%%
%% %CopyrightBegin%
%%
%% Copyright Ericsson AB 2003-2009. All Rights Reserved.
%%
%% The contents of this file are subject to the Erlang Public License,
%% Version 1.1, (the "License"); you may not use this file except in
%% compliance with the License. You should have received a copy of the
%% Erlang Public License along with this software. If not, it can be
%% retrieved online at http://www.erlang.org/.
%%
%% Software distributed under the License is distributed on an "AS IS"
%% basis, WITHOUT WARRANTY OF ANY KIND, either express or implied. See
%% the License for the specific language governing rights and limitations
%% under the License.
%%
%% %CopyrightEnd%
%%
%% =====================================================================
%% Closure analysis of Core Erlang programs.
%%
%% Copyright (C) 2001-2002 Richard Carlsson
%%
%% Author contact: [email protected]
%% =====================================================================
%% TODO: might need a "top" (`any') element for any-length value lists.
-module(cerl_closurean).
-export([analyze/1, annotate/1]).
%% The following functions are exported from this module since they
%% are also used by Dialyzer (file dialyzer/src/dialyzer_dep.erl)
-export([is_escape_op/2, is_escape_op/3, is_literal_op/2, is_literal_op/3]).
-import(cerl, [ann_c_apply/3, ann_c_fun/3, ann_c_var/2, apply_args/1,
apply_op/1, atom_val/1, bitstr_size/1, bitstr_val/1,
binary_segments/1, c_letrec/2, c_seq/2, c_tuple/1,
c_nil/0, call_args/1, call_module/1, call_name/1,
case_arg/1, case_clauses/1, catch_body/1, clause_body/1,
clause_guard/1, clause_pats/1, cons_hd/1, cons_tl/1,
fun_body/1, fun_vars/1, get_ann/1, is_c_atom/1,
let_arg/1, let_body/1, let_vars/1, letrec_body/1,
letrec_defs/1, module_defs/1, module_defs/1,
module_exports/1, pat_vars/1, primop_args/1,
primop_name/1, receive_action/1, receive_clauses/1,
receive_timeout/1, seq_arg/1, seq_body/1, set_ann/2,
try_arg/1, try_body/1, try_vars/1, try_evars/1,
try_handler/1, tuple_es/1, type/1, values_es/1]).
-import(cerl_trees, [get_label/1]).
%% ===========================================================================
-type label() :: integer() | 'top' | 'external' | 'external_call'.
-type ordset(X) :: [X]. % XXX: TAKE ME OUT
-type labelset() :: ordset(label()).
-type outlist() :: [labelset()] | 'none'.
-type escapes() :: labelset().
%% ===========================================================================
%% annotate(Tree) -> {Tree1, OutList, Outputs, Escapes, Dependencies, Parents}
%%
%% Tree = cerl:cerl()
%%
%% Analyzes `Tree' (see `analyze') and appends terms `{callers,
%% Labels}' and `{calls, Labels}' to the annotation list of each
%% fun-expression node and apply-expression node of `Tree',
%% respectively, where `Labels' is an ordered-set list of labels of
%% fun-expressions in `Tree', possibly also containing the atom
%% `external', corresponding to the dependency information derived
%% by the analysis. Any previous such annotations are removed from
%% `Tree'. `Tree1' is the modified tree; for details on `OutList',
%% `Outputs' , `Dependencies', `Escapes' and `Parents', see
%% `analyze'.
%%
%% Note: `Tree' must be annotated with labels in order to use this
%% function; see `analyze' for details.
