|_}, there are only nine functions (continuous or definable in Haskell) of type Wrap_{_|_} -> Bool_{_|_}. Furthermore, there are eleven continuous functions of type Wrap_{_|_} -> Wrap_{_|_}, and all are definable in Haskell. (Thanks to Stefan O'Rear for pointing that out).

Clearly, the codomain isn't treated opaquely as it was in the last post: in order to count functions, we need to know the ordering on the codomain.

I have some more thinking to do about counting, continuity, and definability before I can make any more assertions about the cardinality of function types.

Update:

Stefan O'Rear pointed out an error in which I claimed certain functions were not definable in Haskell, when they actually are. A better example of a function that is continuous but not definable in Haskell is:

parallel_or _ True = True
parallel_or True _ = True
parallel_or False False = False

C++ and GADTs

I just noticed this post about GADTs in C++. It's more about C++'s warts than GADTs, but it's interesting nonetheless.

I also noticed recently, while preparing for my upcoming talk at the theory lunch (mirror, suspected future permalink, presently 404), that because the constructors in the OOP version of GADTs each get their own class, the class name can function as the constructor tag. This leads to what is called the Curiously Recurring Template Pattern in C++. Example:

Haskell	C++
data Nat t where Z :: Nat ZTag S :: Nat n -> Nat (STag n) data ZTag data STag n	template<class T> class nat {}; class zero : nat<zero> {}; template<class N> class succ : nat<succ<N> > { public: succ<N>(nat<N> x) : pred(x) {} nat<N> pred; };

Of course, the tags in the Haskell version could have the same names as the constructors, since the constructor and type namespaces in Haskell are disjoint, but they would still have distinct meanings, unlike in the C++ code.

Types Feed

I have added to the sidebar a feed for the blogs I read about types. I made this feed using Google reader, which also prepared a blog-like interface and an atom feed. Some of my posts will be appearing in the feed, since I both read and am an inhabitant of Planet Haskell.

If I Had Time

Projects I would work on if I had time:

• On Simulating Dependent Types with Guarded Algebraic Datatypes:
  • Automatic lambda lifting for the Haskell syntax.
  • Automatic printing of Scala and C# syntax
  • A lambdabot plugin for lifting types, using GHC-as-a-library.
• Sized lists:
  • Make a cabal-ized, hackage-d sized lists library.
  • Automatically translate the buggy NoFib benchmarks to using sized lists to see if the bugs get caught by the type checker.
• Write Data.Scannable and instance Foldable ((->) a)

Missing Morphisms

I'm a fan of Data.Foldable, and I wonder if it could be extended to function types by way of Bananas in Space: Extending Fold and Unfold to Exponential Types (mirror) or Boxes Go Bananas: Encoding Higher-Order Abstract Syntax with Parametric Polymorphism (mirror).

Second, I understand that it may not be possible to give simple types to anamorphisms or hylomorphisms (mirror), but I don't see there can't be a Data.Scannable with paramorphisms.

Finally, I'm not even so sure that we can't give all of the bananas, lenses, envelopes and barbed wire generic (but more strongly typed than SYB) interfaces for even non-regular datatypes with initial algebra semantics (code here).

Update:

I think the first and second questions above are worth bothering the Haskell libaries mailing list about, so I did: Paramorphisms / Data.Scanable? (mirror), Catamorphisms for arrows? (mirror). As before, I expect I'll have to push harder (by actually writing some code) if I want any changes made.

Countable Ordinals in Haskell

In Haskell: What happens when you divide infinity by 2?, Eric Kidd asks:

...
2. Is it possible to represent any other (more interesting and/or more correct) definitions of infinity in Haskell?
3. What’s the best way to think about the infinite ordinals?

The code below is my answer to those questions for countable ordinals only. I'm not sure if it makes sense to talk about representing uncoutable sets on the kind of computers we have now: We can talk about infinite countable sets as computable limits of finite objects, but what would an uncountable set look like? They can't be objects of Nat -> Bool, since every object of this type is represented in some finite number of bits. It seems like we are stuck in a universe where Goedel's V=L is true.

