1 (* Title: HOL/Tools/ATP/atp_problem.ML
2 Author: Jia Meng, Cambridge University Computer Laboratory and NICTA
3 Author: Jasmin Blanchette, TU Muenchen
5 Abstract representation of ATP problems and TPTP syntax.
8 signature ATP_PROBLEM =
10 datatype ('a, 'b) ho_term =
11 ATerm of 'a * ('a, 'b) ho_term list |
12 AAbs of ('a * 'b) * ('a, 'b) ho_term
13 datatype quantifier = AForall | AExists
14 datatype connective = ANot | AAnd | AOr | AImplies | AIff
15 datatype ('a, 'b, 'c) formula =
16 AQuant of quantifier * ('a * 'b option) list * ('a, 'b, 'c) formula |
17 AConn of connective * ('a, 'b, 'c) formula list |
20 datatype 'a ho_type = AType of 'a | AFun of 'a ho_type * 'a ho_type
22 datatype format = CNF | CNF_UEQ | FOF | TFF | THF
23 datatype formula_kind = Axiom | Definition | Lemma | Hypothesis | Conjecture
24 datatype 'a problem_line =
25 Decl of string * 'a * 'a ho_type |
26 Formula of string * formula_kind * ('a, 'a ho_type, ('a, 'a ho_type) ho_term) formula
27 * (string, string ho_type) ho_term option * (string, string ho_type) ho_term option
28 type 'a problem = (string * 'a problem_line list) list
34 val tptp_has_type : string
35 val tptp_type_of_types : string
36 val tptp_bool_type : string
37 val tptp_individual_type : string
38 val tptp_fun_type : string
39 val tptp_product_type : string
40 val tptp_forall : string
41 val tptp_ho_forall : string
42 val tptp_exists : string
43 val tptp_ho_exists : string
47 val tptp_implies : string
50 val tptp_not_iff : string
52 val tptp_not_infix : string
53 val tptp_equal : string
54 val tptp_old_equal : string
55 val tptp_false : string
56 val tptp_true : string
57 val tptp_empty_list : string
58 val is_tptp_equal : string -> bool
59 val is_built_in_tptp_symbol : string -> bool
60 val is_tptp_variable : string -> bool
61 val is_tptp_user_symbol : string -> bool
62 val mk_anot : ('a, 'b, 'c) formula -> ('a, 'b, 'c) formula
64 connective -> ('a, 'b, 'c) formula -> ('a, 'b, 'c) formula
65 -> ('a, 'b, 'c) formula
67 bool option -> (bool option -> 'a -> 'b -> 'b) -> connective * 'a list
70 bool option -> (bool option -> 'a -> ('b, 'c, 'd) formula)
71 -> connective * 'a list -> ('b, 'c, 'd) formula
73 bool option -> (bool option -> 'c -> 'd -> 'd) -> ('a, 'b, 'c) formula
75 val formula_map : ('c -> 'd) -> ('a, 'b, 'c) formula -> ('a, 'b, 'd) formula
76 val is_format_typed : format -> bool
77 val tptp_lines_for_atp_problem : format -> string problem -> string list
78 val ensure_cnf_problem :
79 (string * string) problem -> (string * string) problem
80 val filter_cnf_ueq_problem :
81 (string * string) problem -> (string * string) problem
82 val declare_undeclared_syms_in_atp_problem :
83 string -> string -> (string * string) problem -> (string * string) problem
84 val nice_atp_problem :
85 bool -> ('a * (string * string) problem_line list) list
86 -> ('a * string problem_line list) list
87 * (string Symtab.table * string Symtab.table) option
90 structure ATP_Problem : ATP_PROBLEM =
98 datatype ('a, 'b) ho_term =
99 ATerm of 'a * ('a, 'b) ho_term list |
100 AAbs of ('a * 'b) * ('a, 'b) ho_term
101 datatype quantifier = AForall | AExists
102 datatype connective = ANot | AAnd | AOr | AImplies | AIff
103 datatype ('a, 'b, 'c) formula =
104 AQuant of quantifier * ('a * 'b option) list * ('a, 'b, 'c) formula |
105 AConn of connective * ('a, 'b, 'c) formula list |
108 datatype 'a ho_type = AType of 'a | AFun of 'a ho_type * 'a ho_type
110 datatype format = CNF | CNF_UEQ | FOF | TFF | THF
111 datatype formula_kind = Axiom | Definition | Lemma | Hypothesis | Conjecture
112 datatype 'a problem_line =
113 Decl of string * 'a * 'a ho_type |
114 Formula of string * formula_kind * ('a, 'a ho_type, ('a, 'a ho_type) ho_term) formula
115 * (string, string ho_type) ho_term option * (string, string ho_type) ho_term option
116 type 'a problem = (string * 'a problem_line list) list
118 (* official TPTP syntax *)
123 val tptp_has_type = ":"
124 val tptp_type_of_types = "$tType"
125 val tptp_bool_type = "$o"
126 val tptp_individual_type = "$i"
127 val tptp_fun_type = ">"
128 val tptp_product_type = "*"
129 val tptp_forall = "!"
