Source file arith.ml
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open Format
open Options
open Sig
module A = Xliteral
module Sy = Symbols
module E = Expr
module Z = Numbers.Z
module Q = Numbers.Q
let is_mult h = Sy.equal (Sy.Op Sy.Mult) h
let mod_symb = Sy.name "@mod"
module Type (X:Sig.X) : Polynome.T with type r = X.r = struct
include
Polynome.Make(struct
include X
module Ac = Ac.Make(X)
let mult v1 v2 =
X.ac_embed
{
distribute = true;
h = Sy.Op Sy.Mult;
t = X.type_info v1;
l = let l2 = match X.ac_extract v1 with
| Some { h; l; _ } when Sy.equal h (Sy.Op Sy.Mult) -> l
| _ -> [v1, 1]
in Ac.add (Sy.Op Sy.Mult) (v2,1) l2
}
end)
end
module Shostak
(X : Sig.X)
(P : Polynome.EXTENDED_Polynome with type r = X.r) = struct
type t = P.t
type r = P.r
let name = "arith"
module Debug = struct
let solve_aux r1 r2 =
if debug_arith () then
fprintf fmt "[arith:solve-aux] we solve %a=%a@." X.print r1 X.print r2
let solve_one r1 r2 sbs =
if debug_arith () then
begin
fprintf fmt "[arith:solve-one] solving %a = %a yields:@."
X.print r1 X.print r2;
let c = ref 0 in
List.iter
(fun (p,v) ->
incr c;
fprintf fmt " %d) %a |-> %a@." !c X.print p X.print v) sbs
end
end
let is_mine_symb sy _ty =
let open Sy in
match sy with
| Int _ | Real _ -> true
| Op (Plus | Minus | Mult | Div | Modulo
| Float | Fixed | Abs_int | Abs_real | Sqrt_real
| Sqrt_real_default | Sqrt_real_excess
| Real_of_int | Int_floor | Int_ceil
| Max_int | Max_real | Min_int | Min_real
| Pow_real_int | Pow_real_real | Integer_log2
| Integer_round) -> true
| _ -> false
let empty_polynome ty = P.create [] Q.zero ty
let is_mine p = match P.is_monomial p with
| Some (a,x,b) when Q.equal a Q.one && Q.sign b = 0 -> x
| _ -> P.embed p
let embed r = match P.extract r with
| Some p -> p
| _ -> P.create [Q.one, r] Q.zero (X.type_info r)
let mk_modulo md t1 t2 p2 ctx =
let zero = E.int "0" in
let c1 = E.mk_builtin ~is_pos:true A.LE [zero; md] in
let c2 =
match P.is_const p2 with
| Some n2 ->
let an2 = Q.abs n2 in
assert (Q.is_int an2);
let t2 = E.int (Q.to_string an2) in
E.mk_builtin ~is_pos:true A.LT [md; t2]
| None ->
E.mk_builtin ~is_pos:true A.LT [md; t2]
in
let k = E.fresh_name Ty.Tint in
let t3 = E.mk_term (Sy.Op Sy.Mult) [t2;k] Ty.Tint in
let t3 = E.mk_term (Sy.Op Sy.Plus) [t3;md] Ty.Tint in
let c3 = E.mk_eq ~iff:false t1 t3 in
c3 :: c2 :: c1 :: ctx
let mk_euc_division p p2 t1 t2 ctx =
match P.to_list p2 with
| [], coef_p2 ->
let md = E.mk_term (Sy.Op Sy.Modulo) [t1;t2] Ty.Tint in
let r, ctx' = X.make md in
let rp =
P.mult_const (Q.div Q.one coef_p2) (embed r) in
P.sub p rp, ctx' @ ctx
| _ -> assert false
let exact_sqrt_or_Exit q =
let c = Q.sign q in
if c < 0 then raise Exit;
let n = Q.num q in
let d = Q.den q in
let s_n, _ = Z.sqrt_rem n in
assert (Z.sign s_n >= 0);
if not (Z.equal (Z.mult s_n s_n) n) then raise Exit;
let s_d, _ = Z.sqrt_rem d in
assert (Z.sign s_d >= 0);
if not (Z.equal (Z.mult s_d s_d) d) then raise Exit;
let res = Q.from_zz s_n s_d in
assert (Q.equal (Q.mult res res) q);
res
