let (<) : int -> int -> bool = (<) let (>) : int -> int -> bool = (>) let (=) : int -> int -> bool = (=) type 'a bdd = False | True | Split of int * 'a * 'a bdd * 'a bdd * 'a bdd let empty = False module type S = sig type elem include Custom.T val get: t -> (elem list * elem list) list val empty : t val full : t val cup : t -> t -> t val cap : t -> t -> t val diff : t -> t -> t val atom : elem -> t val is_empty : t -> bool val iter: (elem-> unit) -> t -> unit val compute: empty:'b -> full:'b -> cup:('b -> 'b -> 'b) -> cap:('b -> 'b -> 'b) -> diff:('b -> 'b -> 'b) -> atom:(elem -> 'b) -> t -> 'b val trivially_disjoint: t -> t -> bool end module Make(X : Custom.T) = struct type elem = X.t type t = elem bdd let rec equal a b = (a == b) || match (a,b) with | Split (h1,x1, p1,i1,n1), Split (h2,x2, p2,i2,n2) -> (h1 == h2) && (equal p1 p2) && (equal i1 i2) && (equal n1 n2) && (X.equal x1 x2) | _ -> false (* Idea: add a mutable "unique" identifier and set it to the minimum of the two when egality ... *) let rec compare a b = if (a == b) then 0 else match (a,b) with | Split (h1,x1, p1,i1,n1), Split (h2,x2, p2,i2,n2) -> if h1 < h2 then -1 else if h1 > h2 then 1 else let c = X.compare x1 x2 in if c <> 0 then c else let c = compare p1 p2 in if c <> 0 then c else let c = compare i1 i2 in if c <> 0 then c else compare n1 n2 | True,_ -> -1 | _, True -> 1 | False,_ -> -1 | _,False -> 1 let rec hash = function | True -> 1 | False -> 0 | Split(h, _,_,_,_) -> h let compute_hash x p i n = (X.hash x) + 17 * (hash p) + 257 * (hash i) + 16637 * (hash n) let rec check = function | True | False -> () | Split (h,x,p,i,n) -> assert (h = compute_hash x p i n); (match p with Split (_,y,_,_,_) -> assert (X.compare x y < 0) | _ -> ()); (match i with Split (_,y,_,_,_) -> assert (X.compare x y < 0) | _ -> ()); (match n with Split (_,y,_,_,_) -> assert (X.compare x y < 0) | _ -> ()); X.check x; check p; check i; check n let atom x = let h = X.hash x + 17 in (* partial evaluation of compute_hash... *) Split (h, x,True,False,False) let is_empty t = (t == False) let rec iter f = function | Split (_, x, p,i,n) -> f x; iter f p; iter f i; iter f n | _ -> () let rec dump ppf = function | False -> Format.fprintf ppf "⊥" | True -> Format.fprintf ppf "⊤" | Split (_,x, p,i,n) -> Format.fprintf ppf "@[@[%a@][@[%a,@\n%a,@\n%a@]]@]" X.dump x dump p dump i dump n let rec print f ppf = function | True -> Format.fprintf ppf "Any" | False -> Format.fprintf ppf "Empty" | Split (_, x, p,i, n) -> let flag = ref false in let b () = if !flag then Format.fprintf ppf " | " else flag := true in (match p with | True -> b(); Format.fprintf ppf "%a" f x | False -> () | _ -> b (); Format.fprintf ppf "%a & @[(%a)@]" f x (print f) p ); (match i with | True -> assert false; | False -> () | _ -> b(); print f ppf i); (match n with | True -> b (); Format.fprintf ppf "@[~%a@]" f x | False -> () | _ -> b (); Format.fprintf ppf "@[~%a@] & @[(%a)@]" f x (print f) n) let print a f = function | True -> [ fun ppf -> Format.fprintf ppf "%s" a ] | False -> [] | c -> [ fun ppf -> print f ppf c ] let rec get_aux rev accu pos neg = function | True -> let x = if rev then (List.rev pos, List.rev neg) else (pos,neg) in x :: accu | False -> accu | Split (_,x, p,i,n) -> (*OPT: can avoid creating this list cell when pos or neg =False *) let accu = get_aux rev accu (x::pos) neg p in let accu = get_aux rev accu pos (x::neg) n in let accu = get_aux rev accu pos neg i in accu let get x = get_aux