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[sledge] Factor Fol.Fml out into a separate module

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Reviewed By: jvillard

Differential Revision: D24306059

fbshipit-source-id: 7e58c1f04
master
Josh Berdine 4 years ago committed by Facebook GitHub Bot
parent e8b94baae2
commit f29f5cfb6b
4 changed files with 315 additions and 243 deletions
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259
sledge/src/fol.ml
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@ -250,254 +250,27 @@ let one = _Z Z.one
* Formulas
*)
(** Sets of formulas *)
module rec Fmls : sig
include Set.S with type elt := Fml.fml
include Propositional.Make (struct
include Trm
val t_of_sexp : Sexp.t -> t
end = struct
module T = struct
type t = Fml.fml [@@deriving compare, equal, sexp]
end
include Set.Make (T)
include Provide_of_sexp (T)
end
let zero = zero
(** Formulas, built from literals with predicate symbols from various
theories, and propositional constants and connectives. Denote sets of
structures. *)
and Fml : sig
type fml = private
(* propositional constants *)
| Tt
(* equality *)
| Eq of trm * trm
(* arithmetic *)
| Eq0 of trm (** [Eq0(x)] iff x = 0 *)
| Pos of trm (** [Pos(x)] iff x > 0 *)
(* propositional connectives *)
| Not of fml
| And of {pos: Fmls.t; neg: Fmls.t}
| Or of {pos: Fmls.t; neg: Fmls.t}
| Iff of fml * fml
| Cond of {cnd: fml; pos: fml; neg: fml}
(* uninterpreted literals *)
| Lit of Predsym.t * trm array
[@@deriving compare, equal, sexp]
val mk_Tt : unit -> fml
val _Eq : trm -> trm -> fml
val _Eq0 : trm -> fml
val _Pos : trm -> fml
val _Not : fml -> fml
val _And : pos:Fmls.t -> neg:Fmls.t -> fml
val _Or : pos:Fmls.t -> neg:Fmls.t -> fml
val _Iff : fml -> fml -> fml
val _Cond : fml -> fml -> fml -> fml
val _Lit : Predsym.t -> trm array -> fml
val and_ : fml -> fml -> fml
val or_ : fml -> fml -> fml
end = struct
type fml =
| Tt
| Eq of trm * trm
| Eq0 of trm
| Pos of trm
| Not of fml
| And of {pos: Fmls.t; neg: Fmls.t}
| Or of {pos: Fmls.t; neg: Fmls.t}
| Iff of fml * fml
| Cond of {cnd: fml; pos: fml; neg: fml}
| Lit of Predsym.t * trm array
[@@deriving compare, equal, sexp]
let invariant f =
let@ () = Invariant.invariant [%here] f [%sexp_of: fml] in
match f with
(* formulas are in negation-normal form *)
| Not (Not _ | And _ | Or _ | Cond _) -> assert false
(* conjunction and disjunction formulas are: *)
| And {pos; neg} | Or {pos; neg} ->
(* not "zero" (the negation of their unit) *)
assert (Fmls.disjoint pos neg) ;
(* not singleton *)
assert (Fmls.cardinal pos + Fmls.cardinal neg > 1)
(* conditional formulas are in "positive condition" form *)
| Cond {cnd= Not _ | Or _} -> assert false
| _ -> ()
let sort_fml x y = if compare_fml x y <= 0 then (x, y) else (y, x)
(** Some normalization is necessary for [embed_into_fml] (defined below)
to be left inverse to [embed_into_cnd]. Essentially
[0 ≠ (p ? 1 : 0)] needs to normalize to [p], by way of
[0 ≠ (p ? 1 : 0)] ==> [(p ? 0 ≠ 1 : 0 ≠ 0)] ==> [(p ? tt : ff)]
==> [p]. *)
let tt = Tt |> check invariant
let ff = Not Tt |> check invariant
let mk_Tt () = tt
