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Danila Fedorin d8ab3f2226 Continue working on the Alloy blog post

2023-05-04 21:01:31 -07:00

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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:

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:

The inside of class C. The class itself doesn't define any methods, so nothing is found.
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:

The inside of C.doSomething will be searched. doSomething doesn't declare anything, so the search will come up empty.
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.

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 foos 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 used 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:

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:

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 about the release of Alloy 6, 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.

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.

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 and predicates, 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.

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.

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.

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!

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:

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:

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:

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".

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.

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.

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:

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.

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.

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.

module M {
    
}

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

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The Problem at Hand

Modeling Flags and Bitsets in Alloy

Scratch Work

20 KiB

Raw Blame History