LRijkstra.ml 35.3 KB
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(* The purpose of this algorithm is to find, for each pair of a state [s]
   and a terminal symbol [z] such that looking at [z] in state [s] causes
   an error, a minimal path (starting in some initial state) that actually
   triggers this error. *)

(* This is potentially useful for grammar designers who wish to better
   understand the properties of their grammar, or who wish to produce a
   list of all possible syntax errors (or, at least, one syntax error in
   each automaton state where an error may occur). *)

(* The problem seems rather tricky. One might think that it suffices to
   compute shortest paths in the automaton, and to use [Analysis.minimal] to
   replace each non-terminal symbol in a path with a minimal word that this
   symbol generates. One can indeed do so, but this yields only a lower bound
   on the actual shortest path to the error at [s, z]. Indeed, two
   difficulties arise:

   - Some states have a default reduction. Thus, they will not trigger
     an error, even though they should. The error is triggered in some
     other state, after reduction takes place.

   - If the grammar has conflicts, conflict resolution removes some
     (shift or reduce) actions, hence may suppress the shortest path. *)

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(* We explicitly choose to ignore the [error] token. Thus, we disregard any
   reductions or transitions that take place when the lookahead symbol is
   [error]. As a result, any state whose incoming symbol is [error] is found
   unreachable. It would be too complicated to have to create a first error in
   order to be able to take certain transitions or drop certain parts of the
   input. *)

(* We never work with the terminal symbol [#] either. This symbol never
   appears in the maps returned by [Lr1.transitions] and [Lr1.reductions].
   Thus, in principle, we work with ``real'' terminal symbols only. However,
   we encode [any] as [#] -- see below. *)
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(* ------------------------------------------------------------------------ *)

(* To delay the side effects performed by this module, we wrap everything in
   in a big functor without arguments. *)

module Run (X : sig end) = struct

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open Grammar
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(* ------------------------------------------------------------------------ *)

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(* Because of our encoding of terminal symbols as 8-bit characters, this
   algorithm supports at most 256 terminal symbols. *)
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let () =
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  if Terminal.n > 256 then
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    Error.error [] (Printf.sprintf
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      "--list-errors supports at most 256 terminal symbols.\n\
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       The grammar has %d terminal symbols." Terminal.n
    )

(* ------------------------------------------------------------------------ *)

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(* Build a module that represents words as (hash-consed) strings. Note:
   this functor application has a side effect (it allocates memory, and
   more importantly, it may fail). *)

module W = Terminal.Word(struct end)
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(* ------------------------------------------------------------------------ *)

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(* The [error] token may appear in the maps returned by [Lr1.transitions]
   and [Lr1.reductions], so we sometimes need to explicitly check for it. *)
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let non_error z =
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  not (Terminal.equal z Terminal.error)
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(* We introduce a pseudo-terminal symbol [any]. It is used in several places
   later on, in particular in the field [fact.lookahead], to encode the
   absence of a lookahead hypothesis -- i.e., any terminal symbol will do. *)

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(* We choose to encode [any] as [#]. There is no risk of confusion, since we
   do not use [#] anywhere. Thus, the assertion [Terminal.real z] implies
   [z <> any]. *)

let any =
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  Terminal.sharp
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(* ------------------------------------------------------------------------ *)

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(* We begin with a number of auxiliary functions that provide information
   about the LR(1) automaton. These functions could perhaps be moved to a
   separate module. We keep them here, for the moment, because they are not
   used anywhere else. *)

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(* [reductions_on s z] is the list of reductions permitted in state [s] when
   the lookahead symbol is [z]. This is a list of zero or one elements. This
   does not take default reductions into account. *)
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let reductions_on s z : Production.index list =
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  assert (Terminal.real z);
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  try
    TerminalMap.find z (Lr1.reductions s)
  with Not_found ->
    []

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(* [has_reduction s z] tells whether state [s] is willing to reduce some
   production (and if so, which one) when the lookahead symbol is [z]. It
   takes a possible default reduction into account. *)
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let has_reduction s z : Production.index option =
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  assert (Terminal.real z);
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  match Invariant.has_default_reduction s with
  | Some (prod, _) ->
      Some prod
  | None ->
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      match reductions_on s z with
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      | prod :: prods ->
          assert (prods = []);
          Some prod
      | [] ->
          None

