(* ||M|| This file is part of HELM, an Hypertextual, Electronic ||A|| Library of Mathematics, developed at the Computer Science ||T|| Department, University of Bologna, Italy. ||I|| ||T|| HELM is free software; you can redistribute it and/or ||A|| modify it under the terms of the GNU General Public License \ / version 2 or (at your option) any later version. \ / This software is distributed as is, NO WARRANTY. V_______________________________________________________________ *) (* $Id: index.mli 9822 2009-06-03 15:37:06Z tassi $ *) module Superposition (B : Terms.Blob) = struct module IDX = Index.Index(B) module Unif = FoUnif.Founif(B) module Subst = FoSubst (*.Subst(B)*) module Order = Orderings.Orderings(B) module Utils = FoUtils.Utils(B) module Pp = Pp.Pp(B) exception Success of B.t Terms.bag * int * B.t Terms.unit_clause (* let debug s = prerr_endline s;;*) let debug _ = ();; let rec list_first f = function | [] -> None | x::tl -> match f x with Some _ as x -> x | _ -> list_first f tl ;; let first_position pos ctx t f = let rec aux pos ctx = function | Terms.Leaf _ as t -> f t pos ctx | Terms.Var _ -> None | Terms.Node l as t-> match f t pos ctx with | Some _ as x -> x | None -> let rec first pre post = function | [] -> None | t :: tl -> let newctx = fun x -> ctx (Terms.Node (pre@[x]@post)) in match aux (List.length pre :: pos) newctx t with | Some _ as x -> x | None -> if post = [] then None (* tl is also empty *) else first (pre @ [t]) (List.tl post) tl in first [] (List.tl l) l in aux pos ctx t ;; let all_positions pos ctx t f = let rec aux pos ctx = function | Terms.Leaf _ as t -> f t pos ctx | Terms.Var _ -> [] | Terms.Node l as t-> let acc, _, _ = List.fold_left (fun (acc,pre,post) t -> (* Invariant: pre @ [t] @ post = l *) let newctx = fun x -> ctx (Terms.Node (pre@[x]@post)) in let acc = aux (List.length pre :: pos) newctx t @ acc in if post = [] then acc, l, [] else acc, pre @ [t], List.tl post) (f t pos ctx, [], List.tl l) l in acc in aux pos ctx t ;; let vars_of_term t = let rec aux acc = function | Terms.Leaf _ -> acc | Terms.Var i -> if (List.mem i acc) then acc else i::acc | Terms.Node l -> List.fold_left aux acc l in aux [] t ;; let build_clause bag filter rule t subst vl id id2 pos dir = let proof = Terms.Step(rule,id,id2,dir,pos,subst) in let t = Subst.apply_subst subst t in if filter t then let literal = match t with | Terms.Node [ Terms.Leaf eq ; ty; l; r ] when B.eq B.eqP eq -> let o = Order.compare_terms l r in Terms.Equation (l, r, ty, o) | t -> Terms.Predicate t in let bag, uc = Utils.add_to_bag bag (0, literal, vars_of_term t, proof) in Some (bag, uc) else ((*prerr_endline ("Filtering: " ^ Pp.pp_foterm t);*)None) ;; (* ============ simplification ================= *) let demod table varlist subterm pos context = let cands = IDX.DT.retrieve_generalizations table subterm in list_first (fun (dir, (id,lit,vl,_)) -> match lit with | Terms.Predicate _ -> assert false | Terms.Equation (l,r,_,o) -> let side, newside = if dir=Terms.Left2Right then l,r else r,l in try let subst, varlist = Unif.unification (varlist@vl) varlist subterm side in if o = Terms.Incomparable then let side = Subst.apply_subst subst side in let newside = Subst.apply_subst subst newside in let o = Order.compare_terms newside side in (* Riazanov, pp. 45 (ii) *) if o = Terms.Lt then Some (context newside, subst, varlist, id, pos, dir) else ((*prerr_endline ("Filtering: " ^ Pp.pp_foterm side ^ " =(< || =)" ^ Pp.pp_foterm newside ^ " coming from " ^ Pp.pp_unit_clause uc );*)None) else Some (context newside, subst, varlist, id, pos, dir) with FoUnif.UnificationFailure _ -> None) (IDX.ClauseSet.elements cands) ;; let demodulate_once ~jump_to_right