{-# OPTIONS_GHC -fglasgow-exts #-}

{- See:
  http://en.wikipedia.org/wiki/Ordinal_arithmetic
  http://en.wikipedia.org/wiki/Large_countable_ordinals

  To add later:
  http://citeseer.ist.psu.edu/487179.html
-}


module Countable where

data Finite = Z
           | S !Finite

{- A countable ordinal is 0, a successor, or a limit of a countable
  sequence. This definition is weak, as the countable sequence should
  be strictly increasing, but that's hard to express. One way to do so
  would be to interpret (Limit f) as the limit of
  (sum i = 0 to n) (1 + f i). Note: this addition is not commutative.
  This would complicate things, however, so I'll leave it for a later
  extension.
-}
data Ordinal = Zero
            | Succ !Ordinal
            | Limit (Finite -> Ordinal) deriving (Show)

{- I believe that there is no computable Eq or Ord instance for Ordinals.
  Comments?
 
  We make an approximation here by taking the nth value of the limit.
  This calculation is very slow, and, of course, will equate a lot of
  ordinals that are not actually equal.
-}

nth :: Ordinal -> Finite -> Integer
nth Zero _ = 0
nth (Succ x) n = 1+(nth x n)
nth (Limit f) n = nth (f n) n

instance Show (Finite -> Ordinal) where
   show f = show [f Z,
                  f (S Z),
                  f (S (S Z)),
                  f (S (S (S Z)))]

instance Eq Ordinal where
   x == y = (show x) == (show y)

{- The basic ordinal arithmetic. Addition and multiplication are not
  commutative.
-}
addOrdinal :: Ordinal -> Ordinal -> Ordinal
addOrdinal x Zero = x
addOrdinal x (Succ y) = Succ (addOrdinal x y)
addOrdinal x (Limit y) = Limit (\z -> addOrdinal x (y z))

multiplyOrdinal :: Ordinal -> Ordinal -> Ordinal
multiplyOrdinal x Zero = Zero
multiplyOrdinal x (Succ y) = addOrdinal (multiplyOrdinal x y) x
multiplyOrdinal x (Limit y) = Limit (\z -> multiplyOrdinal x (y z))

expOrdinal :: Ordinal -> Ordinal -> Ordinal
expOrdinal x Zero = Succ Zero
expOrdinal x (Succ y) = multiplyOrdinal (expOrdinal x y) x
expOrdinal x (Limit y) = Limit (\z -> expOrdinal x (y z))

-- The first infinite ordinal
finite :: Finite -> Ordinal
finite Z = Zero
finite (S x) = Succ (finite x)

omega = Limit finite

-- Apply a f some number of times, starting at z
apply Zero f z = z
apply (Succ n) f z = f (apply n f z)
apply (Limit g) f z = Limit (\x -> apply (g x) f z)

-- With apply, we have more succinct definitions of arithmetic
addOrdinal' x y = apply y Succ x
multiplyOrdinal' x y = apply y (flip addOrdinal x) Zero
expOrdinal' x y = apply y (flip multiplyOrdinal x) (Succ Zero)

-- Enumerates the fixed points of a function.
fix Zero f = apply omega f Zero
fix (Succ n) f = apply omega f (Succ (fix n f))
fix (Limit g) f = Limit (\x -> fix (g x) f)

-- Some larger ordinals:
epsilon n = fix n (expOrdinal omega)

-- The Veblen heirarchy:
veblen Zero b = expOrdinal omega b
veblen (Succ a) b = fix b (veblen a)
veblen (Limit a) b = Limit (\x -> veblen (a x) b)

-- Feferman-Schutte ordinals
gamma n = fix n (flip veblen Zero)

Quotient Types for Information Hiding

Imagine you're writing a datatype for representing sets. You want to hide the internal representation of sets from clients, but you also want to allow clients to perform any operations as efficiently as if the internal representation were exposed. This is not easy, and it's why Data.Set has 36 functions, when all that are really needed to ensure the meaning of a set ("These elements are distinct") are toList and fromList. Of course, writing member :: Ord a => a -> Set a -> Bool as

member x y = Data.List.elem x (toList y)

is just awful, but the price we pay for efficient member is having to open up the module and break the abstraction if one of the 34 other functions doesn't do exactly what we want. In addition to this pain, there's some danger: functions like mapMonotonic are not really safe, and cannot be made so.

The papers I've been reading about quotient types:

• Bart Jacobs' Quotients in Simple Type Theory
• Aleksey Nogin's Quotient Types: A Modular Approach. (mirror 1, mirror 2)

We could alleviate all of this mess with quotient types. Quotient types allow modules to expose their internals with no danger of breaking abstractions. The quotients are the same as the mathematical quotients in abstract algebra, where they are used frequently. Back on the type theory side, the elimination rule for typing quotients depends not just on types, but on terms, and so requires a type system with dependent types. That's a shame, since dependent types are so tricky to typecheck. It would be great if there were some form of lightweight quotient types that didn't require the full power of dependent types.