-spec annotate(cerl:cerl()) ->
{cerl:cerl(), outlist(), dict(), escapes(), dict(), dict()}.
annotate(Tree) ->
{Xs, Out, Esc, Deps, Par} = analyze(Tree),
F = fun (T) ->
case type(T) of
'fun' ->
L = get_label(T),
X = case dict:find(L, Deps) of
{ok, X1} -> X1;
error -> set__new()
end,
set_ann(T, append_ann(callers,
set__to_list(X),
get_ann(T)));
apply ->
L = get_label(T),
X = case dict:find(L, Deps) of
{ok, X1} -> X1;
error -> set__new()
end,
set_ann(T, append_ann(calls,
set__to_list(X),
get_ann(T)));
_ ->
%%% set_ann(T, []) % debug
T
end
end,
{cerl_trees:map(F, Tree), Xs, Out, Esc, Deps, Par}.
append_ann(Tag, Val, [X | Xs]) ->
if tuple_size(X) >= 1, element(1, X) =:= Tag ->
append_ann(Tag, Val, Xs);
true ->
[X | append_ann(Tag, Val, Xs)]
end;
append_ann(Tag, Val, []) ->
[{Tag, Val}].
%% =====================================================================
%% analyze(Tree) -> {OutList, Outputs, Escapes, Dependencies, Parents}
%%
%% Tree = cerl()
%% OutList = [LabelSet] | none
%% Outputs = dict(Label, OutList)
%% Escapes = LabelSet
%% Dependencies = dict(Label, LabelSet)
%% LabelSet = ordset(Label)
%% Label = integer() | top | external | external_call
%% Parents = dict(Label, Label)
%%
%% Analyzes a module or an expression represented by `Tree'.
%%
%% The returned `OutList' is a list of sets of labels of
%% fun-expressions which correspond to the possible closures in the
%% value list produced by `Tree' (viewed as an expression; the
%% "value" of a module contains its exported functions). The atom
%% `none' denotes missing or conflicting information.
%%
%% The atom `external' in any label set denotes any possible
%% function outside `Tree', including those in `Escapes'. The atom
%% `top' denotes the top-level expression `Tree'.
%%
%% `Outputs' is a mapping from the labels of fun-expressions in
%% `Tree' to corresponding lists of sets of labels of
%% fun-expressions (or the atom `none'), representing the possible
%% closures in the value lists returned by the respective
%% functions.
%%
%% `Dependencies' is a similar mapping from the labels of
%% fun-expressions and apply-expressions in `Tree' to sets of
%% labels of corresponding fun-expressions which may contain call
%% sites of the functions or be called from the call sites,
%% respectively. Any such label not defined in `Dependencies'
%% represents an unreachable function or a dead or faulty
%% application.
%%
%% `Escapes' is the set of labels of fun-expressions in `Tree' such
%% that corresponding closures may be accessed from outside `Tree'.
%%
%% `Parents' is a mapping from labels of fun-expressions in `Tree'
%% to the corresponding label of the nearest containing
%% fun-expression or top-level expression. This can be used to
%% extend the dependency graph, for certain analyses.
%%
%% Note: `Tree' must be annotated with labels (as done by the
%% function `cerl_trees:label/1') in order to use this function.
%% The label annotation `{label, L}' (where L should be an integer)
%% must be the first element of the annotation list of each node in
%% the tree. Instances of variables bound in `Tree' which denote
%% the same variable must have the same label; apart from this,
%% labels should be unique. Constant literals do not need to be
%% labeled.
-record(state, {vars, out, dep, work, funs, par}).
%% Note: In order to keep our domain simple, we assume that all remote
%% calls and primops return a single value, if any.
%% We use the terms `closure', `label', `lambda' and `fun-expression'
%% interchangeably. The exact meaning in each case can be grasped from
%% the context.
%%
%% Rules:
%% 1) The implicit top level lambda escapes.
%% 2) A lambda returned by an escaped lambda also escapes.
%% 3) An escaped lambda can be passed an external lambda as argument.
%% 4) A lambda passed as argument to an external lambda also escapes.
%% 5) An argument passed to an unknown operation escapes.
%% 6) A call to an unknown operation can return an external lambda.
%%
%% Escaped lambdas become part of the set of external lambdas, but this
%% does not need to be represented explicitly.
%% We wrap the given syntax tree T in a fun-expression labeled `top',
%% which is initially in the set of escaped labels. `top' will be
%% visited at least once.