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(* [can_reduce s prod] indicates whether state [s] is able to reduce
   production [prod] (either as a default reduction, or as a normal
   reduction). *)

let can_reduce s prod =
  match Invariant.has_default_reduction s with
  | Some (prod', _) when prod = prod' ->
      true
  | _ ->
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      TerminalMap.fold (fun z prods accu ->
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        (* A reduction on [#] is always a default reduction. (See [lr1.ml].) *)
        assert (not (Terminal.equal z Terminal.sharp));
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        accu || non_error z && List.mem prod prods
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      ) (Lr1.reductions s) false

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(* [causes_an_error s z] tells whether state [s] will initiate an error on the
   lookahead symbol [z]. *)
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let causes_an_error s z : bool =
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  assert (Terminal.real z);
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  match Invariant.has_default_reduction s with
  | Some _ ->
      false
  | None ->
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      reductions_on s z = [] &&
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      not (SymbolMap.mem (Symbol.T z) (Lr1.transitions s))

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(* [foreach_terminal f] applies the function [f] to every terminal symbol in
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   turn, except [error] and [#]. *)
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let foreach_terminal =
  Terminal.iter_real
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(* [foreach_terminal_not_causing_an_error s f] applies the function [f] to
   every terminal symbol [z] such that [causes_an_error s z] is false. This
   could be implemented in a naive manner using [foreach_terminal] and
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   [causes_an_error]. This implementation is significantly more efficient. *)
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let foreach_terminal_not_causing_an_error s f =
  match Invariant.has_default_reduction s with
  | Some _ ->
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      (* There is a default reduction. No symbol causes an error. *)
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      foreach_terminal f
  | None ->
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      (* Enumerate every terminal symbol [z] for which there is a
         reduction. *)
      TerminalMap.iter (fun z _ ->
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        (* A reduction on [#] is always a default reduction. (See [lr1.ml].) *)
        assert (not (Terminal.equal z Terminal.sharp));
        if non_error z then
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          f z
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      ) (Lr1.reductions s);
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      (* Enumerate every terminal symbol [z] for which there is a
         transition. *)
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      SymbolMap.iter (fun sym _ ->
        match sym with
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        | Symbol.T z ->
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            assert (not (Terminal.equal z Terminal.sharp));
            if non_error z then
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              f z
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        | Symbol.N _ ->
            ()
      ) (Lr1.transitions s)

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(* Let us say a state [s] is solid if its incoming symbol is a terminal symbol
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   (or if it has no incoming symbol at all, i.e., it is an initial state). It
   is fragile if its incoming symbol is a non-terminal symbol. *)
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let is_solid s =
  match Lr1.incoming_symbol s with
  | None
  | Some (Symbol.T _) ->
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    true
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  | Some (Symbol.N _) ->
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    false
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(* ------------------------------------------------------------------------ *)

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(* Suppose [s] is a state that carries an outgoing edge labeled with a
   non-terminal symbol [nt]. We are interested in finding out how this edge
   can be taken. In order to do that, we must determine how, by starting in
   [s], one can follow a path that corresponds to (the right-hand side of) a
   production [prod] associated with [nt]. There are in general several such
   productions. The paths that they determine in the automaton form a "star".
   We represent the star rooted at [s] as a trie. For every state [s], the
   star rooted at [s] is constructed in advance, before the algorithm runs.
   While the algorithm runs, a point in the trie (that is, a sub-trie) tells
   us where we come form, where we are, and which production(s) we are hoping
   to reduce in the future. *)

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module Trie : sig

  type trie
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  (* [star s] creates a (new) trie whose source is [s], populated with its
     branches. (There is one branch for every production [prod] associated
     with every non-terminal symbol [nt] for which [s] carries an outgoing
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     edge.) If the star turns out to be trivial then [None] is returned. *)
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  val star: Lr1.node -> trie option

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  (* After [star s] has been called, [size (Lr1.number s)] reports the size
     of the trie that has been constructed for state [s]. *)
  val size: int -> int

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  (* Every (sub-)trie has a unique identity. (One can think of it as its
     address.) [compare] compares the identity of two tries. This can be
     used, e.g., to set up a map whose keys are tries. *)
  val compare: trie -> trie -> int

