Source file CfgCompiler.ml
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open Sigs
open Cil_types
open Lang
let dkey = Wp_parameters.register_category "cfg_compiler"
let dumpkey = Wp_parameters.register_category "cfg_compiler_dump"
type mode = [
| `Tree
| `Bool_Backward
| `Bool_Forward
]
module type Cfg =
sig
module S : Sigma
module Node : sig
type t
module Map : Qed.Idxmap.S with type key = t
module Set : Qed.Idxset.S with type elt = t
module Hashtbl : Hashtbl.S with type key = t
val pp: Format.formatter -> t -> unit
val create: unit -> t
val equal: t -> t -> bool
end
type node = Node.t
val node : unit -> node
module C :
sig
type t
val equal : t -> t -> bool
val create : S.t -> F.pred -> t
val get : t -> F.pred
val reads : t -> S.domain
val relocate : S.t -> t -> t
end
module P :
sig
type t
val pretty : Format.formatter -> t -> unit
val create : S.t Node.Map.t -> F.pred -> t
val get: t -> F.pred
val reads : t -> S.domain Node.Map.t
val nodes : t -> Node.Set.t
val relocate : S.t Node.Map.t -> t -> t
val to_condition: t -> (C.t * Node.t option) option
end
module T :
sig
type t
val pretty : Format.formatter -> t -> unit
(** Bundle an equation with the sigma sequence that created it. *)
val create : S.t Node.Map.t -> F.term -> t
val get: t -> F.term
val reads : t -> S.domain Node.Map.t
val relocate : S.t Node.Map.t -> t -> t
val init : Node.Set.t -> (S.t Node.Map.t -> F.term) -> t
val init' : Node.t -> (S.t -> F.term) -> t
end
module E : sig
type t
val pretty: Format.formatter -> t -> unit
val create : S.t sequence -> F.pred -> t
val get : t -> F.pred
val reads : t -> S.domain
val writes : t -> S.domain
val relocate : S.t sequence -> t -> t
end
type cfg
val dump_env: name:string -> cfg -> unit
val output_dot: out_channel -> ?checks:P.t Bag.t -> cfg -> unit
val nop : cfg
val add_tmpnode: node -> cfg
val concat : cfg -> cfg -> cfg
val meta : ?stmt:stmt -> ?descr:string -> node -> cfg
val goto : node -> node -> cfg
val branch : node -> C.t -> node -> node -> cfg
val guard : node -> C.t -> node -> cfg
val guard' : node -> C.t -> node -> cfg
val either : node -> node list -> cfg
val implies : node -> (C.t * node) list -> cfg
val effect : node -> E.t -> node -> cfg
val assume : P.t -> cfg
val havoc : node -> effects:node sequence -> node -> cfg
val compile : ?name:string -> ?mode:mode -> node -> Node.Set.t -> S.domain Node.Map.t ->
cfg -> F.pred Node.Map.t * S.t Node.Map.t * Conditions.sequence
end
module Cfg (S:Sigma) : Cfg with module S = S =
struct
module S = S
module Node : sig
type t
module Map : Qed.Idxmap.S with type key = t
module Set : Qed.Idxset.S with type elt = t
module Hashtbl : FCHashtbl.S with type key = t
val tag: t -> int
val compare: t -> t -> int
val equal: t -> t -> bool
val pp: Format.formatter -> t -> unit
val create: unit -> t
val node_internal: unit -> t
end
= struct
type t = int
module I = struct type t = int let id x = x end
module Map = Qed.Idxmap.Make(I)
module Set = Qed.Idxset.Make(I)
module Hashtbl = Datatype.Int.Hashtbl
let tag = I.id
let compare = Datatype.Int.compare
let equal = Datatype.Int.equal
let pp fmt n =
if n>=0 then Format.pp_print_int fmt n
else Format.fprintf fmt "int%i" (-n)
let node_compter = ref (-1)
let create () =
incr node_compter;
!node_compter
let node_internal_compter = ref 0
let node_internal () =
decr node_internal_compter;
!node_internal_compter
end
let node = Node.create
let identify sigma ~src ~tgt =
S.iter2
(fun _chunk u v ->
match u,v with
| Some x , Some y -> F.Subst.add sigma (F.e_var x) (F.e_var y)
| _ -> ())
src tgt
module E = struct
type t = S.t sequence * F.pred
let pretty fmt (_seq,p) = Format.fprintf fmt "effect: @[%a@]" F.pp_pred p
let get : t -> F.pred = snd
let create seq p = seq,p
let relocate tgt (src,p) =
let sigma = Lang.sigma () in
identify sigma ~src:src.pre ~tgt:tgt.pre ;
identify sigma ~src:src.post ~tgt:tgt.post ;
tgt , F.p_subst sigma p
let reads (seq,_) = S.domain seq.pre
let writes (seq,_) = S.writes seq
end
module C = struct
type t = S.t * F.pred
let get = snd
let create seq p = seq,p
let relocate tgt (src,p) =
let sigma = Lang.sigma () in
identify sigma ~src ~tgt ;
tgt , F.p_subst sigma p
let reads (src,_) = S.domain src
let equal (s1,p1) (s2,p2) =
let sigma = Lang.sigma () in
identify sigma ~src:s1 ~tgt:s2 ;
F.eqp (F.p_subst sigma p1) p2
end
module P = struct
type t = S.t Node.Map.t * F.pred
let pretty fmt (m,f) =
Format.fprintf fmt "%a(%a)"
F.pp_pred f (Pretty_utils.pp_iter2 Node.Map.iter ~between:",@ " Node.pp (fun _ _ -> ())) m
let get = snd
let create smap p = smap,p
let relocate tgt (src,p) =
let sigma = Lang.sigma () in
Node.Map.iter2
(fun n src tgt ->
match src,tgt with
| Some src , Some tgt -> identify sigma ~src ~tgt
| Some _, None ->
invalid_arg (Format.asprintf "P.relocate: tgt is smaller than src at %a" Node.pp n)
| _ -> ())
src tgt ;
let tgt = Node.Map.inter (fun _ _ tgt -> tgt) src tgt in
tgt , F.p_subst sigma p
let reads (smap,_) = Node.Map.map (fun _ s -> S.domain s) smap
let nodes (smap,_) = Node.Map.fold (fun k _ acc -> Node.Set.add k acc) smap Node.Set.empty
