---
title: "Proving My Compiler Code Incorrect With Alloy"
date: 2023-05-02T22:48:52-07:00
tags: ["Compilers", "Alloy"]
draft: true
---

{{< todo >}}
Intro section and disclaimer
{{< /todo >}}

I work as a compiler developer on the [Chapel](https://chapel-lang.org) team.

### The Problem at Hand
One of the things that a language like Chapel has to do is called _resolution_,
which is the process of figuring out what each identifier, like `x`, refers to,
and what its type is. Even the first part of that is pretty complicated, what
with public and private variables, methods (which can be decalred outside
of their receiver type in Chapel), and more...

Scope resolution in Chapel is further complicated by the fact that the same
scope might need to be searched multiple times, in different contexts. Let
me start with a few examples to illustrate what I mean. Here's the first
program:

```Chapel {linenos=true}
module M {
    class C {}

    // A regular procedure (not a method)
    proc foo() {}

    // A method on C.
    proc C.foo() {}

    // Another method on C.
    proc C.doSomething() {
        foo();
    }
}
```

If you don't know Chapel (and you probably don't!) this program already merits a fair
bit of explanation. A _module_ in Chapel (declared via a `module` keyword)
is just a collection of definitions. Such definitions could include variables,
methods, classes and more. Putting them in a module helps group them.

{{< todo >}}
Write the rest of this explanation.
{{< /todo >}}

The interesting part of the snippet is the body of the `doSomething` method.
It has a call to `foo`: but which `foo` is it referring to? There are two:
the regular procedure (non-method) `foo`, declared on line 5, and the
method `C.foo` declared on line 8. In Chapel, the rules dictate that when
such a situation arises, and a fitting method is found, the method is
preferred to the non-method. In the rewritten version of the Chapel compiler,
titled Dyno, this disambiguation is achieved by first searching the scopes
visible from the class `C` for methods only. In this particular example,
the two scopes searched will be:

1. The inside of class `C`. The class itself doesn't define any methods, so
   nothing is found.
2. The module in which `C` is defined (`M` in this case). This module does
   have a method, the one on line 8, so that one is returned.

Only if methods are not found are non-methods considered. In this situation,
the search order will be as follows:

1. The inside of `C.doSomething` will be searched. `doSomething` doesn't declare
   anything, so the search will come up empty.
2. The module in which `C.doSomething` is defined (`M` again) will be searched. This time,
   both methods and non-methods will be considered. Since we're considering
   a hypothetical situation in which the method `C.foo` isn't there (otherwise
   it would've been found earlier), the only thing that will be found will
   be the non-method `foo`.

Notice that we've already had to search the module `M` twice, looking for
different things each time. First, we were looking for only method, but
later, we were looking for anything. However, this isn't as complicated as
things can get. The simplifying aspect of this program is that both `doSomething`
and `C` are defined inside the class `C`, and therefore have access to its
private methods and procedures. If we extracted `C.doSomething` into its
own separate module, the program would look like this.

```Chapel {linenos=true}
module M1 {
    class C {}

    // A regular procedure (not a method)
    proc foo() {}

    // A method on C.
    proc C.foo() {}
}
module M2 {
    use super.M1;

    // Another method on C.
    proc C.doSomething() {
        foo();
    }
}
```

Since `doSomething` is now in another module, it can't just access the `foo`s from
`M1` willy-nilly. There are a few ways to get the things that were declared in another module out
and make use of them. I opted for a `use` statement, which, in its simplest form,
just brings all the declarations inside the `use`d module into the current scope. Thus,
the `use` statement on line 11 would bring all things declared in `M1` into
the scope inside `M2`. There's a catch, though: since `M2` is not declared
inside `M1`, a `use` statement will not be able to bring in _private_ symbols
from `M1` (they're private for a reason!). So, this time, when searching the scope
for `M1`, we will have to search only for public symbols. That's another,
different way of searching `M1`. So far, we've seen three:

* Search `M1` for any symbol.
* Search `M1` for methods only.
* Search `M1` for public symbols only.

