Commit f268a31a authored by POTTIER Francois's avatar POTTIER Francois

Refactoring: split off [LRijkstraCore].

Also, move some auxiliary functions to [Default].
parent 2afd7e7c
......@@ -11,10 +11,10 @@
(* *)
(******************************************************************************)
(* 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 module implements [--list-errors]. Its purpose 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
......@@ -29,21 +29,9 @@
difficulties arise, including the fact that reductions are subject to a
lookahead hypothesis; the fact that some states have a default reduction,
hence will never trigger an error; the fact that conflict resolution
removes some (shift or reduce) actions, hence may suppress the shortest
x removes some (shift or reduce) actions, hence may suppress the shortest
path. *)
(* 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. *)
(* NOTE: THIS FILE IS COMPILED WITH -noassert BY DEFAULT. If you would like
the assertions to be tested at runtime, change that in the file _tags.
The performance impact of the assertions is about 10%. *)
......@@ -54,14 +42,18 @@
in a big functor. The functor also serves to pass verbosity parameters. *)
module Run (X : sig
(* If [verbose] is set, produce various messages on [stderr]. *)
val verbose: bool
(* If [statistics] is defined, it is interpreted as the name of
a file to which one line of statistics is appended. *)
val statistics: string option
end) = struct
open Grammar
open Default
(* ------------------------------------------------------------------------ *)
......@@ -79,805 +71,15 @@ let start =
(* ------------------------------------------------------------------------ *)
(* Because of our encoding of terminal symbols as 8-bit characters, this
algorithm supports at most 256 terminal symbols. *)
let () =
if Terminal.n > 256 then
Error.error []
"--list-errors supports at most 256 terminal symbols.\n\
The grammar has %d terminal symbols." Terminal.n
(* ------------------------------------------------------------------------ *)
(* Produce a warning if the grammar uses the [error] pseudo-token. *)
let () =
if grammar_uses_error_token then
Error.warning []
"--list-errors ignores all productions that involve the error token."
(* ------------------------------------------------------------------------ *)
(* 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)
(* ------------------------------------------------------------------------ *)
(* We introduce a pseudo-terminal symbol [any]. It is used in several places
later on, in particular in the [lookahead] field of a fact, to encode the
absence of a lookahead hypothesis -- i.e., any terminal symbol will do. *)
(* 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 =
Terminal.sharp
(* ------------------------------------------------------------------------ *)
(* 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. *)
(* [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. *)
let reductions_on s z : Production.index list =
assert (Terminal.real z);
try
TerminalMap.find z (Lr1.reductions s)
with Not_found ->
[]
(* [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. *)
let has_reduction s z : Production.index option =
assert (Terminal.real z);
match Default.has_default_reduction s with
| Some (prod, _) ->
Some prod
| None ->
match reductions_on s z with
| prod :: prods ->
assert (prods = []);
Some prod
| [] ->
None
(* [causes_an_error s z] tells whether state [s] will initiate an error on the
lookahead symbol [z]. *)
let causes_an_error s z : bool =
assert (Terminal.real z);
match Default.has_default_reduction s with
| Some _ ->
false
| None ->
reductions_on s z = [] &&
not (SymbolMap.mem (Symbol.T z) (Lr1.transitions s))
(* [foreach_terminal f] applies the function [f] to every terminal symbol in
turn, except [error] and [#]. *)
let foreach_terminal =
Terminal.iter_real
(* [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
[causes_an_error]. This implementation is significantly more efficient. *)
let foreach_terminal_not_causing_an_error s f =
match Default.has_default_reduction s with
| Some _ ->
(* There is a default reduction. No symbol causes an error. *)
foreach_terminal f
| None ->
(* Enumerate every terminal symbol [z] for which there is a
reduction. *)
TerminalMap.iter (fun z _ ->
(* A reduction on [#] is always a default reduction. (See [lr1.ml].) *)
assert (not (Terminal.equal z Terminal.sharp));
if Terminal.non_error z then
f z
) (Lr1.reductions s);
(* Enumerate every terminal symbol [z] for which there is a
transition. *)
SymbolMap.iter (fun sym _ ->
match sym with
| Symbol.T z ->
assert (not (Terminal.equal z Terminal.sharp));
if Terminal.non_error z then
f z
| Symbol.N _ ->
()
) (Lr1.transitions s)
(* Let us say a state [s] is solid if its incoming symbol is a terminal symbol
(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. *)
let is_solid s =
match Lr1.incoming_symbol s with
| None
| Some (Symbol.T _) ->
true
| Some (Symbol.N _) ->
false
(* ------------------------------------------------------------------------ *)
(* Instantiate [Trie]. This allocates fresh mutable state, but otherwise has
no effect. The construction of the tries actually takes place when
[Trie.stars] is invoked below. *)
module Trie =
Trie.Make(struct end)
(* ------------------------------------------------------------------------ *)
(* 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 position], by consuming [word], under the assumption
that the next input symbol is [lookahead]. *)
(* We allow [lookahead] to be [any] so as to indicate that this fact does
not have a lookahead assumption. *)
(*
type fact = {
position: Trie.trie;
word: W.word;
lookahead: Terminal.t (* may be [any] *)
}
*)
(* To save memory (and therefore time), we encode a fact in a single OCaml
integer value. This is made possible by the fact that tries, words, and
terminal symbols are represented as (or can be encoded as) integers.
