In this Agda file, I give an introduction to Lawvere theories and Monads. It's not exactly a "gentle" introduction because:
- I'm still trying to grok these things myself,
- I've to live up to my reputation of writing unfathomable gagaposts.
We'll start with the State monad and, briefly, a Tick monad. The game is the following: I'll present them syntactically -- through a signature -- and semantically -- through a collection of equations. I'll then show how the usual monad is related to this presentation.
I (Thomas Williams) am currently doing an internship with Pierre Dagand and Didier Rémy on ornaments, a way to relate datatypes sharing a common structure. Ornaments come from the scary world of dependent types, and the goal of my internship is to implement them in the more familiar setting of ML datatypes. We've produced a draft where we present how ornaments can be useful in practice in an ML-like programming language.
I worked last September on better parallelization for OCamlbuild, but didn't finish the thing (I couldn't get something mergeable in time for ICFP) and got distracted with tons on other things to do since. I just uploaded the state of my branch:
format6 type is the basis of the hackish but damn
convenient, type-safe way in which OCaml handles format strings:
Printf.printf "%d) %s -> %f\n" 3 "foo" 5.12
The first argument of
printf in this example is not a string, but
a format string (they share the same literal syntax, but have
different type, and there is a small hack in the type-inference engine
to make this possible). It's type, which you can get by asking
("%d) %s -> %f\n" : (_,_,_,_,_,_) format6) in a toplevel,
is a mouthful (and I renamed the variables for better readability):
(int -> string -> float -> 'f, 'b, 'c, 'd, 'd, 'f) format6
What do those six arcane parameters mean? In the process of reviewing Benoît Vaugon's work on using GADTs to re-implement format functions, I've finally felt the need (after years of delighted oblivion) to understand each of those parameters. And that came after several days of hacking on easier-to-understand ten-parameters GADT functions; hopefully most of our readers will never need this information, but the probability that I need to refresh my memory in the future is strong enough for me to write this blog post.
This is just to let you know that you can now try Mezzo online! Mezzo runs in your browser, thanks to js_of_ocaml and a few hacks to glue the type-checker to the web interface. There's a few examples, and you can type-check and/or interpret your programs (currently, only the graph traversal actually does something observable).
There's a lot of room for improvement (currently, all files are loaded during initialization, for instance), but it believe it's nice enough already that I can release it. It is known to work in the latest versions of Firefox, Chrome and Internet Explorer.
Lately, it seems all I've been doing is writing. There's been that paper about the implementation of Mezzo, which I'm supposed to push out at some point. There's that ICFP submission (blog post pending) which we've been writing with some other cool kids from the lab. That's not counting the various presentations, research statements, and other misc. documents that I've been busy with...
Last week, I was (again!) writing, this time working on my part for a journal version of our ICFP paper, and I was trying to write a small tutorial with longer examples and a better introduction to Mezzo. After explaining tail-recursive concatenation for the n-th time, I decided to try out explaining a new example. Here's the result.
The example is about an interesting programming pattern in Mezzo, which I dub "control-flow inversion". The idea is not new: we've been using it for iterators, but is was somehow buried under the details of how iterators work. The present blog post tries to present it in a more self-contained way. It somehow showcases both the strengths and weaknesses of Mezzo as it stands right now. Hopefully we'll improve the shortcomings soon.
The following construction is based upon a note scribbled by Conor McFermat^W McBride on a draft of my PhD thesis a few months ago. Upon seeing an earlier version of this file, Stevan Andjelkovic (aka Mr Freemonadic) urged me to borrow one of his free monads to write my recursive functions.
The tl;dr: we intensionally characterise a class of Bove-Capretta predicates (using a custom-made universe of datatypes) with the guarantee that these predicates are "collapsible", ie. have no run-time cost. We then write a generic program that computes such a predicate from the recursive definition of a function. Using this framework, we can write a recursive function without being bothered by that administrative termination-checker and prove its termination after the fact.