let default_sqrt_or_Exit q =
let c = Q.sign q in
if c < 0 then raise Exit;
match Q.sqrt_default q with
| None -> raise Exit
| Some res -> assert (Q.compare (Q.mult res res) q <= 0); res
let excess_sqrt_or_Exit q =
let c = Q.sign q in
if c < 0 then raise Exit;
match Q.sqrt_excess q with
| None -> raise Exit
| Some res -> assert (Q.compare (Q.mult res res) q >= 0); res
let mk_partial_interpretation_1 aux_func coef p_acc ty t x =
let r_x, _ = X.make x in
try
match P.to_list (embed r_x) with
| [], d ->
let d = aux_func d in
P.add_const (Q.mult coef d) p_acc
| _ -> raise Exit
with Exit ->
let a = X.term_embed t in
P.add (P.create [coef, a] Q.zero ty) p_acc
let mk_partial_interpretation_2 aux_func coef p_acc ty t x y =
let px = embed (fst (X.make x)) in
let py = embed (fst (X.make y)) in
try
match P.is_const px, P.is_const py with
| Some c_x, Some c_y ->
P.add_const (Q.mult coef (aux_func c_x c_y)) p_acc
| _ ->
P.add (P.create [coef, (X.term_embed t)] Q.zero ty) p_acc
with Exit ->
P.add (P.create [coef, (X.term_embed t)] Q.zero ty) p_acc
let rec mke coef p t ctx =
let { E.f = sb ; xs; ty; _ } =
match E.term_view t with
| E.Not_a_term _ -> assert false
| E.Term tt -> tt
in
match sb, xs with
| (Sy.Int n | Sy.Real n) , _ ->
let c = Q.mult coef (Q.from_string (Hstring.view n)) in
P.add_const c p, ctx
| Sy.Op Sy.Mult, [t1;t2] ->
let p1, ctx = mke coef (empty_polynome ty) t1 ctx in
let p2, ctx = mke Q.one (empty_polynome ty) t2 ctx in
if Options.no_NLA() && P.is_const p1 == None && P.is_const p2 == None
then
let tau = E.mk_term (Sy.name ~kind:Sy.Ac "@*") [t1; t2] ty in
let xtau, ctx' = X.make tau in
P.add p (P.create [coef, xtau] Q.zero ty), List.rev_append ctx' ctx
else
P.add p (P.mult p1 p2), ctx
| Sy.Op Sy.Div, [t1;t2] ->
let p1, ctx = mke Q.one (empty_polynome ty) t1 ctx in
let p2, ctx = mke Q.one (empty_polynome ty) t2 ctx in
if Options.no_NLA() &&
(P.is_const p2 == None ||
(ty == Ty.Tint && P.is_const p1 == None)) then
let tau = E.mk_term (Sy.name "@/") [t1; t2] ty in
let xtau, ctx' = X.make tau in
P.add p (P.create [coef, xtau] Q.zero ty), List.rev_append ctx' ctx
else
let p3, ctx =
try
let p, approx = P.div p1 p2 in
if approx then mk_euc_division p p2 t1 t2 ctx
else p, ctx
with Division_by_zero | Polynome.Maybe_zero ->
P.create [Q.one, X.term_embed t] Q.zero ty, ctx
in
P.add p (P.mult_const coef p3), ctx
| Sy.Op Sy.Plus , [t1;t2] ->
let p2, ctx = mke coef p t2 ctx in
mke coef p2 t1 ctx
| Sy.Op Sy.Minus , [t1;t2] ->
let p2, ctx = mke (Q.minus coef) p t2 ctx in
mke coef p2 t1 ctx
| Sy.Op Sy.Modulo , [t1;t2] ->
let p1, ctx = mke Q.one (empty_polynome ty) t1 ctx in
let p2, ctx = mke Q.one (empty_polynome ty) t2 ctx in
if Options.no_NLA() &&
(P.is_const p1 == None || P.is_const p2 == None)
then
let tau = E.mk_term (Sy.name "@%") [t1; t2] ty in
let xtau, ctx' = X.make tau in
P.add p (P.create [coef, xtau] Q.zero ty), List.rev_append ctx' ctx
else
let p3, ctx =
try P.modulo p1 p2, ctx
with e ->
let t = E.mk_term mod_symb [t1; t2] Ty.Tint in
let ctx = match e with