130 val tptp_ho_forall = "!!"
131 val tptp_exists = "?"
132 val tptp_ho_exists = "??"
136 val tptp_implies = "=>"
139 val tptp_not_iff = "<~>"
141 val tptp_not_infix = "!"
143 val tptp_old_equal = "equal"
144 val tptp_false = "$false"
145 val tptp_true = "$true"
146 val tptp_empty_list = "[]"
148 fun is_tptp_equal s = (s = tptp_equal orelse s = tptp_old_equal)
149 fun is_built_in_tptp_symbol s =
150 s = tptp_old_equal orelse not (Char.isAlpha (String.sub (s, 0)))
151 fun is_tptp_variable s = Char.isUpper (String.sub (s, 0))
152 val is_tptp_user_symbol = not o (is_tptp_variable orf is_built_in_tptp_symbol)
154 fun raw_polarities_of_conn ANot = (SOME false, NONE)
155 | raw_polarities_of_conn AAnd = (SOME true, SOME true)
156 | raw_polarities_of_conn AOr = (SOME true, SOME true)
157 | raw_polarities_of_conn AImplies = (SOME false, SOME true)
158 | raw_polarities_of_conn AIff = (NONE, NONE)
159 fun polarities_of_conn NONE = K (NONE, NONE)
160 | polarities_of_conn (SOME pos) =
161 raw_polarities_of_conn #> not pos ? pairself (Option.map not)
163 fun mk_anot (AConn (ANot, [phi])) = phi
164 | mk_anot phi = AConn (ANot, [phi])
165 fun mk_aconn c phi1 phi2 = AConn (c, [phi1, phi2])
167 fun aconn_fold pos f (ANot, [phi]) = f (Option.map not pos) phi
168 | aconn_fold pos f (AImplies, [phi1, phi2]) =
169 f (Option.map not pos) phi1 #> f pos phi2
170 | aconn_fold pos f (AAnd, phis) = fold (f pos) phis
171 | aconn_fold pos f (AOr, phis) = fold (f pos) phis
172 | aconn_fold _ f (_, phis) = fold (f NONE) phis
174 fun aconn_map pos f (ANot, [phi]) = AConn (ANot, [f (Option.map not pos) phi])
175 | aconn_map pos f (AImplies, [phi1, phi2]) =
176 AConn (AImplies, [f (Option.map not pos) phi1, f pos phi2])
177 | aconn_map pos f (AAnd, phis) = AConn (AAnd, map (f pos) phis)
178 | aconn_map pos f (AOr, phis) = AConn (AOr, map (f pos) phis)
179 | aconn_map _ f (c, phis) = AConn (c, map (f NONE) phis)
181 fun formula_fold pos f =
183 fun aux pos (AQuant (_, _, phi)) = aux pos phi
184 | aux pos (AConn conn) = aconn_fold pos aux conn
185 | aux pos (AAtom tm) = f pos tm
188 fun formula_map f (AQuant (q, xs, phi)) = AQuant (q, xs, formula_map f phi)
189 | formula_map f (AConn (c, phis)) = AConn (c, map (formula_map f) phis)
190 | formula_map f (AAtom tm) = AAtom (f tm)
192 val is_format_typed = member (op =) [TFF, THF]
194 fun string_for_kind Axiom = "axiom"
195 | string_for_kind Definition = "definition"
196 | string_for_kind Lemma = "lemma"
197 | string_for_kind Hypothesis = "hypothesis"
198 | string_for_kind Conjecture = "conjecture"
200 fun strip_tff_type (AFun (AType s, ty)) = strip_tff_type ty |>> cons s
201 | strip_tff_type (AFun (AFun _, _)) =
202 raise Fail "unexpected higher-order type in first-order format"
203 | strip_tff_type (AType s) = ([], s)
205 fun string_for_type THF ty =
207 fun aux _ (AType s) = s
208 | aux rhs (AFun (ty1, ty2)) =
209 aux false ty1 ^ " " ^ tptp_fun_type ^ " " ^ aux true ty2
210 |> not rhs ? enclose "(" ")"
212 | string_for_type TFF ty =
213 (case strip_tff_type ty of