%%
%% We create a separate function labeled `external', defined as:
%% "'external'/1 = fun (Escape) -> do apply 'external'/1(apply Escape())
%% 'external'/1", which will represent any and all functions outside T,
%% and which returns itself, and contains a recursive call; this models
%% rules 2 and 4 above. It will be revisited if the set of escaped
%% labels changes, or at least once. Its parameter `Escape' is a
%% variable labeled `escape', which will hold the set of escaped labels.
%% initially it contains `top' and `external'.
-spec analyze(cerl:cerl()) -> {outlist(), dict(), escapes(), dict(), dict()}.
analyze(Tree) ->
%% Note that we use different name spaces for variable labels and
%% function/call site labels, so we can reuse some names here. We
%% assume that the labeling of Tree only uses integers, not atoms.
External = ann_c_var([{label, external}], {external, 1}),
Escape = ann_c_var([{label, escape}], 'Escape'),
ExtBody = c_seq(ann_c_apply([{label, loop}], External,
[ann_c_apply([{label, external_call}],
Escape, [])]),
External),
ExtFun = ann_c_fun([{label, external}], [Escape], ExtBody),
%%% io:fwrite("external fun:\n~s.\n",
%%% [cerl_prettypr:format(ExtFun, [noann])]),
Top = ann_c_var([{label, top}], {top, 0}),
TopFun = ann_c_fun([{label, top}], [], Tree),
%% The "start fun" just makes the initialisation easier. It will not
%% be marked as escaped, and thus cannot be called.
StartFun = ann_c_fun([{label, start}], [],
c_letrec([{External, ExtFun}, {Top, TopFun}],
c_nil())),
%%% io:fwrite("start fun:\n~s.\n",
%%% [cerl_prettypr:format(StartFun, [noann])]),
%% Gather a database of all fun-expressions in Tree and initialise
%% all their outputs and parameter variables. Bind all module- and
%% letrec-defined variables to their corresponding labels.
Funs0 = dict:new(),
Vars0 = dict:new(),
Out0 = dict:new(),
Empty = empty(),
F = fun (T, S = {Fs, Vs, Os}) ->
case type(T) of
'fun' ->
L = get_label(T),
As = fun_vars(T),
{dict:store(L, T, Fs),
bind_vars_single(As, Empty, Vs),
dict:store(L, none, Os)};
letrec ->
{Fs, bind_defs(letrec_defs(T), Vs), Os};
module ->
{Fs, bind_defs(module_defs(T), Vs), Os};
_ ->
S
end
end,
{Funs, Vars, Out} = cerl_trees:fold(F, {Funs0, Vars0, Out0},
StartFun),
%% Initialise Escape to the minimal set of escaped labels.
Vars1 = dict:store(escape, from_label_list([top, external]), Vars),
%% Enter the fixpoint iteration at the StartFun.
St = loop(StartFun, start, #state{vars = Vars1,
out = Out,
dep = dict:new(),
work = init_work(),
funs = Funs,
par = dict:new()}),
%%% io:fwrite("dependencies: ~p.\n",
%%% [[{X, set__to_list(Y)}
%%% || {X, Y} <- dict:to_list(St#state.dep)]]),
{dict:fetch(top, St#state.out),
tidy_dict([start, top, external], St#state.out),
dict:fetch(escape, St#state.vars),
tidy_dict([loop], St#state.dep),
St#state.par}.
tidy_dict([X | Xs], D) ->
tidy_dict(Xs, dict:erase(X, D));
tidy_dict([], D) ->
D.
loop(T, L, St0) ->
%%% io:fwrite("analyzing: ~w.\n", [L]),
%%% io:fwrite("work: ~w.\n", [St0#state.work]),
Xs0 = dict:fetch(L, St0#state.out),
{Xs, St1} = visit(fun_body(T), L, St0),
{W, M} = case equal(Xs0, Xs) of
true ->
{St1#state.work, St1#state.out};
false ->
%%% io:fwrite("out (~w) changed: ~w <- ~w.\n",
%%% [L, Xs, Xs0]),
M1 = dict:store(L, Xs, St1#state.out),
case dict:find(L, St1#state.dep) of
{ok, S} ->
{add_work(set__to_list(S), St1#state.work),
M1};
error ->
{St1#state.work, M1}
end
end,
St2 = St1#state{out = M},
case take_work(W) of
{ok, L1, W1} ->
T1 = dict:fetch(L1, St2#state.funs),
loop(T1, L1, St2#state{work = W1});
none ->
St2
end.
visit(T, L, St) ->
case type(T) of
literal ->
{[empty()], St};
var ->
%% If a variable is not already in the store here, we
%% initialize it to empty().
