(* Title: HOL/Tools/ATP/atp_problem.ML
Author: Jia Meng, Cambridge University Computer Laboratory and NICTA
Author: Jasmin Blanchette, TU Muenchen
Abstract representation of ATP problems and TPTP syntax.
*)
signature ATP_PROBLEM =
sig
datatype 'a fo_term = ATerm of 'a * 'a fo_term list
datatype quantifier = AForall | AExists
datatype connective = ANot | AAnd | AOr | AImplies | AIf | AIff | ANotIff
datatype ('a, 'b, 'c) formula =
AQuant of quantifier * ('a * 'b option) list * ('a, 'b, 'c) formula |
AConn of connective * ('a, 'b, 'c) formula list |
AAtom of 'c
datatype 'a ho_type = AType of 'a | AFun of 'a ho_type * 'a ho_type
datatype format = CNF | CNF_UEQ | FOF | TFF | THF
datatype formula_kind = Axiom | Definition | Lemma | Hypothesis | Conjecture
datatype 'a problem_line =
Decl of string * 'a * 'a ho_type |
Formula of string * formula_kind * ('a, 'a ho_type, 'a fo_term) formula
* string fo_term option * string fo_term option
type 'a problem = (string * 'a problem_line list) list
val tptp_cnf : string
val tptp_fof : string
val tptp_tff : string
val tptp_thf : string
val tptp_has_type : string
val tptp_type_of_types : string
val tptp_bool_type : string
val tptp_individual_type : string
val tptp_fun_type : string
val tptp_product_type : string
val tptp_forall : string
val tptp_exists : string
val tptp_not : string
val tptp_and : string
val tptp_or : string
val tptp_implies : string
val tptp_if : string
val tptp_iff : string
val tptp_not_iff : string
val tptp_app : string
val tptp_not_infix : string
val tptp_equal : string
val tptp_old_equal : string
val tptp_false : string
val tptp_true : string
val tptp_empty_list : string
val is_tptp_equal : string -> bool
val is_built_in_tptp_symbol : string -> bool
val is_tptp_variable : string -> bool
val is_tptp_user_symbol : string -> bool
val mk_anot : ('a, 'b, 'c) formula -> ('a, 'b, 'c) formula
val mk_aconn :
connective -> ('a, 'b, 'c) formula -> ('a, 'b, 'c) formula
-> ('a, 'b, 'c) formula
val aconn_fold :
bool option -> (bool option -> 'a -> 'b -> 'b) -> connective * 'a list
-> 'b -> 'b
val aconn_map :
bool option -> (bool option -> 'a -> ('b, 'c, 'd) formula)
-> connective * 'a list -> ('b, 'c, 'd) formula
val formula_fold :
bool option -> (bool option -> 'c -> 'd -> 'd) -> ('a, 'b, 'c) formula
-> 'd -> 'd
val formula_map : ('c -> 'd) -> ('a, 'b, 'c) formula -> ('a, 'b, 'd) formula
val is_format_typed : format -> bool
val tptp_strings_for_atp_problem : format -> string problem -> string list
val ensure_cnf_problem :
(string * string) problem -> (string * string) problem
val filter_cnf_ueq_problem :
(string * string) problem -> (string * string) problem
val declare_undeclared_syms_in_atp_problem :
string -> string -> (string * string) problem -> (string * string) problem
val nice_atp_problem :
bool -> ('a * (string * string) problem_line list) list
-> ('a * string problem_line list) list
* (string Symtab.table * string Symtab.table) option
end;
structure ATP_Problem : ATP_PROBLEM =
struct
open ATP_Util
(** ATP problem **)
datatype 'a fo_term = ATerm of 'a * 'a fo_term list
datatype quantifier = AForall | AExists
datatype connective = ANot | AAnd | AOr | AImplies | AIf | AIff | ANotIff
datatype ('a, 'b, 'c) formula =
AQuant of quantifier * ('a * 'b option) list * ('a, 'b, 'c) formula |
AConn of connective * ('a, 'b, 'c) formula list |
AAtom of 'c
datatype 'a ho_type = AType of 'a | AFun of 'a ho_type * 'a ho_type
datatype format = CNF | CNF_UEQ | FOF | TFF | THF
datatype formula_kind = Axiom | Definition | Lemma | Hypothesis | Conjecture
datatype 'a problem_line =
Decl of string * 'a * 'a ho_type |
Formula of string * formula_kind * ('a, 'a ho_type, 'a fo_term) formula
* string fo_term option * string fo_term option
type 'a problem = (string * 'a problem_line list) list
(* official TPTP syntax *)
val tptp_cnf = "cnf"
val tptp_fof = "fof"
val tptp_tff = "tff"
val tptp_thf = "thf"
val tptp_has_type = ":"
val tptp_type_of_types = "$tType"
val tptp_bool_type = "$o"
val tptp_individual_type = "$i"
val tptp_fun_type = ">"
val tptp_product_type = "*"
val tptp_forall = "!"