The break-the-abstraction vs. lose-efficiency-&-safety issue reminds me of the difference between GADTs and lightweight static capabilities: GADTs are verbose, but they allow the client to safely and efficently operate on datatypes in ways that aren't covered by the library.

How Many Functions are There of Type Bool -> Bool?

This post is incorrect. See the corrections.

Or, rather, how many continuous functions are there of type Bool_{_|_} -> Bool_{_|_}? I count 11, which seems like a strange number. The trick is to remember that if f is continuous, then f _|_ =/= _|_ implies that for all x, f x = f _|_. This gives us only two functions which take _|_ to non-_|_:

one = const True
two = const False

The rest of the continuous functions fill the space Bool -> Bool_{_|_}. Since Bool_{_|_} is of size 3, there are 3² remaining functions, which gives us 9+2 = 11 functions.

We can generalize to say that A_{_|_} -> B_{_|_} has |B| + (|B|+1)^|A| inhabitants.

Bug Reporting

A few months ago, I posted a quandary to haskell-cafe about seemingly ineffable types in GHC: for a certain term, the type inferred by GHCi could not simply be listed in the source code, as this would cause the type checker to complain. I didn't get any bites on my post, and I chalked it up to some complications I didn't understand and no one had time to explain, and I went on to thinking about other things.

Last year, I graded a C++ assignment for a class at the local university (and my alma mater). We required the use of std::set, and a student noticed that he could modify the value of a member of a std::set in place, and the compiler wouldn't complain.

Both of these turn out to be bugs, in GHC and in the C++ standard. The lessons I've learned about programming languages:

1. If something doesn't make sense, investigate. If it turns out you just don't understand something that is the way it is for complicated reasons, you still will have learned something useful.
2. The squeaky wheel gets the grease. I asked the Haskell Cafe about the GHC infelicity, but didn't inquire at the GHC-specific mailing lists, and I didn't keep asking when nobody responded.
3. When you see something funny looking, don't just ask people why it's funny looking. Instead, push things to their logical conclusion, and find a test case that's obviously wrong. This will give people a reason to listen to you.

Foralls, Redexes, and Type Lambdas

In my last post about universally quantified types, I was asking why GHC disallows datatypes like Evil while allowing datatypes like Fine:

data OK (x :: * -> *) where
   OK :: OK x
type Fine = OK Maybe
type Evil = OK (forall (f  :: * -> *) . f)

This seems more confusing when Evil' is just fine:

data OK' (x :: *) where
   OK' :: OK' x
type Fine' = OK' Int
type Evil' = OK' (forall (f  :: *) . f)

Ken Shan answered my question: In the static semantics of the typed lambda calculus, only forms like (Λt:k.r) @ s are redexes. This makes sense to me, but it seems like it's so for simplicity reasons. I expect that this makes the metatheory simpler.

Unfortunately, this makes some things in GHC a pain, since GHC doesn't have type lambdas. SPJ suggests that this is because it makes unification for type inference impossible, but I don't yet understand why. The feature list for EHC indicates that it does support type lambdas, though I haven't tested this.

Update: (29 Jan 2007)

A History of Haskell points to A system of constructor classes (mirror) regarding unification and type lambdas.

Foralls, Kinds with Arrows, and Impredicativity (?)

I can't figure out why the kinding rule for universal quantification in F_ω is:

           C, x : k1 |- y : *
(1)    -----------------------------
       C |- (\forall x : k1 . y) : *

That is, why the type inside the universal quantification must be of kind *.

I would expect the rule to be

            C, x : k1 |- y : k2
(1')    ------------------------------
        C |- (\forall x : k1 . y) : k2

Using (1), I can still get

              C, x : k1 |- y : * -> *
(2)    ------------------------------------------------------
       C |- (\Lambda z : * . \forall x : k1 . y @ z) : * -> *

by eta-expansion, so using (1) instead of (1') doesn't seem to reduce the available power, it just makes the notation more annoying.

I suspect that the reason for (1) instead of (1') has something to do with impredicativity, since (1) is only valid in impredicative systems anyway.

