paulson@1588: (* Title: search paulson@1588: ID: $Id$ paulson@1588: Author: Lawrence C Paulson and Norbert Voelker paulson@1588: paulson@1588: Search tacticals paulson@1588: *) paulson@1588: paulson@1588: signature SEARCH = paulson@1588: sig paulson@1588: val trace_DEPTH_FIRST : bool ref paulson@1588: val DEPTH_FIRST : (thm -> bool) -> tactic -> tactic paulson@1588: val DEPTH_SOLVE : tactic -> tactic paulson@1588: val DEPTH_SOLVE_1 : tactic -> tactic paulson@1588: val ITER_DEEPEN : (thm->bool) -> (int->tactic) -> tactic paulson@1588: val THEN_ITER_DEEPEN : tactic -> (thm->bool) -> (int->tactic) -> tactic paulson@1588: paulson@1588: val has_fewer_prems : int -> thm -> bool paulson@1588: val IF_UNSOLVED : tactic -> tactic paulson@1588: val trace_BEST_FIRST : bool ref paulson@1588: val BEST_FIRST : (thm -> bool) * (thm -> int) -> tactic -> tactic paulson@1588: val THEN_BEST_FIRST : tactic -> (thm->bool) * (thm->int) -> tactic paulson@1588: -> tactic paulson@1588: val trace_ASTAR : bool ref paulson@1588: val ASTAR : (thm -> bool) * (int->thm->int) -> tactic -> tactic paulson@1588: val THEN_ASTAR : tactic -> (thm->bool) * (int->thm->int) -> tactic paulson@1588: -> tactic paulson@1588: val BREADTH_FIRST : (thm -> bool) -> tactic -> tactic paulson@1588: end; paulson@1588: paulson@1588: structure Search : SEARCH = paulson@1588: struct paulson@1588: paulson@1588: (**** Depth-first search ****) paulson@1588: paulson@1588: val trace_DEPTH_FIRST = ref false; paulson@1588: paulson@1588: (*Searches until "satp" reports proof tree as satisfied. paulson@1588: Suppresses duplicate solutions to minimize search space.*) paulson@1588: fun DEPTH_FIRST satp tac = paulson@1588: let val tac = tracify trace_DEPTH_FIRST tac paulson@1588: fun depth used [] = None paulson@1588: | depth used (q::qs) = paulson@1588: case Sequence.pull q of paulson@1588: None => depth used qs paulson@1588: | Some(st,stq) => paulson@1588: if satp st andalso not (gen_mem eq_thm (st, used)) paulson@1588: then Some(st, Sequence.seqof paulson@1588: (fn()=> depth (st::used) (stq::qs))) paulson@1588: else depth used (tac st :: stq :: qs) paulson@1588: in traced_tac (fn st => depth [] ([Sequence.single st])) end; paulson@1588: paulson@1588: paulson@1588: paulson@1588: (*Predicate: Does the rule have fewer than n premises?*) paulson@1588: fun has_fewer_prems n rule = (nprems_of rule < n); paulson@1588: paulson@1588: (*Apply a tactic if subgoals remain, else do nothing.*) paulson@1588: val IF_UNSOLVED = COND (has_fewer_prems 1) all_tac; paulson@1588: paulson@1588: (*Tactical to reduce the number of premises by 1. paulson@1588: If no subgoals then it must fail! *) paulson@1588: fun DEPTH_SOLVE_1 tac = STATE paulson@1588: (fn st => paulson@1588: (case nprems_of st of paulson@1588: 0 => no_tac paulson@1588: | n => DEPTH_FIRST (has_fewer_prems n) tac)); paulson@1588: paulson@1588: (*Uses depth-first search to solve ALL subgoals*) paulson@1588: val DEPTH_SOLVE = DEPTH_FIRST (has_fewer_prems 1); paulson@1588: paulson@1588: paulson@1588: paulson@1588: (**** Iterative deepening ****) paulson@1588: paulson@1588: fun has_vars (Var _) = true paulson@1588: | has_vars (Abs (_,_,t)) = has_vars t paulson@1588: | has_vars (f$t) = has_vars f orelse has_vars t paulson@1588: | has_vars _ = false; paulson@1588: paulson@1588: (*Counting of primitive inferences is APPROXIMATE, as the step tactic paulson@1588: may perform >1 inference*) paulson@1588: paulson@1588: (*Pruning of rigid ancestor to prevent backtracking*) paulson@1588: fun prune (new as (k', np':int, rgd', stq), qs) = paulson@1588: let fun prune_aux (qs, []) = new::qs paulson@1588: | prune_aux (qs, (k,np,rgd,q)::rqs) = paulson@1588: if np'+1 = np andalso rgd then paulson@1588: (if !trace_DEPTH_FIRST then paulson@1588: writeln ("Pruning " ^ paulson@1588: string_of_int (1+length rqs) ^ " levels") paulson@1588: else (); paulson@1588: (*Use OLD k: zero-cost solution; see Stickel, p 365*) paulson@1588: (k, np', rgd', stq) :: qs) paulson@1588: else prune_aux ((k,np,rgd,q)::qs, rqs) paulson@1588: fun take ([], rqs) = ([], rqs) paulson@1588: | take (arg as ((k,np,rgd,stq)::qs, rqs)) = paulson@1588: if np' < np then take (qs, (k,np,rgd,stq)::rqs) paulson@1588: else arg paulson@1588: in prune_aux (take (qs, [])) end; paulson@1588: paulson@1588: paulson@1588: (*Depth-first iterative deepening search for a state that satisfies satp paulson@1588: tactic tac0 sets up the initial goal queue, while tac1 searches it. paulson@1588: The solution sequence is redundant: the cutoff heuristic makes it impossible paulson@1588: to suppress solutions arising from earlier searches, as the accumulated cost paulson@1588: (k) can be wrong.*) paulson@1588: fun THEN_ITER_DEEPEN tac0 satp tac1 = traced_tac (fn st => paulson@1588: let val countr = ref 0 paulson@1588: and tf = tracify trace_DEPTH_FIRST (tac1 1) paulson@1588: and qs0 = tac0 st paulson@1588: (*bnd = depth bound; inc = estimate of increment required next*) paulson@1588: fun depth (bnd,inc) [] = paulson@1588: (writeln (string_of_int (!countr) ^ paulson@1588: " inferences so far. Searching to depth " ^ paulson@1588: string_of_int bnd); paulson@1588: (*larger increments make it run slower for the hard problems*) paulson@1588: depth (bnd+inc, 10)) [(0, 1, false, qs0)] paulson@1588: | depth (bnd,inc) ((k,np,rgd,q)::qs) = paulson@1588: if k>=bnd then depth (bnd,inc) qs paulson@1588: else paulson@1588: case (countr := !countr+1; paulson@1588: if !trace_DEPTH_FIRST then paulson@1588: writeln (string_of_int np ^ paulson@1588: implode (map (fn _ => "*") qs)) paulson@1588: else (); paulson@1588: Sequence.pull q) of paulson@1588: None => depth (bnd,inc) qs paulson@1588: | Some(st,stq) => paulson@1588: if satp st (*solution!*) paulson@1588: then Some(st, Sequence.seqof paulson@1588: (fn()=> depth (bnd,inc) ((k,np,rgd,stq)::qs))) paulson@1588: paulson@1588: else paulson@1588: let val np' = nprems_of st paulson@1588: (*rgd' calculation assumes tactic operates on subgoal 1*) paulson@1588: val rgd' = not (has_vars (hd (prems_of st))) paulson@1588: val k' = k+np'-np+1 (*difference in # of subgoals, +1*) paulson@1588: in if k'+np' >= bnd paulson@2143: then depth (bnd, Int.min(inc, k'+np'+1-bnd)) qs paulson@1588: else if np' < np (*solved a subgoal; prune rigid ancestors*) paulson@1588: then depth (bnd,inc) paulson@1588: (prune ((k', np', rgd', tf st), (k,np,rgd,stq) :: qs)) paulson@1588: else depth (bnd,inc) ((k', np', rgd', tf st) :: paulson@1588: (k,np,rgd,stq) :: qs) paulson@1588: end paulson@1588: in depth (0,5) [] end); paulson@1588: paulson@1588: val ITER_DEEPEN = THEN_ITER_DEEPEN all_tac; paulson@1588: paulson@1588: paulson@1588: (*** Best-first search ***) paulson@1588: paulson@1588: val trace_BEST_FIRST = ref false; paulson@1588: paulson@1588: (*Insertion into priority queue of states *) paulson@1588: fun insert (nth: int*thm, []) = [nth] paulson@1588: | insert ((m,th), (n,th')::nths) = paulson@1588: if n some_of_list l)); paulson@1588: paulson@1588: paulson@1588: (*Best-first search for a