val tptp_exists = "?"
val tptp_not = "~"
val tptp_and = "&"
val tptp_or = "|"
val tptp_implies = "=>"
val tptp_if = "<="
val tptp_iff = "<=>"
val tptp_not_iff = "<~>"
val tptp_app = "@"
val tptp_not_infix = "!"
val tptp_equal = "="
val tptp_old_equal = "equal"
val tptp_false = "$false"
val tptp_true = "$true"
val tptp_empty_list = "[]"
fun is_tptp_equal s = (s = tptp_equal orelse s = tptp_old_equal)
fun is_built_in_tptp_symbol s =
s = tptp_old_equal orelse not (Char.isAlpha (String.sub (s, 0)))
fun is_tptp_variable s = Char.isUpper (String.sub (s, 0))
val is_tptp_user_symbol = not o (is_tptp_variable orf is_built_in_tptp_symbol)
fun mk_anot (AConn (ANot, [phi])) = phi
| mk_anot phi = AConn (ANot, [phi])
fun mk_aconn c phi1 phi2 = AConn (c, [phi1, phi2])
fun aconn_fold pos f (ANot, [phi]) = f (Option.map not pos) phi
| aconn_fold pos f (AImplies, [phi1, phi2]) =
f (Option.map not pos) phi1 #> f pos phi2
| aconn_fold pos f (AAnd, phis) = fold (f pos) phis
| aconn_fold pos f (AOr, phis) = fold (f pos) phis
| aconn_fold _ f (_, phis) = fold (f NONE) phis
fun aconn_map pos f (ANot, [phi]) = AConn (ANot, [f (Option.map not pos) phi])
| aconn_map pos f (AImplies, [phi1, phi2]) =
AConn (AImplies, [f (Option.map not pos) phi1, f pos phi2])
| aconn_map pos f (AAnd, phis) = AConn (AAnd, map (f pos) phis)
| aconn_map pos f (AOr, phis) = AConn (AOr, map (f pos) phis)
| aconn_map _ f (c, phis) = AConn (c, map (f NONE) phis)
fun formula_fold pos f =
let
fun aux pos (AQuant (_, _, phi)) = aux pos phi
| aux pos (AConn conn) = aconn_fold pos aux conn
| aux pos (AAtom tm) = f pos tm
in aux pos end
fun formula_map f (AQuant (q, xs, phi)) = AQuant (q, xs, formula_map f phi)
| formula_map f (AConn (c, phis)) = AConn (c, map (formula_map f) phis)
| formula_map f (AAtom tm) = AAtom (f tm)
val is_format_typed = member (op =) [TFF, THF]
fun string_for_kind Axiom = "axiom"
| string_for_kind Definition = "definition"
| string_for_kind Lemma = "lemma"
| string_for_kind Hypothesis = "hypothesis"
| string_for_kind Conjecture = "conjecture"
fun strip_tff_type (AFun (AType s, ty)) = strip_tff_type ty |>> cons s
| strip_tff_type (AFun (AFun _, _)) =
raise Fail "unexpected higher-order type in first-order format"
| strip_tff_type (AType s) = ([], s)
fun string_for_type THF ty =
let
fun aux _ (AType s) = s
| aux rhs (AFun (ty1, ty2)) =
aux false ty1 ^ " " ^ tptp_fun_type ^ " " ^ aux true ty2
|> not rhs ? enclose "(" ")"
in aux true ty end
| string_for_type TFF ty =
(case strip_tff_type ty of
([], s) => s
| ([s'], s) => s' ^ " " ^ tptp_fun_type ^ " " ^ s
| (ss, s) =>
"(" ^ space_implode (" " ^ tptp_product_type ^ " ") ss ^ ") " ^
tptp_fun_type ^ " " ^ s)
| string_for_type _ _ = raise Fail "unexpected type in untyped format"
fun string_for_term _ (ATerm (s, [])) = s
| string_for_term format (ATerm (s, ts)) =
if s = tptp_empty_list then
(* used for lists in the optional "source" field of a derivation *)
"[" ^ commas (map (string_for_term format) ts) ^ "]"
else if is_tptp_equal s then
space_implode (" " ^ tptp_equal ^ " ") (map (string_for_term format) ts)
