(* Title: Pure/search.ML
ID: $Id$
Author: Lawrence C Paulson and Norbert Voelker
Search tacticals.
*)
infix 1 THEN_MAYBE THEN_MAYBE';
signature SEARCH =
sig
val DEEPEN : int*int -> (int->int->tactic) -> int -> int -> tactic
val THEN_MAYBE : tactic * tactic -> tactic
val THEN_MAYBE' : ('a -> tactic) * ('a -> tactic) -> ('a -> tactic)
val trace_DEPTH_FIRST : bool ref
val DEPTH_FIRST : (thm -> bool) -> tactic -> tactic
val DEPTH_SOLVE : tactic -> tactic
val DEPTH_SOLVE_1 : tactic -> tactic
val ITER_DEEPEN : (thm->bool) -> (int->tactic) -> tactic
val THEN_ITER_DEEPEN : tactic -> (thm->bool) -> (int->tactic) -> tactic
val has_fewer_prems : int -> thm -> bool
val IF_UNSOLVED : tactic -> tactic
val SOLVE : tactic -> tactic
val DETERM_UNTIL_SOLVED: tactic -> tactic
val trace_BEST_FIRST : bool ref
val BEST_FIRST : (thm -> bool) * (thm -> int) -> tactic -> tactic
val THEN_BEST_FIRST : tactic -> (thm->bool) * (thm->int) -> tactic
-> tactic
val trace_ASTAR : bool ref
val ASTAR : (thm -> bool) * (int->thm->int) -> tactic -> tactic
val THEN_ASTAR : tactic -> (thm->bool) * (int->thm->int) -> tactic
-> tactic
val BREADTH_FIRST : (thm -> bool) -> tactic -> tactic
val QUIET_BREADTH_FIRST : (thm -> bool) -> tactic -> tactic
end;
(** Instantiation of heaps for best-first search **)
(*total ordering on theorems, allowing duplicates to be found*)
structure ThmHeap =
HeapFun (type elem = int * thm
val ord = Library.prod_ord Library.int_ord
(Term.term_ord o Library.pairself (#prop o Thm.rep_thm)));
structure Search : SEARCH =
struct
(**** Depth-first search ****)
val trace_DEPTH_FIRST = ref false;
(*Searches until "satp" reports proof tree as satisfied.
Suppresses duplicate solutions to minimize search space.*)
fun DEPTH_FIRST satp tac =
let val tac = tracify trace_DEPTH_FIRST tac
fun depth used [] = NONE
| depth used (q::qs) =
case Seq.pull q of
NONE => depth used qs
| SOME(st,stq) =>
if satp st andalso not (gen_mem eq_thm (st, used))
then SOME(st, Seq.make
(fn()=> depth (st::used) (stq::qs)))
else depth used (tac st :: stq :: qs)
in traced_tac (fn st => depth [] ([Seq.single st])) end;
(*Predicate: Does the rule have fewer than n premises?*)
fun has_fewer_prems n rule = (nprems_of rule < n);
(*Apply a tactic if subgoals remain, else do nothing.*)
val IF_UNSOLVED = COND (has_fewer_prems 1) all_tac;
(*Force a tactic to solve its goal completely, otherwise fail *)
fun SOLVE tac = tac THEN COND (has_fewer_prems 1) all_tac no_tac;
(*Force repeated application of tactic until goal is solved completely *)
val DETERM_UNTIL_SOLVED = DETERM_UNTIL (has_fewer_prems 1);
(*Execute tac1, but only execute tac2 if there are at least as many subgoals
as before. This ensures that tac2 is only applied to an outcome of tac1.*)
fun (tac1 THEN_MAYBE tac2) st =
(tac1 THEN COND (has_fewer_prems (nprems_of st)) all_tac tac2) st;
fun (tac1 THEN_MAYBE' tac2) x = tac1 x THEN_MAYBE tac2 x;
(*Tactical to reduce the number of premises by 1.