state that satisfies satp (incl initial state) paulson@1588: Function sizef estimates size of problem remaining (smaller means better). paulson@1588: tactic tac0 sets up the initial priority queue, while tac1 searches it. *) paulson@1588: fun THEN_BEST_FIRST tac0 (satp, sizef) tac1 = paulson@1588: let val tac = tracify trace_BEST_FIRST tac1 paulson@1588: fun pairsize th = (sizef th, th); paulson@1588: fun bfs (news,nprfs) = paulson@1588: (case partition satp news of paulson@1588: ([],nonsats) => next(foldr insert paulson@1588: (map pairsize nonsats, nprfs)) paulson@1588: | (sats,_) => some_of_list sats) paulson@1588: and next [] = None paulson@1588: | next ((n,prf)::nprfs) = paulson@1588: (if !trace_BEST_FIRST paulson@1588: then writeln("state size = " ^ string_of_int n ^ paulson@1588: " queue length =" ^ string_of_int (length nprfs)) paulson@1588: else (); paulson@1588: bfs (Sequence.list_of_s (tac prf), nprfs)) paulson@1588: fun btac st = bfs (Sequence.list_of_s (tac0 st), []) paulson@1588: in traced_tac btac end; paulson@1588: paulson@1588: (*Ordinary best-first search, with no initial tactic*) paulson@1588: val BEST_FIRST = THEN_BEST_FIRST all_tac; paulson@1588: paulson@1588: (*Breadth-first search to satisfy satpred (including initial state) paulson@1588: SLOW -- SHOULD NOT USE APPEND!*) paulson@1588: fun BREADTH_FIRST satpred tac = paulson@1588: let val tacf = Sequence.list_of_s o tac; paulson@1588: fun bfs prfs = paulson@1588: (case partition satpred prfs of paulson@1588: ([],[]) => [] paulson@1588: | ([],nonsats) => paulson@1588: (prs("breadth=" ^ string_of_int(length nonsats) ^ "\n"); paulson@1588: bfs (flat (map tacf nonsats))) paulson@1588: | (sats,_) => sats) paulson@1588: in (fn st => Sequence.s_of_list (bfs [st])) end; paulson@1588: paulson@1588: paulson@1588: (* Author: Norbert Voelker, FernUniversitaet Hagen paulson@1588: Remarks: Implementation of A*-like proof procedure by modification paulson@1588: of the existing code for BEST_FIRST and best_tac so that the paulson@1588: current level of search is taken into account. paulson@1588: *) paulson@1588: paulson@1588: (*Insertion into priority queue of states, marked with level *) paulson@1588: fun insert_with_level (lnth: int*int*thm, []) = [lnth] paulson@1588: | insert_with_level ((l,m,th), (l',n,th')::nths) = paulson@1588: if n some_of_list l)); paulson@1588: paulson@1588: val trace_ASTAR = ref false; paulson@1588: paulson@1588: fun THEN_ASTAR tac0 (satp, costf) tac1 = paulson@1588: let val tf = tracify trace_ASTAR tac1; paulson@1588: fun bfs (news,nprfs,level) = paulson@1588: let fun cost thm = (level, costf level thm, thm) paulson@1588: in (case partition satp news of paulson@1588: ([],nonsats) paulson@1588: => next (foldr insert_with_level (map cost nonsats, nprfs)) paulson@1588: | (sats,_) => some_of_list sats) paulson@1588: end and paulson@1588: next [] = None paulson@1588: | next ((level,n,prf)::nprfs) = paulson@1588: (if !trace_ASTAR paulson@1588: then writeln("level = " ^ string_of_int level ^ paulson@1588: " cost = " ^ string_of_int n ^ paulson@1588: " queue length =" ^ string_of_int (length nprfs)) paulson@1588: else (); paulson@1588: bfs (Sequence.list_of_s (tf prf), nprfs,level+1)) paulson@1588: fun tf st = bfs (Sequence.list_of_s (tac0 st), [], 0) paulson@1588: in traced_tac tf end; paulson@1588: paulson@1588: (*Ordinary ASTAR, with no initial tactic*) paulson@1588: val ASTAR = THEN_ASTAR all_tac; paulson@1588: paulson@1588: end; paulson@1588: paulson@1588: open Search;