215 | ([s'], s) => s' ^ " " ^ tptp_fun_type ^ " " ^ s
217 "(" ^ space_implode (" " ^ tptp_product_type ^ " ") ss ^ ") " ^
218 tptp_fun_type ^ " " ^ s)
219 | string_for_type _ _ = raise Fail "unexpected type in untyped format"
221 fun string_for_term _ (ATerm (s, [])) = s
222 | string_for_term format (ATerm (s, ts)) =
223 if s = tptp_empty_list then
224 (* used for lists in the optional "source" field of a derivation *)
225 "[" ^ commas (map (string_for_term format) ts) ^ "]"
226 else if is_tptp_equal s then
227 space_implode (" " ^ tptp_equal ^ " ") (map (string_for_term format) ts)
228 |> format = THF ? enclose "(" ")"
230 let val ss = map (string_for_term format) ts in
232 "(" ^ space_implode (" " ^ tptp_app ^ " ") (s :: ss) ^ ")"
234 s ^ "(" ^ commas ss ^ ")"
236 | string_for_term THF (AAbs ((s, ty), tm)) =
237 "(^[" ^ s ^ ":" ^ string_for_type THF ty ^ "] : " ^ string_for_term THF tm ^ ")"
238 | string_for_term _ _ = raise Fail "unexpected term in first-order format"
240 fun string_for_quantifier AForall = tptp_forall
241 | string_for_quantifier AExists = tptp_exists
243 fun string_for_connective ANot = tptp_not
244 | string_for_connective AAnd = tptp_and
245 | string_for_connective AOr = tptp_or
246 | string_for_connective AImplies = tptp_implies
247 | string_for_connective AIff = tptp_iff
249 fun string_for_bound_var format (s, ty) =
250 s ^ (if format = TFF orelse format = THF then
251 " " ^ tptp_has_type ^ " " ^
252 string_for_type format (ty |> the_default (AType tptp_individual_type))
256 fun string_for_formula format (AQuant (q, xs, phi)) =
257 string_for_quantifier q ^
258 "[" ^ commas (map (string_for_bound_var format) xs) ^ "] : " ^
259 string_for_formula format phi
261 | string_for_formula format
262 (AConn (ANot, [AAtom (ATerm ("=" (* tptp_equal *), ts))])) =
263 space_implode (" " ^ tptp_not_infix ^ tptp_equal ^ " ")
264 (map (string_for_term format) ts)
265 |> format = THF ? enclose "(" ")"
266 | string_for_formula format (AConn (c, [phi])) =
267 string_for_connective c ^ " " ^
268 (string_for_formula format phi |> format = THF ? enclose "(" ")")
270 | string_for_formula format (AConn (c, phis)) =
271 space_implode (" " ^ string_for_connective c ^ " ")
272 (map (string_for_formula format) phis)
274 | string_for_formula format (AAtom tm) = string_for_term format tm
277 ATerm ("inference", ATerm ("isabelle", []) :: replicate 2 (ATerm ("[]", [])))
279 fun string_for_format CNF = tptp_cnf
280 | string_for_format CNF_UEQ = tptp_cnf
281 | string_for_format FOF = tptp_fof
282 | string_for_format TFF = tptp_tff
283 | string_for_format THF = tptp_thf
285 fun string_for_problem_line format (Decl (ident, sym, ty)) =
286 string_for_format format ^ "(" ^ ident ^ ", type,\n " ^ sym ^ " : " ^
287 string_for_type format ty ^ ").\n"
288 | string_for_problem_line format (Formula (ident, kind, phi, source, info)) =
289 string_for_format format ^ "(" ^ ident ^ ", " ^ string_for_kind kind ^
290 ",\n (" ^ string_for_formula format phi ^ ")" ^
291 (case (source, info) of
293 | (SOME tm, NONE) => ", " ^ string_for_term format tm
