Merge pull request #228 from tmandry/traits-slg

Add traits SLG solver section
2018-11-20 14:08:02 -05:00 · 2018-11-20 14:08:02 -05:00 · 56a4e40e20
parent 16da1bc2a6 f5cb41ca90
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--- a/src/traits/slg.md
+++ b/src/traits/slg.md
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-# The SLG solver
+# The On-Demand SLG solver
-TODO: <https://github.com/rust-lang-nursery/chalk/blob/master/chalk-engine/src/README.md>
+Given a set of program clauses (provided by our [lowering rules][lowering])
 and a query, we need to return the result of the query and the value of any
 type variables we can determine. This is the job of the solver.
 For example, `exists<T> { Vec<T>: FromIterator<u32> }` has one solution, so
 its result is `Unique; substitution [?T := u32]`. A solution also comes with
 a set of region constraints, which we'll ignore in this introduction.
 [lowering]: ./lowering-rules.html
 ## Goals of the Solver
 ### On demand
 There are often many, or even infinitely many, solutions to a query. For
 example, say we want to prove that `exists<T> { Vec<T>: Debug }` for _some_
 type `?T`. Our solver should be capable of yielding one answer at a time, say
 `?T = u32`, then `?T = i32`, and so on, rather than iterating over every type
 in the type system. If we need more answers, we can request more until we are
 done. This is similar to how Prolog works.
 *See also: [The traditional, interactive Prolog query][pq]*
 [pq]: ./canonical-queries.html#the-traditional-interactive-prolog-query
 ### Breadth-first
 `Vec<?T>: Debug` is true if `?T: Debug`. This leads to a cycle: `[Vec<u32>,
 Vec<Vec<u32>>, Vec<Vec<Vec<u32>>>]`, and so on all implement `Debug`. Our
 solver ought to be breadth first and consider answers like `[Vec<u32>: Debug,
 Vec<i32>: Debug, ...]` before it recurses, or we may never find the answer
 we're looking for.
 ### Cachable
 To speed up compilation, we need to cache results, including partial results
 left over from past solver queries.
 ## Description of how it works
 The basis of the solver is the [`Forest`] type. A *forest* stores a
 collection of *tables* as well as a *stack*. Each *table* represents
 the stored results of a particular query that is being performed, as
 well as the various *strands*, which are basically suspended
 computations that may be used to find more answers. Tables are
 interdependent: solving one query may require solving others.
 [`Forest`]: https://rust-lang-nursery.github.io/chalk/doc/chalk_engine/forest/struct.Forest.html
 ### Walkthrough
 Perhaps the easiest way to explain how the solver works is to walk
 through an example. Let's imagine that we have the following program:
 ```rust,ignore
 trait Debug { }
 struct u32 { }
 impl Debug for u32 { }
 struct Rc<T> { }
 impl<T: Debug> Debug for Rc<T> { }
 struct Vec<T> { }
 impl<T: Debug> Debug for Vec<T> { }
 ```
 Now imagine that we want to find answers for the query `exists<T> { Rc<T>:
 Debug }`. The first step would be to u-canonicalize this query; this is the
 act of giving canonical names to all the unbound inference variables based on
 the order of their left-most appearance, as well as canonicalizing the
 universes of any universally bound names (e.g., the `T` in `forall<T> { ...
 }`). In this case, there are no universally bound names, but the canonical
 form Q of the query might look something like:
 ```text
 Rc<?0>: Debug
 ```
 where `?0` is a variable in the root universe U0. We would then go and
 look for a table with this canonical query as the key: since the forest is
 empty, this lookup will fail, and we will create a new table T0,
 corresponding to the u-canonical goal Q.
 **Ignoring negative reasoning and regions.** To start, we'll ignore
 the possibility of negative goals like `not { Foo }`. We'll phase them
 in later, as they bring several complications.
 **Creating a table.** When we first create a table, we also initialize
 it with a set of *initial strands*. A "strand" is kind of like a
 "thread" for the solver: it contains a particular way to produce an
 answer. The initial set of strands for a goal like `Rc<?0>: Debug`
 (i.e., a "domain goal") is determined by looking for *clauses* in the
 environment. In Rust, these clauses derive from impls, but also from
 where-clauses that are in scope. In the case of our example, there
 would be three clauses, each coming from the program. Using a
 Prolog-like notation, these look like:
 ```text
 (u32: Debug).
