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(* Title: net
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ID: $Id$
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory
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Copyright 1993 University of Cambridge
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Discrimination nets: a data structure for indexing items
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From the book
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E. Charniak, C. K. Riesbeck, D. V. McDermott.
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Artificial Intelligence Programming.
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(Lawrence Erlbaum Associates, 1980). [Chapter 14]
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*)
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signature NET =
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sig
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type key
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type 'a net
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exception DELETE and INSERT
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val delete: (key list * 'a) * 'a net * ('a*'a -> bool) -> 'a net
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val delete_term: (term * 'a) * 'a net * ('a*'a -> bool) -> 'a net
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val empty: 'a net
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val insert: (key list * 'a) * 'a net * ('a*'a -> bool) -> 'a net
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val insert_term: (term * 'a) * 'a net * ('a*'a -> bool) -> 'a net
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val lookup: 'a net * key list -> 'a list
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val match_term: 'a net -> term -> 'a list
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val key_of_term: term -> key list
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val unify_term: 'a net -> term -> 'a list
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end;
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functor NetFun () : NET =
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struct
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datatype key = CombK | VarK | AtomK of string;
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(*Only 'loose' bound variables could arise, since Abs nodes are skipped*)
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fun string_of_bound i = "*B*" ^ chr i;
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(*Keys are preorder lists of symbols -- constants, Vars, bound vars, ...
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Any term whose head is a Var is regarded entirely as a Var;
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abstractions are also regarded as Vars (to cover eta-conversion)
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*)
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fun add_key_of_terms (t, cs) =
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let fun rands (f$t, cs) = CombK :: rands (f, add_key_of_terms(t, cs))
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| rands (Const(c,_), cs) = AtomK c :: cs
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| rands (Free(c,_), cs) = AtomK c :: cs
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| rands (Bound i, cs) = AtomK (string_of_bound i) :: cs
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in case (head_of t) of
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Var _ => VarK :: cs
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| Abs (_,_,t) => VarK :: cs
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| _ => rands(t,cs)
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end;
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(*convert a term to a key -- a list of keys*)
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fun key_of_term t = add_key_of_terms (t, []);
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(*Trees indexed by key lists: each arc is labelled by a key.
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Each node contains a list of items, and arcs to children.
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Keys in the association list (alist) are stored in ascending order.
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The empty key addresses the entire net.
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Lookup functions preserve order in items stored at same level.
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*)
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datatype 'a net = Leaf of 'a list
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| Net of {comb: 'a net,
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var: 'a net,
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alist: (string * 'a net) list};
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val empty = Leaf[];
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val emptynet = Net{comb=empty, var=empty, alist=[]};
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(*** Insertion into a discrimination net ***)
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exception INSERT; (*duplicate item in the net*)
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(*Adds item x to the list at the node addressed by the keys.
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Creates node if not already present.
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eq is the equality test for items.
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The empty list of keys generates a Leaf node, others a Net node.
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*)
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fun insert ((keys,x), net, eq) =
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let fun ins1 ([], Leaf xs) =
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if gen_mem eq (x,xs) then raise INSERT else Leaf(x::xs)
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| ins1 (keys, Leaf[]) = ins1 (keys, emptynet) (*expand empty...*)
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| ins1 (CombK :: keys, Net{comb,var,alist}) =
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Net{comb=ins1(keys,comb), var=var, alist=alist}
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| ins1 (VarK :: keys, Net{comb,var,alist}) =
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Net{comb=comb, var=ins1(keys,var), alist=alist}
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| ins1 (AtomK a :: keys, Net{comb,var,alist}) =
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let fun newpair net = (a, ins1(keys,net))
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fun inslist [] = [newpair empty]
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| inslist((b: string, netb) :: alist) =
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if a=b then newpair netb :: alist
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else if a<b then (*absent, ins1ert in alist*)
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newpair empty :: (b,netb) :: alist
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else (*a>b*) (b,netb) :: inslist alist
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in Net{comb=comb, var=var, alist= inslist alist} end
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in ins1 (keys,net) end;
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fun insert_term ((t,x), net, eq) = insert((key_of_term t, x), net, eq);
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(*** Deletion from a discrimination net ***)
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exception DELETE; (*missing item in the net*)
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(*Create a new Net node if it would be nonempty*)
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fun newnet {comb=Leaf[], var=Leaf[], alist=[]} = empty
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| newnet {comb,var,alist} = Net{comb=comb, var=var, alist=alist};
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(*add new (b,net) pair to the alist provided net is nonempty*)
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fun conspair((b, Leaf[]), alist) = alist
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| conspair((b, net), alist) = (b, net) :: alist;
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(*Deletes item x from the list at the node addressed by the keys.
