Category Archives: Computation

The new and improved Programming languages zoo

It is my pleasure to announce the new and improved Programming languages Zoo, a potpourri of miniature but fully functioning programming language implementations. The new zoo has a decent web site, it is now hosted on GitHub, and the source code was cleaned up. Many thanks to Matija Pretnar for all the work.

The purpose of the zoo is to demonstrate design and implementation techniques, from dirty practical details to lofty theoretical considerations:

  • functional, declarative, object-oriented, and procedural languages
  • source code parsing with a parser generator
  • recording of source code positions
  • pretty-printing of values
  • interactive shell (REPL) and non-interactive file processing
  • untyped, statically and dynamically typed languages
  • type checking and type inference
  • subtyping, parametric polymorphism, and other kinds of type systems
  • eager and lazy evaluation strategies
  • recursive definitions
  • exceptions
  • interpreters and compilers
  • abstract machine

There is still a lot of room for improvement and new languages. Contributions are welcome!

 

A Brown-Palsberg self-interpreter for Gödel’s System T

In a paper accepted at POPL 2016 Matt Brown and Jens Palsberg constructed a self-interpreter for System $F_\omega$, a strongly normalizing typed $\lambda$-calculus. This came as a bit of a surprise as it is “common knowledge” that total programming languages do not have self-interpreters.

Thinking about what they did I realized that their conditions allow a self-interpreter for practically any total language expressive enough to encode numbers and pairs. In the PDF note accompanying this post I give such a self-interpreter for Gödel’s System T, the weakest such calculus. It is clear from the construction that I abused the definition given by Brown and Palsberg. Their self-interpreter has good structural properties which mine obviously lacks. So what we really need is a better definition of self-interpreters, one that captures the desired structural properties. Frank Pfenning and Peter Lee called such properties reflexivity, but only at an informal level. Can someone suggest a good definition?

Note: self-interpreter-for-T.pdf

The troublesome reflection rule (TYPES 2015 slides)

Here are the slides of my TYPES 2015 talk “The troublesome reflection rule” with fairly detailed presenter notes. The meeting is  taking place in Tallinn, Estonia – a very cool country in many senses (it’s not quite spring yet even though we’re in the second half of May, and it’s the country that gave us Skype).

Download slides: The troublesome reflection rule (TYPES 2015) [PDF].

Seemingly impossible constructive proofs

In the post Seemingly impossible functional programs, I wrote increasingly efficient Haskell programs to realize the mathematical statement

$\forall p : X \to 2. (\exists x:X.p(x)=0) \vee (\forall x:X.p(x)=1)$

for $X=2^\mathbb{N}$, the Cantor set of infinite binary sequences, where $2$ is the set of binary digits. Then in the post A Haskell monad for infinite search in finite time I looked at ways of systematically constructing such sets $X$ with corresponding Haskell realizers of the above omniscience principle.

In this post I give examples of infinite sets $X$ and corresponding constructive proofs of their omniscience that are intended to be valid in Bishop mathematics, and which I have formalized in Martin-Löf type theory in Agda notation. This rules out the example $X=2^\mathbb{N}$, as discussed below, but includes many interesting infinite examples. I also look at ways of constructing new omniscient sets from given ones. Such sets include, in particular, ordinals, for which we can find minimal witnesses if any witness exists.

Agda is a dependently typed functional programming language based on Martin-Löf type theory. By the Curry-Howard correspondence, Agda is also a language for formulating mathematical theorems (types) and writing down their proofs (programs). Agda acts as a thorough referee, only accepting correct theorems and proofs. Moreover, Agda can run your proofs. Here is a graph of the main Agda modules for this post, and here is a full graph with all modules.

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