In Dyno, there are even more different ways of searching a single scope, and
some of them are mixes of others (one might consider, for instance, searching
for only public methods). To represent the various search configurations,
the Dyno team came up with using a bitfield of _flags_, each of which indicated
a necessary condition for a symbol to be returned. A bitfield with flags set
for two properties (like "public" and "method") requires that both such
properties be found on each symbol that's returned from a scope. This led to
C++ code along the lines of:

```C++
auto allPublicSymbols = Flags::PUBLIC;
auto allPublicMethods = Flags::PUBLIC | Flags::METHOD;
```

It also turned out convenient to add negative versions of each flag
(`NOT_PUBLIC` for private symbols, `NOT_METHOD` for regular old procedures
and other definitions, and so on. So, some other possible flag combinations
include:

```C++
auto allNonMethods = Flags::NOT_METHOD;
auto privateMethods = Flags::NOT_PUBLIC | Flags::METHOD;
```

Given these flags, there are some situations in which checking a scope a
second time is redundant, in that it is guaranteed to find no additional
symbols. For instance, if you search a scope for all public symbols, and
then subsequently search for all public methods, you will only find
duplicates -- after all, all public methods are public symbols. Most
generally, this occurs when a second search has all the flags from a
previous search, and maybe more. In math lingo, if the set of flags checked
the first time is a subset of the set of flags checked the second time,
it's guaranteed not to find anything new.

In Dyno, we like to avoid additional work when we can. To do so, we track
which scopes have already been searched, and avoid searching them again.
Since what comes up from a search depends on the flags, we store the flags
alongside the scopes we've checked. __If we find that the previously-checked
bitfield is a subset of the current biset, we just skip the search__.

But then, what if it _isn't_ a subset? Another concern here is avoiding
duplicate results (it's easier to check for duplicate definitions if you
know a symbol is only returned from a search once). So, another feature of
Dyno's scope search is an additional bitfield of what to _exclude_, which we
set to be the previous search's filter. So if the first search looked for
symbols matching description \\(A\\), and the second search is supposed to
look for symbols matching description \\(B\\), __then really we do a search
for \\(A \\land \\lnot B\\) (that is, \\(A\\) and not \\(B\\))__.

{{< dialog >}}
{{< message "question" "reader" >}}
Hold on, why do you need a whole another bitfield? There are already negated
versions of each flag available. Can't you just add those to the filter?
{{< /message >}}
{{< message "answer" "Daniel" >}}
Good question. The difference is a little bit tricky. If we just negated
each flag, we'd turn an expression like \(A \land B\) into \(\lnot A \land
\lnot B\). However, according to De Morgan's laws, the proper negation of
\(A \land B\) is \(\lnot A \lor \lnot B\) (notice the use of "or" instead
of "and"). On the other hand, using an "exclude" bitfield negates the whole
conjunction, rather than the individual flags, and so gives us the result
we need.
{{< /message >}}
{{< /dialog >}}

One last thing: what happens if there were two previous searches? What we
need is to to somehow combine the two filters into one. Taking a cue from
a previous example, in which "public" was followed by "public methods", we
can observe that since the second search has additional flags, it's more
restrictive, and thus guaranteed to not find anything. __So we try to create
the least restrictive bitfield possible, by taking an intersection of the
flags used.__

Actually, that last point is not quite correct in every possible case
(taking the intersection is not always the right thing to do).
However, running the code through our test suite, we did not notice any
cases in which it misbehaved. So, noting the potential issue in a comment,
we moved on to other things.

That is, until I decided that it was time to add another possible flag to
the bitfield. At that point, sitting and trying to reason about the
possible cases, I realized that it would be much nicer to describe this
mathematically, and have a model checker generate outlandish scenarios for
me. Having at some point seen [Hillel Wayne's
post](https://www.hillelwayne.com/post/alloy6/) about the release of
[Alloy 6](https://alloytools.org/), I thought I'd give it a go. I'd never
touched alloy before this, so be warned: this is what I came up with on my
own attempt.

### Modeling Flags and Bitsets in Alloy
Flags are represented on the C++ side as an `enum` (with custom indexing so
as to make each flag be exactly one bit). I checked, and it looked like
Alloy had an `enum` feature, too! I started off by making an enum of the
flags I wanted to play with.

{{< codelines "Alloy" "dyno-alloy/DynoAlloy.als" 1 1 >}}

We haven't seen the `MethodOrField` flag, but it's an important one. It
turns out that it's much more common to look for anything that could be
part of a class, rather than just its methods. This flag is itself an "or"
of two properties (something being a method and something being a class
field). Note that this is not the same as having two flags, `Method` and
`Field`, and always including them together (because that would be an
"and", not an "or").