bag (id, literal, vl, pr) table = (* debug ("Demodulating : " ^ (Pp.pp_unit_clause (id, literal, vl, pr)));*) match literal with | Terms.Predicate t -> assert false | Terms.Equation (l,r,ty,_) -> let left_position = if jump_to_right then None else first_position [2] (fun x -> Terms.Node [ Terms.Leaf B.eqP; ty; x; r ]) l (demod table vl) in match left_position with | Some (newt, subst, varlist, id2, pos, dir) -> begin match build_clause bag (fun _ -> true) Terms.Demodulation newt subst varlist id id2 pos dir with | None -> assert false | Some x -> Some (x,false) end | None -> match first_position [3] (fun x -> Terms.Node [ Terms.Leaf B.eqP; ty; l; x ]) r (demod table vl) with | None -> None | Some (newt, subst, varlist, id2, pos, dir) -> match build_clause bag (fun _ -> true) Terms.Demodulation newt subst varlist id id2 pos dir with | None -> assert false | Some x -> Some (x,true) ;; let rec demodulate ~jump_to_right bag clause table = match demodulate_once ~jump_to_right bag clause table with | None -> bag, clause | Some ((bag, clause),r) -> demodulate ~jump_to_right:r bag clause table ;; let demodulate bag clause table = demodulate ~jump_to_right:false bag clause table ;; (* move away *) let is_identity_clause ~unify = function | _, Terms.Equation (_,_,_,Terms.Eq), _, _ -> true | _, Terms.Equation (l,r,_,_), vl, proof when unify -> (try ignore(Unif.unification vl [] l r); true with FoUnif.UnificationFailure _ -> false) | _, Terms.Equation (_,_,_,_), _, _ -> false | _, Terms.Predicate _, _, _ -> assert false ;; let build_new_clause bag maxvar filter rule t subst vl id id2 pos dir = let maxvar, vl, relocsubst = Utils.relocate maxvar vl in let subst = Subst.concat relocsubst subst in match build_clause bag filter rule t subst vl id id2 pos dir with | Some (bag, c) -> Some ((bag, maxvar), c) | None -> None ;; let fold_build_new_clause bag maxvar id rule filter res = let (bag, maxvar), res = HExtlib.filter_map_acc (fun (bag, maxvar) (t,subst,vl,id2,pos,dir) -> build_new_clause bag maxvar filter rule t subst vl id id2 pos dir) (bag, maxvar) res in bag, maxvar, res ;; let rewrite_eq ~unify l r ty vl table = let retrieve = if unify then IDX.DT.retrieve_unifiables else IDX.DT.retrieve_generalizations in let lcands = retrieve table l in let rcands = retrieve table r in let f b c = let id, dir, l, r, vl = match c with | (d, (id,Terms.Equation (l,r,ty,_),vl,_))-> id, d, l, r, vl |_ -> assert false in let reverse = (dir = Terms.Left2Right) = b in let l, r, proof_rewrite_dir = if reverse then l,r,Terms.Left2Right else r,l, Terms.Right2Left in (id,proof_rewrite_dir,Terms.Node [ Terms.Leaf B.eqP; ty; l; r ], vl) in let cands1 = List.map (f true) (IDX.ClauseSet.elements lcands) in let cands2 = List.map (f false) (IDX.ClauseSet.elements rcands) in let t = Terms.Node [ Terms.Leaf B.eqP; ty; l; r ] in let locked_vars = if unify then [] else vl in let rec aux = function | [] -> None | (id2,dir,c,vl1)::tl -> try let subst,vl1 = Unif.unification (vl@vl1) locked_vars c t in Some (id2, dir, subst) with FoUnif.UnificationFailure _ -> aux tl in aux (cands1 @ cands2) ;; let is_subsumed ~unify bag maxvar (id, lit, vl, _) table = match lit with | Terms.Predicate _ -> assert false | Terms.Equation (l,r,ty,_) -> match rewrite_eq ~unify l r ty vl table with | None -> None | Some (id2, dir, subst) -> let id_t = Terms.Node [ Terms.Leaf B.eqP; ty; r; r ] in build_new_clause bag maxvar (fun _ -> true) Terms.Superposition id_t subst [] id id2 [2] dir ;; (* id refers to a clause proving contextl l = contextr r *) let rec deep_eq ~unify l r ty pos contextl contextr table acc = match acc with | None -> None | Some(bag,maxvar,(id,lit,vl,p),subst) -> let l = Subst.apply_subst