| Division_by_zero | Polynome.Maybe_zero -> ctx
| Polynome.Not_a_num -> mk_modulo t t1 t2 p2 ctx
| _ -> assert false
in
P.create [Q.one, X.term_embed t] Q.zero ty, ctx
in
P.add p (P.mult_const coef p3), ctx
| Sy.Op Sy.Float, [prec; exp; mode; x] ->
let aux_func e =
let res, _, _ = Fpa_rounding.float_of_rational prec exp mode e in
res
in
mk_partial_interpretation_1 aux_func coef p ty t x, ctx
| Sy.Op Sy.Integer_round, [mode; x] ->
let aux_func = Fpa_rounding.round_to_integer mode in
mk_partial_interpretation_1 aux_func coef p ty t x, ctx
| Sy.Op (Sy.Abs_int | Sy.Abs_real) , [x] ->
mk_partial_interpretation_1 Q.abs coef p ty t x, ctx
| Sy.Op Sy.Sqrt_real, [x] ->
mk_partial_interpretation_1 exact_sqrt_or_Exit coef p ty t x, ctx
| Sy.Op Sy.Sqrt_real_default, [x] ->
mk_partial_interpretation_1 default_sqrt_or_Exit coef p ty t x, ctx
| Sy.Op Sy.Sqrt_real_excess, [x] ->
mk_partial_interpretation_1 excess_sqrt_or_Exit coef p ty t x, ctx
| Sy.Op Sy.Real_of_int, [x] ->
mk_partial_interpretation_1 (fun d -> d) coef p ty t x, ctx
| Sy.Op Sy.Int_floor, [x] ->
mk_partial_interpretation_1 Q.floor coef p ty t x, ctx
| Sy.Op Sy.Int_ceil, [x] ->
mk_partial_interpretation_1 Q.ceiling coef p ty t x, ctx
| Sy.Op (Sy.Max_int | Sy.Max_real), [x;y] ->
let aux_func c d = if Q.compare c d >= 0 then c else d in
mk_partial_interpretation_2 aux_func coef p ty t x y, ctx
| Sy.Op (Sy.Min_int | Sy.Min_real), [x;y] ->
let aux_func c d = if Q.compare c d <= 0 then c else d in
mk_partial_interpretation_2 aux_func coef p ty t x y, ctx
| Sy.Op Sy.Integer_log2, [x] ->
let aux_func q =
if Q.compare_to_0 q <= 0 then raise Exit;
Q.from_int (Fpa_rounding.integer_log_2 q)
in
mk_partial_interpretation_1 aux_func coef p ty t x, ctx
| Sy.Op Sy.Pow_real_int, [x; y] ->
let aux_func (c : Q.t) (d : Q.t) =
assert (Q.is_int d);
let n = match Z.to_machine_int (Q.to_z d) with
| Some n -> n
| None -> raise Exit
in
let sz = Z.numbits (Q.num c) + Z.numbits (Q.den c) in
if sz <> 0 && abs n > 100_000 / sz then raise Exit;
Q.power c n
in
mk_partial_interpretation_2 aux_func coef p ty t x y, ctx
| Sy.Op Sy.Pow_real_real, [x; y] ->
let aux_func (c : Q.t) (d : Q.t) =
if not (Q.is_int d) then raise Exit;
let n = match Z.to_machine_int (Q.to_z d) with
| Some n -> n
| None -> raise Exit
in
let sz = Z.numbits (Q.num c) + Z.numbits (Q.den c) in
if sz <> 0 && abs n > 100_000 / sz then raise Exit;
Q.power c n
in
mk_partial_interpretation_2 aux_func coef p ty t x y, ctx
| Sy.Op Sy.Fixed, _ ->
assert false
| _ ->
let a, ctx' = X.make t in
let ctx = ctx' @ ctx in
match P.extract a with
| Some p' -> P.add p (P.mult_const coef p'), ctx
| _ -> P.add p (P.create [coef, a] Q.zero ty), ctx
let make t =
Options.tool_req 4 "TR-Arith-Make";
let ty = E.type_info t in
let p, ctx = mke Q.one (empty_polynome ty) t [] in
is_mine p, ctx
let rec expand p n acc =
assert (n >=0);
if n = 0 then acc else expand p (n-1) (p::acc)
let unsafe_ac_to_arith { l = rl; t = ty; _ } =
let mlt = List.fold_left (fun l (r,n) -> expand (embed r)n l) [] rl in
