add ats post
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---
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title: a little ats program
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date: 2022-09-16
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tags: [computer, ATS, cool languages]
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show-toc: true
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...
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[ats] is a language that is a little infamous among type system likers, for its
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complexity, its weird syntax, and its thin documentation. well i won't be beat,
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so here's a little implementation of simply-typed lambda calculus. right now it
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doesn't actually use many ats-specific features, since i just didn't end up
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needing them.
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[ats]: http://www.ats-lang.org
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it does leak memory without gc though, so maybe i should have.
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oh well one thing at a time
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# installation, et cetera
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instead of trying to install ats myself, i just used [nix], since i already
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have it installed as the least annoying package manager for idris. i just run
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`nix shell nixpkgs#ats2` and i'm in a shell with `patscc` and `patsopt`
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available. if you want to try building it yourself, that's, uh, your choice.
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[nix]: https://nixos.org
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to compile the program, i wrote a makefile. just a small one!
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```makefile
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ATSCC := patscc
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ATSFLAGS := \
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-DATS_MEMALLOC_GCBDW -lgc \
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-atsccomp \
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'gcc -I$${PATSHOME} -I$${PATSHOME}/ccomp/runtime \
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-L$${PATSHOME}/ccomp/atslib/lib -std=gnu99'
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%: %.dats
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$(ATSCC) $< $(ATSFLAGS) -o $@
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```
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(in some installs the compiler is just called `atscc` for some reason.)
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the first line of `ATSFLAGS` tells it which allocator to use. in this case,
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`libgc`. the rest tells it to use `-std=gnu99`, so that it can find `alloca()`;
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the rest is the same as the default.
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one last thing. the first two lines of the program have to be this. i didn't
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look into it too hard but they seem to define... something... and if you forget
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to include them the c output by ats isn't even syntactically correct. ouch.
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```ats
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#include "share/atspre_define.hats"
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#include "share/atspre_staload.hats"
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```
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# types
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let's start by defining the syntax for types. it's going to be a single base
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type, and functions.
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$$
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\newcommand\Typ{\mathit{typ}}
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\newcommand\OR{\mathrel|}
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\Typ ::= \bullet \OR \Typ_1 \to \Typ_2
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$$
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so obviously, my first attempt was to just write this:
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```ats
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datatype typ = Base | Arr of (typ, typ)
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```
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i'm going to need to check two types for equality, so let's do that next.
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```ats
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fun eq_typ (a : typ, b : typ) .<a>. :<> bool =
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case (a, b) of
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| (Base(), Base()) => true
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| (Arr(a1, a2), Arr(b1, b2)) => eq_typ(a1, b1) && eq_typ(a2, b2)
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| (_, _) => false
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```
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alas: (it is talking about the `.<a>.`)
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```
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/home/niss/ats/lam.dats: 149(line=7, offs=33) -- 150(line=7, offs=34):
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error(2): the static identifier [a] is unrecognized.
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```
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surprise! as far as i can tell ats only allows termination checking to be
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based on _static_ (compile time) information. so i need to index the
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datatype with, uh, something. when we define functions on syntax like this,
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they basically always recurse on a subterm. so the height of the syntax tree
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will do. what i _actually_ have are these more messy definitions.
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```ats
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datatype typ(int) =
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| Base(1)
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| {h1, h2 : nat} Arr(1 + max(h1, h2)) of (typ(h1), typ(h2))
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typedef typ = [h : nat] typ(h)
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```
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the base type $\bullet$ has height 1, and an arrow type has a height one more
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than the max height of its subterms. `nat` is a version of `int` constrained to
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be at least zero, rather than a separate type, so that's why it says `int` one
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place and `nat` the other. the typechecker has a linear arithmetic solver for
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reasoning about inequalities and expressions like `1 + max(h1, h2)`.
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i don't want to be talking about "types of height $h$" unless i really have to,
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though, so there's a definition for just _a_ type too. the syntax `[x : t1] t2`
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is an existential type that is automatically wrapped and unwrapped, so in this
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case it is a `typ(h)` for some, unknown, `h`. also, names can be overloaded by
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their arity so `typ(h)` and `typ` can both be defined in the same scope.
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so now `eq_typ` has to have a more intricate type to talk about height. the
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definition is still the same though.