|> format = THF ? enclose "(" ")"
else
let val ss = map (string_for_term format) ts in
if format = THF then
"(" ^ space_implode (" " ^ tptp_app ^ " ") (s :: ss) ^ ")"
else
s ^ "(" ^ commas ss ^ ")"
end
fun string_for_quantifier AForall = tptp_forall
| string_for_quantifier AExists = tptp_exists
fun string_for_connective ANot = tptp_not
| string_for_connective AAnd = tptp_and
| string_for_connective AOr = tptp_or
| string_for_connective AImplies = tptp_implies
| string_for_connective AIf = tptp_if
| string_for_connective AIff = tptp_iff
| string_for_connective ANotIff = tptp_not_iff
fun string_for_bound_var format (s, ty) =
s ^ (if format = TFF orelse format = THF then
" " ^ tptp_has_type ^ " " ^
string_for_type format (ty |> the_default (AType tptp_individual_type))
else
"")
fun string_for_formula format (AQuant (q, xs, phi)) =
string_for_quantifier q ^
"[" ^ commas (map (string_for_bound_var format) xs) ^ "] : " ^
string_for_formula format phi
|> enclose "(" ")"
| string_for_formula format
(AConn (ANot, [AAtom (ATerm ("=" (* tptp_equal *), ts))])) =
space_implode (" " ^ tptp_not_infix ^ tptp_equal ^ " ")
(map (string_for_term format) ts)
|> format = THF ? enclose "(" ")"
| string_for_formula format (AConn (c, [phi])) =
string_for_connective c ^ " " ^
(string_for_formula format phi |> format = THF ? enclose "(" ")")
|> enclose "(" ")"
| string_for_formula format (AConn (c, phis)) =
space_implode (" " ^ string_for_connective c ^ " ")
(map (string_for_formula format) phis)
|> enclose "(" ")"
| string_for_formula format (AAtom tm) = string_for_term format tm
val default_source =
ATerm ("inference", ATerm ("isabelle", []) :: replicate 2 (ATerm ("[]", [])))
fun string_for_format CNF = tptp_cnf
| string_for_format CNF_UEQ = tptp_cnf
| string_for_format FOF = tptp_fof
| string_for_format TFF = tptp_tff
| string_for_format THF = tptp_thf
fun string_for_problem_line format (Decl (ident, sym, ty)) =
string_for_format format ^ "(" ^ ident ^ ", type,\n " ^ sym ^ " : " ^
string_for_type format ty ^ ").\n"
| string_for_problem_line format (Formula (ident, kind, phi, source, info)) =
string_for_format format ^ "(" ^ ident ^ ", " ^ string_for_kind kind ^
",\n (" ^ string_for_formula format phi ^ ")" ^
(case (source, info) of
(NONE, NONE) => ""
| (SOME tm, NONE) => ", " ^ string_for_term format tm
| (_, SOME tm) =>
", " ^ string_for_term format (source |> the_default default_source) ^
", " ^ string_for_term format tm) ^ ").\n"
fun tptp_strings_for_atp_problem format problem =
"% This file was generated by Isabelle (most likely Sledgehammer)\n\
\% " ^ timestamp () ^ "\n" ::
maps (fn (_, []) => []
| (heading, lines) =>
"\n% " ^ heading ^ " (" ^ string_of_int (length lines) ^ ")\n" ::
map (string_for_problem_line format) lines)
problem
(** CNF UEQ (Waldmeister) **)
fun is_problem_line_negated (Formula (_, _, AConn (ANot, _), _, _)) = true
| is_problem_line_negated _ = false
fun is_problem_line_cnf_ueq
(Formula (_, _, AAtom (ATerm ((s, _), _)), _, _)) = is_tptp_equal s
| is_problem_line_cnf_ueq _ = false
fun open_conjecture_term (ATerm ((s, s'), tms)) =