Notice also that the list of flags doesn't include the negative versions.
Since the negative versions are one-for-one with the positive ones, I
instead chose to represent bitfields as simply two sets: one set of
"positive" flags, in which the presence of e.g. `Method` indicates that the
`METHOD` flag was set, and one set of "negative" flags, in which the
presence of `Method` indicates that `NOT_METHOD` was set. This way, I'm
guaranteed that there's a positive and negative version of each flag,
automatically. Here's how I wrote that in Alloy.

{{< codelines "Alloy" "dyno-alloy/DynoAlloy.als" 6 9 >}}

This definition (a _signature_ in Alloy terms) specifies what a bitfield is
like, but not any operations on it. My next order of business is to define
some common functionality on bitfields. Alloy is all about
[relations](https://en.wikipedia.org/wiki/Relation_(mathematics)) and
[predicates](https://en.wikipedia.org/wiki/Predicate_(mathematical_logic)),
so for all of these, I had to effectively write something that _checks_ if
some condition holds for some arguments. This might seem abstract; as an
example, here's `bitfieldEmpty`, which checks that a bitfield has no flags
set.

{{< codelines "Alloy" "dyno-alloy/DynoAlloy.als" 27 29 >}}

The `#` operator in Alloy is used to check the size of a set. So, to check
if a bitfield is empty, I simply check if there are neither positive nor
negative flags. Probably the most unusual aspect of this piece of code is
that equality is written as `=`, as opposed to `==` like in most common
languages. This is because, like I said, alloy is all about relations and
predicates, and not at all about imperative manipulation of data. So,
there's no need to reserve `=` for assignment.

The next step from here is a predicate that accepts two arguments,
`bitfieldEqual`. As its name suggests, this predicate accepts two
bitfields, and makes sure they have exactly the same flags set. 

{{< codelines "Alloy" "dyno-alloy/DynoAlloy.als" 31 33 >}}

So far, this has been pretty similar to just writing boolean functions in a
language like C++. However, the similarity is only superficial. An easy way
to see that is to try to determine the _intersection_ of two bitfields --
that's the operation we will be having to model, since the Dyno
implementation uses `&` to combine filter sets. In a language like C++, you
might write a function like the following, in which you accept two bitfield
arguments and return a new bitfield.

```C++
Bitfield intersection(Bitfield b1, Bitfield b2) { /* ... */ }
```

However, in Alloy, you can't create a new bitfield, nor return something
from a `pred` that isn't a boolean. Instead, you describe how the inputs
will be related to the output. So, to model a binary function, you end up
with a three-parameter predicate: two inputs, and one output. But how
_does_ the output of a bitfield intersection connect to the two operands
being intersected? Well, its two flag sets will be intersections of the
flag sets of the inputs!

{{< codelines "Alloy" "dyno-alloy/DynoAlloy.als" 35 38 >}}

Next, let's talk about what flags _do_. They are used to include and
exclude symbols based on certain properties. One property is being a
method: a `METHOD` flag requires this property, whereas a `NOT_METHOD` flag
ensures that a symbol does not have it. Another property is being a public
definition: if a symbol isn't public, it'll be ignored by searches with the
`PUBLIC` flag set. Just like a bitfield can have multiple flags, a symbol
can have multiple properties (e.g., a public method). Unlike our bitfields,
though, we won't be modeling symbols as having both positive and negative
properties. That is to say, we won't have a "not public" property: the
absence of the "public" property will be enough to make something private.
Here's the alloy definition for everything I just said:

{{< codelines "Alloy" "dyno-alloy/DynoAlloy.als" 60 64 >}}

Now, we can specify how flags in a bitfield relates to properties on a
symbol. We can do so by saying which flags match which properties. The
`Method` flag, for instance, will be satisfied by the `PMethod` property.
The `MethodOrField` flag is more lenient, and will be satisfied by either
`PMethod` or `PField`. Here's a predicate `flagMatchesProperty` that
encodes all the flag-property combinations:

{{< codelines "Alloy" "dyno-alloy/DynoAlloy.als" 66 70 >}}

A bitfield matching a symbol is a little bit more complicated. Said
informally, the condition for a bitfield matching a symbol is twofold: 

* Every single positive flag, like `METHOD`, must be satisfied by a
  property on the symbol.
* None of the negative flags, like `NOT_METHOD`, must be satisfied by a
  property on the symbol (that is to say, if `Method` is in the negative
  flags set, then the symbol must not have `PMethod` property). It is more
  conveniently to formulate this -- equivalently -- as follows: for each
  negative flag, there most not be a property that satisfies it.