L1 = get_label(T),
Vars = St#state.vars,
case dict:find(L1, Vars) of
{ok, X} ->
{[X], St};
error ->
X = empty(),
St1 = St#state{vars = dict:store(L1, X, Vars)},
{[X], St1}
end;
'fun' ->
%% Must revisit the fun also, because its environment might
%% have changed. (We don't keep track of such dependencies.)
L1 = get_label(T),
St1 = St#state{work = add_work([L1], St#state.work),
par = set_parent([L1], L, St#state.par)},
{[singleton(L1)], St1};
values ->
visit_list(values_es(T), L, St);
cons ->
{Xs, St1} = visit_list([cons_hd(T), cons_tl(T)], L, St),
{[join_single_list(Xs)], St1};
tuple ->
{Xs, St1} = visit_list(tuple_es(T), L, St),
{[join_single_list(Xs)], St1};
'let' ->
{Xs, St1} = visit(let_arg(T), L, St),
Vars = bind_vars(let_vars(T), Xs, St1#state.vars),
visit(let_body(T), L, St1#state{vars = Vars});
seq ->
{_, St1} = visit(seq_arg(T), L, St),
visit(seq_body(T), L, St1);
apply ->
{Xs, St1} = visit(apply_op(T), L, St),
{As, St2} = visit_list(apply_args(T), L, St1),
case Xs of
[X] ->
%% We store the dependency from the call site to the
%% called functions
Ls = set__to_list(X),
Out = St2#state.out,
Xs1 = join_list([dict:fetch(Lx, Out) || Lx <- Ls]),
St3 = call_site(Ls, L, As, St2),
L1 = get_label(T),
D = dict:store(L1, X, St3#state.dep),
{Xs1, St3#state{dep = D}};
none ->
{none, St2}
end;
call ->
M = call_module(T),
F = call_name(T),
{_, St1} = visit(M, L, St),
{_, St2} = visit(F, L, St1),
{Xs, St3} = visit_list(call_args(T), L, St2),
remote_call(M, F, Xs, St3);
primop ->
As = primop_args(T),
{Xs, St1} = visit_list(As, L, St),
primop_call(atom_val(primop_name(T)), length(Xs), Xs, St1);
'case' ->
{Xs, St1} = visit(case_arg(T), L, St),
visit_clauses(Xs, case_clauses(T), L, St1);
'receive' ->
X = singleton(external),
{Xs1, St1} = visit_clauses([X], receive_clauses(T), L, St),
{_, St2} = visit(receive_timeout(T), L, St1),
{Xs2, St3} = visit(receive_action(T), L, St2),
{join(Xs1, Xs2), St3};
'try' ->
{Xs1, St1} = visit(try_arg(T), L, St),
X = singleton(external),
Vars = bind_vars(try_vars(T), [X], St1#state.vars),
{Xs2, St2} = visit(try_body(T), L, St1#state{vars = Vars}),
Evars = bind_vars(try_evars(T), [X, X, X], St2#state.vars),
{Xs3, St3} = visit(try_handler(T), L, St2#state{vars = Evars}),
{join(join(Xs1, Xs2), Xs3), St3};
'catch' ->
{_, St1} = visit(catch_body(T), L, St),
{[singleton(external)], St1};
binary ->
{_, St1} = visit_list(binary_segments(T), L, St),
{[empty()], St1};
bitstr ->
%% The other fields are constant literals.
{_, St1} = visit(bitstr_val(T), L, St),
{_, St2} = visit(bitstr_size(T), L, St1),
{none, St2};
letrec ->
%% All the bound funs should be revisited, because the
%% environment might have changed.