let nodes_list (smap,_) = Node.Map.fold (fun k _ acc -> k::acc) smap []
let to_condition (m,p) =
let l = Node.Map.fold (fun k e acc -> (k,e)::acc) m [] in
match l with
| [] -> Some ((S.create (),p), None)
| [n,s] -> Some ((s,p), Some n)
| _ -> None
end
module T = struct
type t = S.t Node.Map.t * F.term
let pretty fmt (m,f) =
Format.fprintf fmt "%a(%a)"
F.pp_term f (Pretty_utils.pp_iter2 Node.Map.iter ~between:",@ " Node.pp (fun _ _ -> ())) m
let get = snd
let create smap t = smap,t
let reads (smap,_) = Node.Map.map (fun _ s -> S.domain s) smap
let relocate tgt (src,p) =
let sigma = Lang.sigma () in
Node.Map.iter2
(fun _ src tgt ->
match src,tgt with
| Some src , Some tgt -> identify sigma ~src ~tgt
| Some _, None -> invalid_arg "T.relocate: tgt is smaller than src"
| _ -> ())
src tgt ;
let tgt = Node.Map.inter (fun _ _ tgt -> tgt) src tgt in
tgt , F.e_subst sigma p
let init node_set f =
let node_map = Node.Set.fold (fun x m ->
Node.Map.add x (S.create ()) m
) node_set Node.Map.empty
in
let t = f node_map in
(node_map,t)
let init' node f =
let src = S.create () in
let t = f src in
let node_map =
Node.(Map.add node src Map.empty)
in
(node_map,t)
end
type node = Node.t
type without_bindings = Without_Bindings
type with_bindings = With_Bindings
let _ = Without_Bindings
let _ = With_Bindings
type ('havoc,_) edge =
| Goto of node
| Branch of C.t * node option * node option
| Either of node list
| Implies of (C.t * node) list
| Effect of E.t * node
| Havoc of 'havoc * node
| Binding : Passive.t * node -> ('havoc,with_bindings) edge
(** Binding used for sigma merging *)
type data =
| Meta of stmt option * string option
type ('havoc, 'bindings) env = {
succs : ('havoc, 'bindings) edge Node.Map.t;
datas : data Bag.t Node.Map.t;
assumes : P.t Bag.t;
tmpnodes : Node.Set.t;
}
type pre_env = (node sequence, without_bindings) env
type restricted_env = (S.domain, without_bindings) env
type localised_env = (S.domain, with_bindings) env
type cfg = pre_env
let iter_succs : type a b. (Node.t -> unit) -> (a,b) edge -> unit = fun f -> function
| Goto n2 | Effect(_,n2) | Havoc(_,n2) -> f n2
| Branch(_,n2a,n2b) ->
let f' = function None -> () | Some x -> f x in
f' n2a; f' n2b
| Either l -> List.iter f l
| Implies l -> List.iter (fun (_,a) -> f a) l
| Binding (_,n2) -> f n2
let iter_succs_e f cfg n =
match Node.Map.find n cfg.succs with
| exception Not_found -> ()
| e -> iter_succs f e
let succs : type a b. (a,b) env -> Node.t -> Node.t list =
fun cfg n ->
match Node.Map.find n cfg.succs with
| exception Not_found -> []
| Goto n2 | Effect(_,n2) | Havoc(_,n2)
| Branch(_,Some n2,None)
| Branch(_,None,Some n2) -> [n2]
| Binding (_,n2) -> [n2]
| Branch(_,Some n1,Some n2) -> [n1;n2]
| Branch(_,None,None) -> []
| Either l -> l
| Implies l -> List.map snd l
let pretty_edge : type a. Format.formatter -> (_,a) edge -> unit = fun fmt edge ->
match edge with
| Goto(n) -> Format.fprintf fmt "goto(%a)" Node.pp n
| Branch(c,n1,n2) -> Format.fprintf fmt "branch(%a,%a,%a)"
Lang.F.pp_pred (C.get c)
(Pretty_utils.pp_opt Node.pp) n1 (Pretty_utils.pp_opt Node.pp) n2
| Either l -> Format.fprintf fmt "either(%a)" (Pretty_utils.pp_list ~sep:",@ " Node.pp) l
| Implies l -> Format.fprintf fmt "implies(%a)"
(Pretty_utils.pp_list ~sep:",@ " (fun fmt (c,a) ->
Format.fprintf fmt "%a=>%a" Lang.F.pp_pred (C.get c) Node.pp a)) l
| Effect(_,n) -> Format.fprintf fmt "effect(%a)" Node.pp n
| Havoc(_,n) -> Format.fprintf fmt "havoc(%a)" Node.pp n
| Binding(_,n) -> Format.fprintf fmt "binding(%a)" Node.pp n
let pretty_data fmt = function
| Meta(s_opt,str_opt) ->
Format.fprintf fmt "Meta(%a,%a)"
(Pretty_utils.pp_opt ~none:"None" Cil_datatype.Stmt.pretty_sid) s_opt
(Pretty_utils.pp_opt ~none:"None" Format.pp_print_string) str_opt
let pretty_env : type a. Format.formatter -> (_,a) env -> unit =
fun fmt env ->
Context.bind Lang.F.context_pp (Lang.F.env Lang.F.Vars.empty) (fun () ->
Format.fprintf fmt
"@[<v>@[<3>@[succs:@]@ %a@]@,@[<3>@[datas:@]@ %a@]@,@[<3>@[assumes:@]@ %a@]@]@."
(Pretty_utils.pp_iter2 ~between:"->@," ~sep:",@ " Node.Map.iter Node.pp pretty_edge) env.succs
(Pretty_utils.pp_iter2 ~between:"->@," ~sep:",@ " Node.Map.iter Node.pp
(Pretty_utils.pp_iter Bag.iter pretty_data)) env.datas
(Pretty_utils.pp_iter ~sep:",@ " Bag.iter P.pretty) env.assumes
) ()
let dump_edge : type a. node -> Format.formatter -> (_, a) edge -> unit =
fun n fmt edge ->
let pp_edge ?(label="") n' =
Format.fprintf fmt " %a -> %a [ label=\"%s\" ] ;@." Node.pp n Node.pp n' label
in
begin match edge with
| Goto n1 -> pp_edge n1
| Branch (_, n1, n2)->
Option.iter pp_edge n1;
Option.iter pp_edge n2
| Either ns -> List.iter pp_edge ns
| Implies ns -> List.iter (fun (_,a) -> pp_edge a) ns
| Effect (e, n') ->
pp_edge ~label:(Format.asprintf "%a" E.pretty e) n'
| Havoc (_, n') -> pp_edge ~label:"havoc" n'
| Binding (_,n') -> pp_edge ~label:"binding" n'
end
let dump_node : data Bag.t -> Format.formatter -> node -> unit =
fun datas fmt n ->
Format.fprintf fmt " %a [ label=\"%a\n%a\" ] ;@."