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```ats
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fun eq_typ {ha : nat} .<ha>. (a : typ(ha), b : typ) :<> bool =
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case (a, b) of
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| (Base(), Base()) => true
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| (Arr(a1, a2), Arr(b1, b2)) => eq_typ(a1, b1) && eq_typ(a2, b2)
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| (_, _) => false
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overload = with eq_typ
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```
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since i only actually need to care about one of the arguments' height for the
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termination proof, the other can still just be a `typ`. the last line lets me
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write `a = b` instead of `eq_typ(a, b)`, which is nice.
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next, i want to be able to print types. ideally this would be a function that
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returns a string, but string stuff involves pointer weirdness i haven't worked
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out yet. so it just prints. the only two precedence levels in these types are
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"left of an arrow" and "not left of an arrow", so that can just be a `bool`.
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unfortunately `print` has no information about its termination behaviour so
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there's no point bothering with that stuff here either. :confused:
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```ats
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fun print_typ (parens : bool, t : typ) : void =
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case t of
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| Base() => print "•"
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| Arr(a, b) => begin
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if parens then print("(");
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print_typ(true, a);
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print(" → ");
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print_typ(false, b);
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if parens then print(")");
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end
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fn print_typ (t : typ) : void = print_typ(false, t)
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overload print with print_typ
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```
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# terms
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types are done, now let's move on to terms. we just want names, lambdas, and
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application.
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$$
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\newcommand\Term{\mathit{term}}
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\newcommand\Var{\mathit{var}}
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\Term ::= \Var \OR \lambda \Var : \Typ. \Term \OR \Term_1 \; \Term_2
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$$
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for the same reason as before, terms are indexed by their height, but _also_ by
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the number of variables in scope, since i want de bruijn indices to be
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intrinsically well-scoped. why even have a type system otherwise. so a
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`term(n, h)` has $n$ variables and height $h$.
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```ats
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datatype term(int, int) =
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| {n, i : nat | i < n} Var(n, 1) of int(i)
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| {n, h : nat} Lam(n, 1 + h) of (typ, term(1 + n, h))
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| {n, h1, h2 : nat} App(n, 1 + max(h1, h2)) of (term(n, h1), term(n, h2))
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typedef term(n : int) = [h : nat] term(n, h)
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```
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a variable is [a de bruijn index][db], which is a natural less than `n`. to
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express this we need to bridge the static world, where `n` lives, and the
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dynamic, where `Var`'s field lives. to do this i use a singleton type
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`int(i)`, whose only value is `i` itself. with this type, you can constraint the
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static `i`, and in turn the value of the field.
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[db]: https://en.wikipedia.org/wiki/De_Bruijn_index
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# contexts
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```ats
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typedef ctx(n: int) = list(typ, n)
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```
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a typing context is just a list of types of the appropriate length. since
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there's no binding in types here, that's all we need.
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looking up into a context is just list lookup, with some constraints to make it
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work.
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```ats
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fun lookup {n, i : nat | i < n} .<n>. (g : ctx(n), i : nat(i)) :<> typ =
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let val+ list_cons(x, g) = g in
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if i = 0 then x else lookup(g, i - 1)
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end
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```
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from the constraints $n \ge 0, i \ge 0, i < n$, the compiler can see that, as
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$n \ge 1$, the list is nonempty and `list_cons` _is_ the only case. it can also
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propagate the $i \ne 0$ condition from the `else` to see the recursive call is
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ok too, and this function is total.
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:::aside
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while writing this post i discovered that this is actually a restatement of
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the type of `xs[i]` in the standard library and i could have just used that.
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oops!
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:::
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# typechecker
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i'm going to write the type checker using exceptions for error handling, because
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using an error monad without `do` notation was just too annoying. but from the
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outside it is still going to use a `result` type.
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```ats
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datatype result(t@ype, t@ype) =
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| {a, e : t@ype} Ok(a, e) of a
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| {a, e : t@ype} Err(a, e) of e
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```
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the funky looking `t@ype` refers to a type of unknown size, as opposed to `type`
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which can only be instantiated with pointer-sized types. that will not come up
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again here but "`t@ype`" is definitely the funniest part of ats's syntax.
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the errors we need here are:
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- a function takes an argument of type $A$, but is given one of type $B$;
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- a term of a non-arrow type is being applied.