List.fold_left P.mult (P.create [] Q.one ty) mlt
let rec number_of_vars l =
List.fold_left (fun acc (r, n) -> acc + n * nb_vars_in_alien r) 0 l
and nb_vars_in_alien r =
match P.extract r with
| Some p ->
let l, _ = P.to_list p in
List.fold_left (fun acc (_, x) -> max acc (nb_vars_in_alien x)) 0 l
| None ->
begin
match X.ac_extract r with
| Some ac when is_mult ac.h ->
number_of_vars ac.l
| _ -> 1
end
let max_list_ = function
| [] -> 0
| [ _, x ] -> nb_vars_in_alien x
| (_, x) :: l ->
let acc = nb_vars_in_alien x in
List.fold_left (fun acc (_, x) -> max acc (nb_vars_in_alien x)) acc l
let contains_a_fresh_alien xp =
List.exists
(fun x ->
match X.term_extract x with
| Some t, _ -> E.is_fresh t
| _ -> false
) (X.leaves xp)
let has_ac p kind =
List.exists
(fun (_, x) ->
match X.ac_extract x with Some ac -> kind ac | _ -> false)
(fst (P.to_list p))
let color ac =
match ac.l with
| [(_, 1)] -> assert false
| _ ->
let p = unsafe_ac_to_arith ac in
if not ac.distribute then
if has_ac p (fun ac -> is_mult ac.h) then X.ac_embed ac
else is_mine p
else
let xp = is_mine p in
if contains_a_fresh_alien xp then
let l, _ = P.to_list p in
let mx = max_list_ l in
if mx = 0 || mx = 1 || number_of_vars ac.l > mx then is_mine p
else X.ac_embed ac
else xp
let type_info p = P.type_info p
module SX = Set.Make(struct type t = r let compare = X.hash_cmp end)
let leaves p = P.leaves p
let subst x t p =
let p = P.subst x (embed t) p in
let ty = P.type_info p in
let l, c = P.to_list p in
let p =
List.fold_left
(fun p (ai, xi) ->
let xi' = X.subst x t xi in
let p' = match P.extract xi' with
| Some p' -> P.mult_const ai p'
| _ -> P.create [ai, xi'] Q.zero ty
in
P.add p p')
(P.create [] c ty) l
in
is_mine p
let compare x y = P.compare (embed x) (embed y)
let equal p1 p2 = P.equal p1 p2
let hash = P.hash
let mod_sym a b =
let m = Q.modulo a b in
let m =
if Q.sign m < 0 then
if Q.compare m (Q.minus b) >= 0 then Q.add m b else assert false
else
if Q.compare m b <= 0 then m else assert false
in
if Q.compare m (Q.div b (Q.from_int 2)) < 0 then m else Q.sub m b
let map_monomes f l ax =
List.fold_left
(fun acc (a,x) ->
let a = f a in if Q.sign a = 0 then acc else (a, x) :: acc)
[ax] l
let apply_subst sb v =
is_mine (List.fold_left (fun v (x, p) -> embed (subst x p v)) v sb)
let subst_bigger x l =
List.fold_left
(fun (l, sb) (b, y) ->
if X.ac_extract y != None && X.str_cmp y x > 0 then
let k = X.term_embed (E.fresh_name Ty.Tint) in
(b, k) :: l, (y, embed k)::sb
else (b, y) :: l, sb)
([], []) l
let is_mine_p = List.map (fun (x,p) -> x, is_mine p)
let = function
| [] -> assert false
| [c] -> c, []
| (a, x) :: s ->
List.fold_left
(fun ((a, x), l) (b, y) ->
if Q.compare (Q.abs a) (Q.abs b) <= 0 then
(a, x), ((b, y) :: l)
else (b, y), ((a, x):: l)) ((a, x),[]) s
let rec omega l b =
let (a, x), l = extract_min l in
let l, sbs = subst_bigger x l in
let p = P.create l b Ty.Tint in
assert (Q.sign a <> 0);
if Q.equal a Q.one then
let p = P.mult_const Q.m_one p in
(x, is_mine p) :: (is_mine_p sbs)