Node.pp n Node.pp n (Pretty_utils.pp_iter ~sep:"\n" Bag.iter pretty_data) datas
let dump_succ : type a. (_, a) env -> Format.formatter -> node -> (_, a) edge -> unit =
fun env fmt n e ->
let datas = try Node.Map.find n env.datas with Not_found -> Bag.empty in
Format.fprintf fmt "%a\n%a@\n" (dump_node datas) n (dump_edge n) e
let dump_assume : Format.formatter -> P.t -> unit =
let count = ref 0 in
fun fmt p ->
incr count;
Format.fprintf fmt " subgraph cluster_%d {@\n" !count;
Format.fprintf fmt " color=\"palegreen\";@\n";
Node.Map.iter
(fun n _ -> Format.fprintf fmt " %a;\n" Node.pp n)
(P.reads p);
Format.fprintf fmt " label=\"%a\";" Lang.F.pp_pred (P.get p);
Format.fprintf fmt " }@."
let escape fmt = Pretty_utils.ksfprintf (fun s -> String.escaped s) fmt
let output_dot : type a b. out_channel -> ?checks:_ -> (a,b) env -> unit =
fun cout ?(checks=Bag.empty) env ->
let count = let c = ref max_int in fun () -> decr c; !c in
let module E = struct
type t = Graph.Graphviz.DotAttributes.edge list
let default = []
let compare x y = assert (x == y); 0
end
in
let module V = struct
type t =
| Node of Node.t
| Assume of int * Lang.F.pred
| Check of int * Lang.F.pred
let tag = function | Node i -> Node.tag i | Assume (i,_) -> i | Check (i,_) -> i
let pp fmt = function | Node i -> Node.pp fmt i | Assume (i,_) -> Format.fprintf fmt "ass%i" i
| Check (i,_) -> Format.fprintf fmt "chk%i" i
let equal x y = (tag x) = (tag y)
let compare x y = Stdlib.compare (tag x) (tag y)
let hash x = tag x
end in
let module G = Graph.Imperative.Digraph.ConcreteBidirectionalLabeled (V)(E) in
let module Dot = Graph.Graphviz.Dot(struct
let graph_attributes _g = [`Fontname "fixed"]
let default_vertex_attributes _g = [`Shape `Circle]
let vertex_name v = Format.asprintf "cp%a" V.pp v
let vertex_attributes = function
| V.Node n -> [`Label (escape "%a" Node.pp n)]
| V.Assume (_,p) -> [`Style `Dashed; `Label (escape "%a" Lang.F.pp_pred p)]
| V.Check (_,p) -> [`Style `Dotted; `Label (escape "%a" Lang.F.pp_pred p)]
let get_subgraph _ = None
let default_edge_attributes _g = []
let edge_attributes ((_,e,_):G.E.t) : Graph.Graphviz.DotAttributes.edge list = e
include G
end) in
let g = G.create () in
let add_edge n1 l n2 = G.add_edge_e g (V.Node n1,l,V.Node n2) in
let add_edges : type a b. Node.t -> (a,b) edge -> unit = fun n1 -> function
| Goto n2 -> add_edge n1 [] n2
| Branch((_,c),n2,n2') ->
let aux s = function
| None -> ()
| Some n -> add_edge n1 [`Label (escape "%s%a" s Lang.F.pp_pred c)] n
in
aux "" n2; aux "!" n2'
| Either l -> List.iter (add_edge n1 []) l
| Implies l ->
List.iter (fun (c,n) -> add_edge n1 [`Label (escape "%a" Lang.F.pp_pred (C.get c))] n) l
| Effect ((_,e),n2) ->
add_edge n1 [`Label (escape "%a" Lang.F.pp_pred e)] n2
| Havoc (_,n2) -> add_edge n1 [`Label (escape "havoc")] n2
| Binding (_,n2) -> add_edge n1 [`Label (escape "binding")] n2
in
Node.Map.iter add_edges env.succs;
Bag.iter (fun (m,p) ->
let n1 = V.Assume(count (), p) in
let assume_label = [`Style `Dashed ] in
Node.Map.iter (fun n2 _ -> G.add_edge_e g (n1,assume_label,V.Node n2)) m
) env.assumes;
Bag.iter (fun (m,p) ->
let n1 = V.Check(count (), p) in
let label = [`Style `Dotted ] in
Node.Map.iter (fun n2 _ -> G.add_edge_e g (V.Node n2,label,n1)) m
) checks;
Dot.output_graph cout g
let dump_env : type a. name:string -> (_, a) env -> unit = fun ~name env ->
let file = (Filename.get_temp_dir_name ()) ^ "/cfg_" ^ name in
let fout = open_out (file ^ ".dot") in
if false then begin
let out = Format.formatter_of_out_channel fout in
Format.fprintf out "digraph %s {@\n" name;
Format.fprintf out " rankdir = TB ;@\n";
Format.fprintf out " node [ style = filled, shape = circle ] ;@\n";
Node.Map.iter (dump_succ env out) env.succs;
Bag.iter (dump_assume out) env.assumes;
Format.fprintf out "}@.";
end
else begin
output_dot fout env;
end;
close_out fout;
ignore (Sys.command
(Printf.sprintf "dot -Tpdf %s.dot > %s.pdf" file file));
Wp_parameters.debug ~dkey:dumpkey "Saving dump %s into %s.pdf" name file
let env_union env1 env2 =
{
succs = Node.Map.union
(fun _ _v1 _v2 -> invalid_arg "A node has more than one successor")
env1.succs env2.succs;
datas = Node.Map.union (fun _ -> Bag.concat) env1.datas env2.datas;
assumes = Bag.concat env1.assumes env2.assumes;
tmpnodes = Node.Set.union env1.tmpnodes env2.tmpnodes;
}
let new_env ?(succs=Node.Map.empty) ?(datas=Node.Map.empty) ?(assumes=Bag.empty)
?(tmpnodes=Node.Set.empty) () =
{succs; datas; assumes; tmpnodes}
let nop = new_env ()
let add_tmpnode node = new_env ~tmpnodes:(Node.Set.singleton node) ()
let concat a b = env_union a b
let meta ?stmt ?descr n =
let data = Meta(stmt,descr) in
new_env ~datas:(Node.Map.add n (Bag.elt data) (Node.Map.empty)) ()
let edge n e =
new_env ~succs:(Node.Map.add n e (Node.Map.empty)) ()
let goto node_orig node_target =
edge node_orig (Goto(node_target))