If no subgoals then it must fail! *)
fun DEPTH_SOLVE_1 tac st = st |>
(case nprems_of st of
0 => no_tac
| n => DEPTH_FIRST (has_fewer_prems n) tac);
(*Uses depth-first search to solve ALL subgoals*)
val DEPTH_SOLVE = DEPTH_FIRST (has_fewer_prems 1);
(**** Iterative deepening with pruning ****)
fun has_vars (Var _) = true
| has_vars (Abs (_,_,t)) = has_vars t
| has_vars (f$t) = has_vars f orelse has_vars t
| has_vars _ = false;
(*Counting of primitive inferences is APPROXIMATE, as the step tactic
may perform >1 inference*)
(*Pruning of rigid ancestor to prevent backtracking*)
fun prune (new as (k', np':int, rgd', stq), qs) =
let fun prune_aux (qs, []) = new::qs
| prune_aux (qs, (k,np,rgd,q)::rqs) =
if np'+1 = np andalso rgd then
(if !trace_DEPTH_FIRST then
tracing ("Pruning " ^
string_of_int (1+length rqs) ^ " levels")
else ();
(*Use OLD k: zero-cost solution; see Stickel, p 365*)
(k, np', rgd', stq) :: qs)
else prune_aux ((k,np,rgd,q)::qs, rqs)
fun take ([], rqs) = ([], rqs)
| take (arg as ((k,np,rgd,stq)::qs, rqs)) =
if np' < np then take (qs, (k,np,rgd,stq)::rqs)
else arg
in prune_aux (take (qs, [])) end;
(*Depth-first iterative deepening search for a state that satisfies satp
tactic tac0 sets up the initial goal queue, while tac1 searches it.
The solution sequence is redundant: the cutoff heuristic makes it impossible
to suppress solutions arising from earlier searches, as the accumulated cost
(k) can be wrong.*)
fun THEN_ITER_DEEPEN tac0 satp tac1 = traced_tac (fn st =>
let val countr = ref 0
and tf = tracify trace_DEPTH_FIRST (tac1 1)
and qs0 = tac0 st
(*bnd = depth bound; inc = estimate of increment required next*)
fun depth (bnd,inc) [] =
(tracing (string_of_int (!countr) ^
" inferences so far. Searching to depth " ^
string_of_int bnd);
(*larger increments make it run slower for the hard problems*)
depth (bnd+inc, 10)) [(0, 1, false, qs0)]
| depth (bnd,inc) ((k,np,rgd,q)::qs) =
if k>=bnd then depth (bnd,inc) qs
else
case (countr := !countr+1;
if !trace_DEPTH_FIRST then
tracing (string_of_int np ^
implode (map (fn _ => "*") qs))
else ();
Seq.pull q) of
NONE => depth (bnd,inc) qs
| SOME(st,stq) =>
if satp st (*solution!*)
then SOME(st, Seq.make
(fn()=> depth (bnd,inc) ((k,np,rgd,stq)::qs)))
else
let val np' = nprems_of st
(*rgd' calculation assumes tactic operates on subgoal 1*)
val rgd' = not (has_vars (hd (prems_of st)))
val k' = k+np'-np+1 (*difference in # of subgoals, +1*)
in if k'+np' >= bnd
then depth (bnd, Int.min(inc, k'+np'+1-bnd)) qs
else if np' < np (*solved a subgoal; prune rigid ancestors*)
then depth (bnd,inc)
(prune ((k', np', rgd', tf st), (k,np,rgd,stq) :: qs))
else depth (bnd,inc) ((k', np', rgd', tf st) ::
(k,np,rgd,stq) :: qs)
end
in depth (0,5) [] end);
val ITER_DEEPEN = THEN_ITER_DEEPEN all_tac;
(*Simple iterative deepening tactical. It merely "deepens" any search tactic
using increment "inc" up to limit "lim". *)
fun DEEPEN (inc,lim) tacf m i =
let fun dpn m st =
st |> (if has_fewer_prems i st then no_tac
else if m>lim then
(warning "Search depth limit exceeded: giving up";
no_tac)
else (warning ("Search depth = " ^ string_of_int m);
tacf m i ORELSE dpn (m+inc)))
in dpn m end;
(*** Best-first search ***)
val trace_BEST_FIRST = ref false;
(*For creating output sequence*)
fun some_of_list [] = NONE
| some_of_list (x::l) = SOME (x, Seq.make (fn () => some_of_list l));
(*Check for and delete duplicate proof states*)
fun deleteAllMin prf heap =
if ThmHeap.is_empty heap then heap
else if eq_thm (prf, #2 (ThmHeap.min heap))
then deleteAllMin prf (ThmHeap.delete_min heap)
else heap;
(*Best-first search for a state that satisfies satp (incl initial state)
Function sizef estimates size of problem remaining (smaller means better).
