Call by Name (CBN) is dual to Call By Value (CBV)

Probably one of the best papers I've read on the relationship between CBN, CBV and the Curry-Howard correspondance is the paper Call-by-value is dual to call-by-name by Wadler. The calculus he develops for describing the relationship shows an obvious schematic duality that is very visually appealing. After reading the paper that I mentioned earlier on Socially Responsive, Environmentally Friendly Logic (which shall henceforth be called SREF Logic), it struck me that it would be interesting to see what a CPS (Continuation-passing Style) like construction looks like in SREF logic, so I went back to the Wadler paper to see if I could figure out how to mimic the technique for multi-player logic. It looks like the formulation by Wadler comes out directly by thinking about logic as a two player game! I'm excited to see what happens with n-player logic. This has been a big diversion from what I'm actually suppose to be working on but I didn't want to forget about it :).

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Decidable Equality in Agda

So I've been playing with typing various things in System-F which previously I had left with auxiliary well-formedness conditions. This includes substitutions and contexts, both of which are interesting to have well typed versions of. Since I've been learning Agda, it seemed sensible to carry out this work in that language, as there is nothing like a problem to help you learn a language. In the course of proving properties, I ran into the age old problem of showing that equivalence is decidable between two objects. In this particular case, I need to be able to show the decidability of equality over types in System F in order to have formation rules for variable contexts. We'd like a context Γ to have (x:A) only if (x:B) does not occur in Γ when (A ≠ B). For us to have statements about whether two types are equal or not, we're going to need to be able to decide if that's true using a terminating procedure. And so we arrive at our story. In Coq, equality is ...

Total Functional Programming

Recently on Lambda the Ultimate there was a post called "Total Functional Programming". The title didn't catch my eye particularly, but I tend to read the abstract of any paper that is posted there that doesn't sound terribly boring. I've found that this is a fairly good strategy since I tend to get very few false positives this way and I'm too busy for false negatives. The paper touches directly and indirectly on subjects I've posted about here before. The idea is basically to eschew the current dogma that programming languages should be Turing-complete, and run with the alternative to the end of supplying "Total Functional Programming" . At first glance this might seem to be a paper aimed at "hair-shirt" wearing functional programming weenies. My reading was somewhat different. Most hobbiest mathematicians have some passing familiarity with "Turing's Halting Problem" and the related "Goedel's Inco...

Managing state in Prolog monadically, using DCGs.

Prolog is a beautiful language which makes a lot of irritating rudimentary rule application and search easy. I have found it is particularly nice when trying to deal with compilers which involve rule based transformation from a source language L to a target language L'. However, the management of these rules generally requires keeping track of a context, and this context has to be explicitly threaded through the entire application, which involves a lot of irritating and error prone sequence variables. This often leads to your code looking something a bit like this: compile(seq(a,b),(ResultA,ResultB),S0,S2) :- compile(a,ResultA,S0,S1), compile(b,ResultB,S1,S2). While not the worst thing, I've found it irritating and ugly, and I've made a lot of mistakes with incorrectly sequenced variables. It's much easier to see sequence made explicitly textually in the code. While they were not designed for this task, but rather for parsing, DCGs turn out to be a conveni...

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