295 ", " ^ string_for_term format (source |> the_default default_source) ^
296 ", " ^ string_for_term format tm) ^ ").\n"
297 fun tptp_lines_for_atp_problem format problem =
298 "% This file was generated by Isabelle (most likely Sledgehammer)\n\
299 \% " ^ timestamp () ^ "\n" ::
300 maps (fn (_, []) => []
301 | (heading, lines) =>
302 "\n% " ^ heading ^ " (" ^ string_of_int (length lines) ^ ")\n" ::
303 map (string_for_problem_line format) lines)
307 (** CNF (Metis) and CNF UEQ (Waldmeister) **)
309 fun is_problem_line_negated (Formula (_, _, AConn (ANot, _), _, _)) = true
310 | is_problem_line_negated _ = false
312 fun is_problem_line_cnf_ueq (Formula (_, _, AAtom (ATerm ((s, _), _)), _, _)) =
314 | is_problem_line_cnf_ueq _ = false
316 fun open_conjecture_term (ATerm ((s, s'), tms)) =
317 ATerm (if is_tptp_variable s then (s |> Name.desymbolize false, s')
318 else (s, s'), tms |> map open_conjecture_term)
319 | open_conjecture_term _ = raise Fail "unexpected higher-order term"
320 fun open_formula conj =
322 (* We are conveniently assuming that all bound variable names are
323 distinct, which should be the case for the formulas we generate. *)
324 fun opn (pos as SOME true) (AQuant (AForall, _, phi)) = opn pos phi
325 | opn (pos as SOME false) (AQuant (AExists, _, phi)) = opn pos phi
326 | opn pos (AConn (ANot, [phi])) = mk_anot (opn (Option.map not pos) phi)
327 | opn pos (AConn (c, [phi1, phi2])) =
328 let val (pos1, pos2) = polarities_of_conn pos c in
329 AConn (c, [opn pos1 phi1, opn pos2 phi2])
331 | opn _ (AAtom t) = AAtom (t |> conj ? open_conjecture_term)
333 in opn (SOME (not conj)) end
334 fun open_formula_line (Formula (ident, kind, phi, source, info)) =
335 Formula (ident, kind, open_formula (kind = Conjecture) phi, source, info)
336 | open_formula_line line = line
338 fun negate_conjecture_line (Formula (ident, Conjecture, phi, source, info)) =
339 Formula (ident, Hypothesis, mk_anot phi, source, info)
340 | negate_conjecture_line line = line
342 exception CLAUSIFY of unit
344 (* This "clausification" only expands syntactic sugar, such as "phi => psi" to
345 "~ phi | psi" and "phi <=> psi" to "~ phi | psi" and "~ psi | phi". We don't
346 attempt to distribute conjunctions over disjunctions. *)
347 fun clausify_formula pos (phi as AAtom _) = [phi |> not pos ? mk_anot]
348 | clausify_formula pos (AConn (ANot, [phi])) = clausify_formula (not pos) phi
349 | clausify_formula true (AConn (AOr, [phi1, phi2])) =
350 (phi1, phi2) |> pairself (clausify_formula true)
351 |> uncurry (map_product (mk_aconn AOr))
352 | clausify_formula false (AConn (AAnd, [phi1, phi2])) =
353 (phi1, phi2) |> pairself (clausify_formula false)
354 |> uncurry (map_product (mk_aconn AOr))
355 | clausify_formula true (AConn (AImplies, [phi1, phi2])) =
356 clausify_formula true (AConn (AOr, [mk_anot phi1, phi2]))
357 | clausify_formula true (AConn (AIff, phis)) =
358 clausify_formula true (AConn (AImplies, phis)) @
359 clausify_formula true (AConn (AImplies, rev phis))
360 | clausify_formula _ _ = raise CLAUSIFY ()