  (* [source t] returns the source state of the (sub-)trie [t]. This is
     the root of the star of which [t] is a sub-trie. In other words, this
     tells us "where we come from". *)
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  val source: trie -> Lr1.node
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  (* [current t] returns the current state of the (sub-)trie [t]. This is
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     the root of the sub-trie [t]. In other words, this tells us "where
     we are". *)
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  val current: trie -> Lr1.node
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  (* [accepts prod t] tells whether the current state of the trie [t] is
     the end of a branch associated with production [prod]. If so, this
     means that we have successfully followed a path that corresponds to
     the right-hand side of production [prod]. *)
  val accepts: Production.index -> trie -> bool
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  (* [step sym t] is the immediate sub-trie of [t] along the symbol [sym].
     This function raises [Not_found] if [t] has no child labeled [sym]. *)
  val step: Symbol.t -> trie -> trie

  (* [verbose()] outputs debugging & performance information. *)
  val verbose: unit -> unit

end = struct
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  (* A trie has the following structure. *)

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  type trie = {
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    (* A unique identity, used by [compare]. The trie construction code
       ensures that these numbers are indeed unique: see [fresh], [insert],
       [star]. *)
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    identity: int;
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    (* The root state of this star: "where we come from". *)
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    source: Lr1.node;
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    (* The current state, i.e., the root of this sub-trie: "where we are". *)
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    current: Lr1.node;
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    (* The productions that we can reduce in the current state. In other
       words, if this list is nonempty, then the current state is the end
       of one (or several) branches. It can nonetheless have children. *)
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    productions: Production.index list;
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    (* The children, or sub-tries. *)
    transitions: trie SymbolMap.t
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  }
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  (* This counter is used by [mktrie] to produce unique identities. *)
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  let c = ref 0

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  (* This smart constructor creates a new trie with a unique identity. *)
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  let mktrie source current productions transitions =
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    let identity = Misc.postincrement c in
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    { identity; source; current; productions; transitions }
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  exception DeadBranch

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  let rec insert w prod t =
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    match w with
    | [] ->
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        (* We check whether the current state [t.current] is able to reduce
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           production [prod]. (If [prod] cannot be reduced, the reduction
           action must have been suppressed by conflict resolution.) If not,
           then this branch is dead. This test is superfluous (i.e., it would
           be OK to conservatively assume that [prod] can be reduced) but
           allows us to build a slightly smaller star in some cases. *)
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        if can_reduce t.current prod then
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          (* We consume (update) the trie [t], so there is no need to allocate
             a new stamp. (Of course we could allocate a new stamp, but I prefer
             to be precise.) *)
          { t with productions = prod :: t.productions }
        else
          raise DeadBranch
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    | (Symbol.T t) :: _ when not (non_error t) ->
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         raise DeadBranch
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    | a :: w ->
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        (* Check if there is a transition labeled [a] out of [t.current]. If
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           there is, we add a child to the trie [t]. If there isn't, then it
           must have been removed by conflict resolution. (Indeed, it must be
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           present in a canonical automaton.) We could in this case return an
           unchanged sub-trie. We can do slightly better: we abort the whole
           insertion, so as to return an unchanged toplevel trie. *)
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        match SymbolMap.find a (Lr1.transitions t.current) with
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        | successor ->
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            (* Find our child at [a], or create it. *)
            let t' =
              try
                SymbolMap.find a t.transitions
              with Not_found ->
                mktrie t.source successor [] SymbolMap.empty
            in
            (* Update the child [t']. *)
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            let t' = insert w prod t' in
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            (* Update [t]. Again, no need to allocate a new stamp. *)
            { t with transitions = SymbolMap.add a t' t.transitions }
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        | exception Not_found ->
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            raise DeadBranch
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  (* [insert prod t] inserts a new branch, corresponding to production
     [prod], into the trie [t]. This function consumes its argument,
     which should no longer be used afterwards. *)
  let insert prod t =
    let w = Array.to_list (Production.rhs prod) in
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    let save = !c in
    try
      insert w prod t
    with DeadBranch ->
      c := save;
      t
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  (* [fresh s] creates a new empty trie whose source is [s]. *)
  let fresh source =
    mktrie source source [] SymbolMap.empty

  let star s =
    SymbolMap.fold (fun sym _ accu ->
      match sym with
      | Symbol.T _ ->
          accu
      | Symbol.N nt ->
          Production.foldnt nt accu insert
    ) (Lr1.transitions s) (fresh s)