let branch node_orig predicate node_target_then node_target_else =
edge node_orig (Branch(predicate,
Some node_target_then,
Some node_target_else))
let guard node_orig predicate node_target_then =
edge node_orig (Branch(predicate,
Some node_target_then,
None))
let guard' node_orig predicate node_target_else =
edge node_orig (Branch(predicate,
None,
Some node_target_else
))
let either node = function
| [] -> nop
| [dest] -> goto node dest
| node_list -> edge node (Either(node_list))
let implies node = function
| [] -> nop
| [g,dest] -> guard node g dest
| node_list -> edge node (Implies(node_list))
let effect node1 e node2 =
edge node1 (Effect(e, node2))
let assume (predicate:P.t) =
if F.is_ptrue (P.get predicate) = Qed.Logic.Yes
then nop
else new_env ~assumes:(Bag.elt predicate) ()
let havoc node1 ~effects:node_seq node2 =
edge node1 (Havoc(node_seq,node2))
let option_bind ~f = function
| None -> None
| Some x -> f x
let union_opt_or union d1 d2 =
match d1, d2 with
| Some d1, Some d2 -> Some (union d1 d2)
| (Some _ as d), None | None, (Some _ as d) -> d
| None, None -> None
let union_opt_and union d1 d2 =
match d1, d2 with
| Some d1, Some d2 -> Some (union d1 d2)
| _ -> None
let add_only_if_alive union d1 = function
| None -> None
| Some d2 -> Some (union d1 d2)
(** return None when post is not accessible from this node *)
let rec effects : type a. (_,a) env -> node -> node -> S.domain option =
fun env post node ->
if node = post
then Some S.empty
else
match Node.Map.find node env.succs with
| exception Not_found -> None
| Goto (node2) ->
effects env post node2
| Branch (_, node2, node3) ->
union_opt_or S.union
(option_bind ~f:(effects env post) node2)
(option_bind ~f:(effects env post) node3)
| Either (l) ->
(List.fold_left
(fun acc node2 -> union_opt_or S.union
acc (effects env post node2))
None l)
| Implies (l) ->
(List.fold_left
(fun acc (_,node2) -> union_opt_or S.union
acc (effects env post node2))
None l)
| Effect (effect , node2) ->
add_only_if_alive S.union
(E.writes effect)
(effects env post node2)
| Havoc (m, node2) ->
union_opt_and S.union
(effects env m.post m.pre)
(effects env post node2)
| Binding (_,node2) ->
effects env post node2
(** restrict a cfg to the nodes accessible from the pre post given,
and compute havoc effect *)
let restrict (cfg:pre_env) pre posts : restricted_env =
let rec walk acc node : restricted_env option =
if Node.Map.mem node acc.succs then Some acc
else
let new_env edge = new_env ~succs:(Node.Map.add node edge (Node.Map.empty)) () in
let r = match Node.Map.find node cfg.succs with
| exception Not_found -> None
| (Goto (node2) | Effect (_ , node2)) as edge ->
union_opt_and env_union
(Some (new_env edge))
(walk acc node2)
| Branch (pred, node2, node3) ->
let f acc node =
match option_bind ~f:(walk acc) node with
| None -> None, acc
| Some acc -> node, acc in
let node2, acc = f acc node2 in
let node3, acc = f acc node3 in
if node2 = None && node3 = None then None
else Some (env_union acc (new_env (Branch(pred, node2, node3))))
| Either (l) ->
let acc,l = List.fold_left
(fun ((acc,l) as old) node2 ->
match walk acc node2 with
| None -> old
| Some acc -> (acc,node2::l))
(acc,[]) l in
if l = [] then None
else Some (env_union acc (new_env (Either (List.rev l))))
| Implies (l) ->
let acc,l = List.fold_left
(fun ((acc,l) as old) ((_,node2) as e) ->
match walk acc node2 with
| None -> old
| Some acc -> (acc,e::l))
(acc,[]) l in
if l = [] then None
else Some (env_union acc (new_env (Implies (List.rev l))))
| Havoc (m, node2) ->
match effects cfg m.post m.pre with
| None -> None
| Some eff ->
union_opt_and env_union
(Some (new_env (Havoc(eff,node2))))
(walk acc node2)
in
if Node.Set.mem node posts && r = None
then Some acc
else r
in
match walk (new_env ()) pre with
| None -> (new_env ())
| Some acc ->
{ succs = acc.succs;
datas = Node.Map.inter (fun _ _ v -> v) acc.succs cfg.datas;
assumes = Bag.filter (fun (seq,_) ->
Node.Map.subset (fun _ _ _ -> true)
seq acc.succs) cfg.assumes;
tmpnodes = cfg.tmpnodes;
}
(** succ is decreasing for this order *)
let topological (type a) (type b) (cfg:(a,b) env) =
let module G = struct
type t = (a,b) env
module V = struct let hash = Hashtbl.hash include Node end
let iter_vertex f cfg =
let h = Node.Hashtbl.create 10 in
let replace n = Node.Hashtbl.replace h n () in
Node.Map.iter (fun k _ -> replace k; iter_succs_e replace cfg k) cfg.succs;
Node.Hashtbl.iter (fun k () -> f k) h
let iter_succ = iter_succs_e
end in
let module T = Graph.Topological.Make(G) in
let h = Node.Hashtbl.create 10 in
let h' = Datatype.Int.Hashtbl.create 10 in
let c = ref (-1) in
let l = ref [] in
T.iter (fun n -> l := n::!l; incr c; Node.Hashtbl.add h n !c; Datatype.Int.Hashtbl.add h' !c n) cfg;
h,h',List.rev !l
(** topo_list: elements in topological order
topo_order: post-order mapping
nb: number of elements *)
let idoms topo_list topo_order nb ~pred ~is_after =