This admittedly horrible hack allows us to save roughly a factor of 2
in space, and to gain 10% in time. *)
type fact = int
let dummy : fact =
-1 (* should never be accessed! *)
(* Encoding and decoding facts. *)
(* We encode [position|word|lookahead] in a single word of memory. *)
(* The lookahead symbol fits in 8 bits. *)
(* In the largest grammars that we have seen, the number of unique words is
about 3.10^5, so a word should fit in about 19 bits (2^19 = 524288). In the
largest grammars that we have seen, the total star size is about 64000, so a
trie should fit in about 17 bits (2^17 = 131072). *)
(* On a 64-bit machine, we have ample space in a 63-bit word! We allocate 30
bits for [word] and the rest (i.e., 25 bits) for [position]. *)
(* On a 32-bit machine, we are a bit more cramped! In Menhir's own fancy-parser,
the number of terminal symbols is 27, the number of unique words is 566, and
the total star size is 546. We allocate 12 bits for [word] and 11 bits for
[position]. This is better than refusing to work altogether, but still not
great. A more satisfactory approach might be to revert to heap allocation of
facts when in 32-bit mode, but that would make the code somewhat ugly. *)
let w_lookahead =
8
let w_word =
if Sys.word_size < 64 then 12 else 30
let w_position =
Sys.word_size - 1 - (w_word + w_lookahead) (* 25, on a 64-bit machine *)
let identity (fact : fact) : int =
assert (fact <> dummy);
fact lsr (w_word + w_lookahead)
let position (fact : fact) : Trie.trie =
assert (fact <> dummy);
Trie.decode (identity fact)
let word (fact : fact) : W.word =
assert (fact <> dummy);
(fact lsr w_lookahead) land (1 lsl w_word - 1)
let lookahead (fact : fact) : Terminal.t =
Terminal.i2t (fact land (1 lsl w_lookahead - 1))
let mkfact position (word : W.word) lookahead =
let position : int = Trie.encode position
and word : int = word
and lookahead : int = Terminal.t2i lookahead in
assert (0 <= position && 0 <= word && 0 <= lookahead);
assert (lookahead < 1 lsl w_lookahead);
if position < 1 lsl w_position && word < 1 lsl w_word then
(* [lsl] binds tighter than [lor] *)
(position lsl w_word lor word) lsl w_lookahead lor lookahead
else
let advice =
if Sys.word_size < 64 then
"Please use a 64-bit machine."
else
"Please report this error to Menhir's developers."
in
Error.error []
"an internal limit was exceeded.\n\
Sys.word_size = %d. Position = %d. Word = %d.\n\
%s%!"
Sys.word_size position word advice
let mkfact p w l =
let fact = mkfact p w l in
assert (word fact == w); (* round-trip property *)
assert (lookahead fact == l); (* round-trip property *)
assert (position fact == p); (* round-trip property *)
fact
(* Two invariants reduce the number of facts that we consider:
1. If [lookahead] is a real terminal symbol [z] (i.e., not [any]),
then [z] does not cause an error in the [current] state.