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  (* [nontrivial t] tests whether the trie [t] has any branches, i.e.,
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     contains at least one sub-trie whose [productions] field is nonempty.
     Trivia: a trie of size greater than 1 is necessarily nontrivial, but the
     converse is not true: a nontrivial trie can have size 1. (This occurs
     when all productions have zero length.) *)
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  let nontrivial t =
    not (t.productions = [] && SymbolMap.is_empty t.transitions)

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  (* Redefine [star] to include a [nontrivial] test and to record the size
     of the newly built trie. *)

  let size =
    Array.make Lr1.n (-1)

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  let star s =
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    let initial = !c in
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    let t = star s in
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    let final = !c in
    size.(Lr1.number s) <- final - initial;
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    if nontrivial t then
      Some t
    else
      None

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  let size s =
    assert (size.(s) >= 0);
    size.(s)

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  let compare t1 t2 =
    Pervasives.compare (t1.identity : int) t2.identity
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  let source t =
    t.source

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  let current t =
    t.current
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  let accepts prod t =
    List.mem prod t.productions

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  let step a t =
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    SymbolMap.find a t.transitions (* careful: may raise [Not_found] *)
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  let verbose () =
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    Printf.fprintf stderr "Cumulated star size: %d\n%!" !c
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end

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(* ------------------------------------------------------------------------ *)

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(* The main algorithm, [LRijkstra], accumulates facts. A fact is a triple of a
   position (that is, a sub-trie), a word, and a lookahead assumption. Such a
   fact means that this position can be reached, from the source state
   [Trie.source fact.position], by consuming [fact.word], under the assumption
   that the next input symbol is [fact.lookahead]. *)

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(* We allow [fact.lookahead] to be [any] so as to indicate that this fact does
   not have a lookahead assumption. *)
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type fact = {
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  position: Trie.trie;
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  word: W.word;
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  lookahead: Terminal.t (* may be [any] *)
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}

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(* Accessors. *)

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let source fact =
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  Trie.source fact.position
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let current fact =
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  Trie.current fact.position
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(* Two invariants reduce the number of facts that we consider:

   1. If [fact.lookahead] is a terminal symbol [z] (i.e., not [any]), then
      [z] does not cause an error in the current state [current fact]. It
      would be useless to consider a fact that violates this property; it
      cannot possibly lead to a successful reduction.

   2. [fact.lookahead] is [any] iff the current state [current fact] is
      solid. This sounds rather reasonable (when a state is entered
      by shifting, it is entered regardless of which symbol follows)
      and simplifies the implementation of the sub-module [T].

*)

let invariant1 fact =
  fact.lookahead = any || not (causes_an_error (current fact) fact.lookahead)

let invariant2 fact =
  (fact.lookahead = any) = is_solid (current fact)

(* [compatible z a] checks whether the terminal symbol [a] satisfies the
   lookahead assumption [z] -- which can be [any]. *)

let compatible z a =
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  assert (non_error z);
  assert (Terminal.real a);
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  z = any || z = a

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(* ------------------------------------------------------------------------ *)
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(* As in Dijkstra's algorithm, a priority queue contains the facts that await
   examination. The length of [fact.word] serves as the priority of a fact.
   This guarantees that we discover shortest paths. (We never insert into the
   queue a fact whose priority is less than the priority of the last fact
   extracted out of the queue.) *)

(* [LowIntegerPriorityQueue] offers very efficient operations (essentially
   constant time, for a small constant). It exploits the fact that priorities
   are low nonnegative integers. *)

module Q = LowIntegerPriorityQueue

let q =
  Q.create()

(* We never insert into the queue a fact that immediately causes an error,
   i.e., a fact such that [causes_an_error (current fact) fact.lookahead]
   holds. In practice, this convention allows reducing the number of facts
   that go through the queue by a factor of two. *)

(* In principle, there is no need to insert the fact into the queue if [T]
   already stores a comparable fact. We could perform this test in [add].
   However, a quick experiment suggests that this is not worthwhile. The run
   time augments (because membership in [T] is tested twice, upon inserting
   and upon extracting) and the memory consumption does not seem to go down
   significantly. *)

let add fact =
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  (* [fact.lookahead] can be [any] *)
  assert (non_error fact.lookahead);
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  assert (invariant1 fact);
  assert (invariant2 fact);
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  (* The length of [fact.word] serves as the priority of this fact. *)
  Q.add q fact (W.length fact.word)