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```ats
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datatype err =
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| Clash of (typ, typ)
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| NotArr of typ
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fn print_err (e : err) : void = … // incredibly uninteresting
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overload print with print_err
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typedef typ_res = result(typ, err)
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fn print_res (r : typ_res) : void = …
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overload print with print_res
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```
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so onto the typechecker. like i said, it's going to use a local exception for
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errors, but then catch it on the outside so that it has a pure type.
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```ats
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fn typ_of {n : nat} (g : ctx n, t : term(n)) :<> typ_res =
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let
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exception ERR of err
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fun go {n, h : nat} .<h>. (g : ctx(n), t : term(n, h)) :<!exn> typ =
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case t of
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| Var(i) => lookup(g, i)
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| Lam(a, t) => Arr(a, go(list_cons(a, g), t))
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| App(s, t) => begin
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case go(g, s) of
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| Arr(a1, b) =>
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let val a2 = go(g, t) in
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if a1 = a2 then b else $raise ERR(Clash(a1, a2))
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end
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| b => $raise ERR(NotArr(b))
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end
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in
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$effmask_exn(try Ok(go(g, t)) with ~ERR(e) => Err(e))
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end
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fn typ_of (t : term(0)) :<> typ_res = typ_of(list_nil, t)
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```
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most of this is fine. a bit noisy i guess. but there are a couple of new things:
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- in the type of `go`, the `:<!exn>` part means it can throw exceptions, but has
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no other effects.
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- `$effmask_exn(e)` hides the `exn` effect in `e`, because ats doesn't
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distinguish different exceptions in effects so it can't tell that `ERR` is the
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only one that is ever raised.
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- exceptions are linear, and the pattern `~ERR(e)` is consuming the outermost
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constructor.
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- exceptions are the only reason we needed `alloca` in this program
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:woozy_face:
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# the end
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ok, a quick `main` function, and that's it!
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```ats
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implement main0() = begin
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print("type of [λx:•. x] is: ");
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print(typ_of(Lam(Base, Var(0))));
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print("\ntype of [λx:•. x x] is: ");
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print(typ_of(Lam(Base, App(Var(0), Var(0)))));
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print("\n")
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end
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```
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and it works, at least on these two examples.
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```
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type of [λx:•. x] is: • → •
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type of [λx:•. x x] is: expected an arrow type, but got [•]
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```
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uh. thank's for reading. maybe i'll write another one if i think of another
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nice little program that actually uses more of ats's weirdness.
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--------------------------------------------------------------------------------
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<details>
