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% FILE : sup-inf.ml %
% DESCRIPTION : SUP-INF method for deciding a subset of Presburger %
% arithmetic (R.E.Shostak, JACM Vol.24 No.4 Pages 529-543) %
% %
% READS FILES : <none> %
% WRITES FILES : <none> %
% %
% AUTHOR : R.J.Boulton %
% DATE : 4th March 1991 %
% %
% LAST MODIFIED : R.J.Boulton %
% DATE : 2nd July 1992 %
%****************************************************************************%
%============================================================================%
% SUP-INF algorithm %
%============================================================================%
%----------------------------------------------------------------------------%
% Datatype for representing the bounds of a normalised expression %
%----------------------------------------------------------------------------%
rectype bound = Bound of rat # (string # rat) list
| Max_bound of bound list
| Min_bound of bound list
| Pos_inf
| Neg_inf;;
%----------------------------------------------------------------------------%
% Datatype for representing the bounds of an non-normalised expression %
%----------------------------------------------------------------------------%
rectype internal_bound = Ibound of bound
| Mult_ibound of rat # internal_bound
| Plus_ibound of internal_bound # internal_bound
| Max_ibound of internal_bound list
| Min_ibound of internal_bound list;;
%----------------------------------------------------------------------------%
% solve_ineqs : %
% (int # (string # int) list) list -> %
% string -> %
% ((rat # (string # rat) list) list # (rat # (string # rat) list) list) %
%----------------------------------------------------------------------------%
letrec solve_ineqs ineqs var =
if (null ineqs)
then ([],[])
else let (const,bind) = hd ineqs
and (restl,restr) = solve_ineqs (tl ineqs) var
in (let i = snd (assoc var bind)
in let const' = Rat (const,(-i))
and bind' = map (I # (\n. Rat (n,(-i))))
(filter (\(name,_) . not (name = var)) bind)
in if (i < 0)
then (((const',bind').restl),restr)
else (restl,((const',bind').restr)))
? (restl,restr);;
%----------------------------------------------------------------------------%
% UPPER : (int # (string # int) list) list -> string -> bound %
%----------------------------------------------------------------------------%
let UPPER s x =
let uppers = fst (solve_ineqs s x)
in if (null uppers)
then Pos_inf
else if (null (tl uppers))
then Bound (hd uppers)
else Min_bound (map Bound uppers);;
%----------------------------------------------------------------------------%
% LOWER : (int # (string # int) list) list -> string -> bound %
%----------------------------------------------------------------------------%
let LOWER s x =
let lowers = snd (solve_ineqs s x)
in if (null lowers)
then Neg_inf
else if (null (tl lowers))
then Bound (hd lowers)
else Max_bound (map Bound lowers);;
%----------------------------------------------------------------------------%
% SIMP_mult : rat -> bound -> bound %
%----------------------------------------------------------------------------%
letrec SIMP_mult r b =
case b
of (Bound (const,bind)) .
(Bound (rat_mult r const,map (I # (rat_mult r)) bind))
| (Max_bound bl) .
(if ((Numerator r) < 0)
then (Min_bound (map (SIMP_mult r) bl))
else (Max_bound (map (SIMP_mult r) bl)))
| (Min_bound bl) .
(if ((Numerator r) < 0)
then (Max_bound (map (SIMP_mult r) bl))
else (Min_bound (map (SIMP_mult r) bl)))
| Pos_inf . (if ((Numerator r) < 0) then Neg_inf else Pos_inf)
| Neg_inf . (if ((Numerator r) < 0) then Pos_inf else Neg_inf);;
%----------------------------------------------------------------------------%
% sum_bindings : %
% (string # rat) list -> (string # rat) list -> (string # rat) list %
%----------------------------------------------------------------------------%
letrec sum_bindings bind1 bind2 =
if (null bind1) then bind2
if (null bind2) then bind1
else (let (name1,coeff1) = hd bind1
and (name2,coeff2) = hd bind2
in if (name1 = name2) then
(let coeff = rat_plus coeff1 coeff2
and bind = sum_bindings (tl bind1) (tl bind2)
in if ((Numerator coeff) = 0)
then bind
else (name1,coeff).bind)
if (string_less name1 name2) then
(name1,coeff1).(sum_bindings (tl bind1) bind2)
else (name2,coeff2).(sum_bindings bind1 (tl bind2)));;
%----------------------------------------------------------------------------%
% SIMP_plus : bound -> bound -> bound %
%----------------------------------------------------------------------------%
letrec SIMP_plus b1 b2 =
(case (b1,b2)
of (Bound (const1,bind1),Bound (const2,bind2)) .