362 fun clausify_formula_line (Formula (ident, kind, phi, source, info)) =
364 val (n, phis) = phi |> try (clausify_formula true) |> these |> `length
366 map2 (fn phi => fn j =>
367 Formula (ident ^ replicate_string (j - 1) "x", kind, phi, source,
371 | clausify_formula_line _ = []
373 fun ensure_cnf_problem_line line =
374 line |> open_formula_line |> negate_conjecture_line |> clausify_formula_line
376 fun ensure_cnf_problem problem =
377 problem |> map (apsnd (maps ensure_cnf_problem_line))
379 fun filter_cnf_ueq_problem problem =
381 |> map (apsnd (map open_formula_line
382 #> filter is_problem_line_cnf_ueq
383 #> map negate_conjecture_line))
386 val conjs = problem |> maps snd |> filter is_problem_line_negated
387 in if length conjs = 1 then problem else [] end)
390 (** Symbol declarations **)
392 (* TFF allows implicit declarations of types, function symbols, and predicate
393 symbols (with "$i" as the type of individuals), but some provers (e.g.,
394 SNARK) require explicit declarations. The situation is similar for THF. *)
396 val atype_of_types = AType (`I tptp_type_of_types)
397 val bool_atype = AType (`I tptp_bool_type)
398 val individual_atype = AType (`I tptp_individual_type)
400 fun default_type pred_sym =
402 fun typ 0 = if pred_sym then bool_atype else individual_atype
403 | typ ary = AFun (individual_atype, typ (ary - 1))
406 fun add_declared_syms_in_problem_line (Decl (_, sym, _)) = insert (op =) sym
407 | add_declared_syms_in_problem_line _ = I
408 fun declared_syms_in_problem problem =
409 fold (fold add_declared_syms_in_problem_line o snd) problem []
411 fun undeclared_syms_in_problem declared problem =
414 if member (op =) declared name then I else AList.default (op =) (name, ty)
415 fun do_type (AFun (ty1, ty2)) = fold do_type [ty1, ty2]
416 | do_type (AType name) = do_sym name (K atype_of_types)
417 fun do_term pred_sym (ATerm (name as (s, _), tms)) =
418 is_tptp_user_symbol s
419 ? do_sym name (fn _ => default_type pred_sym (length tms))
420 #> fold (do_term false) tms
421 | do_term _ (AAbs ((_, ty), tm)) = do_type ty #> do_term false tm
422 fun do_formula (AQuant (_, xs, phi)) =
423 fold do_type (map_filter snd xs) #> do_formula phi
424 | do_formula (AConn (_, phis)) = fold do_formula phis
425 | do_formula (AAtom tm) = do_term true tm
426 fun do_problem_line (Decl (_, _, ty)) = do_type ty
427 | do_problem_line (Formula (_, _, phi, _, _)) = do_formula phi
429 fold (fold do_problem_line o snd) problem []
430 |> filter_out (is_built_in_tptp_symbol o fst o fst)
433 fun declare_undeclared_syms_in_atp_problem prefix heading problem =
435 fun decl_line (x as (s, _), ty) = Decl (prefix ^ s, x, ty ())
436 val declared = problem |> declared_syms_in_problem
438 problem |> undeclared_syms_in_problem declared
439 |> sort_wrt (fst o fst)
441 in (heading, decls) :: problem end
445 fun empty_name_pool readable_names =
446 if readable_names then SOME (Symtab.empty, Symtab.empty) else NONE
448 fun pool_fold f xs z = pair z #> fold_rev (fn x => uncurry (f x)) xs
450 pool_fold (fn x => fn ys => fn pool => f x pool |>> (fn y => y :: ys)) xs []