 (Rc<T>: Debug) :- (T: Debug).
 (Vec<T>: Debug) :- (T: Debug).
 ```
 To create our initial strands, then, we will try to apply each of
 these clauses to our goal of `Rc<?0>: Debug`. The first and third
 clauses are inapplicable because `u32` and `Vec<?0>` cannot be unified
 with `Rc<?0>`. The second clause, however, will work.
 **What is a strand?** Let's talk a bit more about what a strand *is*. In the code, a strand
 is the combination of an inference table, an _X-clause_, and (possibly)
 a selected subgoal from that X-clause. But what is an X-clause
 ([`ExClause`], in the code)? An X-clause pulls together a few things:
 - The current state of the goal we are trying to prove;
 - A set of subgoals that have yet to be proven;
 - There are also a few things we're ignoring for now:
  - delayed literals, region constraints
 The general form of an X-clause is written much like a Prolog clause,
 but with somewhat different semantics. Since we're ignoring delayed
 literals and region constraints, an X-clause just looks like this:
 ```text
 G :- L
 ```
 where G is a goal and L is a set of subgoals that must be proven.
 (The L stands for *literal* -- when we address negative reasoning, a
 literal will be either a positive or negative subgoal.) The idea is
 that if we are able to prove L then the goal G can be considered true.
 In the case of our example, we would wind up creating one strand, with
 an X-clause like so:
 ```text
 (Rc<?T>: Debug) :- (?T: Debug)
 ```
 Here, the `?T` refers to one of the inference variables created in the
 inference table that accompanies the strand. (I'll use named variables
 to refer to inference variables, and numbered variables like `?0` to
 refer to variables in a canonicalized goal; in the code, however, they
 are both represented with an index.)
 For each strand, we also optionally store a *selected subgoal*. This
 is the subgoal after the turnstile (`:-`) that we are currently trying
 to prove in this strand. Initally, when a strand is first created,
 there is no selected subgoal.
 [`ExClause`]: https://rust-lang-nursery.github.io/chalk/doc/chalk_engine/struct.ExClause.html
 **Activating a strand.** Now that we have created the table T0 and
 initialized it with strands, we have to actually try and produce an answer.
 We do this by invoking the [`ensure_root_answer`] operation on the table:
 specifically, we say `ensure_root_answer(T0, A0)`, meaning "ensure that there
 is a 0th answer A0 to query T0".
 Remember that tables store not only strands, but also a vector of cached
 answers. The first thing that [`ensure_root_answer`] does is to check whether
 answer A0 is in this vector. If so, we can just return immediately. In this
 case, the vector will be empty, and hence that does not apply (this becomes
 important for cyclic checks later on).
 When there is no cached answer, [`ensure_root_answer`] will try to produce one.
 It does this by selecting a strand from the set of active strands -- the
 strands are stored in a `VecDeque` and hence processed in a round-robin
 fashion. Right now, we have only one strand, storing the following X-clause
 with no selected subgoal:
 ```text
 (Rc<?T>: Debug) :- (?T: Debug)
 ```
 When we activate the strand, we see that we have no selected subgoal,
 and so we first pick one of the subgoals to process. Here, there is only
 one (`?T: Debug`), so that becomes the selected subgoal, changing
 the state of the strand to:
 ```text
 (Rc<?T>: Debug) :- selected(?T: Debug, A0)
 ```
 Here, we write `selected(L, An)` to indicate that (a) the literal `L`
 is the selected subgoal and (b) which answer `An` we are looking for. We
 start out looking for `A0`.
 [`ensure_root_answer`]:  https://rust-lang-nursery.github.io/chalk/doc/chalk_engine/forest/struct.Forest.html#method.ensure_root_answer
 **Processing the selected subgoal.** Next, we have to try and find an
 answer to this selected goal. To do that, we will u-canonicalize it
 and try to find an associated table. In this case, the u-canonical
 form of the subgoal is `?0: Debug`: we don't have a table yet for
 that, so we can create a new one, T1. As before, we'll initialize T1
 with strands. In this case, there will be three strands, because all
 the program clauses are potentially applicable. Those three strands
 will be:
 - `(u32: Debug) :-`, derived from the program clause `(u32: Debug).`.
