author | paulson |
Tue, 20 Jun 2000 11:54:07 +0200 | |
changeset 9094 | 530e7a33b3dd |
parent 8149 | 941afb897532 |
child 9411 | 0d5a171db2f0 |
permissions | -rw-r--r-- |
1588 | 1 |
(* Title: search |
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ID: $Id$ |
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Author: Lawrence C Paulson and Norbert Voelker |
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Search tacticals |
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*) |
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infix 1 THEN_MAYBE THEN_MAYBE'; |
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signature SEARCH = |
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sig |
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val DEEPEN : int*int -> (int->int->tactic) -> int -> int -> tactic |
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val THEN_MAYBE : tactic * tactic -> tactic |
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val THEN_MAYBE' : ('a -> tactic) * ('a -> tactic) -> ('a -> tactic) |
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val trace_DEPTH_FIRST : bool ref |
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val DEPTH_FIRST : (thm -> bool) -> tactic -> tactic |
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val DEPTH_SOLVE : tactic -> tactic |
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val DEPTH_SOLVE_1 : tactic -> tactic |
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val ITER_DEEPEN : (thm->bool) -> (int->tactic) -> tactic |
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val THEN_ITER_DEEPEN : tactic -> (thm->bool) -> (int->tactic) -> tactic |
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val has_fewer_prems : int -> thm -> bool |
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val IF_UNSOLVED : tactic -> tactic |
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val SOLVE : tactic -> tactic |
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val DETERM_UNTIL_SOLVED: tactic -> tactic |
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val trace_BEST_FIRST : bool ref |
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val BEST_FIRST : (thm -> bool) * (thm -> int) -> tactic -> tactic |
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val THEN_BEST_FIRST : tactic -> (thm->bool) * (thm->int) -> tactic |
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-> tactic |
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val trace_ASTAR : bool ref |
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val ASTAR : (thm -> bool) * (int->thm->int) -> tactic -> tactic |
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val THEN_ASTAR : tactic -> (thm->bool) * (int->thm->int) -> tactic |
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-> tactic |
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val BREADTH_FIRST : (thm -> bool) -> tactic -> tactic |
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val QUIET_BREADTH_FIRST : (thm -> bool) -> tactic -> tactic |
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end; |
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(** Instantiation of leftist heaps for best-first search **) |
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(*Termless makes the ordering total, allowing duplicates to be found*) |
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structure MeasureThm : ORDERED = |
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struct |
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type T = int * thm |
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fun eq ((m,th):T, (m',th')) = (m=m' andalso eq_thm(th,th')) |
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fun lt ((m,th):T, (m',th')) = |
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(m<m' orelse |
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m=m' andalso Term.termless (#prop(rep_thm th), #prop(rep_thm th'))) |
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fun leq (mth, mth') = not (lt (mth',mth)) |
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end; |
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structure MeasureThmHeap = LeftistHeap(MeasureThm); |
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structure Search : SEARCH = |
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struct |
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(**** Depth-first search ****) |
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val trace_DEPTH_FIRST = ref false; |
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(*Searches until "satp" reports proof tree as satisfied. |
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Suppresses duplicate solutions to minimize search space.*) |
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fun DEPTH_FIRST satp tac = |
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let val tac = tracify trace_DEPTH_FIRST tac |
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fun depth used [] = None |
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| depth used (q::qs) = |
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case Seq.pull q of |
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None => depth used qs |
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| Some(st,stq) => |
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if satp st andalso not (gen_mem eq_thm (st, used)) |
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then Some(st, Seq.make |
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(fn()=> depth (st::used) (stq::qs))) |
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else depth used (tac st :: stq :: qs) |
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in traced_tac (fn st => depth [] ([Seq.single st])) end; |
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(*Predicate: Does the rule have fewer than n premises?*) |
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fun has_fewer_prems n rule = (nprems_of rule < n); |
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(*Apply a tactic if subgoals remain, else do nothing.*) |
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val IF_UNSOLVED = COND (has_fewer_prems 1) all_tac; |
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(*Force a tactic to solve its goal completely, otherwise fail *) |
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fun SOLVE tac = tac THEN COND (has_fewer_prems 1) all_tac no_tac; |
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(*Force repeated application of tactic until goal is solved completely *) |
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val DETERM_UNTIL_SOLVED = DETERM_UNTIL (has_fewer_prems 1); |
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(*Execute tac1, but only execute tac2 if there are at least as many subgoals |
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as before. This ensures that tac2 is only applied to an outcome of tac1.*) |
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fun (tac1 THEN_MAYBE tac2) st = |
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(tac1 THEN COND (has_fewer_prems (nprems_of st)) all_tac tac2) st; |
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fun (tac1 THEN_MAYBE' tac2) x = tac1 x THEN_MAYBE tac2 x; |
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(*Tactical to reduce the number of premises by 1. |
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If no subgoals then it must fail! *) |
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fun DEPTH_SOLVE_1 tac st = st |> |
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(case nprems_of st of |
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0 => no_tac |
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| n => DEPTH_FIRST (has_fewer_prems n) tac); |