(** classification of terms as either semantically equal or disequal, or
if semantic relationship is unknown, as either syntactically less than
or greater than *)
type compare_semantic_syntactic = SemEq | SemDq | SynLt | SynGt
let compare_semantic_syntactic d e =
let eval_eq d e =
match (d, e) with
| Z y, Z z -> if Z.equal y z then SemEq else SemDq
| Q q, Q r -> if Q.equal q r then SemEq else SemDq
| _ ->
let ord = compare_trm d e in
if ord < 0 then SynLt else if ord = 0 then SemEq else SynGt
let _Eq0 x =
( match compare_semantic_syntactic zero x with
(* 0 = 0 ==> tt *)
| SemEq -> tt
(* 0 = N ==> ff for N ≢ 0 *)
| SemDq -> ff
| SynLt | SynGt -> Eq0 x )
|> check invariant
let _Eq x y =
( if x == zero then _Eq0 y
else if y == zero then _Eq0 x
else
match compare_semantic_syntactic x y with
| SemEq -> tt
| SemDq -> ff
| SynLt -> Eq (x, y)
| SynGt -> Eq (y, x) )
|> check invariant
let _Pos x =
( match x with
| Z z -> if Z.gt z Z.zero then tt else ff
| Q q -> if Q.gt q Q.zero then tt else ff
| x -> Pos x )
|> check invariant
let _Lit p xs = Lit (p, xs) |> check invariant
let rec _Not p =
( match p with
| Not x -> x
| And {pos; neg} -> Or {pos= neg; neg= pos}
| Or {pos; neg} -> And {pos= neg; neg= pos}
| Cond {cnd; pos; neg} -> Cond {cnd; pos= _Not pos; neg= _Not neg}
| Tt | Eq _ | Eq0 _ | Pos _ | Lit _ | Iff _ -> Not p )
|> check invariant
let _Join cons zero ~pos ~neg =
if not (Fmls.disjoint pos neg) then zero
else if Fmls.is_empty neg then
match Fmls.only_elt pos with Some p -> p | _ -> cons ~pos ~neg
else if Fmls.is_empty pos then
match Fmls.only_elt neg with Some n -> _Not n | _ -> cons ~pos ~neg
else cons ~pos ~neg
let _And ~pos ~neg = _Join (fun ~pos ~neg -> And {pos; neg}) ff ~pos ~neg
let _Or ~pos ~neg = _Join (fun ~pos ~neg -> Or {pos; neg}) tt ~pos ~neg
let join _Cons zero split_pos_neg p q =
( if equal_fml p zero || equal_fml q zero then zero
else
let pp, pn = split_pos_neg p in
if Fmls.is_empty pp && Fmls.is_empty pn then q
else
let qp, qn = split_pos_neg q in
if Fmls.is_empty qp && Fmls.is_empty qn then p
else
let pos = Fmls.union pp qp in
let neg = Fmls.union pn qn in
_Cons ~pos ~neg )
|> check invariant
| Z y, Z z -> Some (Z.equal y z)
| Q q, Q r -> Some (Q.equal q r)
| _ -> None
let and_ p q =
join _And ff
(function
| And {pos; neg} -> (pos, neg)
| Not p -> (Fmls.empty, Fmls.of_ p)
| p -> (Fmls.of_ p, Fmls.empty) )
p q
let or_ p q =
join _Or tt
(function
| Or {pos; neg} -> (pos, neg)
| Not p -> (Fmls.empty, Fmls.of_ p)
| p -> (Fmls.of_ p, Fmls.empty) )
p q
type equal_or_opposite = Equal | Opposite | Unknown
let rec equal_or_opposite p q =
match (p, q) with
| p, Not p' | Not p', p -> if equal_fml p p' then Opposite else Unknown
| And {pos= ap; neg= an}, Or {pos= op; neg= on}
|Or {pos= op; neg= on}, And {pos= ap; neg= an}
when Fmls.equal ap on && Fmls.equal an op ->
Opposite
| Cond {cnd= c; pos= p; neg= n}, Cond {cnd= c'; pos= p'; neg= n'} ->
if equal_fml c c' then
match equal_or_opposite p p' with
| Opposite -> (
match equal_or_opposite n n' with
| Opposite -> Opposite
| _ -> Unknown )
| Equal -> if equal_fml n n' then Equal else Unknown