Ls = [get_label(F) || {_, F} <- letrec_defs(T)],
St1 = St#state{work = add_work(Ls, St#state.work),
par = set_parent(Ls, L, St#state.par)},
visit(letrec_body(T), L, St1);
module ->
%% All the exported functions escape, and can thus be passed
%% any external closures as arguments. We regard a module as
%% a tuple of function variables in the body of a `letrec'.
visit(c_letrec(module_defs(T), c_tuple(module_exports(T))),
L, St)
end.
visit_clause(T, Xs, L, St) ->
Vars = bind_pats(clause_pats(T), Xs, St#state.vars),
{_, St1} = visit(clause_guard(T), L, St#state{vars = Vars}),
visit(clause_body(T), L, St1).
%% We assume correct value-list typing.
visit_list([T | Ts], L, St) ->
{Xs, St1} = visit(T, L, St),
{Xs1, St2} = visit_list(Ts, L, St1),
X = case Xs of
[X1] -> X1;
none -> none
end,
{[X | Xs1], St2};
visit_list([], _L, St) ->
{[], St}.
visit_clauses(Xs, [T | Ts], L, St) ->
{Xs1, St1} = visit_clause(T, Xs, L, St),
{Xs2, St2} = visit_clauses(Xs, Ts, L, St1),
{join(Xs1, Xs2), St2};
visit_clauses(_, [], _L, St) ->
{none, St}.
bind_defs([{V, F} | Ds], Vars) ->
bind_defs(Ds, dict:store(get_label(V), singleton(get_label(F)),
Vars));
bind_defs([], Vars) ->
Vars.
bind_pats(Ps, none, Vars) ->
bind_pats_single(Ps, empty(), Vars);
bind_pats(Ps, Xs, Vars) ->
if length(Xs) =:= length(Ps) ->
bind_pats_list(Ps, Xs, Vars);
true ->
bind_pats_single(Ps, empty(), Vars)
end.
bind_pats_list([P | Ps], [X | Xs], Vars) ->
bind_pats_list(Ps, Xs, bind_vars_single(pat_vars(P), X, Vars));
bind_pats_list([], [], Vars) ->
Vars.
bind_pats_single([P | Ps], X, Vars) ->
bind_pats_single(Ps, X, bind_vars_single(pat_vars(P), X, Vars));
bind_pats_single([], _X, Vars) ->
Vars.
bind_vars(Vs, none, Vars) ->
bind_vars_single(Vs, empty(), Vars);
bind_vars(Vs, Xs, Vars) ->
if length(Vs) =:= length(Xs) ->
bind_vars_list(Vs, Xs, Vars);
true ->
bind_vars_single(Vs, empty(), Vars)
end.
bind_vars_list([V | Vs], [X | Xs], Vars) ->
bind_vars_list(Vs, Xs, dict:store(get_label(V), X, Vars));
bind_vars_list([], [], Vars) ->
Vars.
bind_vars_single([V | Vs], X, Vars) ->
bind_vars_single(Vs, X, dict:store(get_label(V), X, Vars));
bind_vars_single([], _X, Vars) ->
Vars.
%% This handles a call site - adding dependencies and updating parameter
%% variables with respect to the actual parameters. The 'external'
%% function is handled specially, since it can get an arbitrary number
%% of arguments, which must be unified into a single argument.
call_site(Ls, L, Xs, St) ->
%%% io:fwrite("call site: ~w -> ~w (~w).\n", [L, Ls, Xs]),
{D, W, V} = call_site(Ls, L, Xs, St#state.dep, St#state.work,
St#state.vars, St#state.funs),
St#state{dep = D, work = W, vars = V}.