subst l in let r = Subst.apply_subst subst r in try let subst1,vl1 = Unif.unification vl [] l r in let lit = match lit with Terms.Predicate _ -> assert false | Terms.Equation (l,r,ty,o) -> Terms.Equation (FoSubst.apply_subst subst1 l, FoSubst.apply_subst subst1 r, ty, o) in Some(bag,maxvar,(id,lit,vl1,p),Subst.concat subst1 subst) with FoUnif.UnificationFailure _ -> match rewrite_eq ~unify l r ty vl table with | Some (id2, dir, subst1) -> let newsubst = Subst.concat subst1 subst in let id_t = FoSubst.apply_subst newsubst (Terms.Node[Terms.Leaf B.eqP;ty;contextl r;contextr r]) in (match build_new_clause bag maxvar (fun _ -> true) Terms.Superposition id_t subst1 [] id id2 (pos@[2]) dir with | Some ((bag, maxvar), c) -> Some(bag,maxvar,c,newsubst) | None -> assert false) | None -> match l,r with | Terms.Node (a::la), Terms.Node (b::lb) when a = b && List.length la = List.length lb -> let acc,_,_,_ = List.fold_left2 (fun (acc,pre,postl,postr) a b -> let newcl = fun x -> contextl(Terms.Node (pre@(x::postl))) in let newcr = fun x -> contextr(Terms.Node (pre@(x::postr))) in let newpos = List.length pre::pos in let footail l = if l = [] then [] else List.tl l in (deep_eq ~unify a b ty newpos newcl newcr table acc,pre@[b], footail postl, footail postr)) (acc,[a],List.tl la,List.tl lb) la lb in acc | Terms.Var _, _ | _, Terms.Var _ -> assert false | _,_ -> None ;; let rec orphan_murder bag acc i = match Terms.M.find i bag with | (_,_,_,Terms.Exact _),discarded -> (discarded,acc) | (_,_,_,Terms.Step (_,i1,i2,_,_,_)),true -> (true,acc) | (_,_,_,Terms.Step (_,i1,i2,_,_,_)),false -> if (List.mem i acc) then (false,acc) else match orphan_murder bag acc i1 with | (true,acc) -> (true,acc) | (false,acc) -> let (res,acc) = orphan_murder bag acc i2 in if res then res,acc else res,i::acc ;; let orphan_murder bag cl = let (id,_,_,_) = cl in let (res,_) = orphan_murder bag [] id in if res then debug "Orphan murdered"; res ;; (* demodulate and check for subsumption *) let simplify table maxvar bag clause = if is_identity_clause ~unify:false clause then bag,None (* else if orphan_murder bag clause then bag,None *) else let bag, clause = demodulate bag clause table in if is_identity_clause ~unify:false clause then bag,None else match is_subsumed ~unify:false bag maxvar clause table with | None -> bag, Some clause | Some _ -> bag, None ;; let simplify table maxvar bag clause = match simplify table maxvar bag clause with | bag, None -> let (id,_,_,_) = clause in Terms.M.add id (clause,true) bag, None | bag, Some clause -> bag, Some clause ;; let one_pass_simplification new_clause (alist,atable) bag maxvar = match simplify atable maxvar bag new_clause with | bag,None -> bag,None (* new_clause has been discarded *) | bag,(Some clause) -> let ctable = IDX.index_unit_clause IDX.DT.empty clause in let bag, alist, atable = List.fold_left (fun (bag, alist, atable) c -> match simplify ctable maxvar bag c with |bag,None -> (bag,alist,atable) (* an active clause as been discarded *) |bag,Some c1 -> bag, c :: alist, IDX.index_unit_clause atable c) (bag,[],IDX.DT.empty) alist in bag, Some (clause, (alist,atable)) ;; let simplification_step ~new_cl cl (alist,atable) bag maxvar new_clause = let atable1 = if new_cl then atable else IDX.index_unit_clause atable cl in (* Simplification of new_clause with : * * - actives and cl if new_clause is not cl * * - only actives otherwise *) match simplify atable1 maxvar bag new_clause with | bag,None -> bag,(Some cl, None) (* new_clause has been discarded *) | bag,Some clause -> (* Simplification of each active clause with clause * * which is the simplified form of new_clause *) let ctable = IDX.index_unit_clause IDX.DT.empty