Lambda the Ultimate

Here's a little light reading for your day-after-Labor-Day (or whatever yesterday was where you live): Unchecked Exceptions can be Strictly More Powerful than Call/CC, Mark Lillibridge and Olivier Danvy, 1999, Higher-Order and Symbolic Computation.

We demonstrate that in the context of statically-typed purely-functional lambda calculi without recursion, unchecked exceptions (e.g., SML exceptions) can be strictly more powerful than call/cc. More precisely, we prove that a natural extension of the simply-typed lambda calculus with unchecked exceptions is strictly more powerful than all known sound extensions of Girard’s Fω (a superset of the simply-typed lambda calculus) with call/cc. This result is established by showing that the first language is Turing complete while the later languages permit only a subset of the recursive functions to be written.

I have to say that on seeing the title I was surprised: I cut my functional teeth on Scheme and every baby Schemer sucks up the knowledge that call/cc lets you create all manner of flow control including exceptions. But, as the paper makes clear, that's not necessarily the case in a statically-typed context.

Edit: Citeseerx was not responding very well, here's an alternative URL for the paper.

Relational Parametricity and Units of Measure, Andrew J. Kennedy, POPL 1997.

Type systems for programming languages with numeric types can be extended to support the checking of units of measure. Quantification over units then introduces a new kind of parametric polymorphism with a corresponding Reynolds-style representation independence principle: that the behaviour of programs is invariant under changes to the units used. We prove this `dimensional invariance' result and describe four consequences.

The first is that the type of an expression can be used to derive equations which describe its properties with respect to scaling (akin to Wadler's `theorems for free' for
System F). Secondly there are certain types which are inhabited only by trivial terms. For example, we prove that a fully polymorphic square root function cannot
be written using just the usual arithmetic primitives. Thirdly we exhibit interesting isomorphisms between types and for first-order types relate these to the central theorem of classical dimensional analysis. Finally we suggest that for any expression whose behaviour is dimensionally invariant there exists some equivalent expression whose type reflects this behaviour, a consequence of which would be a full abstraction result for a model of the language.

There's a new release of F# coming out with support for measure types, and so I thought I'd post a link to Andrew's paper about the subject. Now, if you've done any physics or engineering, you're probably familiar with the fact that units can sometimes really strongly constrain what form your equations can take. If you studied dimensional analysis more carefully than I did, you might even have learned that this is a consequence of the Buckingham pi theorem -- you can prove that if you have an equation with n variables involving k physical units, you can recast it as an equation with (n - k) dimension-free variables. Kennedy shows that the analogue of this theorem for programs in his language is a form of parametricity result at first order, which is quite slick.

As a follow up to the previous post, check out how Chung-chieh Shan applied regions to a seemingly unrelated problem. His post begins by explaining how automatic (numerical) partial differentiation can be implemented, and goes on to show how to use regions to avoid mixing-up the variables being differentiated.

Oleg Kiselyov and Chung-chieh Shan. Lightweight Monadic Regions. Haskell'08.

We present Haskell libraries that statically ensure the safe use of resources such as file handles. We statically prevent accessing an already closed handle or forgetting to close it. The libraries can be trivially extended to other resources such as database connections and graphic contexts...

Region annotations are part of an expression's inferred type. Our new Haskell encoding of monadic regions as monad transformers needs no witness terms. It assures timely deallocation even when resources have markedly different lifetimes and the identity of the longest-living resource is determined only dynamically.

For contrast, we also implement a Haskell library for manual resource management, where deallocation is explicit and safety is assured by a form of linear types. We implement the linear typing in Haskell with the help of phantom types and a parameterized monad to statically track the type-state of resources.

I am starting to think we need a department for effect systems and related topics (though we managed without a monads department!)...

You'll probably want to read the code, so go ahead. The code makes it plain which features of the type system are needed to achieve the end result.

Parametric Higher-Order Abstract Syntax for Mechanized Semantics

We present parametric higher-order abstract syntax (PHOAS), a new approach to formalizing the syntax of programming languages in computer proof assistants based on type theory. Like higher-order abstract syntax (HOAS), PHOAS uses the meta language's binding constructs to represent the object language's binding constructs. Unlike HOAS, PHOAS types are definable in general-purpose type theories that support traditional functional programming, like Coq's Calculus of Inductive Constructions. We walk through how Coq can be used to develop certified, executable program transformations over several statically-typed functional programming languages formalized with PHOAS; that is, each transformation has a machine-checked proof of type preservation and semantic preservation. Our examples include CPS translation and closure conversion for simply-typed lambda calculus, CPS translation for System F, and translation from a language with ML-style pattern matching to a simpler language with no variable-arity binding constructs. By avoiding the syntactic hassle associated with first-order representation techniques, we achieve a very high degree of proof automation.