(Bound (rat_plus const1 const2,sum_bindings bind1 bind2))
| (Bound _,Max_bound bl) . (Max_bound (map (SIMP_plus b1) bl))
| (Bound _,Min_bound bl) . (Min_bound (map (SIMP_plus b1) bl))
| (Bound _,Pos_inf) . Pos_inf
| (Bound _,Neg_inf) . Neg_inf
| (Max_bound bl,_) . (Max_bound (map (\b. SIMP_plus b b2) bl))
| (Min_bound bl,_) . (Min_bound (map (\b. SIMP_plus b b2) bl))
| (Pos_inf,Pos_inf) . Pos_inf
| (Pos_inf,Neg_inf) . fail
| (Pos_inf,_) . (SIMP_plus b2 b1)
| (Neg_inf,Neg_inf) . Neg_inf
| (Neg_inf,Pos_inf) . fail
| (Neg_inf,_) . (SIMP_plus b2 b1)
) ? failwith `SIMP_plus`;;
%----------------------------------------------------------------------------%
% SIMP : internal_bound -> bound %
%----------------------------------------------------------------------------%
letrec SIMP ib =
case ib
of (Ibound b) . b
| (Mult_ibound (r,ib')) . (SIMP_mult r (SIMP ib'))
| (Plus_ibound (ib1,ib2)) . (SIMP_plus (SIMP ib1) (SIMP ib2))
| (Max_ibound ibl) . (Max_bound (map SIMP ibl))
| (Min_ibound ibl) . (Min_bound (map SIMP ibl));;
%----------------------------------------------------------------------------%
% SUPP : (string # bound) -> bound %
% INFF : (string # bound) -> bound %
%----------------------------------------------------------------------------%
letrec SUPP (x,y) =
case y
of (Bound (_,[])) . y
| Pos_inf . y
| Neg_inf . y
| (Min_bound bl) . (Min_bound (map (\y. SUPP (x,y)) bl))
| (Bound (const,bind)) .
(let b = snd (assoc x bind) ? rat_zero
and bind' = filter (\p. not (fst p = x)) bind
in if ((null bind') & (const = rat_zero) & (b = rat_one))
then Pos_inf
else let b' = rat_minus rat_one b
in if (Numerator b' < 0) then Pos_inf
if (Numerator b' > 0) then
(Bound (rat_div const b',
map (I # (\r. rat_div r b')) bind'))
else if (not (null bind')) then Pos_inf
if (Numerator const < 0) then Neg_inf
else Pos_inf)
| (_) . failwith `SUPP`;;
letrec INFF (x,y) =
case y
of (Bound (_,[])) . y
| Pos_inf . y
| Neg_inf . y
| (Max_bound bl) . (Max_bound (map (\y. INFF (x,y)) bl))
| (Bound (const,bind)) .
(let b = snd (assoc x bind) ? rat_zero
and bind' = filter (\p. not (fst p = x)) bind
in if ((null bind') & (const = rat_zero) & (b = rat_one))
then Neg_inf
else let b' = rat_minus rat_one b
in if (Numerator b' < 0) then Neg_inf
if (Numerator b' > 0) then
(Bound (rat_div const b',
map (I # (\r. rat_div r b')) bind'))
else if (not (null bind')) then Neg_inf
if (Numerator const > 0) then Pos_inf
else Neg_inf)
| (_) . failwith `INFF`;;
%----------------------------------------------------------------------------%
% occurs_in_bound : string -> bound -> bool %
%----------------------------------------------------------------------------%
letrec occurs_in_bound v b =
case b
of (Bound (_,bind)) . (mem v (map fst bind))
| (Max_bound bl) .
(itlist (\x y. x or y) (map (occurs_in_bound v) bl) false)
| (Min_bound bl) .
(itlist (\x y. x or y) (map (occurs_in_bound v) bl) false)
| (_) . false;;
%----------------------------------------------------------------------------%
% occurs_in_ibound : string -> internal_bound -> bool %
%----------------------------------------------------------------------------%
letrec occurs_in_ibound v ib =
case ib
of (Ibound b) . (occurs_in_bound v b)
| (Mult_ibound (_,ib')) . (occurs_in_ibound v ib')
| (Plus_ibound (ib1,ib2)) .
((occurs_in_ibound v ib1) or (occurs_in_ibound v ib2))
| (Max_ibound ibl) .
(itlist (\x y. x or y) (map (occurs_in_ibound v) ibl) false)
| (Min_ibound ibl) .
(itlist (\x y. x or y) (map (occurs_in_ibound v) ibl) false);;
%----------------------------------------------------------------------------%
% SUP : (int # (string # int) list) list -> %
% (bound # (string list)) -> %
% internal_bound %
% INF : (int # (string # int) list) list -> %
% (bound # (string list)) -> %
% internal_bound %
%----------------------------------------------------------------------------%
letrec SUP s (J,H) =
case J
of (Bound (_,[])) . (Ibound J)
| Pos_inf . (Ibound J)
| Neg_inf . (Ibound J)
| (Min_bound bl) . (Min_ibound (map (\j. SUP s (j,H)) bl))
| (Bound (const,bind)) .