| Unknown -> Unknown
else Unknown
| _ -> if equal_fml p q then Equal else Unknown
let is_negative = function Not _ | Or _ -> true | _ -> false
let _Iff p q =
( match (p, q) with
| Tt, p | p, Tt -> p
| Not Tt, p | p, Not Tt -> _Not p
| _ -> (
match equal_or_opposite p q with
| Equal -> tt
| Opposite -> ff
| Unknown ->
let p, q = sort_fml p q in
Iff (p, q) ) )
|> check invariant
let eval_eq0 = function
| Z z -> Some (Z.equal Z.zero z)
| Q q -> Some (Q.equal Q.zero q)
| _ -> None
let _Cond cnd pos neg =
( match (cnd, pos, neg) with
(* (tt ? p : n) ==> p *)
| Tt, _, _ -> pos
(* (ff ? p : n) ==> n *)
| Not Tt, _, _ -> neg
(* (c ? tt : ff) ==> c *)
| _, Tt, Not Tt -> cnd
(* (c ? ff : tt) ==> ¬c *)
| _, Not Tt, Tt -> _Not cnd
(* (c ? p : ff) ==> c ∧ p *)
| _, _, Not Tt -> and_ cnd pos
(* (c ? ff : n) ==> ¬c ∧ n *)
| _, Not Tt, _ -> and_ (_Not cnd) neg
(* (c ? tt : n) ==> c ∨ n *)
| _, Tt, _ -> or_ cnd neg
(* (c ? p : tt) ==> ¬c ∨ p *)
| _, _, Tt -> or_ (_Not cnd) pos
| _ -> (
match equal_or_opposite pos neg with
(* (c ? p : p) ==> c *)
| Equal -> cnd
(* (c ? p : ¬p) ==> c <=> p *)
| Opposite -> _Iff cnd pos
(* (¬c ? n : p) ==> (c ? p : n) *)
| Unknown when is_negative cnd ->
Cond {cnd= _Not cnd; pos= neg; neg= pos}
(* (c ? p : n) *)
| _ -> Cond {cnd; pos; neg} ) )
|> check invariant
end
let eval_pos = function
| Z z -> Some (Z.gt z Z.zero)
| Q q -> Some (Q.gt q Q.zero)
| _ -> None
end)
open Fml

217
sledge/src/propositional.ml
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@ -0,0 +1,217 @@
(*
* Copyright (c) Facebook, Inc. and its affiliates.
*
* This source code is licensed under the MIT license found in the
* LICENSE file in the root directory of this source tree.
*)
(** Propositional formulas *)
include Propositional_intf
open Ses
module Make (Trm : TERM) = struct
open Trm
(** Sets of formulas *)
module rec Fmls : (FORMULA_SET with type elt := Fml.fml) = struct
module T = struct
type t = Fml.fml [@@deriving compare, equal, sexp]
end
include Set.Make (T)
include Provide_of_sexp (T)
end
(** Formulas, built from literals with predicate symbols from various
theories, and propositional constants and connectives. Denote sets of
structures. *)
and Fml : (FORMULA with type trm := Trm.trm with type fmls := Fmls.t) =
struct
type fml =
| Tt
| Eq of trm * trm
| Eq0 of trm
| Pos of trm
| Not of fml
| And of {pos: Fmls.t; neg: Fmls.t}
| Or of {pos: Fmls.t; neg: Fmls.t}
| Iff of fml * fml
| Cond of {cnd: fml; pos: fml; neg: fml}
| Lit of Predsym.t * trm array
[@@deriving compare, equal, sexp]
let invariant f =
let@ () = Invariant.invariant [%here] f [%sexp_of: fml] in
match f with
(* formulas are in negation-normal form *)
| Not (Not _ | And _ | Or _ | Cond _) -> assert false
(* conjunction and disjunction formulas are: *)
| And {pos; neg} | Or {pos; neg} ->
(* not "zero" (the negation of their unit) *)
assert (Fmls.disjoint pos neg) ;
(* not singleton *)
assert (Fmls.cardinal pos + Fmls.cardinal neg > 1)
(* conditional formulas are in "positive condition" form *)
| Cond {cnd= Not _ | Or _} -> assert false
| _ -> ()
let sort_fml x y = if compare_fml x y <= 0 then (x, y) else (y, x)