(* Construct the [star] of every state [s]. Initialize the priority queue. *)

let () =
  Lr1.iter (fun s ->
    match Trie.star s with
    | Some trie ->
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        (* TEMPORARY weird *)
        if is_solid s then
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          add {
            position = trie;
            word = W.epsilon;
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            lookahead = any
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          }
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        else
          foreach_terminal_not_causing_an_error s (fun z ->
            add {
              position = trie;
              word = W.epsilon;
              lookahead = z
            }
          )
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    | None ->
        ()
  )

(* ------------------------------------------------------------------------ *)

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(* The first symbol of the input, [W.first fact.word fact.lookahead], plays a
   special role. Indeed, for every position, for every first symbol, and for
   every lookahead symbol, we keep track of at most one fact. Thus, the total
   number of facts accumulated by the algorithm is at most [T.n^2], where [T]
   is the total size of the tries that we have constructed, and [n] is the
   number of terminal symbols. (This number can be quite large. [T] can be in
   the tens of thousands, and [n] can be over one hundred. These figures lead
   to a theoretical upper bound of 100M. In practice, for T=25K and n=108, we
   observe that the algorithm gathers about 7M facts.) *)

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module T : sig
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  (* [register fact] registers the fact [fact]. It returns [true] if this fact
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     is new, i.e., no fact concerning the same triple of [position], [a], and
     [z] was previously known. *)
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  val register: fact -> bool

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  (* [query current z f] enumerates all known facts whose current state is
     [current] and whose lookahead assumption is [z]. *)
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  val query: Lr1.node -> Terminal.t -> (fact -> unit) -> unit
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  val verbose: unit -> unit
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end = struct
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  (* This module implements a set of facts. Two facts are considered equal
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     (for the purposes of this set) if they have the same [position], [a], and
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     [z] fields. The [word] is not considered. Indeed, we are not interested
     in keeping track of several words that produce the same effect. Only the
     shortest such word is of interest. *)

  (* We need to query the set of facts in two ways. In [register], we need to
     test whether a fact is in the set. In [query], we need to find all facts
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     that match a pair [current, z]. For this reason, we use a two-level table.
     The first level is a matrix indexed by [current] and [z]. At the second
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     level, we find sets of facts. *)
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(**)

  module M =
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    MySet.Make(struct
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      type t = fact
      let compare fact1 fact2 =
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        let c = Trie.compare fact1.position fact2.position in
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        if c <> 0 then c else
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        let a1 = W.first fact1.word fact1.lookahead
        and a2 = W.first fact2.word fact2.lookahead in
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        (* note: [a1] and [a2] can be [any] here *)
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        Terminal.compare a1 a2
    end)
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  let table = (* a pretty large table... *)
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    Array.make (Lr1.n * Terminal.n) M.empty
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  (* TEMPORARY this space is wasted for solid states *)
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  let index current z =
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    Terminal.n * (Lr1.number current) + Terminal.t2i z
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  let count = ref 0

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  let register fact =
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    let current = current fact in
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    let z = fact.lookahead in
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    assert (non_error z);
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    (* [z] is [any] iff [current] is solid. *)
    assert ((z = any) = is_solid current);
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    let i = index current z in
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    let m = table.(i) in
    (* We crucially rely on the fact that [M.add] guarantees not to
       change the set if an ``equal'' fact already exists. Thus, a
       later, longer path is ignored in favor of an earlier, shorter
       path. *)
    let m' = M.add fact m in
    m != m' && begin
      incr count;
      table.(i) <- m';
      true
    end

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  let query current z f =
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    assert (z <> any);
    (* if [current] is solid then the facts that concern it are stored
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       under [any], not under [z] *)
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    let i = index current (if is_solid current then any else z) in
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    let m = table.(i) in
    M.iter f m
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  let verbose () =
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    Printf.fprintf stderr "T stores %d facts.\n%!" !count

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end

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(* ------------------------------------------------------------------------ *)