let a = Array.make nb (-1) in
let iter n =
let first,preds = match pred n with
| [] -> topo_order n, []
| f::p -> topo_order f, List.map topo_order p
in
let rec find_common n1 n2 =
if n1 = n2 then n1
else if is_after n1 n2 then find_common a.(n1) n2
else find_common n1 a.(n2) in
let idom = List.fold_left find_common first preds in
a.(topo_order n) <- idom
in
List.iter iter topo_list;
a
let find_def ~def x t = try Node.Map.find x t with Not_found -> def
let rec remove_dumb_gotos (env:restricted_env) : Node.t Node.Map.t * restricted_env =
let add_map m acc = Node.Map.fold (fun n _ acc -> Node.Set.add n acc) m acc in
let used_nodes =
Bag.fold_left (fun acc p -> add_map (P.reads p) acc) Node.Set.empty env.assumes
in
let used_nodes = add_map env.datas used_nodes in
let how_many_preds = Node.Hashtbl.create 10 in
let incr_how_many_preds n =
Node.Hashtbl.replace how_many_preds n (succ (Node.Hashtbl.find_def how_many_preds n 0))
in
let subst =
Node.Map.fold (fun n e acc ->
iter_succs incr_how_many_preds e;
match (e:(_,without_bindings) edge) with
| Goto n' when not (Node.Set.mem n used_nodes) ->
Node.Map.add n n' acc
| Goto _
| Branch (_,_,_)
| Either _
| Implies _
| Effect (_,_)
| Havoc (_,_) -> acc)
env.succs Node.Map.empty
in
let subst =
let rec compress n =
match Node.Map.find n subst with
| exception Not_found -> n
| n -> compress n
in
Node.Map.map (fun _ n' -> compress n') subst
in
let find n = find_def ~def:n n subst in
let to_remove = Node.Hashtbl.create 10 in
Node.Map.iter (fun _ e ->
match (e:(_,without_bindings) edge) with
| Either [a;b] when Node.Hashtbl.find how_many_preds a = 1 &&
Node.Hashtbl.find how_many_preds b = 1 &&
not (Node.Set.mem a used_nodes) &&
not (Node.Set.mem b used_nodes) &&
Node.Set.mem a env.tmpnodes &&
Node.Set.mem b env.tmpnodes &&
true
->
begin
let find_opt k m =
match Node.Map.find k m with
| exception Not_found -> None
| v -> Some v
in
match find_opt a env.succs, find_opt b env.succs with
| Some Branch(c,Some n1, None), Some Branch(c',None, Some n2)
| Some Branch(c,None, Some n2), Some Branch(c',Some n1,None) when C.equal c c' ->
let n1 = find n1 in
let n2 = find n2 in
let br = Branch(c,Some n1, Some n2) in
Node.Hashtbl.add to_remove a br;
Node.Hashtbl.add to_remove b br
| _ -> ()
end
| Goto _
| Branch (_,_,_)
| Effect (_,_)
| Either _
| Implies _
| Havoc (_,_) -> ()
) env.succs;
let succs = Node.Map.mapq (fun n e ->
match (e:(_,without_bindings) edge) with
| _ when Node.Hashtbl.mem to_remove n -> None
| Goto _ when not (Node.Set.mem n used_nodes) -> None
| Goto n' ->
let n'' = find n' in
if Node.equal n' n'' then Some e
else Some (Goto n'')
| Branch (c,n1,n2) ->
let n1' = Option.map find n1 in
let n2' = Option.map find n2 in
if Option.equal Node.equal n1 n1' && Option.equal Node.equal n2 n2'
then Some e
else Some (Branch(c,n1',n2'))
| Either l ->
let l' = List.map find l in
let l' = List.sort_uniq Node.compare l' in
begin match l' with
| [] -> assert false
| [a] -> Some (Goto a)
| [a;_] when Node.Hashtbl.mem to_remove a ->
let br = Node.Hashtbl.find to_remove a in
Some br
| l' -> Some (Either l')
end
| Implies l ->
let l' = List.map (fun (g,n) -> (g,find n)) l in
Some (Implies l')
| Effect (ef,n') ->
let n'' = find n' in
if Node.equal n' n'' then Some e
else Some (Effect(ef,n''))
| Havoc (h,n') ->
let n'' = find n' in
if Node.equal n' n'' then Some e
else Some (Havoc(h,n''))
)
env.succs
in
let env = {env with succs} in
if Node.Map.is_empty subst
then subst, env
else
let subst', env = remove_dumb_gotos env in
let subst = Node.Map.map (fun _ n' -> find_def ~def:n' n' subst') subst in
Node.Map.merge (fun _ a b ->
match a, b with
| Some _, Some _ -> assert false
| Some x, None | None, Some x -> Some x
| None, None -> assert false
) subst subst', env
let allocate domain sigma =
S.Chunk.Set.iter (fun chunk -> ignore (S.get sigma chunk)) domain
let domains (env : restricted_env) reads pre : localised_env * S.t Node.Map.t =
let visited = ref Node.Map.empty in
let new_succs = ref Node.Map.empty in
let add_edge node edge = new_succs := Node.Map.add node edge !new_succs in
let add_binding_edge n (p: Passive.t) =
if Passive.is_empty p then n
else
let n' = Node.node_internal () in
add_edge n' (Binding(p,n));
n'
in
let rec aux node : S.t =
try Node.Map.find node !visited
with Not_found ->
let dom = find_def ~def:S.empty node reads in
let ret =
match Node.Map.find node env.succs with
| exception Not_found ->
let s1 = S.create () in
allocate dom s1;
s1
| Goto (node2) ->
let s1 = S.copy (aux node2) in
allocate dom s1;
add_edge node (Goto node2);
s1
| Branch (pred, node2, node3) ->
let dom = (S.union (C.reads pred) dom) in
begin match node2, node3 with
| (None, Some next) | (Some next, None) ->
let s1 = S.copy (aux next) in
allocate dom s1;
let pred = C.relocate s1 pred in
add_edge node (Branch(pred,node2,node3));
s1
| Some node2, Some node3 ->
let s2 = aux node2 in