It would be useless to consider a fact that violates this property;
this cannot possibly lead to a successful reduction. In practice,
this refinement allows reducing the number of facts that go through
the queue by a factor of two.
2. [lookahead] is [any] iff the [current] state 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 [F].
*)
let invariant1 position _word lookahead =
let current = Trie.current position in
lookahead = any || not (causes_an_error current lookahead)
let invariant2 position _word lookahead =
let current = Trie.current position in
(lookahead = any) = is_solid current
(* [compatible z a] checks whether the terminal symbol [a] satisfies the
lookahead assumption [z] -- which can be [any]. *)
let compatible z a =
assert (Terminal.non_error z);
assert (Terminal.real a);
z = any || z = a
(* ------------------------------------------------------------------------ *)
(* As in Dijkstra's algorithm, a priority queue contains the facts that await
examination. The length of [word fact] 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 dummy
(* In principle, there is no need to insert the fact into the queue if [F]
already stores a comparable fact. We could perform this test in [enqueue].
However, a few experiments suggests that this is not worthwhile. The run
time augments (because membership in [F] is tested twice, upon inserting
and upon extracting) and the memory consumption does not seem to go down
significantly. *)
let enqueue position word lookahead =
(* [lookahead] can be [any], but cannot be [error] *)
assert (Terminal.non_error lookahead);
assert (invariant1 position word lookahead);
assert (invariant2 position word lookahead);
(* The length of [word] serves as the priority of this fact. *)
let priority = W.length word in
(* Encode and enqueue this fact. *)
Q.add q (mkfact position word lookahead) priority
(* ------------------------------------------------------------------------ *)
(* Construct the [star] of every state [s]. Initialize the priority queue. *)
let () =
(* For every state [s], if the trie rooted at [s] is nontrivial, ... *)
Trie.stars (fun s position ->
(* ...then insert an initial fact into the priority queue. *)
(* In order to respect invariants 1 and 2, we must distinguish two
cases. If [s] is solid, then we insert a single fact, whose
lookahead assumption is [any]. Otherwise, we must insert one
initial fact for every terminal symbol [z] that does not cause
an error in state [s]. *)
let word = W.epsilon in
if is_solid s then
enqueue position word any
else
foreach_terminal_not_causing_an_error s (fun z ->
enqueue position word z
)
);
if X.verbose then
Trie.verbose()
(* ------------------------------------------------------------------------ *)
(* The module [F] maintains a set of known facts. *)
(* Three aspects of a fact are of particular interest:
- its position [position], given by [position fact];
- its first symbol [a], given by [W.first (word fact) (lookahead fact)];
- its lookahead assumption [z], given by [lookahead fact].
For every triple of [position], [a], and [z], we store at most one fact,
(whose word has minimal length). Indeed, we are not interested in keeping
track of several words that produce the same effect. Only the shortest such
word is of interest.
(* Run the core reachability analysis, which finds out exactly under
what conditions each nonterminal transition in the automaton can be
taken. *)
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.) *)
module Core =
LRijkstraCore.Run(X)
module F : sig
(* [register fact] registers the fact [fact]. It returns [true] if this fact
is new, i.e., no fact concerning the same triple of [position], [a], and
[z] was previously known. *)
val register: fact -> bool
(* [query current z f] enumerates all known facts whose current state is
[current] and whose lookahead assumption is compatible with [z]. The
symbol [z] must a real terminal symbol, i.e., cannot be [any]. *)
val query: Lr1.node -> Terminal.t -> (fact -> unit) -> unit
(* [size()] returns the number of facts currently stored in the set. *)
val size: unit -> int
(* [verbose()] outputs debugging & performance information. *)
val verbose: unit -> unit
end = struct
(* We need to query the set of facts in two ways. In [register], we must test
whether a proposed triple of [position], [a], [z] already appears in the
set. In [query], we must find all facts that match a pair [current, z],
where [current] is a state. (Note that [position] determines [current], but
the converse is not true: a position contains more information besides the
current state.)