call_site([external | Ls], T, Xs, D, W, V, Fs) ->
D1 = add_dep(external, T, D),
X = join_single_list(Xs),
case bind_arg(escape, X, V) of
{V1, true} ->
%%% io:fwrite("escape changed: ~w <- ~w + ~w.\n",
%%% [dict:fetch(escape, V1), dict:fetch(escape, V),
%%% X]),
{W1, V2} = update_esc(set__to_list(X), W, V1, Fs),
call_site(Ls, T, Xs, D1, add_work([external], W1), V2, Fs);
{V1, false} ->
call_site(Ls, T, Xs, D1, W, V1, Fs)
end;
call_site([L | Ls], T, Xs, D, W, V, Fs) ->
D1 = add_dep(L, T, D),
Vs = fun_vars(dict:fetch(L, Fs)),
case bind_args(Vs, Xs, V) of
{V1, true} ->
call_site(Ls, T, Xs, D1, add_work([L], W), V1, Fs);
{V1, false} ->
call_site(Ls, T, Xs, D1, W, V1, Fs)
end;
call_site([], _, _, D, W, V, _) ->
{D, W, V}.
%% Note that `visit' makes sure all lambdas are visited at least once.
%% For every called function, we add a dependency from the *called*
%% function to the function containing the call site.
add_dep(Source, Target, Deps) ->
case dict:find(Source, Deps) of
{ok, X} ->
case set__is_member(Target, X) of
true ->
Deps;
false ->
%%% io:fwrite("new dep: ~w <- ~w.\n", [Target, Source]),
dict:store(Source, set__add(Target, X), Deps)
end;
error ->
%%% io:fwrite("new dep: ~w <- ~w.\n", [Target, Source]),
dict:store(Source, set__singleton(Target), Deps)
end.
%% If the arity does not match the call, nothing is done here.
bind_args(Vs, Xs, Vars) ->
if length(Vs) =:= length(Xs) ->
bind_args(Vs, Xs, Vars, false);
true ->
{Vars, false}
end.
bind_args([V | Vs], [X | Xs], Vars, Ch) ->
L = get_label(V),
{Vars1, Ch1} = bind_arg(L, X, Vars, Ch),
bind_args(Vs, Xs, Vars1, Ch1);
bind_args([], [], Vars, Ch) ->
{Vars, Ch}.
bind_args_single(Vs, X, Vars) ->
bind_args_single(Vs, X, Vars, false).
bind_args_single([V | Vs], X, Vars, Ch) ->
L = get_label(V),
{Vars1, Ch1} = bind_arg(L, X, Vars, Ch),
bind_args_single(Vs, X, Vars1, Ch1);
bind_args_single([], _, Vars, Ch) ->
{Vars, Ch}.
bind_arg(L, X, Vars) ->
bind_arg(L, X, Vars, false).
bind_arg(L, X, Vars, Ch) ->
X0 = dict:fetch(L, Vars),
X1 = join_single(X, X0),
case equal_single(X0, X1) of
true ->
{Vars, Ch};
false ->
%%% io:fwrite("arg (~w) changed: ~w <- ~w + ~w.\n",
%%% [L, X1, X0, X]),
{dict:store(L, X1, Vars), true}
end.
%% This handles escapes from things like primops and remote calls.
%% escape(none, St) ->
%% St;
escape([X], St) ->
Vars = St#state.vars,
X0 = dict:fetch(escape, Vars),
X1 = join_single(X, X0),
case equal_single(X0, X1) of
true ->
St;
false ->
%%% io:fwrite("escape changed: ~w <- ~w + ~w.\n", [X1, X0, X]),
%%% io:fwrite("updating escaping funs: ~w.\n", [set__to_list(X)]),
Vars1 = dict:store(escape, X1, Vars),
{W, Vars2} = update_esc(set__to_list(set__subtract(X, X0)),
St#state.work, Vars1,
St#state.funs),
St#state{work = add_work([external], W), vars = Vars2}
end.
%% For all escaping lambdas, since they might be called from outside the
%% program, all their arguments may be an external lambda. (Note that we
%% only have to include the `external' label once per escaping lambda.)
%% If the escape set has changed, we need to revisit the `external' fun.
update_esc(Ls, W, V, Fs) ->
update_esc(Ls, singleton(external), W, V, Fs).