false [] [] [] x let rec get' accu pos neg = function | True -> (pos,neg) :: accu | False -> accu | Split (_,x,p,i,n) -> let accu = get' accu (x::pos) neg p in let rec aux l = function | Split (_,x,False,i,n') when equal n n' -> aux (x :: l) i | i -> (* if (List.length l > 1) then (print_int (List.length l); flush stdout); *) let accu = get' accu pos (l :: neg) n in get' accu pos neg i in aux [x] i let get' x = get' [] [] [] x let compute ~empty ~full ~cup ~cap ~diff ~atom b = let rec aux = function | True -> full | False -> empty | Split(_,x, p,i,n) -> let x1 = atom x in let p = cap x1 (aux p) and i = aux i and n = diff (aux n) x1 in cup (cup p i) n in aux b (* Invariant: correct hash value *) let split0 x pos ign neg = Split (compute_hash x pos ign neg, x, pos, ign, neg) let empty = empty let full = True (* Invariants: Split (x, pos,ign,neg) ==> (ign <> True), (pos <> neg) *) let rec has_true = function | [] -> false | True :: _ -> true | _ :: l -> has_true l let rec has_same a = function | [] -> false | b :: l -> (equal a b) || (has_same a l) module type LeafOpts = sig val split0 : elem -> t -> t -> t ->t val merge_leaves : elem -> elem -> bool val cup : elem -> elem -> t val cap : elem -> elem -> t val diff : elem -> elem -> t end let make_ops (module LO : LeafOpts) = let rec split x p i n = if i == True then True else if equal p n then p ++ i else let p = simplify p [i] and n = simplify n [i] in if equal p n then p ++ i else LO.split0 x p i n and simplify a l = match a with | False -> False | True -> if has_true l then False else True | Split (_,x,p,i,n) -> if (has_true l) || (has_same a l) then False else s_aux2 a x p i n [] [] [] l and s_aux2 a x p i n ap ai an = function | [] -> let p = simplify p ap and n = simplify n an and i = simplify i ai in if equal p n then p ++ i else split0 x p i n | b :: l -> s_aux3 a x p i n ap ai an l b and s_aux3 a x p i n ap ai an l = function | False -> s_aux2 a x p i n ap ai an l | True -> assert false | Split (_,x2,p2,i2,n2) as b -> if equal a b then False else let c = X.compare x2 x in if c < 0 then s_aux3 a x p i n ap ai an l i2 else if c > 0 then s_aux2 a x p i n (b :: ap) (b :: ai) (b :: an) l else s_aux2 a x p i n (p2 :: i2 :: ap) (i2 :: ai) (n2 :: i2 :: an) l and ( ++ ) a b = if a == b then a else match (a,b) with | True, _ | _, True -> True | False, a | a, False -> a | Split (_,x1, True,False,False), Split (_,x2, True,False,False) when LO.merge_leaves x1 x2 -> LO.cup x1 x2 | Split (_,x1, p1,i1,n1), Split (_,x2, p2,i2,n2) -> let c = X.compare x1 x2 in if c = 0 then split x1 (p1 ++ p2) (i1 ++ i2) (n1 ++ n2) else if c < 0 then split x1 p1 (i1 ++ b) n1 else split x2 p2 (i2 ++ a) n2 (* seems better not to make ++ and this split mutually recursive; is the invariant still inforced ? *) in let rec ( ** ) a b = if a == b then a else match (a,b) with | True, a | a, True -> a | False, _ | _, False -> False | Split (_,x1, True,False,False), Split (_,x2, True,False,False) when LO.merge_leaves x1 x2 -> LO.cap x1 x2 | Split (_,x1, p1,i1,n1), Split (_,x2, p2,i2,n2) -> let c = X.compare x1 x2 in if c = 0 then split x1 (p1 ** (p2 ++ i2) ++ (p2 ** i1)) (i1 ** i2) (n1 ** (n2 ++ i2) ++ (n2 ** i1)) else if c < 0 then split x1 (p1 ** b) (i1 ** b) (n1 ** b) else split x2 (p2 ** a) (i2 ** a) (n2 ** a) in let rec neg = function | True -> False | False -> True | Split (_,x, p,i,False) -> split x False (neg (i ++ p)) (neg i) | Split (_,x, False,i,n) -> split x (neg i) (neg (i ++ n)) False | Split (_,x, p,False,n) -> split x (neg p) (neg (p ++ n)) (neg n) | Split (_,x, p,i,n) -> split x (neg (i ++ p)) False (neg (i ++ n)) in let rec ( // ) a b = (* if equal a b then False *) if a == b then False else match (a,b) with | False,_ | _, True -> False | a, False -> a | True, b -> neg b | Split (_,x1, True,False,False), Split (_,x2, True,False,False) when LO.merge_leaves x1 x2 -> LO.diff x1 x2 | Split (_,x1, p1,i1,n1), Split (_,x2, p2,i2,n2) -> let c = X.compare x1 x2 in if c = 0 then if (i2 == False) && (n2 == False) then split x1 (p1 // p2) (i1 // p2) (n1 ++ i1) else split x1 ((p1++i1) // (p2++i2)) False ((n1++i1) // (n2++i2)) else if c < 0 then split x1 (p1 // b) (i1 // b) (n1 // b) else split x2 (a // (i2 ++ p2)) False (a // (i2 ++ n2)) in let rec trivially_disjoint a b = if a == b then a == False else match (a,b) with | True, a | a, True -> a == False | False, _ | _, False -> true | Split (_,x1, p1,i1,n1), Split (_,x2, p2,i2,n2) -> let c = X.compare x1 x2 in if c = 0 then (* try expanding -> p1 p2; p1 i2; i1 p2; i1 i2 ... *) trivially_disjoint (p1 ++ i1) (p2 ++ i2) && trivially_disjoint (n1 ++ i1) (n2 ++ i2) else if c < 0 then trivially_disjoint p1 b && trivially_disjoint i1 b && trivially_disjoint n1 b else trivially_disjoint p2 a && trivially_disjoint i2 a && trivially_disjoint n2 a in ( ++ ), ( ** ), ( // ),trivially_disjoint module SimpleOpts : LeafOpts = struct let split0 = split0 let cup _ _ = assert false let cap _ _ = assert false let diff _ _ = assert false let merge_leaves _ _ = false end let cup, cap, diff, trivially_disjoint = make_ops (module SimpleOpts) end module type V = sig module Atom : S include S with type elem = Atom.t Var.var_or_atom and type t = Atom.t Var.var_or_atom bdd val var : Var.t -> t (** returns the union of all leaves in the BDD *) val leafconj: t -> Atom.t val is_empty : t -> bool val extract : t -> [ `Empty | `Full | `Split of (elem * t * t * t) ] end module MakeVar (T : S) = struct module Atom = T module X = Var.Make(Atom) include Make(X) let var v = let compute_hash x p i n = (Var.hash x) + 17 * (hash p) + 257 * (hash i) + 16637 * (hash n) in let a = atom (`Atm T.full) in let h = compute_hash v a False False in (Split (h,`Var v, a, False, False) :> t ) let get x = get_aux true [] [] [] x let leafconj x = let rec aux accu = function | True -> accu | False -> accu | Split (_,`Atm x, True,False,False) -> T.cup x accu | Split (_,`Atm x, _,_,_) -> assert false | Split (_,`Var x, p,i,n) -> let accu = aux accu p in let accu = aux accu n in let accu = aux accu i in accu in (aux T.empty x) let split0 a p i n = match a, p, i, n with `Atm x, True, False, False when T.is_empty x -> False | `Atm _, True, False, False | `Var _, _, _, _ -> split0 a p i n | `Atm _, _, _, _ -> assert false let full = split0 (`Atm T.full) True False False module VarOpts : LeafOpts = struct let split0 = split0 let cup a b = match a, b with `Atm x1, `Atm x2 -> split0 (`Atm (T.cup x1 x2)) True False False | _ -> assert false let cap a b = match a, b with `Atm x1, `Atm x2 -> split0 (`Atm (T.cap x1 x2)) True False False | _ -> assert false let diff a b = match a, b with `Atm x1, `Atm x2 -> split0 (`Atm (T.diff x1 x2)) True False False | _ -> assert false let merge_leaves a b = match a, b with `Atm _, `Atm _ -> true | _ -> false end let cup, cap, diff, trivially_disjoint = make_ops (module VarOpts) let extract = function | False -> `Empty | True -> `Full | Split (_,x,p,i,n) -> `Split (x,p,i,n) end