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Raises DELETE if absent. Collapses the net if possible.
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eq is the equality test for items. *)
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fun delete ((keys, x), net, eq) =
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let fun del1 ([], Leaf xs) =
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if gen_mem eq (x,xs) then Leaf (gen_rem eq (xs,x))
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else raise DELETE
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| del1 (keys, Leaf[]) = raise DELETE
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| del1 (CombK :: keys, Net{comb,var,alist}) =
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newnet{comb=del1(keys,comb), var=var, alist=alist}
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| del1 (VarK :: keys, Net{comb,var,alist}) =
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newnet{comb=comb, var=del1(keys,var), alist=alist}
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| del1 (AtomK a :: keys, Net{comb,var,alist}) =
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let fun newpair net = (a, del1(keys,net))
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fun dellist [] = raise DELETE
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| dellist((b: string, netb) :: alist) =
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if a=b then conspair(newpair netb, alist)
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else if a<b then (*absent*) raise DELETE
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else (*a>b*) (b,netb) :: dellist alist
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in newnet{comb=comb, var=var, alist= dellist alist} end
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in del1 (keys,net) end;
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fun delete_term ((t,x), net, eq) = delete((key_of_term t, x), net, eq);
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(*** Retrieval functions for discrimination nets ***)
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exception OASSOC;
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(*Ordered association list lookup*)
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fun oassoc ([], a: string) = raise OASSOC
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| oassoc ((b,x)::pairs, a) =
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if b<a then oassoc(pairs,a)
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else if a=b then x
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else raise OASSOC;
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(*Return the list of items at the given node, [] if no such node*)
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fun lookup (Leaf(xs), []) = xs
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| lookup (Leaf _, _::_) = [] (*non-empty keys and empty net*)
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| lookup (Net{comb,var,alist}, CombK :: keys) = lookup(comb,keys)
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| lookup (Net{comb,var,alist}, VarK :: keys) = lookup(var,keys)
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| lookup (Net{comb,var,alist}, AtomK a :: keys) =
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lookup(oassoc(alist,a),keys) handle OASSOC => [];
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(*Skipping a term in a net. Recursively skip 2 levels if a combination*)
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fun net_skip (Leaf _, nets) = nets
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| net_skip (Net{comb,var,alist}, nets) =
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foldr net_skip
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(net_skip (comb,[]),
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foldr (fn ((_,net), nets) => net::nets) (alist, var::nets));
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(** Matching and Unification**)
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(*conses the linked net, if present, to nets*)
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fun look1 (alist, a) nets =
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oassoc(alist,a) :: nets handle OASSOC => nets;
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(*Return the nodes accessible from the term (cons them before nets)
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"unif" signifies retrieval for unification rather than matching.
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Var in net matches any term.
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Abs in object (and Var if "unif") regarded as wildcard.
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If not "unif", Var in object only matches a variable in net.*)
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fun matching unif t (net,nets) =
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let fun rands _ (Leaf _, nets) = nets
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| rands t (Net{comb,alist,...}, nets) =
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case t of
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(f$t) => foldr (matching unif t) (rands f (comb,[]), nets)
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| (Const(c,_)) => look1 (alist, c) nets
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| (Free(c,_)) => look1 (alist, c) nets
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| (Bound i) => look1 (alist, string_of_bound i) nets
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in
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case net of
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Leaf _ => nets
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| Net{var,...} =>
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case (head_of t) of
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Var _ => if unif then net_skip (net,nets)
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else var::nets (*only matches Var in net*)
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| Abs(_,_,u) => net_skip (net,nets) (*could match anything*)
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| _ => rands t (net, var::nets) (*var could match also*)
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end;
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val extract_leaves = flat o map (fn Leaf(xs) => xs);
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(*return items whose key could match t*)
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fun match_term net t =
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extract_leaves (matching false t (net,[]));
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(*return items whose key could unify with t*)
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fun unify_term net t =
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extract_leaves (matching true t (net,[]));
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end;
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