(let (rv.bind') = bind
in let (v,r) = rv
in if ((const = rat_zero) & (null bind'))
then (if (r = rat_one) then
(if (mem v H)
then Ibound J
else let Q = UPPER s v
in let Z = SUP s (Q,union H [v])
in Ibound (SUPP (v,SIMP Z)))
if (Numerator r < 0)
then (Mult_ibound
(r,INF s (Bound (rat_zero,[v,rat_one]),H)))
else (Mult_ibound
(r,SUP s (Bound (rat_zero,[v,rat_one]),H)))
)
else let B' = SUP s (Bound (const,bind'),union H [v])
and rvb = Bound (rat_zero,[rv])
in if (occurs_in_ibound v B')
then let J' = SIMP (Plus_ibound (Ibound rvb,B'))
in SUP s (J',H)
else Plus_ibound (SUP s (rvb,H),B'))
| (_) . failwith `SUP`
and INF s (J,H) =
case J
of (Bound (_,[])) . (Ibound J)
| Pos_inf . (Ibound J)
| Neg_inf . (Ibound J)
| (Max_bound bl) . (Max_ibound (map (\j. INF s (j,H)) bl))
| (Bound (const,bind)) .
(let (rv.bind') = bind
in let (v,r) = rv
in if ((const = rat_zero) & (null bind'))
then (if (r = rat_one) then
(if (mem v H)
then Ibound J
else let Q = LOWER s v
in let Z = INF s (Q,union H [v])
in Ibound (INFF (v,SIMP Z)))
if (Numerator r < 0)
then (Mult_ibound
(r,SUP s (Bound (rat_zero,[v,rat_one]),H)))
else (Mult_ibound
(r,INF s (Bound (rat_zero,[v,rat_one]),H)))
)
else let B' = INF s (Bound (const,bind'),union H [v])
and rvb = Bound (rat_zero,[rv])
in if (occurs_in_ibound v B')
then let J' = SIMP (Plus_ibound (Ibound rvb,B'))
in INF s (J',H)
else Plus_ibound (INF s (rvb,H),B'))
| (_) . failwith `INF`;;
%----------------------------------------------------------------------------%
% eval_max_bound : bound list -> bound %
%----------------------------------------------------------------------------%
letrec eval_max_bound bl =
if (null bl) then failwith `eval_max_bound`
if (null (tl bl)) then (hd bl)
else let b = hd bl
and max = eval_max_bound (tl bl)
in case (b,max)
of (Pos_inf,_) . Pos_inf
| (_,Pos_inf) . Pos_inf
| (Neg_inf,_) . max
| (_,Neg_inf) . b
| (Bound (r1,[]),Bound (r2,[])) .
(if (Numerator (rat_minus r1 r2) < 0) then max else b)
| (_) . failwith `eval_max_bound`;;
%----------------------------------------------------------------------------%
% eval_min_bound : bound list -> bound %
%----------------------------------------------------------------------------%
letrec eval_min_bound bl =
if (null bl) then failwith `eval_min_bound`
if (null (tl bl)) then (hd bl)
else let b = hd bl
and min = eval_min_bound (tl bl)
in case (b,min)
of (Pos_inf,_) . min
| (_,Pos_inf) . b
| (_,Neg_inf) . Neg_inf
| (Neg_inf,_) . Neg_inf
| (Bound (r1,[]),Bound (r2,[])) .
(if (Numerator (rat_minus r1 r2) < 0) then b else min)
| (_) . failwith `eval_min_bound`;;
%----------------------------------------------------------------------------%
% eval_bound : bound -> bound %
%----------------------------------------------------------------------------%
letrec eval_bound b =
case b
of (Bound (_,[])) . b
| (Max_bound bl) . (eval_max_bound (map eval_bound bl))
| (Min_bound bl) . (eval_min_bound (map eval_bound bl))
| Pos_inf . b
| Neg_inf . b;;
%----------------------------------------------------------------------------%
% SUP_INF : %
% (int # (string # int) list) list -> (string # bound # bound) list %
%----------------------------------------------------------------------------%
let SUP_INF set =
let vars_of_coeffs coeffsl = setify (flat (map ((map fst) o snd) coeffsl))
in
let vars = vars_of_coeffs set
and make_bound v = Bound (rat_zero,[v,rat_one])
and eval = eval_bound o SIMP
in map (\v. let b = make_bound v
in (v,eval (INF set (b,[])),eval (SUP set (b,[])))) vars;;
|