else if Q.equal a Q.m_one then
(x,is_mine p) :: (is_mine_p sbs)
else
let a, l, b =
if Q.sign a < 0 then
(Q.minus a,
List.map (fun (a,x) -> Q.minus a,x) l, (Q.minus b))
else (a, l, b)
in
omega_sigma sbs a x l b
and omega_sigma sbs a x l b =
let m = Q.add a Q.one in
let sigma = X.term_embed (E.fresh_name Ty.Tint) in
let mm_sigma = (Q.minus m, sigma) in
let l_mod = map_monomes (fun a -> mod_sym a m) l mm_sigma in
let b_mod = Q.minus (mod_sym (Q.minus b) m) in
let p = P.create l_mod b_mod Ty.Tint in
let sbs = (x, p) :: sbs in
let p' = P.add (P.mult_const a p) (P.create l b Ty.Tint) in
let sbs2 = solve_int p' in
let sbs = List.map (fun (x, v) -> x, apply_subst sbs2 v) sbs in
let sbs2 = List.filter (fun (y, _) -> not (X.equal y sigma)) sbs2 in
List.rev_append sbs sbs2
and solve_int p =
if P.is_empty p then raise Not_found;
let pgcd = P.pgcd_numerators p in
let ppmc = P.ppmc_denominators p in
let p = P.mult_const (Q.div ppmc pgcd) p in
let l, b = P.to_list p in
if not (Q.is_int b) then raise Util.Unsolvable;
omega l b
let is_null p =
if Q.sign (snd (P.separate_constant p)) <> 0 then
raise Util.Unsolvable;
[]
let solve_int p =
try solve_int p with Not_found -> is_null p
let solve_real p =
try
let a, x = P.choose p in
let p = P.mult_const (Q.div Q.m_one a) (P.remove x p) in
[x, is_mine p]
with Not_found -> is_null p
let unsafe_ac_to_arith { l = rl; t = ty; _ } =
let mlt = List.fold_left (fun l (r, n) -> expand (embed r) n l) [] rl in
List.fold_left P.mult (P.create [] Q.one ty) mlt
let polynome_distribution p unsafe_mode =
let l, c = P.to_list p in
let ty = P.type_info p in
let pp =
List.fold_left
(fun p (coef, x) ->
match X.ac_extract x with
| Some ac when is_mult ac.h ->
P.add p (P.mult_const coef (unsafe_ac_to_arith ac))
| _ ->
P.add p (P.create [coef,x] Q.zero ty)
) (P.create [] c ty) l
in
if not unsafe_mode && has_ac pp (fun ac -> is_mult ac.h) then p
else pp
let solve_aux r1 r2 unsafe_mode =
Options.tool_req 4 "TR-Arith-Solve";
Debug.solve_aux r1 r2;
let p = P.sub (embed r1) (embed r2) in
let pp = polynome_distribution p unsafe_mode in
let ty = P.type_info p in
let sbs = if ty == Ty.Treal then solve_real pp else solve_int pp in
let sbs = List.fast_sort (fun (a,_) (x,_) -> X.str_cmp x a)sbs in
sbs
let apply_subst r l = List.fold_left (fun r (p,v) -> X.subst p v r) r l
exception Unsafe
let check_pivot_safety p nsbs unsafe_mode =
let q = apply_subst p nsbs in
if X.equal p q then p
else
match X.ac_extract p with
| Some _ when unsafe_mode -> raise Unsafe
| Some ac -> X.ac_embed {ac with distribute = false}
| None -> assert false
let triangular_down sbs unsafe_mode =
List.fold_right
(fun (p,v) nsbs ->
(check_pivot_safety p nsbs unsafe_mode, apply_subst v nsbs) :: nsbs)
sbs []
let is_non_lin pv = match X.ac_extract pv with
| Some { Sig.h; _ } -> is_mult h
| _ -> false
let make_idemp _ _ sbs lvs unsafe_mode =
let sbs = triangular_down sbs unsafe_mode in
let sbs = triangular_down (List.rev sbs) unsafe_mode in
let sbs = List.filter (fun (p,_) -> SX.mem p lvs || is_non_lin p) sbs in