clause in let bag, newa, alist, atable = List.fold_left (fun (bag, newa, alist, atable) c -> match simplify ctable maxvar bag c with |bag,None -> (bag, newa, alist, atable) (* an active clause as been discarded *) |bag,Some c1 -> if (c1 == c) then bag, newa, c :: alist, IDX.index_unit_clause atable c else bag, c1 :: newa, alist, atable) (bag,[],[],IDX.DT.empty) alist in if new_cl then bag, (Some cl, Some (clause, (alist,atable), newa)) else (* if new_clause is not cl, we simplify cl with clause *) match simplify ctable maxvar bag cl with | bag,None -> (* cl has been discarded *) bag,(None, Some (clause, (alist,atable), newa)) | bag,Some cl1 -> bag,(Some cl1, Some (clause, (alist,atable), newa)) ;; let keep_simplified cl (alist,atable) bag maxvar = let rec keep_simplified_aux ~new_cl cl (alist,atable) bag newc = if new_cl then match simplification_step ~new_cl cl (alist,atable) bag maxvar cl with | _,(None, _) -> assert false | bag,(Some _, None) -> bag,None | bag,(Some _, Some (clause, (alist,atable), newa)) -> keep_simplified_aux ~new_cl:(cl!=clause) clause (alist,atable) bag (newa@newc) else match newc with | [] -> bag, Some (cl, (alist,atable)) | hd::tl -> match simplification_step ~new_cl cl (alist,atable) bag maxvar hd with | _,(None,None) -> assert false | bag,(Some _,None) -> keep_simplified_aux ~new_cl cl (alist,atable) bag tl | bag,(None, Some _) -> bag,None | bag,(Some cl1, Some (clause, (alist,atable), newa)) -> let alist,atable = (clause::alist, IDX.index_unit_clause atable clause) in keep_simplified_aux ~new_cl:(cl!=cl1) cl1 (alist,atable) bag (newa@tl) in keep_simplified_aux ~new_cl:true cl (alist,atable) bag [] ;; let are_alpha_eq cl1 cl2 = let get_term (_,lit,_,_) = match lit with | Terms.Predicate _ -> assert false | Terms.Equation (l,r,ty,_) -> Terms.Node [Terms.Leaf B.eqP; ty; l ; r] in try ignore(Unif.alpha_eq (get_term cl1) (get_term cl2)) ; true with FoUnif.UnificationFailure _ -> false ;; (* this is like simplify but raises Success *) let simplify_goal maxvar table bag g_actives clause = let bag, clause = demodulate bag clause table in if (is_identity_clause ~unify:true clause) then raise (Success (bag, maxvar, clause)) else let (id,lit,vl,_) = clause in let l,r,ty = match lit with | Terms.Equation(l,r,ty,_) -> l,r,ty | _ -> assert false in match deep_eq ~unify:true l r ty [] (fun x -> x) (fun x -> x) table (Some(bag,maxvar,clause,Subst.id_subst)) with | None -> if List.exists (are_alpha_eq clause) g_actives then None else Some (bag, clause) | Some (bag,maxvar,cl,subst) -> prerr_endline "Goal subsumed"; raise (Success (bag,maxvar,cl)) (* else match is_subsumed ~unify:true bag maxvar clause table with | None -> if List.exists (are_alpha_eq clause) g_actives then None else Some (bag, clause) | Some ((bag,maxvar),c) -> prerr_endline "Goal subsumed"; raise (Success (bag,maxvar,c))*) ;; (* =================== inference ===================== *) (* this is OK for both the sup_left and sup_right inference steps *) let superposition table varlist subterm pos context = let cands = IDX.DT.retrieve_unifiables table subterm in HExtlib.filter_map (fun (dir, (id,lit,vl,_ (*as uc*))) -> match lit with | Terms.Predicate _ -> assert false | Terms.Equation (l,r,_,o) -> let side, newside = if dir=Terms.Left2Right then l,r else r,l in try let subst, varlist = Unif.unification (varlist@vl) [] subterm side in if o = Terms.Incomparable then let side = Subst.apply_subst subst side in let newside = Subst.apply_subst subst newside in let o = Order.compare_terms side newside in (* XXX: check Riazanov p. 33 (iii) *) if o <> Terms.Lt && o <> Terms.Eq then Some (context newside, subst, varlist, id, pos, dir) else ((*prerr_endline ("Filtering: " ^ Pp.pp_foterm side ^ " =(< || =)" ^ Pp.pp_foterm newside ^ " coming from " ^ Pp.pp_unit_clause uc );*)None) else Some (context newside, subst, varlist, id, pos, dir) with FoUnif.UnificationFailure _ -> None) (IDX.ClauseSet.elements cands) ;; (* Superposes selected equation with equalities in table *) let superposition_with_table bag maxvar (id,selected,vl,_) table = match selected with | Terms.Predicate _ -> assert false | Terms.Equation (l,r,ty,Terms.Lt) -> fold_build_new_clause bag maxvar id Terms.Superposition (fun _ -> true) (all_positions [3] (fun x -> Terms.Node [ Terms.Leaf B.eqP; ty; l; x ]) r (superposition table vl)) | Terms.Equation (l,r,ty,Terms.Gt) -> fold_build_new_clause bag maxvar id Terms.Superposition (fun _ -> true) (all_positions [2] (fun x -> Terms.Node [ Terms.Leaf B.eqP; ty; x; r ]) l (superposition table vl)) | Terms.Equation (l,r,ty,Terms.Incomparable) -> fold_build_new_clause bag maxvar id Terms.Superposition (function (* Riazanov: p.33 condition (iv) *) | Terms.Node [Terms.Leaf eq; ty; l; r ] when B.eq B.eqP eq -> Order.compare_terms l r <> Terms.Eq | _ -> assert false) ((all_positions [3] (fun x -> Terms.Node [ Terms.Leaf B.eqP; ty; l; x ]) r (superposition table vl)) @ (all_positions [2] (fun x -> Terms.Node [ Terms.Leaf B.eqP; ty; x; r ]) l (superposition table vl))) | _ -> assert false ;; (* the current equation is normal w.r.t. demodulation with atable * (and is not the identity) *) let infer_right bag maxvar current (alist,atable) = (* We demodulate actives clause with current until all * * active clauses are reduced w.r.t each other *) (* let bag, (alist,atable) = keep_simplified (alist,atable) bag [current] in *) let ctable = IDX.index_unit_clause IDX.DT.empty current in (* let bag, (alist, atable) = let bag, alist = HExtlib.filter_map_acc (simplify ctable) bag alist in bag, (alist, List.fold_left IDX.index_unit_clause IDX.DT.empty alist) in*) debug "Simplified active clauses with fact"; (* We superpose active clauses with current *) let bag, maxvar, new_clauses = List.fold_left (fun (bag, maxvar, acc) active -> let bag, maxvar, newc = superposition_with_table bag maxvar active ctable in bag, maxvar, newc @ acc) (bag, maxvar, []) alist in debug "First superpositions"; (* We add current to active clauses so that it can be * * superposed with itself *) let alist, atable = current :: alist, IDX.index_unit_clause atable current in debug "Indexed"; let fresh_current, maxvar = Utils.fresh_unit_clause maxvar current in (* We need to put fresh_current into the bag so that all * * variables clauses refer to are known. *) let bag, fresh_current = Utils.add_to_bag bag fresh_current in (* We superpose current with active clauses *) let bag, maxvar, additional_new_clauses = superposition_with_table bag maxvar fresh_current atable in debug "Another superposition"; let new_clauses = new_clauses @ additional_new_clauses in debug (Printf.sprintf "Demodulating %d clauses" (List.length new_clauses)); let bag, new_clauses = HExtlib.filter_map_monad (simplify atable maxvar) bag new_clauses in debug "Demodulated new clauses"; bag, maxvar, (alist, atable), new_clauses ;; let infer_left bag maxvar goal (_alist, atable) = (* We superpose the goal with active clauses *) let bag, maxvar, new_goals = superposition_with_table bag maxvar goal atable in debug "Superposed goal with active clauses"; (* We simplify the new goals with active clauses *) let bag, new_goals = List.fold_left (fun (bag, acc) g -> match simplify_goal maxvar atable bag [] g with | None -> assert false | Some (bag,g) -> bag,g::acc) (bag, []) new_goals in debug "Simplified new goals with active clauses"; bag, maxvar, List.rev new_goals ;; end