%% The external lambda is skipped here - the Escape variable is known to
%% contain `external' from the start.
update_esc([external | Ls], X, W, V, Fs) ->
update_esc(Ls, X, W, V, Fs);
update_esc([L | Ls], X, W, V, Fs) ->
Vs = fun_vars(dict:fetch(L, Fs)),
case bind_args_single(Vs, X, V) of
{V1, true} ->
update_esc(Ls, X, add_work([L], W), V1, Fs);
{V1, false} ->
update_esc(Ls, X, W, V1, Fs)
end;
update_esc([], _, W, V, _) ->
{W, V}.
set_parent([L | Ls], L1, D) ->
set_parent(Ls, L1, dict:store(L, L1, D));
set_parent([], _L1, D) ->
D.
%% Handle primop calls: (At present, we assume that all unknown primops
%% yield exactly one value. This might have to be changed.)
primop_call(F, A, Xs, St0) ->
case is_pure_op(F, A) of
%% XXX: this case is currently not possible -- commented out.
%% true ->
%% case is_literal_op(F, A) of
%% true -> {[empty()], St0};
%% false -> {[join_single_list(Xs)], St0}
%% end;
false ->
St1 = case is_escape_op(F, A) of
true -> escape([join_single_list(Xs)], St0);
false -> St0
end,
case is_literal_op(F, A) of
true -> {none, St1};
false -> {[singleton(external)], St1}
end
end.
%% Handle remote-calls: (At present, we assume that all unknown calls
%% yield exactly one value. This might have to be changed.)
remote_call(M, F, Xs, St) ->
case is_c_atom(M) andalso is_c_atom(F) of
true ->
remote_call_1(atom_val(M), atom_val(F), length(Xs), Xs, St);
false ->
%% Unknown function
{[singleton(external)], escape([join_single_list(Xs)], St)}
end.
remote_call_1(M, F, A, Xs, St0) ->
case is_pure_op(M, F, A) of
true ->
case is_literal_op(M, F, A) of
true -> {[empty()], St0};
false -> {[join_single_list(Xs)], St0}
end;
false ->
St1 = case is_escape_op(M, F, A) of
true -> escape([join_single_list(Xs)], St0);
false -> St0
end,
case is_literal_op(M, F, A) of
true -> {[empty()], St1};
false -> {[singleton(external)], St1}
end
end.
%% Domain: none | [Vs], where Vs = set(integer()).
join(none, Xs2) -> Xs2;
join(Xs1, none) -> Xs1;
join(Xs1, Xs2) ->
if length(Xs1) =:= length(Xs2) ->
join_1(Xs1, Xs2);
true ->
none
end.
join_1([X1 | Xs1], [X2 | Xs2]) ->
[join_single(X1, X2) | join_1(Xs1, Xs2)];
join_1([], []) ->
[].
empty() -> set__new().
singleton(X) -> set__singleton(X).
from_label_list(X) -> set__from_list(X).
join_single(none, Y) -> Y;
join_single(X, none) -> X;
join_single(X, Y) -> set__union(X, Y).
join_list([Xs | Xss]) ->
join(Xs, join_list(Xss));
join_list([]) ->
none.
join_single_list([X | Xs]) ->
join_single(X, join_single_list(Xs));
join_single_list([]) ->
empty().
equal(none, none) -> true;
equal(none, _) -> false;
equal(_, none) -> false;
equal(X1, X2) -> equal_1(X1, X2).
equal_1([X1 | Xs1], [X2 | Xs2]) ->
equal_single(X1, X2) andalso equal_1(Xs1, Xs2);
equal_1([], []) -> true;
equal_1(_, _) -> false.
equal_single(X, Y) -> set__equal(X, Y).
%% Set abstraction for label sets in the domain.
set__new() -> [].
set__singleton(X) -> [X].
set__to_list(S) -> S.
set__from_list(S) -> ordsets:from_list(S).
set__union(X, Y) -> ordsets:union(X, Y).
set__add(X, S) -> ordsets:add_element(X, S).
set__is_member(X, S) -> ordsets:is_element(X, S).
set__subtract(X, Y) -> ordsets:subtract(X, Y).
set__equal(X, Y) -> X =:= Y.