Non-constructive Proofs and Programming

Two excellent examples of programming with non-constructive proofs are given in Simplifying Programs Extracted from Classical Proofs by Yevgeniy Makarov. The examples they give are better than the immediate example I think of: the proof that there are some two irrational numbers x and y such that x^y is rational. This proof is simple to do, but requires some heavy background machinery for rational and irrational numbers. Perhaps I could code this proof by using _|_ for the theorems about rational numbers that I know to be true, but don't wish to prove.

The other two examples of simple non-constructive proofs that I think of are the irrationality of the square root of 2 and the infinitude of the primes. Of course, those are usually stated in non-constructive ways, but they are actually constructive with only a slight bit of tweaking. I haven't read enough of the Makarov paper to see if the simplifications proposed would turn these non-constructive proofs into constructive ones.

Non-constructive Proofs and Productivity

Languages in the Dependent ML family allow users to ensure termination by using a termination metric, a number that is strictly decreasing on recursive calls in the definition of the function.

There is a counterpart to termination for co-recursive structures, where we need to show that a function produces values indefinitely. So, for the Collatz conjecture, we want to show termination for:

collatz n = collatz' n []
collatz' 1 r = 1:r
collatz' n r =
    if n `mod` 2 == 0
    then collatz' (n `div` 2) (n:r)
    else collatz' (3*n + 1) (n:r)

But for the twin prime conjecture we want to show the productivity of:

data Stream a = Stream a (Stream a)
twins = twins' 3
twins' n =
    if (primep n) && (primep (n+2))
    then Stream (n,n+2) (twins' (n+2))
    else twins (n+2)

I don't know if any of the members of the DML family can check productivity. Imagine they could, and imagine we had a non-constructive proof of the twin primes conjecture. Could we use that proof to allow a definition of twins to be marked productive?

Most of the time, we talk about the Curry-Howard correspondence in reference to intuitionist logic and functional programming, but non-constructive proofs of productivity seem like a good way to be able to use more powerful logic in programming, since they don't have to change the implementation of the function they are attached to.

Static Security Assurance From Afar

I work at an e-commerce company. I was thinking today of how typed programming would make our code more reliable, and I began to wonder about using types (or other static assurance methods) to ensure that we don't accidentally reveal information to parties that shouldn't see it. I'm thinking here of the Apollo project (specifically, Translating dependency into parametricity) or Flow Caml. I don't see a way for us to use technology like this when working with other companies.

As an example, we might pass a customer's info to a third party site along with some identifying informaiton about us, to verify the third party that the customer is legit. How can I trust the third party site to not reveal our shared secret to the customer?

I suspect this is a problem for cryptographers, not type theorists. Since much of the information that we deliver to customers is tainted by using secrets to obtain it, static analysis wouldn't help maintain security.

Type (Dis)equality

Type equality is the essential ingredient in guarded algebraic datatypes:

data Z ; data S n ;

data ListGADT a n where
    NilListGADT :: ListGADT a Zero
    ConsListGADT :: a -> ListGADT a n -> ListGADT a (S n)

data TypeEqual a b = . . .
data ListEq a n = NilListEq (TypeEqual n Z)
                | forall m . ConsListEq (TypeEqual n (S m)) a (ListEq a m)

What new power would type disequality bring? It might prevent infinite types.

data WrapList a = forall n . WrapList (ListGADT a n)

cons :: a -> WrapList a -> WrapList a
cons x (WrapList y) = WrapList (Cons x y)

ones = cons 1 ones

Here, the size type inside ones is S (S (S . . . . That's bad. What if we wrote:

data TypeDisEq a b where
    DisEqLess :: DisEq Z (S a)
    DisEqMore :: DisEq (S a) Z
    DisEqInduct :: DisEq a b -> DisEq (S a) (S b)

data ListGADTDisEq a n where
    NilListGADTDisEq :: ListGADTDisEq a Zero
    ConsListGADTDisEq :: a -> ListGADTDisEq a n -> (TypeDisEq n (S n)) -> ListGADTDisEq a (S n)

Now, since the size type inside a ListGADTDisEq can't be one more than itself, it's finite, and we might not be able to form ones. Sadly, that's not true:

data WrapListDisEq a = forall n . WrapListDisEq (ListGADTDisEq a n)

cons' :: a -> WrapListDisEq a -> WrapListDisEq a
cons' x (WrapListDisEq NilListGADTDisEq) = WrapListDisEq (ConsListGADTDisEq x Nil DisEqBase)
cons' x (WrapListDisEq p@(Cons q r s)) = WrapListDisEq (ConsListGADTDisEq x p (DisEqInduct s))

ones' = cons' 1 ones'