(** Some normalization is necessary for [embed_into_fml] (defined below)
to be left inverse to [embed_into_cnd]. Essentially
[0 ≠ (p ? 1 : 0)] needs to normalize to [p], by way of
[0 ≠ (p ? 1 : 0)] ==> [(p ? 0 ≠ 1 : 0 ≠ 0)] ==>
[(p ? tt : ff)] ==> [p]. *)
let tt = Tt |> check invariant
let ff = Not Tt |> check invariant
let mk_Tt () = tt
let bool b = if b then tt else ff
let _Eq0 x =
(match eval_eq0 x with Some b -> bool b | None -> Eq0 x)
|> check invariant
let _Eq x y =
( if x == zero then _Eq0 y
else if y == zero then _Eq0 x
else
match eval_eq x y with
| Some b -> bool b
| None -> (
match Sign.of_int (compare_trm x y) with
| Neg -> Eq (x, y)
| Zero -> tt
| Pos -> Eq (y, x) ) )
|> check invariant
let _Pos x =
(match eval_pos x with Some b -> bool b | None -> Pos x)
|> check invariant
let _Lit p xs = Lit (p, xs) |> check invariant
let rec _Not p =
( match p with
| Not x -> x
| And {pos; neg} -> Or {pos= neg; neg= pos}
| Or {pos; neg} -> And {pos= neg; neg= pos}
| Cond {cnd; pos; neg} -> Cond {cnd; pos= _Not pos; neg= _Not neg}
| Tt | Eq _ | Eq0 _ | Pos _ | Lit _ | Iff _ -> Not p )
|> check invariant
let _Join cons zero ~pos ~neg =
if not (Fmls.disjoint pos neg) then zero
else if Fmls.is_empty neg then
match Fmls.only_elt pos with Some p -> p | _ -> cons ~pos ~neg
else if Fmls.is_empty pos then
match Fmls.only_elt neg with
| Some n -> _Not n
| _ -> cons ~pos ~neg
else cons ~pos ~neg
let _And ~pos ~neg =
_Join (fun ~pos ~neg -> And {pos; neg}) ff ~pos ~neg
let _Or ~pos ~neg = _Join (fun ~pos ~neg -> Or {pos; neg}) tt ~pos ~neg
let join _Cons zero split_pos_neg p q =
( if equal_fml p zero || equal_fml q zero then zero
else
let pp, pn = split_pos_neg p in
if Fmls.is_empty pp && Fmls.is_empty pn then q
else
let qp, qn = split_pos_neg q in
if Fmls.is_empty qp && Fmls.is_empty qn then p
else
let pos = Fmls.union pp qp in
let neg = Fmls.union pn qn in
_Cons ~pos ~neg )
|> check invariant
let and_ p q =
join _And ff
(function
| And {pos; neg} -> (pos, neg)
| Not p -> (Fmls.empty, Fmls.of_ p)
| p -> (Fmls.of_ p, Fmls.empty) )
p q
let or_ p q =
join _Or tt
(function
| Or {pos; neg} -> (pos, neg)
| Not p -> (Fmls.empty, Fmls.of_ p)
| p -> (Fmls.of_ p, Fmls.empty) )
p q
type equal_or_opposite = Equal | Opposite | Unknown
let rec equal_or_opposite p q =
match (p, q) with
| p, Not p' | Not p', p ->
if equal_fml p p' then Opposite else Unknown
| And {pos= ap; neg= an}, Or {pos= op; neg= on}
|Or {pos= op; neg= on}, And {pos= ap; neg= an}
when Fmls.equal ap on && Fmls.equal an op ->
Opposite
| Cond {cnd= c; pos= p; neg= n}, Cond {cnd= c'; pos= p'; neg= n'} ->
if equal_fml c c' then
match equal_or_opposite p p' with
| Opposite -> (
match equal_or_opposite n n' with
| Opposite -> Opposite
| _ -> Unknown )
| Equal -> if equal_fml n n' then Equal else Unknown
| Unknown -> Unknown
else Unknown
| _ -> if equal_fml p q then Equal else Unknown
let is_negative = function Not _ | Or _ -> true | _ -> false
let _Iff p q =
( match (p, q) with
| Tt, p | p, Tt -> p
| Not Tt, p | p, Not Tt -> _Not p
| _ -> (
match equal_or_opposite p q with
| Equal -> tt
| Opposite -> ff