%% A simple but efficient functional queue.
queue__new() -> {[], []}.
queue__put(X, {In, Out}) -> {[X | In], Out}.
queue__get({In, [X | Out]}) -> {ok, X, {In, Out}};
queue__get({[], _}) -> empty;
queue__get({In, _}) ->
[X | In1] = lists:reverse(In),
{ok, X, {[], In1}}.
%% The work list - a queue without repeated elements.
init_work() ->
{queue__new(), sets:new()}.
add_work(Ls, {Q, Set}) ->
add_work(Ls, Q, Set).
%% Note that the elements are enqueued in order.
add_work([L | Ls], Q, Set) ->
case sets:is_element(L, Set) of
true ->
add_work(Ls, Q, Set);
false ->
add_work(Ls, queue__put(L, Q), sets:add_element(L, Set))
end;
add_work([], Q, Set) ->
{Q, Set}.
take_work({Queue0, Set0}) ->
case queue__get(Queue0) of
{ok, L, Queue1} ->
Set1 = sets:del_element(L, Set0),
{ok, L, {Queue1, Set1}};
empty ->
none
end.
%% Escape operators may let their arguments escape. Unless we know
%% otherwise, and the function is not pure, we assume this is the case.
%% Error-raising functions (fault/match_fail) are not considered as
%% escapes (but throw/exit are). Zero-argument functions need not be
%% listed.
-spec is_escape_op(atom(), arity()) -> boolean().
is_escape_op(match_fail, 1) -> false;
is_escape_op(F, A) when is_atom(F), is_integer(A) -> true.
-spec is_escape_op(atom(), atom(), arity()) -> boolean().
is_escape_op(erlang, error, 1) -> false;
is_escape_op(erlang, error, 2) -> false;
is_escape_op(M, F, A) when is_atom(M), is_atom(F), is_integer(A) -> true.
%% "Literal" operators will never return functional values even when
%% found in their arguments. Unless we know otherwise, we assume this is
%% not the case. (More functions can be added to this list, if needed
%% for better precision. Note that the result of `term_to_binary' still
%% contains an encoding of the closure.)
-spec is_literal_op(atom(), arity()) -> boolean().
is_literal_op(match_fail, 1) -> true;
is_literal_op(F, A) when is_atom(F), is_integer(A) -> false.
-spec is_literal_op(atom(), atom(), arity()) -> boolean().
is_literal_op(erlang, '+', 2) -> true;
is_literal_op(erlang, '-', 2) -> true;
is_literal_op(erlang, '*', 2) -> true;
is_literal_op(erlang, '/', 2) -> true;
is_literal_op(erlang, '=:=', 2) -> true;
is_literal_op(erlang, '==', 2) -> true;
is_literal_op(erlang, '=/=', 2) -> true;
is_literal_op(erlang, '/=', 2) -> true;
is_literal_op(erlang, '<', 2) -> true;
is_literal_op(erlang, '=<', 2) -> true;
is_literal_op(erlang, '>', 2) -> true;
is_literal_op(erlang, '>=', 2) -> true;
is_literal_op(erlang, 'and', 2) -> true;
is_literal_op(erlang, 'or', 2) -> true;
is_literal_op(erlang, 'not', 1) -> true;
is_literal_op(erlang, length, 1) -> true;
is_literal_op(erlang, size, 1) -> true;
is_literal_op(erlang, fun_info, 1) -> true;
is_literal_op(erlang, fun_info, 2) -> true;
is_literal_op(erlang, fun_to_list, 1) -> true;
is_literal_op(erlang, throw, 1) -> true;
is_literal_op(erlang, exit, 1) -> true;
is_literal_op(erlang, error, 1) -> true;
is_literal_op(erlang, error, 2) -> true;
is_literal_op(M, F, A) when is_atom(M), is_atom(F), is_integer(A) -> false.
%% Pure functions neither affect the state, nor depend on it.
is_pure_op(F, A) when is_atom(F), is_integer(A) -> false.
is_pure_op(M, F, A) -> erl_bifs:is_pure(M, F, A).
%% =====================================================================