tactic tac0 sets up the initial priority queue, while tac1 searches it. *)
fun THEN_BEST_FIRST tac0 (satp, sizef) tac1 =
let val tac = tracify trace_BEST_FIRST tac1
fun pairsize th = (sizef th, th);
fun bfs (news,nprf_heap) =
(case List.partition satp news of
([],nonsats) => next(foldr ThmHeap.insert
nprf_heap (map pairsize nonsats))
| (sats,_) => some_of_list sats)
and next nprf_heap =
if ThmHeap.is_empty nprf_heap then NONE
else
let val (n,prf) = ThmHeap.min nprf_heap
in if !trace_BEST_FIRST
then tracing("state size = " ^ string_of_int n)
else ();
bfs (Seq.list_of (tac prf),
deleteAllMin prf (ThmHeap.delete_min nprf_heap))
end
fun btac st = bfs (Seq.list_of (tac0 st), ThmHeap.empty)
in traced_tac btac end;
(*Ordinary best-first search, with no initial tactic*)
val BEST_FIRST = THEN_BEST_FIRST all_tac;
(*Breadth-first search to satisfy satpred (including initial state)
SLOW -- SHOULD NOT USE APPEND!*)
fun gen_BREADTH_FIRST message satpred (tac:tactic) =
let val tacf = Seq.list_of o tac;
fun bfs prfs =
(case List.partition satpred prfs of
([],[]) => []
| ([],nonsats) =>
(message("breadth=" ^ string_of_int(length nonsats));
bfs (List.concat (map tacf nonsats)))
| (sats,_) => sats)
in (fn st => Seq.of_list (bfs [st])) end;
val BREADTH_FIRST = gen_BREADTH_FIRST tracing;
val QUIET_BREADTH_FIRST = gen_BREADTH_FIRST (K ());
(* Author: Norbert Voelker, FernUniversitaet Hagen
Remarks: Implementation of A*-like proof procedure by modification
of the existing code for BEST_FIRST and best_tac so that the
current level of search is taken into account.
*)
(*Insertion into priority queue of states, marked with level *)
fun insert_with_level (lnth: int*int*thm, []) = [lnth]
| insert_with_level ((l,m,th), (l',n,th')::nths) =
if n<m then (l',n,th') :: insert_with_level ((l,m,th), nths)
else if n=m andalso eq_thm(th,th')
then (l',n,th')::nths
else (l,m,th)::(l',n,th')::nths;
(*For creating output sequence*)
fun some_of_list [] = NONE
| some_of_list (x::l) = SOME (x, Seq.make (fn () => some_of_list l));
val trace_ASTAR = ref false;
fun THEN_ASTAR tac0 (satp, costf) tac1 =
let val tf = tracify trace_ASTAR tac1;
fun bfs (news,nprfs,level) =
let fun cost thm = (level, costf level thm, thm)
in (case List.partition satp news of
([],nonsats)
=> next (foldr insert_with_level nprfs (map cost nonsats))
| (sats,_) => some_of_list sats)
end and
next [] = NONE
| next ((level,n,prf)::nprfs) =
(if !trace_ASTAR
then tracing("level = " ^ string_of_int level ^
" cost = " ^ string_of_int n ^
" queue length =" ^ string_of_int (length nprfs))
else ();
bfs (Seq.list_of (tf prf), nprfs,level+1))
fun tf st = bfs (Seq.list_of (tac0 st), [], 0)
in traced_tac tf end;
(*Ordinary ASTAR, with no initial tactic*)
val ASTAR = THEN_ASTAR all_tac;
end;
open Search;