To address these needs, we use a two-level table. The first level is a
matrix indexed by [current] and [z]. At the second level, we find sets of
facts, where two facts are considered equal if they have the same triple of
[position], [a], and [z]. In fact, we know at this level that all facts
have the same [z] component, so only [position] and [a] are compared.
Because our facts satisfy invariant 2, [z] is [any] if and only if the
state [current] is solid. This means that we are wasting quite a
lot of space in the matrix (for a solid state, the whole line is empty,
except for the [any] column). *)
(* The level-2 sets. *)
module M =
MySet.Make(struct
type t = fact
let compare fact1 fact2 =
assert (lookahead fact1 = lookahead fact2);
(* Compare the two positions first. This can be done without going
through [Trie.decode], by directly comparing the two integer
identities. *)
let c = Pervasives.compare (identity fact1) (identity fact2) in
assert (c = Trie.compare (position fact1) (position fact2));
if c <> 0 then c else
let z = lookahead fact1 in
let a1 = W.first (word fact1) z
and a2 = W.first (word fact2) z in
(* note: [a1] and [a2] can be [any] here *)
Terminal.compare a1 a2
end)
(* The level-1 matrix. *)
let table =
Array.make (Lr1.n * Terminal.n) M.empty
let index current z =
Terminal.n * (Lr1.number current) + Terminal.t2i z
let count = ref 0
let register fact =
let current = Trie.current (position fact) in
let z = lookahead fact in
let i = index current z in
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
let query current z f =
assert (not (Terminal.equal z any));
(* If the state [current] is solid then the facts that concern it are
stored in the column [any], and all of them are compatible with [z].
Otherwise, they are stored in all columns except [any], and only
those stored in the column [z] are compatible with [z]. *)
let i = index current (if is_solid current then any else z) in
let m = table.(i) in
M.iter f m
let size () =
!count
let verbose () =
Printf.eprintf "F stores %d facts.\n%!" (size())
end
(* ------------------------------------------------------------------------ *)
(* 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.
It maintains a set of quadruples [s, nt, w, z], where such a quadruple means
that in the state [s], the outgoing edge labeled [nt] can be taken by
consuming the word [w], under the assumption that the next symbol is [z].
Again, the terminal symbol [a], given by [W.first w z], plays a role. For
each quadruple [s, nt, a, z], we store at most one quadruple [s, nt, w, z].
Thus, internally, we maintain a mapping of [s, nt, a, z] to [w].
For greater simplicity, we do not allow [z] to be [any] in [register] or
[query]. Allowing it would complicate things significantly, it seems. *)
module E : sig
(* [register s nt w z] records that, in state [s], the outgoing edge labeled
[nt] can be taken by consuming the word [w], if the next symbol is [z].
It returns [true] if this information is new, i.e., if the underlying
quadruple [s, nt, a, z] is new. The symbol [z] cannot be [any]. *)
val register: Lr1.node -> Nonterminal.t -> W.word -> Terminal.t -> bool
(* [query s nt a foreach] enumerates all words [w] and all real symbols [z]
such that, in state [s], the outgoing edge labeled [nt] can be taken by
consuming the word [w], under the assumption that the next symbol is [z],
and the first symbol of the word [w.z] is [a]. The symbol [a] can be [any].