let s3 = aux node3 in
let s1,p2,p3 = S.merge s2 s3 in
allocate dom s1;
let node2' = add_binding_edge node2 p2 in
let node3' = add_binding_edge node3 p3 in
let pred = C.relocate s1 pred in
add_edge node (Branch(pred,Some node2',Some node3'));
s1
| _ -> assert false
end
| Either (l) ->
let s1, pl = S.merge_list (List.map aux l) in
allocate dom s1;
let l = List.map2 add_binding_edge l pl in
add_edge node (Either l);
s1
| Implies (l) ->
let dom =
List.fold_left (fun acc (c,_) -> S.union (C.reads c) acc) dom l
in
let s1, pl = S.merge_list (List.map (fun (_,n) -> aux n) l) in
allocate dom s1;
let l = List.map2 (fun (c,a) b ->
let a = add_binding_edge a b in
let c = C.relocate s1 c in
(c,a)) l pl
in
add_edge node (Implies l);
s1
| Effect (effect , node2) ->
let s2 = aux node2 in
let s1 = S.remove_chunks s2 (E.writes effect) in
allocate dom s1;
allocate (E.reads effect) s1;
let effect = E.relocate {pre=s1;post=s2} effect in
add_edge node (Effect(effect,node2));
s1
| Havoc (eff, node2) ->
let s2 = aux node2 in
let s1 = S.havoc s2 eff in
allocate dom s1;
add_edge node (Havoc(eff,node2));
s1
in
visited := Node.Map.add node ret !visited;
ret
in
ignore (aux pre);
let sigmas = !visited in
let new_env =
{succs = !new_succs;
datas = env.datas;
assumes = Bag.map (fun e -> P.relocate sigmas e) env.assumes;
tmpnodes = env.tmpnodes;
} in
new_env, sigmas
let compute_preds env =
let h = Node.Hashtbl.create 10 in
let add = Node.Hashtbl.add h in
Node.Map.iter (fun n s ->
match s with
| Goto n1 | Havoc (_, n1) | Effect (_,n1) | Binding (_,n1) -> add n1 n
| Branch (_,Some n1,Some n2) ->
add n1 n;
add n2 n
| Branch(_,Some n1,None) -> add n1 n
| Branch(_,None,Some n1) -> add n1 n
| Branch(_,None,None) -> ()
| Either l -> List.iter (fun n1 -> add n1 n) l
| Implies l -> List.iter (fun (_,n1) -> add n1 n) l
) env.succs;
h
let to_sequence_bool ~mode pre posts env : Conditions.sequence * F.pred Node.Map.t =
let preds = Node.Hashtbl.create 10 in
let access n = Node.Hashtbl.memo preds n
(fun _ ->
let v = F.fresh ~basename:"node"
(get_pool ()) Qed.Logic.Bool in
F.p_bool (F.e_var v))
in
let (!.) c = (Conditions.sequence [Conditions.step c]) in
let have_access n = !. (Conditions.Have (access n)) in
let add_cond ?descr ?stmt f cond =
Conditions.append (Conditions.sequence [Conditions.step ?descr ?stmt cond]) f
in
let either = function
| [] -> !. (Conditions.Have F.p_false)
| [a] -> a
| l -> !. (Conditions.Either l)
in
let f = Conditions.empty in
let pre = Conditions.Have (access pre) in
let f = add_cond f pre in
let f = Node.Set.fold
(fun n f -> add_cond f (Conditions.Have (access n)))
posts f in
let f =
Bag.fold_left (fun f p ->
let nodes_are_accessible =
Node.Map.fold (fun n _ acc -> F.p_and (access n) acc)
(P.reads p) F.p_true in
let f' = F.p_imply nodes_are_accessible (P.get p) in
add_cond f (Conditions.Have f')
) f env.assumes in
let to_sequence_basic_backward f =
let predecessors = Node.Map.fold (fun n s acc ->
let add acc n' p =
Node.Map.change (fun _ (n,p) -> function
| None -> Some (Node.Map.add n p Node.Map.empty)
| Some s -> Some (Node.Map.add n p s)) n' (n,p) acc
in
match s with
| Goto n' | Havoc (_, n') -> add acc n' F.p_true
| Branch (c,Some n1,Some n2) ->
let c = P.get c in
add (add acc n1 c) n2 (F.p_not c)
| Branch(c,Some n1,None) -> add acc n1 (P.get c)
| Branch(c,None,Some n1) -> add acc n1 (F.p_not (P.get c))
| Branch(_,None,None) -> acc
| Either l -> List.fold_left (fun acc e -> add acc e F.p_true) acc l
| Implies l -> List.fold_left (fun acc (c,e) -> add acc e (P.get c)) acc l
| Effect (e,n') -> add acc n' (E.get e)
| Binding (b,n') ->
let b = F.p_conj (Passive.conditions b (fun _ -> true)) in
add acc n' b
) env.succs Node.Map.empty
in
Node.Map.fold (fun n' preds f ->
let l = Node.Map.fold (fun n p acc ->
(Conditions.append (have_access n) (!. (Conditions.Have p)))::acc
) preds [] in
let f' =
Conditions.Branch(access n', either l, Conditions.empty)
in
let stmt,descr =
let bag = match Node.Map.find n' env.datas with
| exception Not_found -> Bag.empty
| bag -> bag
in
Bag.fold_left (fun (os,od) b ->
match b with
| Meta(os',od') ->
(if os = None then os' else os),
(if od = None then od' else od)
) (None,None) bag in
add_cond ?stmt ?descr f f'
) predecessors f
in
let to_sequence_basic_forward f =
Node.Map.fold (fun n s f ->
let node_is_accessible = access n in
let f' = match s with
| Goto n' | Havoc (_, n') ->
Conditions.Branch(node_is_accessible,
have_access n',
Conditions.empty)
| Branch (c,Some n1,Some n2) ->
Conditions.Branch(node_is_accessible,
!. (Conditions.Branch((C.get c),
have_access n1,
have_access n2)),
Conditions.empty)
| Branch(c,Some n1,None) ->
Conditions.Branch(node_is_accessible,
Conditions.append
(!. (Conditions.Have (C.get c)))
(have_access n1),
Conditions.empty)
| Branch(c,_,Some n1) ->
Conditions.Branch(node_is_accessible,
Conditions.append
(!. (Conditions.Have (F.p_not (C.get c))))