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(* The module [E] is in charge of recording the non-terminal edges that we have
   discovered, or more precisely, the conditions under which these edges can be
   taken. *)

module E : sig

  (* [register s nt w z] records that, in state [s], the outgoing edge labeled
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     [nt] can be taken by consuming the word [w], if the next symbol is [z].
     It returns [true] if this information is new. *)
  val register: Lr1.node -> Nonterminal.t -> W.word -> Terminal.t -> bool
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  (* [query s nt a z] answers whether, in state [s], the outgoing edge labeled
     [nt] can be taken by consuming some word [w], under the assumption that
     the next symbol is [z], and under the constraint that the first symbol of
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     [w.z] is [a]. *)
  val query: Lr1.node -> Nonterminal.t -> Terminal.t -> Terminal.t -> (W.word -> unit) -> unit
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  val verbose: unit -> unit
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end = struct

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  (* At a high level, we must implement a mapping of [s, nt, a, z] to [w]. In
     practice, we can implement this specification using any combination of
     arrays, hash tables, balanced binary trees, and perfect hashing (i.e.,
     packing several of [s], [nt], [a], [z] in one word.) Here, we choose to
     use an array, indexed by [s], of hash tables, indexed by a key that packs
     [nt], [a], and [z] in one word. According to a quick experiment, the
     final population of the hash table [table.(index s)] seems to be roughly
     [Terminal.n * Trie.size s]. We note that using an initial capacity
     of 0 and relying on the hash table's resizing mechanism has a significant
     cost, which is why we try to guess a good initial capacity. *)
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  module H = Hashtbl
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  let table = (* a pretty large table... *)
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    Array.init (Lr1.n) (fun i ->
      let size = Trie.size i in
      H.create (if size = 1 then 0 else Terminal.n * size)
    )
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  let index s =
    Lr1.number s
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  let pack nt a z : int =
    (Nonterminal.n2i nt lsl 16) lor
    (Terminal.t2i a lsl 8) lor
    (Terminal.t2i z)
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  let count = ref 0

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  let register s nt w z =
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    assert (Terminal.real z);
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    let i = index s in
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    let m = table.(i) in
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    let a = W.first w z in
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    assert (not (causes_an_error s a));
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    let key = pack nt a z in
    if H.mem m key then
      false
    else begin
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      incr count;
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      H.add m key w;
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      true
    end
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  let rec query s nt a z f =
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    assert (Terminal.real z);
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    (* [a] can be [any] *)
    if a <> any then begin
      let i = index s in
      let m = table.(i) in
      let key = pack nt a z in
      match H.find m key with
      | w -> f w
      | exception Not_found -> ()
    end
    else begin
      (* If [a] is [any], we query the table for every concrete [a].
         We can limit ourselves to symbols that do not cause an error
         in state [s]. Those that do certainly do not have an entry;
         see the assertion in [register] above. *)
      foreach_terminal_not_causing_an_error s (fun a ->
        query s nt a z f
      )
        (* TEMPORARY try a scheme that allows a more efficient iteration? *)
    end
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  let verbose () =
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    Printf.fprintf stderr "E stores %d facts.\n%!" !count

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end

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(* ------------------------------------------------------------------------ *)

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let new_edge s nt w z =
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  assert (Terminal.real z);
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  if E.register s nt w z then
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    let sym = Symbol.N nt in
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    T.query s (W.first w z) (fun fact ->
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      assert (compatible fact.lookahead (W.first w z));
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      match Trie.step sym fact.position with
      | position ->
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          assert (not (is_solid (Trie.current position)));
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          if not (causes_an_error (Trie.current position) z) then
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            add {
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              position;
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              word = W.append fact.word w;
              lookahead = z
            }
      | exception Not_found ->
          ()
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    )
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(* [consequences fact] is invoked when we discover a new fact (i.e., one that
   was not previously known). It studies the consequences of this fact. These
   consequences are of two kinds:

   - As in Dijkstra's algorithm, the new fact can be viewed as a newly
   discovered vertex. We study its (currently known) outgoing edges,
   and enqueue new facts in the priority queue.