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<summary>full code</summary>
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```ats
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#include "share/atspre_define.hats"
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#include "share/atspre_staload.hats"
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datatype typ(int) =
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| Base(1)
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| {h1, h2 : nat} Arr(1 + max(h1, h2)) of (typ(h1), typ(h2))
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typedef typ = [h : nat] typ(h)
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fun eq_typ {ha : nat} .<ha>. (a : typ(ha), b : typ) :<> bool =
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case (a, b) of
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| (Base(), Base()) => true
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| (Arr(a1, a2), Arr(b1, b2)) => eq_typ(a1, b1) && eq_typ(a2, b2)
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| (_, _) => false
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overload = with eq_typ
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fun print_typ (parens : bool, t : typ) : void =
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case t of
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| Base() => print "•"
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| Arr(a, b) => begin
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if parens then print("(");
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print_typ(true, a);
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print(" → ");
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print_typ(false, b);
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if parens then print(")");
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end
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fn print_typ (t : typ) : void = print_typ(false, t)
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overload print with print_typ
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datatype term(int, int) =
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| {n, i : nat | i < n} Var(n, 1) of int(i)
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| {n, h : nat} Lam(n, 1 + h) of (typ, term(1 + n, h))
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| {n, h1, h2 : nat} App(n, 1 + max(h1, h2)) of (term(n, h1), term(n, h2))
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typedef term(n : int) = [h : nat] term(n, h)
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typedef ctx(n: int) = list(typ, n)
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fun lookup {n, i : nat | i < n} .<n>. (g : ctx(n), i : int(i)) :<> typ =
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let val+ list_cons(x, g) = g in
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if i = 0 then x else lookup(g, i - 1)
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end
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datatype result(t@ype, t@ype) =
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| {a, e : t@ype} Ok(a, e) of a
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| {a, e : t@ype} Err(a, e) of e
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datatype err =
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| Clash of (typ, typ)
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| NotArr of typ
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fn print_err (e : err) : void =
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case e of
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| Clash(a, b) => begin
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print("clash between types [");
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print(a);
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print("] and [");
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print(b);
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print("]");
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end
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| NotArr(t) => begin
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print("expected an arrow type, but got [");
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print(t);
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print("]");
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end
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overload print with print_err
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typedef typ_res = result(typ, err)
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fn print_res (r : typ_res) : void =
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case r of Ok(t) => print(t) | Err(e) => print(e)
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overload print with print_res
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fn typ_of {n : nat} (g : ctx n, t : term(n)) :<> typ_res =
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let
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exception ERR of err
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fun go {n, h : nat} .<h>. (g : ctx(n), t : term(n, h)) :<!exn> typ =
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case t of
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| Var(i) => lookup(g, i)