(have_access n1),
Conditions.empty)
| Branch(_,None,None) -> assert false
| Either l ->
let l = List.map have_access l in
Conditions.Branch(node_is_accessible, either l, Conditions.empty)
| Implies l ->
let l = List.map
(fun (c,n) -> !. (Conditions.Branch (C.get c, have_access n, Conditions.empty)))
l in
Conditions.Branch(node_is_accessible, Conditions.concat l, Conditions.empty)
| Effect (e,n) ->
Conditions.Branch(node_is_accessible,
Conditions.append
(!. (Conditions.Have (E.get e)))
(have_access n) ,
Conditions.empty)
| Binding (b,n') ->
let b = !. (Conditions.Have(F.p_conj (Passive.conditions b (fun _ -> true)))) in
Conditions.Branch(node_is_accessible,
Conditions.append b (have_access n'),
Conditions.empty) in
let stmt,descr =
let bag = match Node.Map.find n env.datas with
| exception Not_found -> Bag.empty
| bag -> bag
in
Bag.fold_left (fun (os,od) b ->
match b with
| Meta(os',od') ->
(if os = None then os' else os),
(if od = None then od' else od)
) (None,None) bag in
add_cond ?stmt ?descr f f'
) env.succs f
in
let f = match mode with
| `Bool_Backward -> to_sequence_basic_backward f
| `Bool_Forward -> to_sequence_basic_forward f
in
f,Node.Hashtbl.fold Node.Map.add preds Node.Map.empty
module To_tree = struct
type tree = {
c : F.pred (** condition for this tree *) ;
q : tree Queue.t (** childrens *) ;
mutable fact : F.pred list (** facts at this level *) ;
}
[@@@ warning "-32"]
let rec pp_tree
?(pad : (string * string)= ("", ""))
(tree : tree) : unit =
let pd, pc = pad in
Format.printf "%sNode condition: %a @." pd Lang.F.pp_pred tree.c;
Format.printf "%sNode fact:%a@."
pd
(Pretty_utils.pp_list ~sep:"," ~pre:"[" ~suf:"]" Lang.F.pp_pred) tree.fact;
let n = Queue.length tree.q - 1 in
let _ =
Queue.fold (
fun i c ->
let pad =
(pc ^ (if i = n then "`-- " else "|-- "),
pc ^ (if i = n then " " else "| "))
in
pp_tree ~pad c;
i+1
) 0 tree.q
in ()
let pp_idoms fmt a =
Pretty_utils.pp_array ~sep:";@ " (fun fmt i j -> Format.fprintf fmt "%i -> %i" i j)
fmt a
[@@@ warning "+32"]
type env_to_sequence_tree = {
env: localised_env;
pred: Node.t -> Node.t list;
(** predecessors *)
topo_order : Node.t -> int;
(** topological order *)
get_idom_forward: Node.t -> Node.t;
(** Immediate dominator forward *)
get_idom_backward: int -> int;
(** Immediate dominator backward *)
full_conds: Lang.F.pred Node.Hashtbl.t;
(** For each node we are going to compute different formulas
Necessary conditions of the node from start *)
conds: Lang.F.pred Node.Hashtbl.t;
(** Necessary conditions from its forward idiom *)
subtrees: tree Node.Hashtbl.t;
(** To which subtree corresponds this node *)
root: tree;
(** Root the full tree *)
eithers: Lang.F.pred Node.Hashtbl.t Node.Hashtbl.t;
(** Variable used for the non-deterministic choice of either *)
}
let is_after n1 n2 = n1 > n2
let is_before n1 n2 = n1 < n2
let create_env_to_sequence_tree env =
let node_int,int_node,ordered = topological env in
let nb = Node.Hashtbl.length node_int in
let predecessors = compute_preds env in
let pred n = Node.Hashtbl.find_all predecessors n in
let succ n = succs env n in
let topo_order = Node.Hashtbl.find node_int in
let idoms_forward =
idoms ordered topo_order nb ~pred ~is_after in
let idoms_backward =
idoms (List.rev ordered) topo_order nb ~pred:succ ~is_after:is_before in
let get_idom_forward n =
Datatype.Int.Hashtbl.find int_node idoms_forward.(topo_order n) in
{
env;
pred;
topo_order;
get_idom_forward;
get_idom_backward = (fun i -> idoms_backward.(i));
full_conds = Node.Hashtbl.create 10;
conds = Node.Hashtbl.create 10;
subtrees = Node.Hashtbl.create 10;
root = {c = Lang.F.p_true; q = Queue.create (); fact = [] };
eithers = Node.Hashtbl.create 10;
}, ordered
let either env n last =
let h = Node.Hashtbl.memo env.eithers n (fun _ -> Node.Hashtbl.create 10) in
Node.Hashtbl.memo h last (fun _ ->
let v = F.fresh ~basename:"node"
(get_pool ()) Qed.Logic.Bool in
F.p_bool (F.e_var v)
)
(** For a node n *)
let iter env n =
let idom = env.get_idom_forward n in
let rec get_cond acc n' =
if n' = idom then acc
else
let acc = F.p_and acc (Node.Hashtbl.find env.conds n') in
get_cond acc (env.get_idom_forward n')
in
let rec find_frontiere last acc n' =
if (env.topo_order n) <= env.get_idom_backward (env.topo_order n')
then
let cond = get_cond F.p_true n' in
let branch = match Node.Map.find n' env.env.succs with
| exception Not_found -> F.p_true
| Goto _ | Havoc (_, _) -> F.p_true
| Branch (c,Some n'',Some _) when Node.equal n'' last -> C.get c
| Branch (c,Some _,Some n'') when Node.equal n'' last -> F.p_not (C.get c)
| Branch (_,_,_) -> F.p_true
| Either _ -> either env n' last
| Implies l -> List.fold_left (fun acc (c,n) ->
if n = last then C.get c
else acc) F.p_true l
| Effect (_,_) -> F.p_true
| Binding (_,_) -> F.p_true
in
F.p_or acc (F.p_and branch cond)
else
List.fold_left (find_frontiere n') acc (env.pred n')