| Unknown ->
let p, q = sort_fml p q in
Iff (p, q) ) )
|> check invariant
let _Cond cnd pos neg =
( match (cnd, pos, neg) with
(* (tt ? p : n) ==> p *)
| Tt, _, _ -> pos
(* (ff ? p : n) ==> n *)
| Not Tt, _, _ -> neg
(* (c ? tt : ff) ==> c *)
| _, Tt, Not Tt -> cnd
(* (c ? ff : tt) ==> ¬c *)
| _, Not Tt, Tt -> _Not cnd
(* (c ? p : ff) ==> c ∧ p *)
| _, _, Not Tt -> and_ cnd pos
(* (c ? ff : n) ==> ¬c ∧ n *)
| _, Not Tt, _ -> and_ (_Not cnd) neg
(* (c ? tt : n) ==> c ∨ n *)
| _, Tt, _ -> or_ cnd neg
(* (c ? p : tt) ==> ¬c ∨ p *)
| _, _, Tt -> or_ (_Not cnd) pos
| _ -> (
match equal_or_opposite pos neg with
(* (c ? p : p) ==> c *)
| Equal -> cnd
(* (c ? p : ¬p) ==> c <=> p *)
| Opposite -> _Iff cnd pos
(* (¬c ? n : p) ==> (c ? p : n) *)
| Unknown when is_negative cnd ->
Cond {cnd= _Not cnd; pos= neg; neg= pos}
(* (c ? p : n) *)
| _ -> Cond {cnd; pos; neg} ) )
|> check invariant
end
end

11
sledge/src/propositional.mli
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(*
* Copyright (c) Facebook, Inc. and its affiliates.
*
* This source code is licensed under the MIT license found in the
* LICENSE file in the root directory of this source tree.
*)
(** Propositional formulas *)
include module type of Propositional_intf
module Make : MAKE

71
sledge/src/propositional_intf.ml
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(*
* Copyright (c) Facebook, Inc. and its affiliates.
*
* This source code is licensed under the MIT license found in the
* LICENSE file in the root directory of this source tree.
*)
(** Propositional formulas *)
open Ses
module type TERM = sig
type trm [@@deriving compare, equal, sexp]
val zero : trm
val eval_eq : trm -> trm -> bool option
val eval_eq0 : trm -> bool option
val eval_pos : trm -> bool option
end
(** Formulas, built from literals with predicate symbols from various
theories, and propositional constants and connectives. Denote sets of
structures. *)
module type FORMULA = sig
type trm
type fmls
type fml = private
(* propositional constants *)
| Tt
(* equality *)
| Eq of trm * trm
(* arithmetic *)
| Eq0 of trm (** [Eq0(x)] iff x = 0 *)
| Pos of trm (** [Pos(x)] iff x > 0 *)
(* propositional connectives *)
| Not of fml
| And of {pos: fmls; neg: fmls}
| Or of {pos: fmls; neg: fmls}
| Iff of fml * fml
| Cond of {cnd: fml; pos: fml; neg: fml}
(* uninterpreted literals *)
| Lit of Predsym.t * trm array
[@@deriving compare, equal, sexp]
val mk_Tt : unit -> fml
val _Eq : trm -> trm -> fml
val _Eq0 : trm -> fml
val _Pos : trm -> fml
val _Not : fml -> fml
val _And : pos:fmls -> neg:fmls -> fml
val _Or : pos:fmls -> neg:fmls -> fml
val _Iff : fml -> fml -> fml
val _Cond : fml -> fml -> fml -> fml
val _Lit : Predsym.t -> trm array -> fml
val and_ : fml -> fml -> fml
val or_ : fml -> fml -> fml
end
(** Sets of formulas *)
module type FORMULA_SET = sig
include Set.S
val t_of_sexp : Sexp.t -> t
end
module type MAKE = functor (Trm : TERM) -> sig
module rec Fml :
(FORMULA with type trm := Trm.trm with type fmls := Fmls.t)
and Fmls : (FORMULA_SET with type elt := Fml.fml)
end
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