# The on-demand SLG solver 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. ## 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 trait Debug { } struct u32 { } impl Debug for u32 { } struct Rc { } impl Debug for Rc { } struct Vec { } impl Debug for Vec { } ``` Now imagine that we want to find answers for the query `exists { Rc: 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 { ... }`). In this case, there are no universally bound names, but the canonical form Q of the query might look something like: Rc: Debug where `?0` is a variable in the root universe U0. We would then go and look for a table with this 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: 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: ``` (u32: Debug). (Rc: Debug) :- (T: Debug). (Vec: Debug) :- (T: Debug). ``` To create our initial strands, then, we will try to apply each of these clauses to our goal of `Rc: Debug`. The first and third clauses are inapplicable because `u32` and `Vec` cannot be unified with `Rc`. 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; - A set of floundered subgoals (see the section on floundering below); - 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: 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: (Rc: 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. Initially, when a strand is first created, there is no selected subgoal. ### 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_answer` operation on the table: specifically, we say `ensure_answer(T0, A0)`, meaning "ensure that there is a 0th answer". Remember that tables store not only strands, but also a vector of cached answers. The first thing that `ensure_answer` does is to check whether answer 0 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_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: (Rc: 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: (Rc: 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`. ### 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: Debug) :- (?U: Debug)`, derived from the `Vec` impl. - `(Rc: Debug) :- (?U: Debug)`, derived from the `Rc` impl. We can thus summarize the state of the whole forest at this point as follows: ``` Table T0 [Rc: Debug] Strands: (Rc: Debug) :- selected(?T: Debug, A0) Table T1 [?0: Debug] Strands: (u32: Debug) :- (Vec: Debug) :- (?U: Debug) (Rc: 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: ``` Table T1 [?0: Debug] Answers: A0 = [?0 = u32] Strand: (Vec: Debug) :- (?U: Debug) (Rc: 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. 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 incorporates the answer from A0 and removes the subgoal. The resulting state of table T0 is: ``` Table T0 [Rc: Debug] Strands: (Rc: Debug) :- selected(?T: Debug, A1) (Rc: Debug) :- ``` We then immediately activate the strand that incorporated the answer (the `Rc: 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: ``` Table T0 [Rc: Debug] Answer: A0 = [?0 = u32] Strands: (Rc: Debug) :- selected(?T: Debug, A1) Table T1 [?0: Debug] Answers: A0 = [?0 = u32] Strand: (Vec: Debug) :- (?U: Debug) (Rc: 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. ### Floundering The first thing we do when we create a table is to initialize it with a set of strands. These strands represent all the ways that one can solve the table's associated goal. For an ordinary trait, we would effectively create one strand per "relevant impl". But sometimes the goals are too vague for this to be possible; other times, it may be possible but just really inefficient, since all of those strands must be explored. As an example of when it may not be possible, consider a goal like `?T: Sized`. This goal can in principle enumerate **every sized type** that exists -- that includes not only struct/enum types, but also closure types, fn types with arbitrary arity, tuple types with arbitrary arity, and so forth. In other words, there are not only an infinite set of **answers** but actually an infinite set of **strands**. The same applies to auto traits like `Send` as well as "hybrid" traits like `Clone`, which contain *some* auto-generated sets of impls. Another example of floundering comes from negative logic. In general, we cannot process negative subgoals if they have unbound existential variables, such as `not { Vec: Foo }`. This is because we can only enumerate things that *do* match a given trait (or which *are* provable, more generally). We cannot enumerate out possible types `?T` that *are not* provable (there is an infinite set, to be sure). To handle this, we allow tables to enter a **floundered** state. This state begins when we try to create the program clauses for a table -- if that is not possible (e.g., in one of the cases above) then the table enters a floundered state. Attempts to get an answer from a floundered table result in an error (e.g., `RecursiveSearchFail::Floundered`). Whenever a goal results in a floundered result, that goal is placed into a distinct list (the "floundered subgoals"). We then go on and process the rest of the subgoals. Once all the normal subgoals have completed, floundered subgoals are removed from the floundered list and re-attempted: the idea is that we may have accumulated more type information in the meantime. If they continue to flounder, then we stop. Let's look at an example. Imagine that we have: ```rust trait Foo { } trait Bar { } impl Foo for T { } impl Bar for u32 { } impl Bar for i32 { } ``` Now imagine we are trying to prove `?T: Foo`. There is only one impl, so we will create a state like: ``` (?T: Foo) :- (?T: Send), (?T: Bar) ``` When we attempt to solve `?T: Send`, however, that subgoal will flounder, because `Send` is an auto-trait. So it will be put into a floundered list: ``` (?T: Foo) :- (?T: Bar) [ floundered: (?T: Send) ] ``` and we will go on to solve `?T: Bar`. `Bar` is an ordinary trait -- so we can enumerate two strands (one for `u32` and one for `i32`). When we process the first answer, we wind up with: ``` (u32: Foo) :- [ floundered: (u32: Send) ] ``` At this point we can move the floundered subgoal back into the main list and process: ``` (u32: Foo) :- (u32: Send) ``` This time, the goal does not flounder. But how do we detect when it makes sense to move a floundered subgoal into the main list? To handle this, we use a timestamp scheme. We keep a counter as part of the strand -- each time we succeed in solving some subgoal, we increment the counter, as that *may* have provided more information about some type variables. When we move a goal to the floundered list, we also track the current value of the timestamp. Then, when it comes time to move things *from* the floundered list, we can compare if the timestamp has been changed since the goal was marked as floundering. If not, then no new information can have been attempted, and we can mark the current table as being floundered itself. This mechanism allows floundered to propagate up many levels, e.g. in an example like this: ```rust trait Foo { } trait Bar { } trait Baz { } impl Foo for T { } impl Bar for u32 { } impl Bar for i32 { } impl Baz for T { } ``` Here, solving `?T: Baz` will in turn invoke `?T: Sized` -- this floundering state will be propagated up to the `?T: Foo` table. ## Heritage and acronyms This solver implements the SLG solving technique, though extended to accommodate hereditary harrop (HH) predicates, as well as the needs of lazy normalization. Its design is kind of a fusion of [MiniKanren] and the following papers, which I will refer to as EWFS and NTFD respectively: > Efficient Top-Down Computation of Queries Under the Well-formed Semantics > (Chen, Swift, and Warren; Journal of Logic Programming '95) > A New Formulation of Tabled resolution With Delay > (Swift; EPIA '99) [MiniKanren]: http://minikanren.org/ In addition, I incorporated extensions from the following papers, which I will refer to as SA and RR respectively, that describes how to do introduce approximation when processing subgoals and so forth: > Terminating Evaluation of Logic Programs with Finite Three-Valued Models > Riguzzi and Swift; ACM Transactions on Computational Logic 2013 > (Introduces "subgoal abstraction", hence the name SA) > > Radial Restraint > Grosof and Swift; 2013 Another useful paper that gives a kind of high-level overview of concepts at play is the following: > XSB: Extending Prolog with Tabled Logic Programming > (Swift and Warren; Theory and Practice of Logic Programming '10) There are a places where I intentionally diverged from the semantics as described in the papers -- e.g. by more aggressively approximating -- which I marked them with a comment DIVERGENCE. Those places may want to be evaluated in the future. A few other acronyms that I use: - WAM: Warren abstract machine, an efficient way to evaluate Prolog programs. See . - HH: Hereditary harrop predicates. What Chalk deals in. Popularized by Lambda Prolog.