ATerm (if is_tptp_variable s then (s |> Name.desymbolize false, s')
else (s, s'), tms |> map open_conjecture_term)
fun open_formula conj (AQuant (AForall, _, phi)) = open_formula conj phi
| open_formula true (AAtom t) = AAtom (open_conjecture_term t)
| open_formula _ phi = phi
fun open_formula_line (Formula (ident, kind, phi, source, info)) =
Formula (ident, kind, open_formula (kind = Conjecture) phi, source, info)
| open_formula_line line = line
fun negate_conjecture_line (Formula (ident, Conjecture, phi, source, info)) =
Formula (ident, Hypothesis, mk_anot phi, source, info)
| negate_conjecture_line line = line
fun ensure_cnf_problem problem = problem |> map (apsnd (map open_formula_line))
fun filter_cnf_ueq_problem problem =
problem
|> map (apsnd (map open_formula_line
#> filter is_problem_line_cnf_ueq
#> map negate_conjecture_line))
|> (fn problem =>
let
val conjs = problem |> maps snd |> filter is_problem_line_negated
in if length conjs = 1 then problem else [] end)
(** Symbol declarations **)
(* TFF allows implicit declarations of types, function symbols, and predicate
symbols (with "$i" as the type of individuals), but some provers (e.g.,
SNARK) require explicit declarations. The situation is similar for THF. *)
val atype_of_types = AType (`I tptp_type_of_types)
val bool_atype = AType (`I tptp_bool_type)
val individual_atype = AType (`I tptp_individual_type)
fun default_type pred_sym =
let
fun typ 0 = if pred_sym then bool_atype else individual_atype
| typ ary = AFun (individual_atype, typ (ary - 1))
in typ end
fun add_declared_syms_in_problem_line (Decl (_, sym, _)) = insert (op =) sym
| add_declared_syms_in_problem_line _ = I
fun declared_syms_in_problem problem =
fold (fold add_declared_syms_in_problem_line o snd) problem []
fun undeclared_syms_in_problem declared problem =
let
fun do_sym name ty =
if member (op =) declared name then I else AList.default (op =) (name, ty)
fun do_type (AFun (ty1, ty2)) = fold do_type [ty1, ty2]
| do_type (AType name) = do_sym name (K atype_of_types)
fun do_term pred_sym (ATerm (name as (s, _), tms)) =
is_tptp_user_symbol s
? do_sym name (fn _ => default_type pred_sym (length tms))
#> fold (do_term false) tms
fun do_formula (AQuant (_, xs, phi)) =
fold do_type (map_filter snd xs) #> do_formula phi
| do_formula (AConn (_, phis)) = fold do_formula phis
| do_formula (AAtom tm) = do_term true tm
fun do_problem_line (Decl (_, _, ty)) = do_type ty
| do_problem_line (Formula (_, _, phi, _, _)) = do_formula phi
in
fold (fold do_problem_line o snd) problem []
|> filter_out (is_built_in_tptp_symbol o fst o fst)
end
fun declare_undeclared_syms_in_atp_problem prefix heading problem =
let
fun decl_line (x as (s, _), ty) = Decl (prefix ^ s, x, ty ())
val declared = problem |> declared_syms_in_problem
val decls =
problem |> undeclared_syms_in_problem declared
|> sort_wrt (fst o fst)
|> map decl_line
in (heading, decls) :: problem end
(** Nice names **)
fun empty_name_pool readable_names =
if readable_names then SOME (Symtab.empty, Symtab.empty) else NONE
