writing custom type systems for python in prolog


I play with Prolog, I find out I can infer custom type systems in under 50 lines of it and after that I create hatlog to feed it with some Python. In the end I find out my programs ~= 2 parallel programs: type inference / compile-time prolog and the normal program.

prologue (heh)

I was playing with Prolog last weekend and I was wondering how can I apply it for something different than the usual uni-solve-8-queens type of thing. Prolog is often accepted as “that bizarre magic logical programming language from uni”, so I wanted to actually build something real-world-y in it.

Most Prolog implementations have a repl, so it’s pretty easy to just start modeling relationships between .. things. I am very interested in prog languages, so sooner or later I started trying to create rules for a simple language. I started describing the signatures of several “builtin” functions and it was pretty and neat:

push(list, any, list).

It didn’t take long to realize I can do

?- push(list, any, X).
X = list.

?- push(Receiver, Element, list).
Receiver = list,
Element = any.

The builtin Prolog unification gave me type inference for free. This looks even cooler when you realize you can do stuff like

sub(int, int).
?- push(X, any, X), sub(X, int).

You can apply this to a whole “program” and basically type check it and infer all the types.

However list and any don’t help much! So I decided to support generic types in my signatures. Accidentally Prolog’s free variables seemed like a good solution:

push([list, T], T, [list, T]).
?- push(X, z, X).
X = [list, z].
?- push(X, z, [list, v]).

my prolog milk brings all the pythons to the backyard

I had a very simple type inference engine but it worked only on some imaginary facts. Those facts did look suspiciously similar to real programs though. Actually if you just write a simple fibonacci in Python, you can easily map it to rules like these: if is if(Test, IfBlock, ElseBlock, IfType), == is eq(Left, Right, EqType) and + is bin_op(Left, Right, BiType). I mapped several simple functions to prolog rules like that and I realized the type inference was working ok for that too.

So then hatlog was born.

hatlog takes a one-function python program:

bin/hatlog examples/map.py
# A,B::Callable[[A],B] -> List[A] -> List[B]

The cool part is that all the Python-related type rules are isolated in pythonTypeSystem.pl and the generated programs just use that as a library.

That way you can easily write your own custom.pl and use your own swappable custom type system: you can have a simple declarative representation of it just like grammars can represent parsers!

the type system

If you look at pythonTypeSystem.pl you will see ~50 lines of code, but most of the magic is happening from line 29 to 57: the lines before that define some common interfaces and the lines after 57 just contain signatures for several example standard library methods.

However just 1-2 simple relations are enough to describe the rules for most nodes:

z_index([list, X], int, X).
z_index([dict, X, Y], X, Y).
z_index(str, int, str).
z_for(Element, Sequence, _, void) :- sequence(Sequence), element_of(Element, Sequence).

the generated program just uses this rules with some free variables for unknown types:

f(map, [Z0, Z1], X) :-
    z_list(Z3, Z4),
    z_assign(Z2, Z4, Z5),
    z_fcall(Z0, [Z6], Z7),
    z_method_call(Z2, append, [Z7], Z8),
    z_for(Z6, Z1, [Z8], Z9),
    =(Z2, X).

Here prolog originally unifies Z0 with [function, _G2, _G4], Z1 with [list, _G2] and X with [list, _G4]. _G.. are free variables and most of the time we try to bind constants to them. However in this case they can be used as solutins: they correspond to generic types, and you can see they repeat twice: we have unification of generic types.

We do some post-processing work behind the scenes with Prolog and reformat the result to

A,B::Callable[[A],B] -> List[A] -> List[B]

I’ve tried to match the types with mypy’s rules.


In the end, I’ll try to summarize my feelings about that.

Everybody who does PL theory more seriously probably knows that, but I didn’t realize how close type inference and prolog-like systems look. Of course this project is an oversimplification, I support only a subset of Python for a proof of concept.

However now I can imagine a statically typed program like 2 parallel programs:

Is that close to truth? Is there a deeper link between logical programming and type theory? Any experiments like this? I’d love to hear more about this kind of stuff

In the end I enjoyed that, Prolog is actually pretty ok even for writing “normal” code and this very high level type of rules are way better experience than e.g. implementing type inference for Pseudo-Python in Python(for prototyping)