src/Pure/search.ML

tuned;

(*  Title: 	search
    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 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
  end;

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;

(*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
		       writeln ("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) [] = 
	     (writeln (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
		    writeln (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 
				(writeln "Giving up..."; no_tac)
				 else (writeln ("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;

(*Insertion into priority queue of states *)
fun insert (nth: int*thm, []) = [nth]
  | insert ((m,th), (n,th')::nths) = 
      if  n<m then (n,th') :: insert ((m,th), nths)
      else if  n=m andalso eq_thm(th,th')
              then (n,th')::nths
              else (m,th)::(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));


(*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,nprfs) =
	   (case  partition satp news  of
		([],nonsats) => next(foldr insert
					(map pairsize nonsats, nprfs)) 
	      | (sats,_)  => some_of_list sats)
      and next [] = None
        | next ((n,prf)::nprfs) =
	    (if !trace_BEST_FIRST 
	       then writeln("state size = " ^ string_of_int n ^ 
		         "  queue length =" ^ string_of_int (length nprfs))  
               else ();
	     bfs (Seq.list_of (tac prf), nprfs))
      fun btac st = bfs (Seq.list_of (tac0 st),  [])
  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 BREADTH_FIRST satpred tac = 
  let val tacf = Seq.list_of o tac;
      fun bfs prfs =
	 (case  partition satpred prfs  of
	      ([],[]) => []
	    | ([],nonsats) => 
		  (prs("breadth=" ^ string_of_int(length nonsats) ^ "\n");
		   bfs (List.concat (map tacf nonsats)))
	    | (sats,_)  => sats)
  in (fn st => Seq.of_list (bfs [st])) end;


(*  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  partition satp news  of
            ([],nonsats) 
		 => next (foldr insert_with_level (map cost nonsats, nprfs))
          | (sats,_)  => some_of_list sats)
      end and    
      next []  = None
        | next ((level,n,prf)::nprfs)  =
            (if !trace_ASTAR 
               then writeln("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;