fun pool_fold f xs z = pair z #> fold_rev (fn x => uncurry (f x)) xs
fun pool_map f xs =
pool_fold (fn x => fn ys => fn pool => f x pool |>> (fn y => y :: ys)) xs []
val no_qualifiers =
let
fun skip [] = []
| skip (#"." :: cs) = skip cs
| skip (c :: cs) = if Char.isAlphaNum c then skip cs else c :: keep cs
and keep [] = []
| keep (#"." :: cs) = skip cs
| keep (c :: cs) = c :: keep cs
in String.explode #> rev #> keep #> rev #> String.implode end
(* Long names can slow down the ATPs. *)
val max_readable_name_size = 20
(* "equal" is reserved by some ATPs. "op" is also reserved, to avoid the
unreadable "op_1", "op_2", etc., in the problem files. "eq" is reserved to
ensure that "HOL.eq" is correctly mapped to equality (not clear whether this
is still necessary). *)
val reserved_nice_names = [tptp_old_equal, "op", "eq"]
fun readable_name full_name s =
if s = full_name then
s
else
s |> no_qualifiers
|> Name.desymbolize (Char.isUpper (String.sub (full_name, 0)))
|> (fn s =>
if size s > max_readable_name_size then
String.substring (s, 0, max_readable_name_size div 2 - 4) ^
Word.toString (hashw_string (full_name, 0w0)) ^
String.extract (s, size s - max_readable_name_size div 2 + 4,
NONE)
else
s)
|> (fn s => if member (op =) reserved_nice_names s then full_name else s)
fun nice_name (full_name, _) NONE = (full_name, NONE)
| nice_name (full_name, desired_name) (SOME the_pool) =
if is_built_in_tptp_symbol full_name then
(full_name, SOME the_pool)
else case Symtab.lookup (fst the_pool) full_name of
SOME nice_name => (nice_name, SOME the_pool)
| NONE =>
let
val nice_prefix = readable_name full_name desired_name
fun add j =
let
val nice_name =
nice_prefix ^ (if j = 0 then "" else "_" ^ string_of_int j)
in
case Symtab.lookup (snd the_pool) nice_name of
SOME full_name' =>
if full_name = full_name' then (nice_name, the_pool)
else add (j + 1)
| NONE =>
(nice_name,
(Symtab.update_new (full_name, nice_name) (fst the_pool),
Symtab.update_new (nice_name, full_name) (snd the_pool)))
end
in add 0 |> apsnd SOME end
fun nice_term (ATerm (name, ts)) =
nice_name name ##>> pool_map nice_term ts #>> ATerm
fun nice_type (AType name) = nice_name name #>> AType
| nice_type (AFun (ty1, ty2)) = nice_type ty1 ##>> nice_type ty2 #>> AFun
fun nice_formula (AQuant (q, xs, phi)) =
pool_map nice_name (map fst xs)
##>> pool_map (fn NONE => pair NONE
| SOME ty => nice_type ty #>> SOME) (map snd xs)
##>> nice_formula phi
#>> (fn ((ss, ts), phi) => AQuant (q, ss ~~ ts, phi))
| nice_formula (AConn (c, phis)) =
pool_map nice_formula phis #>> curry AConn c
| nice_formula (AAtom tm) = nice_term tm #>> AAtom
fun nice_problem_line (Decl (ident, sym, ty)) =
nice_name sym ##>> nice_type ty #>> (fn (sym, ty) => Decl (ident, sym, ty))
| nice_problem_line (Formula (ident, kind, phi, source, info)) =
nice_formula phi #>> (fn phi => Formula (ident, kind, phi, source, info))
fun nice_problem problem =
pool_map (fn (heading, lines) =>
pool_map nice_problem_line lines #>> pair heading) problem
fun nice_atp_problem readable_names problem =
nice_problem problem (empty_name_pool readable_names)
end;