in
let c, q =
if Node.equal idom n
then
begin
Node.Hashtbl.add env.full_conds n F.p_true;
Node.Hashtbl.add env.conds n F.p_true;
F.p_true, env.root.q
end
else
let c = List.fold_left (find_frontiere n) F.p_false (env.pred n) in
Node.Hashtbl.add env.conds n c;
Node.Hashtbl.add env.full_conds n (F.p_and c (Node.Hashtbl.find env.full_conds idom));
let p = Node.Hashtbl.find env.subtrees idom in
c,p.q
in
let fact = match Node.Map.find n env.env.succs with
| exception Not_found -> F.p_true
| Goto _ | Havoc (_, _) -> F.p_true
| Branch (c,Some _,None) -> C.get c
| Branch (c,None,Some _) -> F.p_not (C.get c)
| Branch (_,_,_) -> F.p_true
| Either _ -> F.p_true
| Implies _ -> F.p_true
| Effect (e,_) -> E.get e
| Binding (b,_) -> F.p_conj (Passive.conditions b (fun _ -> true))
in
let t = {c = c; q = Queue.create (); fact = [fact]} in
Queue.push t q;
Node.Hashtbl.add env.subtrees n t
let add_cond ?descr ?stmt f cond =
match cond with
| Conditions.Have c when F.is_ptrue c = Qed.Logic.Yes -> f
| _ ->
Conditions.append (Conditions.sequence [Conditions.step ?descr ?stmt cond]) f
let access env n = Node.Hashtbl.find env.full_conds n
let get_latest_node env = function
| [] -> env.root
| a::l ->
let n = List.fold_left (fun a e ->
if env.topo_order a < env.topo_order e then e else a
) a l in
Node.Hashtbl.find env.subtrees n
(** Add each assume to the sub-tree corresponding to the latest
node it uses. The assumes are true if all their nodes are
accessible *)
let add_assumes_fact env = Bag.iter (fun p ->
let nodes = P.nodes_list p in
let nodes_are_accessible =
List.fold_left (fun acc n -> F.p_and (access env n) acc) F.p_true nodes in
let f' = F.p_imply nodes_are_accessible (P.get p) in
let t = get_latest_node env nodes in
t.fact <- f' :: t.fact
) env.env.assumes
(** convert the tree to formula *)
let rec convert t f =
let f' =
if t.fact = []
then Conditions.empty
else List.fold_left (fun acc e -> add_cond acc (Conditions.Have e)) Conditions.empty t.fact in
let f' = Queue.fold (fun f' t -> convert t f') f' t.q in
match F.is_ptrue t.c with
| Qed.Logic.Yes -> Conditions.concat [f;f']
| Qed.Logic.No -> f
| Qed.Logic.Maybe ->
add_cond f (Conditions.Branch(t.c,f',Conditions.empty))
let to_sequence_tree _ posts env =
let env,ordered = create_env_to_sequence_tree env in
List.iter (iter env) ordered;
let f = Conditions.empty in
let f = Node.Set.fold
(fun n f -> add_cond f (Conditions.Have (access env n)))
posts f in
let f = Node.Hashtbl.fold (fun _ h f ->
let p = Node.Hashtbl.fold (fun _ t p -> F.p_or p t) h F.p_false in
add_cond f (Conditions.Have p)
) env.eithers f in
add_assumes_fact env;
let f = convert env.root f in
f, Node.Hashtbl.fold Node.Map.add env.full_conds Node.Map.empty
end
let compile : ?name:string -> ?mode:mode -> node -> Node.Set.t -> S.domain Node.Map.t ->
cfg -> F.pred Node.Map.t * S.t Node.Map.t * Conditions.sequence =
fun ?(name="cfg") ?(mode=`Bool_Forward) pre posts user_reads env ->
if Wp_parameters.has_dkey dkey then
Format.printf "@[0) pre:%a post:%a@]@."
Node.pp pre (Pretty_utils.pp_iter ~sep:"@ " Node.Set.iter Node.pp) posts;
if Wp_parameters.has_dkey dkey then
Format.printf "@[1) %a@]@." pretty_env env;
let env = restrict env pre posts in
if Wp_parameters.has_dkey dkey then
Format.printf "@[2) %a@]@." pretty_env env;
if Node.Map.is_empty env.succs then
Node.Map.empty,Node.Map.empty,
Conditions.sequence [Conditions.step (Conditions.Have(F.p_false))]
else
let subst,env =
if true
then remove_dumb_gotos env
else Node.Map.empty, env
in
let pre = find_def ~def:pre pre subst in
let user_reads =
Node.Map.fold
(fun n n' acc ->
match Node.Map.find n user_reads with
| exception Not_found -> acc
| domain ->
let domain' =
try
S.union (Node.Map.find n' acc) domain
with Not_found -> domain
in
Node.Map.add n' domain' acc)
subst user_reads
in
let reads =
Bag.fold_left (fun acc e ->
Node.Map.union
(fun _ -> S.union) acc
(P.reads e))
user_reads env.assumes in
let env, sigmas = domains env reads pre in
if Wp_parameters.has_dkey dkey then
Format.printf "@[3) %a@]@." pretty_env env;
if Wp_parameters.has_dkey dumpkey then
dump_env ~name env;
let f, preds =
match mode with
| `Tree ->
let final_node = node () in
let env =
Node.Set.fold (fun p cfg ->
let s = {pre=S.create();post=S.create()} in
let e = s,Lang.F.p_true in
let goto = effect p e final_node in
concat goto cfg
) posts env
in
To_tree.to_sequence_tree pre posts env
| (`Bool_Backward | `Bool_Forward) as mode ->
to_sequence_bool ~mode pre posts env in
let predssigmas =
Node.Map.merge
(fun _ p s -> Some (Option.value ~default:F.p_false p, Option.value ~default:(S.create ()) s))
preds sigmas in
let predssigmas =
Node.Map.fold (fun n n' acc -> Node.Map.add n (Node.Map.find n' predssigmas) acc )
subst predssigmas
in
let preds =
Node.Map.map(fun _ (x,_) -> x) predssigmas
in
let sigmas =
Node.Map.map(fun _ (_,x) -> x) predssigmas
in
preds,sigmas,f
end