Each of the above two conditions translate quite literally into Alloy:

{{< codelines "Alloy" "dyno-alloy/DynoAlloy.als" 72 75 >}}

We can read line 73 as "for each flag in a bitfield's positive flags, there
must be some property in the symbol that matches it". Similarly, line 74
can be read out loud as "for each flag in the negative flags, no property
in the symbol must match it".

We've written a fair bit of alloy. If you're anything like me, you might be
getting a bit twitchy: how do we even check that any of this works? For
this, we'll need to run our model. We will give Alloy a claim, and ask
it to find a situation in which that claim holds true. The simplest claim
is "there exists a bitfield".

{{< codelines "Alloy" "dyno-alloy/DynoAlloy.als" 77 79 >}}

Executing this model yields a pretty interesting bitfield: one in which every single flag is set -- both the positive and negative versions.

{{< figure src="bitfield_exists.png" caption="Alloy's output satisfying \"a bit field exists\"" >}}

That's a little bit ridiculous: this bitfield will never match anything!
You can't be and not be a method at the same time, for instance. For for a
more interesting example, let's ask for a bitfield that matches some
symbol.

{{< codelines "Alloy" "dyno-alloy/DynoAlloy.als" 81 83 >}}

The output here is pretty interesting too. Alloy finds a symbol and a
bitfield that matches it, but they're both empty. In effect, it said:
"if you don't specify any filters, any private definition will match". Fair
enough, of course, but a curious departure from the previous maximalist
"put in all the flags!" approach.

{{< figure src="matching_bitfield_exists.png" caption="Alloy's output satisfying \"a bit field that matches a symbol exists\"" >}}

Let's try nudge it towards a more interesting case. I'm going to ask for a
filter with one positive and one negative flag, and a symbol with two
properties.

{{< codelines "Alloy" "dyno-alloy/DynoAlloy.als" 85 92 >}}

The results are more interesting this time: we get a filter for private
methods, and a private symbol that was... both a field and a method?

{{< figure src="matching_bitfield_exists_2.png" caption="Alloy's spiced up output satisfying \"a bit field that matches a symbol exists\"" >}}

We never told alloy that a symbol can't be both a field and a method. It
had no idea what the flags meant, just that they exist.
To let Alloy know what we do -- that the two properties are incompatible --
we can use a _fact_. To me, the most natural way of phrasing this is "there
is never a symbol that has both the method and field properties". Alas,
Alloy doesn't have a `never` keyword; it only has `always`. So I opt instead
for an alternative formulation: "there are always zero symbols that are
both methods and fields". In alloy, the claim looks like this:

{{< codelines "Alloy" "dyno-alloy/DynoAlloy.als" 94 99 >}}

Re-running the example program with this fact, alloy spits out a filter for
public non-method symbols, and a symbol that's a public field. Public
fields also aren't a thing in Chapel (all fields in a class are publicly
readable in the current version of the language). Perhaps it's time for
another fact.

{{< codelines "Alloy" "dyno-alloy/DynoAlloy.als" 100 104 >}}

But now, Alloy fails to come up with anything at all. That makes sense: by
restricting the search to a symbol with two properties, and making `PField`
incompatible with the other two possible properties, we've guaranteed that
our symbol would be a public method. But then, we also required a negative
flag in the filter; however, all the flags in the list match a public
method, so making any of them negative would guarantee that our symbol
would not be found. Let's change the example up a bit to only ask for
positive flags.

{{< codelines "Alloy" "dyno-alloy/DynoAlloy.als" 106 112 >}}

This time, alloy gives us a symbol that's a public method, and a filter
that only looks for public methods. Fair enough.

{{< figure src="matching_bitfield_exists_3.png" caption="Alloy's spiced up output satisfying \"a bit field that matches a symbol exists\"" >}}

{{< todo >}}
The rest of the article
{{< /todo >}}

### Scratch Work
{{< todo >}}
This section is temporary
{{< /todo >}}

a small-ish program to illustrate what I mean.

```Chapel
module M {
    
}
```


```Chapel {linenos=true}
module M1 {
    public use super.M2;
}
module M2 {
    private var x = 1;

    module M3 {
        public use super;
    }
}

use M1;
use M2.M3;
writeln(x)
```


 Moreover, a `public use` makes these definitions part
of the module that does the `use` -- that is, `M1` would now contain the
definitions from `M2`. However, since `M1` is not defined inside of `M2`, it
isn't able to access its private variables (like `x` on line 5), so this
particular use statement leaves `M1` just containing (a reference to) `M3`.

The `public use` on line