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