The function [foreach] can be either [foreach_terminal] or of the form
[foreach_terminal_not_causing_an_error _]. It limits the symbols [z] that
are considered. *)
val query: Lr1.node -> Nonterminal.t -> Terminal.t ->
(* foreach: *) ((Terminal.t -> unit) -> unit) ->
(W.word -> Terminal.t -> unit) -> unit
(* [size()] returns the number of edges currently stored in the set. *)
val size: unit -> int
(* [verbose()] outputs debugging & performance information. *)
val verbose: unit -> unit
end = struct
(* 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. *)
module H = Hashtbl
let table =
Array.init Lr1.n (fun i ->
let size = Trie.size i in
H.create (if size = 1 then 0 else Terminal.n * size)
)
let index s =
Lr1.number s
let pack nt a z : int =
(* We rely on the fact that we have at most 256 terminal symbols. *)
(Nonterminal.n2i nt lsl 16) lor
(Terminal.t2i a lsl 8) lor
(Terminal.t2i z)
let count = ref 0
let register s nt w z =
assert (Terminal.real z);
let i = index s in
let m = table.(i) in
let a = W.first w z in
(* Note that looking at [a] in state [s] cannot cause an error. *)
assert (not (causes_an_error s a));
let key = pack nt a z in
if H.mem m key then
false
else begin
incr count;
H.add m key w;
true
end
let rec query s nt a foreach f =
if Terminal.equal a any then begin
(* If [a] is [any], we query the table for every real symbol [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 foreach f
)
end
else
let i = index s in
let m = table.(i) in
foreach (fun z ->
assert (Terminal.real z);
let key = pack nt a z in
match H.find m key with
| w -> f w z
| exception Not_found -> ()
)
let size () =
!count
let verbose () =
Printf.eprintf "E stores %d edges.\n%!" (size())
end
(* ------------------------------------------------------------------------ *)
(* [new_edge s nt w z] is invoked when we discover that in the state [s], the
outgoing edge labeled [nt] can be taken by consuming the word [w], under
the assumption that the next symbol is [z]. We check whether this quadruple
already exists in the set [E]. If not, then we add it, and we compute its
consequences, in the form of new facts, which we insert into the priority
queue for later examination. *)
let new_edge s nt w z =
assert (Terminal.real z);
if E.register s nt w z then
let sym = Symbol.N nt in
(* Query [F] for existing facts which could be extended by following
this newly discovered edge. They must be facts whose current state
is [s] and whose lookahead assumption is compatible with [a]. For
each such fact, ... *)
F.query s (W.first w z) (fun fact ->
assert (compatible (lookahead fact) (W.first w z));
(* ... try to take one step in the trie along an edge labeled [nt]. *)
match Trie.step sym (position fact) with
| position ->
(* This takes us to a new state whose incoming symbol is [nt].
Hence, this state is not solid. In order to satisfy invariant 2,
we must create fact whose lookahead assumption is not [any].
That's fine, since our lookahead assumption is [z]. In order to
satisfy invariant 1, we must check that [z] does not cause an
error in this state. *)
assert (not (is_solid (Trie.current position)));
if not (causes_an_error (Trie.current position) z) then
let word = W.append (word fact) w in
enqueue position word z
| exception Not_found ->
(* Could not take a step in the trie. This means this branch
leads nowhere of interest, and was pruned when the trie
was constructed. *)
()
)
(* ------------------------------------------------------------------------ *)
(* [new_fact 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. This is
the case when the word that took us from [source] to [current]
represents a production of the grammar and [current] is willing 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 new_fact fact =
(* Throughout this rather long function, there is just one [fact]. Let's
name its components right now, so as to avoid accessing them several
times. (That could be costly, as it requires decoding the fact.) *)
let position = position fact
and lookahead = lookahead fact
and word = word fact in
let source = Trie.source position
and current = Trie.current position in
(* 1. View [fact] as a vertex. Examine the transitions out of [current].
For every transition labeled by a symbol [sym] and into a state
[target], ... *)
Lr1.transitions current |> SymbolMap.iter (fun sym target ->
(* ... try to follow this transition in the trie [position],
down to a child which we call [child]. *)
match Trie.step sym position, sym with
| exception Not_found ->
(* Could not take a step in the trie. This means this transition
leads nowhere of interest. *)
()
| child, Symbol.T t ->
(* 1a. The transition exists in the trie, and [sym] is in fact a
terminal symbol [t]. We note that [t] cannot be the [error] token,
because the trie does not have any edges labeled [error]. *)
assert (Lr1.Node.compare (Trie.current child) target = 0);
assert (is_solid target);
assert (Terminal.non_error t);
(* If the lookahead assumption [lookahead] is compatible with
[t], then we derive a new fact, where one more edge has been taken,
and enqueue this new fact for later examination. *)
(* The state [target] is solid, i.e., its incoming symbol is terminal.
This state is always entered without consideration for the next
lookahead symbol. Thus, we can use [any] as the lookahead assumption
in the new fact that we produce. If we did not have [any], we would
have to produce one fact for every possible lookahead symbol. *)
if compatible lookahead t then
let word = W.append word (W.singleton t) in
enqueue child word any
| child, Symbol.N nt ->