I was aware of this some months ago now, but held back commenting on it at Adam's request until it had been accepted for publication, which it now apparently has. This is (one element of) Adam's continued work on LambdaTamer, his Coq-based environment for building certified compilers. There is a new version of LambdaTamer using this parametric higher-order abstract syntax approach. The new version also works in current and future versions of Coq, unlike the previous version. Finally, Adam is apparently working on a paper regarding "type-theoretic denotational semantics for higher order, impure object languages" and is post-docing with Greg Morrisett. The relationship between Adam's work and the YNot project is a bit unclear to me; perhaps either Adam or Greg could help clarify that.

Update: Whoops. I got ahead of myself and neglected to notice that the paper is not actually yet available, although the new version of LambdaTamer is. So at the moment, this story is merely to note that the paper exists and to provide a link to the new LambdaTamer. My apologies to Adam and the LtU readership.

2nd Update: The paper is now available at the link, in either PostScript or PDF form.

Benjamin C. Pierce's presentation slides (in PDF) for his talk on Types Considered Harmful. The talk starts out discussing some of the general advantages and disadvantages of static typing. But the aim of the talk centers on the problems of building a type checker for the Boomerang Programming Languague (an offshoot of harmony).

• Boomerang language design as an example of
  1. the need for very precise types
  2. some of the technical problems they raise
• Contracts as an attractive way of addressing some of these issues

Pierce's work is currently centered on creating a PL for Bidirectional Programming. A work in progress but it's interesting to see the thought process behind language design in real-time.

Flexible types: Robust type inference for first-class polymorphism, by Dan Leijen:

We present HML, a type inference system that supports full firstclass polymorphism where few annotations are needed: only function parameters with a polymorphic type need to be annotated. HML is a simplification of MLF where only flexibly quantified types are used. This makes the types easier to work with from a programmers perspective, and we found that it simplifies the implementation of the type inference algorithm significantly. Still, HML retains much of the expressiveness of MLF, it is robust with respect to small program transformations, and has a simple specification of the type rules with an effective type inference algorithm that infers principal types. A reference implementation of the type system is available at: http://research.microsoft.com/users/daan/pubs.html.

Seems there's been a flurry of activity on first-class polymorphism recently, with HML, FPH, and HM^F as simpler alternatives to full ML^F.

FPH: First-class Polymorphism for Haskell, by Dimitrios Vytiniotis, Stephanie Weirich and Simon Peyton Jones:

Languages supporting polymorphism typically have ad-hoc restrictions on where polymorphic types may occur. Supporting “firstclass” polymorphism, by lifting those restrictions, is obviously desirable, but it is hard to achieve this without sacrificing type inference. We present a new type system for higher-rank and impredicative polymorphism that improves on earlier proposals: it is an extension of Damas-Milner; it relies only on System F types; it has a simple, declarative specification; it is robust to program transformations; and it enjoys a complete and decidable type inference algorithm.

Under Related Work, the authors provide a detailed comparison of their system with ML^F, and HM^F.

HMF: Simple type inference for first-class polymorphism - Daan Leijen, Draft April 8, 2008.

HMF is a conservative extension of Hindley-Milner type inference with first-class polymorphism and regular System F types. The system
distinguishes itself from other proposals with simple type rules and a very simple type inference algorithm that is just a small extension
of the usual Damas-Milner algorithm. Given the relative simplicity and expressive power, we feel that HMF can be a very attractive
type system in practice. There is a reference implementation of the type system available at: http://research.microsoft.com/users/daan/pubs.html.

An excellent paper even in its current draft form. I also placed this under the learning category, because Daan's writing style is lucid enough that the concepts can be understood by relative newcomers to the field of type theory

Algebra of programming using dependent types. S-C. Mu, H-S. Ko, and P. Jansson.

Dependent type theory is rich enough to express that a program satisfies an input/output relational specification, but it could be hard to construct the proof term. On the other hand, squiggolists know very well how to show that one relation is included in another by algebraic reasoning. We demonstrate how to encode functional and relational derivations in a dependently typed programming language. A program is coupled with an algebraic derivation from a specification, whose correctness is guaranteed by the type system.

Code accompanying the paper has been developed into an Agda library AoPA.