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| Lam(a, t) => Arr(a, go(list_cons(a, g), t))
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| App(s, t) => begin
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case go(g, s) of
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| Arr(a1, b) =>
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let val a2 = go(g, t) in
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if a1 = a2 then b else $raise ERR(Clash(a1, a2))
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end
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| b => $raise ERR(NotArr(b))
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end
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in
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$effmask_exn(try Ok(go(g, t)) with ~ERR(e) => Err(e))
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end
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fn typ_of (t : term(0)) :<> typ_res = typ_of(list_nil, t)
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implement main0() = begin
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print("type of [λx:•. x] is: ");
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print(typ_of(Lam(Base, Var(0))));
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print("\ntype of [λx:•. x x] is: ");
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print(typ_of(Lam(Base, App(Var(0), Var(0)))));
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print("\n")
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end
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```
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</details>
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@ -405,3 +405,4 @@ aside {
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:is(.st, .fl, .dv, .sc) { color: hsl(143deg, 100%, 20%); }
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.wa { color: hsl(350deg, 80%, 25%); text-decoration: wavy 1.5px underline; }
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.al { color: hsl(350deg, 80%, 25%); }
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.cn { color: hsl(343deg, 100%, 30%); }
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@ -0,0 +1,210 @@
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<?xml version="1.0" encoding="UTF-8"?>
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<!DOCTYPE language SYSTEM "language.dtd" [
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<!ENTITY LETTER "A-Za-z\xc0-\xd6\xd8-\xf6\xf8-\xff">
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<!ENTITY IDENT "[&LETTER;_@$][&LETTER;0-9_@$']*">
|
||||
<!ENTITY ESC "(\\[ntbr'"\\]|\\[0-9]{3}|\\x[0-9A-Fa-f]{2})">
|
||||
<!ENTITY DEC "([0-9][0-9_]*)">
|
||||
<!ENTITY DECEXP "([eE][+-]?&DEC;)">
|
||||
<!ENTITY HEX "([0-9a-fA-F]+)">
|
||||
]>
|
||||
|
||||
<language name="ATS" section="Sources" extensions="*.sats;*.dats;*.hats"
|
||||
version="1" kateversion="5.53">
|
||||
<highlighting>
|
||||
<list name="decl">
|
||||
<item>abstype</item> <item>abst@ype</item> <item>absprop</item>
|
||||
<item>absview</item> <item>absvtype</item> <item>absvt@ype</item>
|
||||
<item>absviewtype</item> <item>absviewt@ype</item>
|
||||
|
||||
<item>exception</item>
|
||||
|
||||
<item>datatype</item> <item>dataprop</item> <item>dataview</item>
|
||||
<item>dataviewtype</item> <item>datavtype</item> <item>datasort</item>
|
||||
|
||||
<item>typedef</item> <item>propdef</item> <item>viewdef</item>
|
||||
<item>vtypedef</item> <item>viewtypedef</item> <item>sortdef</item>
|
||||
<item>tkindef</item> <item>classdec</item>
|
||||
<item>macdef</item> <item>macrodef</item>
|
||||
|
||||
<item>withtype</item> <item>withprop</item> <item>withview</item>
|
||||
<item>withvtype</item> <item>withviewtype</item>
|
||||
|
||||
<item>val</item> <item>fun</item> <item>fn</item> <item>fnx</item>
|
||||
<item>prfun</item> <item>prfn</item> <item>praxi</item>
|
||||
<item>castfn</item>
|
||||
<item>and</item>
|
||||
|
||||
<item>var</item> <item>prvar</item>
|
||||
|
||||
<item>infix</item> <item>infixl</item> <item>infixr</item>
|
||||
<item>prefix</item> <item>postfix</item> <item>nonfix</item>
|
||||
|
||||
<item>symintr</item> <item>symelim</item>
|
||||
|
||||
<item>sta</item> <item>stacst</item>
|
||||
<item>stadef</item> <item>static</item>
|
||||
</list>
|
||||
|
||||
<list name="meta">
|
||||
<item>extern</item> <item>extype</item> <item>extvar</item>
|
||||
<item>$extern</item> <item>$extype</item> <item>$extkind</item>
|
||||
<item>$extype_struct</item>
|
||||
|
||||
<item>$myfilename</item> <item>$mylocation</item> <item>$myfunction</item>
|
||||
|
||||
<item>import</item> <item>staload</item> <item>dynload</item>
|
||||
</list>
|
||||
|
||||
<list name="preproc">
|
||||
<item>#include</item> <item>#staload</item> <item>#dynload</item>
|
||||
<item>#require</item>
|
||||
|
||||
<item>#if</item> <item>#ifdef</item> <item>#ifndef</item>
|
||||
<item>#then</item>
|
||||
<item>#elif</item> <item>#elifdef</item> <item>#elifndef</item>
|
||||
<item>#else</item> <item>#endif</item>
|
||||
|
||||
<item>#error</item> <item>#prerr</item> <item>#print</item>
|
||||
<item>#assert</item>
|
||||
|
||||
<item>#define</item> <item>#undef</item>
|
||||
|
||||
<item>#pragma</item> <item>#codegen2</item> <item>#codegen3</item>
|
||||
</list>
|
||||
|
||||
<list name="impl">
|
||||