   - Sometimes, a fact can also be viewed as a newly discovered edge.
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   This is the case when the word from [fact.source] to [fact.current]
   represents a production of the grammar and [fact.current] is willing
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   to reduce this production. We record the existence of this edge,
   and re-inspect any previously discovered vertices which are
   interested in this outgoing edge.
*)
(**)

let consequences fact =

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  let current = current fact in
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  (* 1. View [fact] as a vertex. Examine the transitions out of [current]. *)
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  SymbolMap.iter (fun sym s' ->
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    match Trie.step sym fact.position, sym with
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    | exception Not_found -> ()
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    | position, Symbol.T t ->
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        (* [t] cannot be the [error] token, because the trie does not have
           any edges labeled [error]. *)
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        assert (non_error t);
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        (* 1a. There is a transition labeled [t] out of [current]. If
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           the lookahead assumption [fact.lookahead] is compatible with [t],
           then we derive a new fact, where one more edge has been taken. We
           enqueue this new fact for later examination. *)
        (**)

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        if compatible fact.lookahead t then begin
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          let word = W.append fact.word (W.singleton t) in
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          (* assert (Lr1.Node.compare (Trie.current position) s' = 0); *)
          (* [s'] has a terminal incoming symbol. It is always entered
             without consideration for the next lookahead symbol. Thus,
             we use [any] as the lookahead assumption in the new fact
             that we produce. *)
          assert (is_solid (Trie.current position));
          add { position; word; lookahead = any }
        end
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    | position, Symbol.N nt ->
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        (* 1b. There is a transition labeled [nt] out of [current]. We
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           need to know how this nonterminal edge can be taken. We query for a
           word [w] that allows us to take this edge. The answer depends on
           the terminal symbol [z] that comes *after* this word: we try all
           such symbols. Furthermore, we need the first symbol of [w.z] to
           satisfy the lookahead assumption [fact.lookahead], so the answer
           also depends on this assumption. *)
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        (* TEMPORARY it could be that the answer does not depend on [z]...
           (default reduction) *)
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        (**)

        foreach_terminal_not_causing_an_error s' (fun z ->
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          E.query current nt fact.lookahead z (fun w ->
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            assert (compatible fact.lookahead (W.first w z));
            assert (not (is_solid (Trie.current position)));
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            add {
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              position;
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              word = W.append fact.word w;
              lookahead = z
            }
          )
        )
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  ) (Lr1.transitions current);
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  (* 2. View [fact] as a possible edge. This is possible if the path from
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     [fact.source] to [current] represents a production [prod] and
     [current] is willing to reduce this production. We check that
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     [fact.position] accepts [epsilon]. This guarantees that reducing [prod]
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     takes us all the way back to [fact.source]. Thus, this production gives
     rise to an edge labeled [nt] -- the left-hand side of [prod] -- out of
     [fact.source]. This edge is subject to the lookahead assumption
     [fact.lookahead], so we record that. *)
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  (**)

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  if fact.lookahead <> any then begin
    match has_reduction current fact.lookahead with
    | Some prod when Trie.accepts prod fact.position ->
        new_edge (source fact) (Production.nt prod) fact.word fact.lookahead
    | _ ->
        ()
  end
  else begin
    (* Every reduction must be considered. *)
    match Invariant.has_default_reduction current with
    | Some (prod, _) ->
        if Trie.accepts prod fact.position then
          (* TEMPORARY for now, avoid sending [any] into [new_edge] *)
          foreach_terminal (fun z ->
            new_edge (source fact) (Production.nt prod) (fact.word) z
          )
    | None ->
       TerminalMap.iter (fun z prods ->
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         if non_error z then
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           let prod = Misc.single prods in
           if Trie.accepts prod fact.position then
             new_edge (source fact) (Production.nt prod) (fact.word) z
       ) (Lr1.reductions current)
  end
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let level = ref 0
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let extracted, considered =
  ref 0, ref 0

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let done_with_level () =
  Printf.fprintf stderr "Done with level %d.\n" !level;
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  W.verbose();
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  T.verbose();
  E.verbose();
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  Printf.fprintf stderr "Q stores %d facts.\n" (Q.cardinal q);
  Printf.fprintf stderr "%d facts extracted out of Q, of which %d considered.\n%!"
    !extracted !considered
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let discover fact =
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  incr extracted;
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  if T.register fact then begin
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    if W.length fact.word > ! level then begin
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      done_with_level();
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      level := W.length fact.word;
    end;
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    incr considered;
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    consequences fact
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  end
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let () =
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  Trie.verbose();
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  Q.repeat q discover;
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  Time.tick "Running LRijkstra";
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  done_with_level()
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(* ------------------------------------------------------------------------ *)