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(*Uses depth-first search to solve ALL subgoals*) |
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val DEPTH_SOLVE = DEPTH_FIRST (has_fewer_prems 1); |
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(**** Iterative deepening with pruning ****) |
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fun has_vars (Var _) = true |
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| has_vars (Abs (_,_,t)) = has_vars t |
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| has_vars (f$t) = has_vars f orelse has_vars t |
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| has_vars _ = false; |
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(*Counting of primitive inferences is APPROXIMATE, as the step tactic |
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may perform >1 inference*) |
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(*Pruning of rigid ancestor to prevent backtracking*) |
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fun prune (new as (k', np':int, rgd', stq), qs) = |
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let fun prune_aux (qs, []) = new::qs |
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| prune_aux (qs, (k,np,rgd,q)::rqs) = |
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if np'+1 = np andalso rgd then |
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(if !trace_DEPTH_FIRST then |
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writeln ("Pruning " ^ |
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string_of_int (1+length rqs) ^ " levels") |
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else (); |
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(*Use OLD k: zero-cost solution; see Stickel, p 365*) |
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(k, np', rgd', stq) :: qs) |
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else prune_aux ((k,np,rgd,q)::qs, rqs) |
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fun take ([], rqs) = ([], rqs) |
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| take (arg as ((k,np,rgd,stq)::qs, rqs)) = |
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if np' < np then take (qs, (k,np,rgd,stq)::rqs) |
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else arg |
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in prune_aux (take (qs, [])) end; |
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(*Depth-first iterative deepening search for a state that satisfies satp |
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tactic tac0 sets up the initial goal queue, while tac1 searches it. |
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The solution sequence is redundant: the cutoff heuristic makes it impossible |
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to suppress solutions arising from earlier searches, as the accumulated cost |
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(k) can be wrong.*) |
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fun THEN_ITER_DEEPEN tac0 satp tac1 = traced_tac (fn st => |
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let val countr = ref 0 |
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and tf = tracify trace_DEPTH_FIRST (tac1 1) |
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and qs0 = tac0 st |
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(*bnd = depth bound; inc = estimate of increment required next*) |
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fun depth (bnd,inc) [] = |
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(writeln (string_of_int (!countr) ^ |
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" inferences so far. Searching to depth " ^ |
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string_of_int bnd); |
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(*larger increments make it run slower for the hard problems*) |
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depth (bnd+inc, 10)) [(0, 1, false, qs0)] |
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| depth (bnd,inc) ((k,np,rgd,q)::qs) = |
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if k>=bnd then depth (bnd,inc) qs |
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else |
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case (countr := !countr+1; |
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if !trace_DEPTH_FIRST then |
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writeln (string_of_int np ^ |
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implode (map (fn _ => "*") qs)) |
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else (); |
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Seq.pull q) of |
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None => depth (bnd,inc) qs |
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| Some(st,stq) => |
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if satp st (*solution!*) |
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then Some(st, Seq.make |
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(fn()=> depth (bnd,inc) ((k,np,rgd,stq)::qs))) |
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else |
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let val np' = nprems_of st |
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(*rgd' calculation assumes tactic operates on subgoal 1*) |
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val rgd' = not (has_vars (hd (prems_of st))) |
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val k' = k+np'-np+1 (*difference in # of subgoals, +1*) |
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in if k'+np' >= bnd |
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then depth (bnd, Int.min(inc, k'+np'+1-bnd)) qs |
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else if np' < np (*solved a subgoal; prune rigid ancestors*) |
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then depth (bnd,inc) |
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(prune ((k', np', rgd', tf st), (k,np,rgd,stq) :: qs)) |
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else depth (bnd,inc) ((k', np', rgd', tf st) :: |
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(k,np,rgd,stq) :: qs) |
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end |
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in depth (0,5) [] end); |
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val ITER_DEEPEN = THEN_ITER_DEEPEN all_tac; |
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(*Simple iterative deepening tactical. It merely "deepens" any search tactic |
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using increment "inc" up to limit "lim". *) |
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fun DEEPEN (inc,lim) tacf m i = |
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let fun dpn m st = st |> (if has_fewer_prems i st then no_tac |
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else if m>lim then |
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(writeln "Giving up..."; no_tac) |
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else (writeln ("Depth = " ^ string_of_int m); |
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tacf m i ORELSE dpn (m+inc))) |
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in dpn m end; |
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(*** Best-first search ***) |
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val trace_BEST_FIRST = ref false; |
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(*For creating output sequence*) |
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fun some_of_list [] = None |
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| some_of_list (x::l) = Some (x, Seq.make (fn () => some_of_list l)); |