455 | skip (#"." :: cs) = skip cs
456 | skip (c :: cs) = if Char.isAlphaNum c then skip cs else c :: keep cs
458 | keep (#"." :: cs) = skip cs
459 | keep (c :: cs) = c :: keep cs
460 in String.explode #> rev #> keep #> rev #> String.implode end
462 (* Long names can slow down the ATPs. *)
463 val max_readable_name_size = 20
465 (* "equal" is reserved by some ATPs. "op" is also reserved, to avoid the
466 unreadable "op_1", "op_2", etc., in the problem files. "eq" is reserved to
467 ensure that "HOL.eq" is correctly mapped to equality (not clear whether this
468 is still necessary). *)
469 val reserved_nice_names = [tptp_old_equal, "op", "eq"]
471 fun readable_name full_name s =
472 if s = full_name then
476 |> Name.desymbolize (Char.isUpper (String.sub (full_name, 0)))
478 if size s > max_readable_name_size then
479 String.substring (s, 0, max_readable_name_size div 2 - 4) ^
480 Word.toString (hashw_string (full_name, 0w0)) ^
481 String.extract (s, size s - max_readable_name_size div 2 + 4,
485 |> (fn s => if member (op =) reserved_nice_names s then full_name else s)
487 fun nice_name (full_name, _) NONE = (full_name, NONE)
488 | nice_name (full_name, desired_name) (SOME the_pool) =
489 if is_built_in_tptp_symbol full_name then
490 (full_name, SOME the_pool)
491 else case Symtab.lookup (fst the_pool) full_name of
492 SOME nice_name => (nice_name, SOME the_pool)
495 val nice_prefix = readable_name full_name desired_name
499 nice_prefix ^ (if j = 0 then "" else "_" ^ string_of_int j)
501 case Symtab.lookup (snd the_pool) nice_name of
503 if full_name = full_name' then (nice_name, the_pool)
507 (Symtab.update_new (full_name, nice_name) (fst the_pool),
508 Symtab.update_new (nice_name, full_name) (snd the_pool)))
510 in add 0 |> apsnd SOME end
512 fun nice_type (AType name) = nice_name name #>> AType
513 | nice_type (AFun (ty1, ty2)) = nice_type ty1 ##>> nice_type ty2 #>> AFun
514 fun nice_term (ATerm (name, ts)) =
515 nice_name name ##>> pool_map nice_term ts #>> ATerm
516 | nice_term (AAbs ((name, ty), tm)) =
517 nice_name name ##>> nice_type ty ##>> nice_term tm #>> AAbs
518 fun nice_formula (AQuant (q, xs, phi)) =
519 pool_map nice_name (map fst xs)
520 ##>> pool_map (fn NONE => pair NONE
521 | SOME ty => nice_type ty #>> SOME) (map snd xs)
522 ##>> nice_formula phi
523 #>> (fn ((ss, ts), phi) => AQuant (q, ss ~~ ts, phi))
524 | nice_formula (AConn (c, phis)) =
525 pool_map nice_formula phis #>> curry AConn c
526 | nice_formula (AAtom tm) = nice_term tm #>> AAtom
527 fun nice_problem_line (Decl (ident, sym, ty)) =
528 nice_name sym ##>> nice_type ty #>> (fn (sym, ty) => Decl (ident, sym, ty))
529 | nice_problem_line (Formula (ident, kind, phi, source, info)) =
530 nice_formula phi #>> (fn phi => Formula (ident, kind, phi, source, info))
531 fun nice_problem problem =
532 pool_map (fn (heading, lines) =>
533 pool_map nice_problem_line lines #>> pair heading) problem
534 fun nice_atp_problem readable_names problem =
535 nice_problem problem (empty_name_pool readable_names)