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(* The following code validates the fact that an error can be triggered in
   state [s'] by beginning in the initial state [s] and reading the
   sequence of terminal symbols [w]. We use this for debugging purposes. *)

let fail msg =
  Printf.fprintf stderr "coverage: internal error: %s.\n%!" msg;
  false

open ReferenceInterpreter

let validate s s' w : bool =
  match
    ReferenceInterpreter.check_error_path (Lr1.nt_of_entry s) (W.elements w)
  with
  | OInputReadPastEnd ->
      fail "input was read past its end"
  | OInputNotFullyConsumed ->
      fail "input was not fully consumed"
  | OUnexpectedAccept ->
      fail "input was unexpectedly accepted"
  | OK state ->
      Lr1.Node.compare state s' = 0 ||
      fail (
        Printf.sprintf "error occurred in state %d instead of %d"
          (Lr1.number state)
          (Lr1.number s')
      )

(* ------------------------------------------------------------------------ *)

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(* We now wish to determine, given a state [s'] and a terminal symbol [z], a
   minimal path that takes us from some entry state to state [s'] with [z] as
   the next (unconsumed) symbol. *)

(* This can be formulated as a search for a shortest path in a graph. The
   graph is not just the automaton, though. It is a (much) larger graph whose
   vertices are pairs [s, z] and whose edges are obtained by querying the
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   module [E] above. *)
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let forward () =
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  let module A = Astar.Make(struct
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    (* A vertex is a pair [s, z].
       [z] cannot be the [error] token. *)
    type node =
        Lr1.node * Terminal.t

    let equal (s'1, z1) (s'2, z2) =
      Lr1.Node.compare s'1 s'2 = 0 && Terminal.compare z1 z2 = 0

    let hash (s, z) =
      Hashtbl.hash (Lr1.number s, z)

    (* An edge is labeled with a word. *)
    type label =
      W.word

    (* Forward search from every [s, z], where [s] is an initial state. *)
    let sources f =
      foreach_terminal (fun z ->
        ProductionMap.iter (fun _ s ->
          f (s, z)
        ) Lr1.entry
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      )

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    let successors (s, z) edge =
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      assert (Terminal.real z);
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      SymbolMap.iter (fun sym s' ->
        match sym with
        | Symbol.T t ->
            if Terminal.equal z t then
              let w = W.singleton t in
              foreach_terminal (fun z ->
                edge w 1 (s', z)
              )
        | Symbol.N nt ->
           foreach_terminal (fun z' ->
             E.query s nt z z' (fun w ->
               edge w (W.length w) (s', z')
             )
           )
      ) (Lr1.transitions s)

    let estimate _ =
      0

  end) in

  (* Search forward. *)

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  Printf.fprintf stderr "Forward search:\n%!";
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  let seen = ref Lr1.NodeSet.empty in
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  let _, _ = A.search (fun ((s', z), path) ->
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    if causes_an_error s' z && not (Lr1.NodeSet.mem s' !seen) then begin
      seen := Lr1.NodeSet.add s' !seen;
      (* An error can be triggered in state [s'] by beginning in the initial
         state [s] and reading the sequence of terminal symbols [w]. *)
      let (s, _), ws = A.reverse path in
      let w = List.fold_right W.append ws (W.singleton z) in
      Printf.fprintf stderr
        "An error can be reached from state %d to state %d:\n%!"
        (Lr1.number s)
        (Lr1.number s');
      Printf.fprintf stderr "%s\n%!" (W.print w);
      assert (validate s s' w)
    end
  ) in
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  Printf.fprintf stderr "Reachable (forward): %d states\n%!"
    (Lr1.NodeSet.cardinal !seen);
  !seen
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let () =
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  let f = forward() in
  Time.tick "Forward search";
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  let stat = Gc.quick_stat() in
  Printf.fprintf stderr
    "Maximum size reached by the major heap: %dM\n"
    (stat.Gc.top_heap_words * (Sys.word_size / 8) / 1024 / 1024);
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  ignore f
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(* TODO:
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  can we store fewer facts when we hit a default reduction?
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  remove CompletedNatWitness?, revert Fix
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  collect performance data, correlated with star size and alphabet size; draw a graph
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  count the unreachable states and see if they are numerous in practice
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  optionally report several ways of reaching an error in state s
    (with different lookahead tokens) (report all of them?)
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  warn if --list-errors is set AND the grammar uses [error]
  remove $syntaxerror?
  how do we maintain the list of error messages when the grammar evolves?
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  implement a naive semi-algorithm that enumerates all input sentences,
    and evaluate how well (or how badly) it scales
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*)
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end