<item>assume</item> <item>reassume</item>
|
||||
<item>implmnt</item> <item>implement</item>
|
||||
<item>primplmnt</item> <item>primplement</item>
|
||||
<item>overload</item>
|
||||
</list>
|
||||
|
||||
<list name="expr">
|
||||
<item>begin</item> <item>end</item>
|
||||
<item>op</item> <item>as</item>
|
||||
|
||||
<item>case</item> <item>prcase</item> <item>of</item> <item>when</item>
|
||||
<item>scase</item>
|
||||
|
||||
<item>let</item> <item>local</item> <item>in</item> <item>rec</item>
|
||||
<item>if</item> <item>then</item> <item>else</item>
|
||||
<item>ifcase</item> <item>sif</item>
|
||||
<item>where</item>
|
||||
|
||||
<item>for</item> <item>do</item>
|
||||
<item>$break</item> <item>$continue</item>
|
||||
|
||||
<item>lam</item> <item>llam</item> <item>lam@</item>
|
||||
<item>fix</item> <item>fix@</item>
|
||||
|
||||
<item>try</item> <item>with</item> <item>$raise</item>
|
||||
|
||||
<item>$delay</item> <item>$ldelay</item>
|
||||
|
||||
<item>$arrpsz</item> <item>$arrptrsize</item>
|
||||
<item>$tyrep</item> <item>$d2ctype</item>
|
||||
|
||||
<item>$effmask</item>
|
||||
<item>$effmask_ntm</item> <item>$effmask_exn</item>
|
||||
<item>$effmask_ref</item> <item>$effmask_wrt</item>
|
||||
<item>$effmask_all</item>
|
||||
|
||||
<item>$lst</item> <item>$lst_t</item> <item>$lst_vt</item>
|
||||
<item>$list</item> <item>$list_t</item> <item>$list_vt</item>
|
||||
|
||||
<item>$rec</item> <item>$rec_t</item> <item>$rec_vt</item>
|
||||
<item>$record</item> <item>$record_t</item> <item>$record_vt</item>
|
||||
|
||||
<item>$tup</item> <item>$tup_t</item> <item>$tup_vt</item>
|
||||
<item>$tuple</item> <item>$tuple_t</item> <item>$tuple_vt</item>
|
||||
|
||||
<item>$solver_assert</item>
|
||||
<item>$solver_verify</item>
|
||||
|
||||
<item>$vararg</item>
|
||||
<item>$showtype</item>
|
||||
<item>$tempenver</item>
|
||||
</list>
|
||||
|
||||
<list name="prim">
|
||||
<item>type</item> <item>t@ype</item>
|
||||
<item>view</item> <item>prop</item>
|
||||
<item>viewtype</item> <item>viewt@ype</item>
|
||||
<item>vtype</item> <item>vt@ype</item>
|
||||
</list>
|
||||
|
||||
<list name="const">
|
||||
<item>bool</item> <item>true</item> <item>false</item>
|
||||
<item>int</item> <item>nat</item>
|
||||
</list>
|
||||
|
||||
<contexts>
|
||||
<context attribute="default" lineEndContext="#stay" name="default">
|
||||
<Detect2Chars attribute="comment" context="comment-1"
|
||||
char="/" char1="/" />
|
||||
<Detect2Chars attribute="comment" context="comment-c"
|
||||
char="/" char1="*" />
|
||||
<Detect2Chars attribute="comment" context="comment-ml"
|
||||
char="(" char1="*" />
|
||||
|
||||
<keyword attribute="decl" context="#stay" String="decl" />
|
||||
<keyword attribute="meta" context="#stay" String="meta" />
|
||||
<keyword attribute="preproc" context="#stay" String="preproc" />
|
||||
<keyword attribute="impl" context="#stay" String="impl" />
|
||||
<keyword attribute="expr" context="#stay" String="expr" />
|
||||
<keyword attribute="prim" context="#stay" String="prim" />
|
||||
<keyword attribute="const" context="#stay" String="const" />
|
||||
|
||||
<RegExpr attribute="arrowlike" context="#stay"
|
||||
String="[-=<>:]|=/?=>>?|\.<|>\." />
|
||||
|
||||
<DetectChar attribute="string" context="string" char=""" />
|
||||
|
||||
<RegExpr attribute="decint" context="#stay" String="~?&DEC;" />
|
||||
<RegExpr attribute="float" context="#stay"
|
||||
String="~?((&DEC;\.&DEC;?|\.&DEC;)&DECEXP;?|&DEC;&DECEXP;)" />
|
||||
<RegExpr attribute="hexint" context="#stay" String="~?0[xX]&HEX;" />
|
||||
</context>
|
||||
|
||||
<context attribute="comment" lineEndContext="#pop" name="comment-1" />
|
||||
<context attribute="comment" lineEndContext="#stay" name="comment-c">
|
||||
<!-- not nestable -->
|
||||
<Detect2Chars attribute="comment" context="#pop"
|
||||
char="*" char1="/" />
|
||||
</context>
|
||||
<context attribute="comment" lineEndContext="#stay" name="comment-ml">
|
||||
<!-- nestable -->
|
||||
<Detect2Chars attribute="comment" context="comment-ml"
|
||||
char="(" char1="*" />
|
||||
<Detect2Chars attribute="comment" context="#pop"
|
||||
char="*" char1=")" />
|
||||
</context>
|
||||
|
||||
<context attribute="string" lineEndContext="#stay" name="string">
|
||||
<RegExpr attribute="esc" context="#stay" String="&ESC;" />
|
||||
<DetectChar attribute="string" context="#pop" char=""" />
|
||||
</context>
|
||||
</contexts>
|
||||
|
||||
<itemDatas>
|
||||
<itemData name="default" defStyleNum="dsNormal" />
|
||||
<itemData name="comment" defStyleNum="dsComment" />
|
||||
|
||||
<itemData name="decl" defStyleNum="dsDataType" />
|
||||
<itemData name="meta" defStyleNum="dsKeyword" />
|
||||
<itemData name="preproc" defStyleNum="dsPreprocessor" />
|
||||
<itemData name="impl" defStyleNum="dsFunction" />
|
||||
<itemData name="expr" defStyleNum="dsControlFlow" />
|
||||
<itemData name="prim" defStyleNum="dsBuiltIn" />
|
||||
<itemData name="const" defStyleNum="dsConstant" />
|
||||
|
||||
<itemData name="arrowlike" defStyleNum="dsOperator" />
|
||||
|
||||
<itemData name="string" defStyleNum="dsString" />
|
||||
<itemData name="decint" defStyleNum="dsDecVal" />
|
||||
<itemData name="float" defStyleNum="dsFloat" />
|
||||
<itemData name="hexint" defStyleNum="dsBaseN" />
|
||||
<itemData name="esc" defStyleNum="dsSpecialChar" />
|
||||
</itemDatas>
|
||||
</highlighting>
|
||||
</language>
|
||||
Loading…
Reference in New Issue