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(*Check for and delete duplicate proof states*) |
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fun deleteAllMin prf heap = |
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if MeasureThmHeap.isEmpty heap then heap |
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else if eq_thm (prf, #2 (MeasureThmHeap.findMin heap)) |
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then deleteAllMin prf (MeasureThmHeap.deleteMin heap) |
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else heap; |
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(*Best-first search for a state that satisfies satp (incl initial state) |
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Function sizef estimates size of problem remaining (smaller means better). |
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tactic tac0 sets up the initial priority queue, while tac1 searches it. *) |
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fun THEN_BEST_FIRST tac0 (satp, sizef) tac1 = |
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let val tac = tracify trace_BEST_FIRST tac1 |
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fun pairsize th = (sizef th, th); |
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fun bfs (news,nprf_heap) = |
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(case partition satp news of |
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([],nonsats) => next(foldr MeasureThmHeap.insert |
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(map pairsize nonsats, nprf_heap)) |
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| (sats,_) => some_of_list sats) |
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and next nprf_heap = |
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if MeasureThmHeap.isEmpty nprf_heap then None |
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else |
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let val (n,prf) = MeasureThmHeap.findMin nprf_heap |
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in if !trace_BEST_FIRST |
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then writeln("state size = " ^ string_of_int n) |
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else (); |
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bfs (Seq.list_of (tac prf), |
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deleteAllMin prf (MeasureThmHeap.deleteMin nprf_heap)) |
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end |
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fun btac st = bfs (Seq.list_of (tac0 st), MeasureThmHeap.empty) |
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in traced_tac btac end; |
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(*Ordinary best-first search, with no initial tactic*) |
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val BEST_FIRST = THEN_BEST_FIRST all_tac; |
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(*Breadth-first search to satisfy satpred (including initial state) |
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SLOW -- SHOULD NOT USE APPEND!*) |
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fun gen_BREADTH_FIRST message satpred (tac:tactic) = |
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let val tacf = Seq.list_of o tac; |
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fun bfs prfs = |
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(case partition satpred prfs of |
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([],[]) => [] |
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| ([],nonsats) => |
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(message("breadth=" ^ string_of_int(length nonsats)); |
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bfs (List.concat (map tacf nonsats))) |
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| (sats,_) => sats) |
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in (fn st => Seq.of_list (bfs [st])) end; |
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val BREADTH_FIRST = gen_BREADTH_FIRST writeln; |
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val QUIET_BREADTH_FIRST = gen_BREADTH_FIRST (K ()); |
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(* Author: Norbert Voelker, FernUniversitaet Hagen |
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Remarks: Implementation of A*-like proof procedure by modification |
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of the existing code for BEST_FIRST and best_tac so that the |
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current level of search is taken into account. |
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*) |
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(*Insertion into priority queue of states, marked with level *) |
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fun insert_with_level (lnth: int*int*thm, []) = [lnth] |
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| insert_with_level ((l,m,th), (l',n,th')::nths) = |
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if n<m then (l',n,th') :: insert_with_level ((l,m,th), nths) |
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else if n=m andalso eq_thm(th,th') |
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then (l',n,th')::nths |
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else (l,m,th)::(l',n,th')::nths; |
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(*For creating output sequence*) |
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fun some_of_list [] = None |
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| some_of_list (x::l) = Some (x, Seq.make (fn () => some_of_list l)); |
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val trace_ASTAR = ref false; |
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fun THEN_ASTAR tac0 (satp, costf) tac1 = |
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let val tf = tracify trace_ASTAR tac1; |
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fun bfs (news,nprfs,level) = |
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let fun cost thm = (level, costf level thm, thm) |
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in (case partition satp news of |
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([],nonsats) |
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=> next (foldr insert_with_level (map cost nonsats, nprfs)) |
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| (sats,_) => some_of_list sats) |
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end and |
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next [] = None |
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| next ((level,n,prf)::nprfs) = |
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(if !trace_ASTAR |
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then writeln("level = " ^ string_of_int level ^ |
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" cost = " ^ string_of_int n ^ |
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" queue length =" ^ string_of_int (length nprfs)) |
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else (); |
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bfs (Seq.list_of (tf prf), nprfs,level+1)) |
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fun tf st = bfs (Seq.list_of (tac0 st), [], 0) |
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in traced_tac tf end; |
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(*Ordinary ASTAR, with no initial tactic*) |
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val ASTAR = THEN_ASTAR all_tac; |
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end; |
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open Search; |