  - Note: This strand has no subgoals.
 - `(Vec<?U>: Debug) :- (?U: Debug)`, derived from the `Vec` impl.
 - `(Rc<?U>: Debug) :- (?U: Debug)`, derived from the `Rc` impl.
 We can thus summarize the state of the whole forest at this point as
 follows:
 ```text
 Table T0 [Rc<?0>: Debug]
  Strands:
    (Rc<?T>: Debug) :- selected(?T: Debug, A0)
 Table T1 [?0: Debug]
  Strands:
    (u32: Debug) :-
    (Vec<?U>: Debug) :- (?U: Debug)
    (Rc<?V>: Debug) :- (?V: Debug)
 ```
 **Delegation between tables.** Now that the active strand from T0 has
 created the table T1, it can try to extract an answer. It does this
 via that same `ensure_answer` operation we saw before. In this case,
 the strand would invoke `ensure_answer(T1, A0)`, since we will start
 with the first answer. This will cause T1 to activate its first
 strand, `u32: Debug :-`.
 This strand is somewhat special: it has no subgoals at all. This means
 that the goal is proven. We can therefore add `u32: Debug` to the set
 of *answers* for our table, calling it answer A0 (it is the first
 answer). The strand is then removed from the list of strands.
 The state of table T1 is therefore:
 ```text
 Table T1 [?0: Debug]
  Answers:
    A0 = [?0 = u32]
  Strand:
    (Vec<?U>: Debug) :- (?U: Debug)
    (Rc<?V>: Debug) :- (?V: Debug)
 ```
 Note that I am writing out the answer A0 as a substitution that can be
 applied to the table goal; actually, in the code, the goals for each
 X-clause are also represented as substitutions, but in this exposition
 I've chosen to write them as full goals, following [NFTD].
 [NFTD]: ./bibliography.html#slg
 Since we now have an answer, `ensure_answer(T1, A0)` will return `Ok`
 to the table T0, indicating that answer A0 is available. T0 now has
 the job of incorporating that result into its active strand. It does
 this in two ways. First, it creates a new strand that is looking for
 the next possible answer of T1. Next, it incorpoates the answer from
 A0 and removes the subgoal. The resulting state of table T0 is:
 ```text
 Table T0 [Rc<?0>: Debug]
  Strands:
    (Rc<?T>: Debug) :- selected(?T: Debug, A1)
    (Rc<u32>: Debug) :-
 ```
 We then immediately activate the strand that incorporated the answer
 (the `Rc<u32>: Debug` one). In this case, that strand has no further
 subgoals, so it becomes an answer to the table T0. This answer can
 then be returned up to our caller, and the whole forest goes quiescent
 at this point (remember, we only do enough work to generate *one*
 answer). The ending state of the forest at this point will be:
 ```text
 Table T0 [Rc<?0>: Debug]
  Answer:
    A0 = [?0 = u32]
  Strands:
    (Rc<?T>: Debug) :- selected(?T: Debug, A1)
 Table T1 [?0: Debug]
  Answers:
    A0 = [?0 = u32]
  Strand:
    (Vec<?U>: Debug) :- (?U: Debug)
    (Rc<?V>: Debug) :- (?V: Debug)
 ```
 Here you can see how the forest captures both the answers we have
 created thus far *and* the strands that will let us try to produce
 more answers later on.
 ## See also
 - [chalk_solve README][readme], which contains links to papers used and
  acronyms referenced in the code
 - This section is a lightly adapted version of the blog post [An on-demand
  SLG solver for chalk][slg-blog]
 - [Negative Reasoning in Chalk][negative-reasoning-blog] explains the need
  for negative reasoning, but not how the SLG solver does it
 [readme]: https://github.com/rust-lang-nursery/chalk/blob/239e4ae4e69b2785b5f99e0f2b41fc16b0b4e65e/chalk-engine/src/README.md
 [slg-blog]: http://smallcultfollowing.com/babysteps/blog/2018/01/31/an-on-demand-slg-solver-for-chalk/
 [negative-reasoning-blog]: http://aturon.github.io/blog/2017/04/24/negative-chalk/