List.iter
(fun (p, _) ->
if not (SX.mem p lvs) then (assert (is_non_lin p); raise Unsafe)
)sbs;
sbs
let solve_one pb r1 r2 lvs unsafe_mode =
let sbt = solve_aux r1 r2 unsafe_mode in
let sbt = make_idemp r1 r2 sbt lvs unsafe_mode in
Debug.solve_one r1 r2 sbt;
{pb with sbt = List.rev_append sbt pb.sbt}
let solve r1 r2 pb =
let lvs = List.fold_right SX.add (X.leaves r1) SX.empty in
let lvs = List.fold_right SX.add (X.leaves r2) lvs in
try
if debug_arith () then
fprintf fmt "[arith] Try solving with unsafe mode.@.";
solve_one pb r1 r2 lvs true
with Unsafe ->
try
if debug_arith () then
fprintf fmt "[arith] Cancel unsafe solving mode. Try safe mode@.";
solve_one pb r1 r2 lvs false
with Unsafe ->
assert false
let make t =
if Options.timers() then
try
Timers.exec_timer_start Timers.M_Arith Timers.F_make;
let res = make t in
Timers.exec_timer_pause Timers.M_Arith Timers.F_make;
res
with e ->
Timers.exec_timer_pause Timers.M_Arith Timers.F_make;
raise e
else make t
let solve r1 r2 pb =
if Options.timers() then
try
Timers.exec_timer_start Timers.M_Arith Timers.F_solve;
let res = solve r1 r2 pb in
Timers.exec_timer_pause Timers.M_Arith Timers.F_solve;
res
with e ->
Timers.exec_timer_pause Timers.M_Arith Timers.F_solve;
raise e
else solve r1 r2 pb
let print = P.print
let fully_interpreted sb =
match sb with
| Sy.Op (Sy.Plus | Sy.Minus) -> true
| _ -> false
let _ = None, false
let abstract_selectors p acc =
let p, acc = P.abstract_selectors p acc in
is_mine p, acc
let assign_value =
let cpt_int = ref Q.m_one in
let cpt_real = ref Q.m_one in
let max_constant distincts acc =
List.fold_left
(fun acc x ->
match P.is_const (embed x) with None -> acc | Some c -> Q.max c acc)
acc distincts
in
fun r distincts eq ->
if P.is_const (embed r) != None then None
else
if List.exists
(fun (t,x) ->
let symb, ty = match E.term_view t with
| E.Not_a_term _ -> assert false
| E.Term tt -> tt.E.f, tt.E.ty
in
is_mine_symb symb ty && X.leaves x == []
) eq
then None
else
let term_of_cst, cpt = match X.type_info r with
| Ty.Tint -> E.int, cpt_int
| Ty.Treal -> E.real, cpt_real
| _ -> assert false
in
cpt := Q.add Q.one (max_constant distincts !cpt);
Some (term_of_cst (Q.to_string !cpt), true)
let pprint_const_for_model =
let pprint_positive_const c =
let num = Q.num c in
let den = Q.den c in
if Z.is_one den then Z.to_string num
else Format.sprintf "(/ %s %s)" (Z.to_string num) (Z.to_string den)
in
fun r ->
match P.is_const (embed r) with
| None -> assert false
| Some c ->
let sg = Q.sign c in
if sg = 0 then "0"
else if sg > 0 then pprint_positive_const c
else Format.sprintf "(- %s)" (pprint_positive_const (Q.abs c))
let choose_adequate_model t r l =
if debug_interpretation() then
fprintf fmt "[arith] choose_adequate_model for %a@." E.print t;
let l = List.filter (fun (_, r) -> P.is_const (embed r) != None) l in
let r =
match l with
| [] ->
assert (P.is_const (embed r) != None);
r
| (_,r)::l ->
List.iter (fun (_,x) -> assert (X.equal x r)) l;
r
in
r, pprint_const_for_model r
end