(* ========================================================================= *)
(* MATCHING AND UNIFICATION FOR SETS OF FIRST ORDER LOGIC TERMS *)
(* Copyright (c) 2001 Joe Leslie-Hurd, distributed under the BSD License *)
(* ========================================================================= *)
structure TermNet :> TermNet =
struct
open Useful;
(* ------------------------------------------------------------------------- *)
(* Anonymous variables. *)
(* ------------------------------------------------------------------------- *)
val anonymousName = Name.fromString "_";
val anonymousVar = Term.Var anonymousName;
(* ------------------------------------------------------------------------- *)
(* Quotient terms. *)
(* ------------------------------------------------------------------------- *)
datatype qterm =
Var
| Fn of NameArity.nameArity * qterm list;
local
fun cmp [] = EQUAL
| cmp (q1_q2 :: qs) =
if Portable.pointerEqual q1_q2 then cmp qs
else
case q1_q2 of
(Var,Var) => EQUAL
| (Var, Fn _) => LESS
| (Fn _, Var) => GREATER
| (Fn f1, Fn f2) => fnCmp f1 f2 qs
and fnCmp (n1,q1) (n2,q2) qs =
case NameArity.compare (n1,n2) of
LESS => LESS
| EQUAL => cmp (zip q1 q2 @ qs)
| GREATER => GREATER;
in
fun compareQterm q1_q2 = cmp [q1_q2];
fun compareFnQterm (f1,f2) = fnCmp f1 f2 [];
end;
fun equalQterm q1 q2 = compareQterm (q1,q2) = EQUAL;
fun equalFnQterm f1 f2 = compareFnQterm (f1,f2) = EQUAL;
fun termToQterm (Term.Var _) = Var
| termToQterm (Term.Fn (f,l)) = Fn ((f, length l), List.map termToQterm l);
local
fun qm [] = true
| qm ((Var,_) :: rest) = qm rest
| qm ((Fn _, Var) :: _) = false
| qm ((Fn (f,a), Fn (g,b)) :: rest) =
NameArity.equal f g andalso qm (zip a b @ rest);
in
fun matchQtermQterm qtm qtm' = qm [(qtm,qtm')];
end;
local
fun qm [] = true
| qm ((Var,_) :: rest) = qm rest
| qm ((Fn _, Term.Var _) :: _) = false
| qm ((Fn ((f,n),a), Term.Fn (g,b)) :: rest) =
Name.equal f g andalso n = length b andalso qm (zip a b @ rest);
in
fun matchQtermTerm qtm tm = qm [(qtm,tm)];
end;
local
fun qn qsub [] = SOME qsub
| qn qsub ((Term.Var v, qtm) :: rest) =
(case NameMap.peek qsub v of
NONE => qn (NameMap.insert qsub (v,qtm)) rest
| SOME qtm' => if equalQterm qtm qtm' then qn qsub rest else NONE)
| qn _ ((Term.Fn _, Var) :: _) = NONE
| qn qsub ((Term.Fn (f,a), Fn ((g,n),b)) :: rest) =
if Name.equal f g andalso length a = n then qn qsub (zip a b @ rest)
else NONE;
in
fun matchTermQterm qsub tm qtm = qn qsub [(tm,qtm)];
end;
local
fun qv Var x = x
| qv x Var = x
| qv (Fn (f,a)) (Fn (g,b)) =
let
val _ = NameArity.equal f g orelse raise Error "TermNet.qv"
in
Fn (f, zipWith qv a b)
end;
fun qu qsub [] = qsub
| qu qsub ((Var, _) :: rest) = qu qsub rest
| qu qsub ((qtm, Term.Var v) :: rest) =
let
val qtm =
case NameMap.peek qsub v of NONE => qtm | SOME qtm' => qv qtm qtm'
in
qu (NameMap.insert qsub (v,qtm)) rest
end
| qu qsub ((Fn ((f,n),a), Term.Fn (g,b)) :: rest) =
if Name.equal f g andalso n = length b then qu qsub (zip a b @ rest)
else raise Error "TermNet.qu";
in
fun unifyQtermQterm qtm qtm' = total (qv qtm) qtm';
fun unifyQtermTerm qsub qtm tm = total (qu qsub) [(qtm,tm)];
end;
local
fun qtermToTerm Var = anonymousVar
| qtermToTerm (Fn ((f,_),l)) = Term.Fn (f, List.map qtermToTerm l);
in
val ppQterm = Print.ppMap qtermToTerm Term.pp;
end;
(* ------------------------------------------------------------------------- *)
(* A type of term sets that can be efficiently matched and unified. *)
(* ------------------------------------------------------------------------- *)
type parameters = {fifo : bool};
datatype 'a net =
Result of 'a list
| Single of qterm * 'a net
| Multiple of 'a net option * 'a net NameArityMap.map;
datatype 'a termNet = Net of parameters * int * (int * (int * 'a) net) option;
(* ------------------------------------------------------------------------- *)
(* Basic operations. *)
(* ------------------------------------------------------------------------- *)
fun new parm = Net (parm,0,NONE);
local
fun computeSize (Result l) = length l
| computeSize (Single (_,n)) = computeSize n
| computeSize (Multiple (vs,fs)) =
NameArityMap.foldl
(fn (_,n,acc) => acc + computeSize n)
(case vs of SOME n => computeSize n | NONE => 0)
fs;
in
fun netSize NONE = NONE
| netSize (SOME n) = SOME (computeSize n, n);
end;
fun size (Net (_,_,NONE)) = 0
| size (Net (_, _, SOME (i,_))) = i;
fun null net = size net = 0;
fun singles qtms a = List.foldr Single a qtms;
local
fun pre NONE = (0,NONE)
| pre (SOME (i,n)) = (i, SOME n);
fun add (Result l) [] (Result l') = Result (l @ l')
| add a (input1 as qtm :: qtms) (Single (qtm',n)) =
if equalQterm qtm qtm' then Single (qtm, add a qtms n)
else add a input1 (add n [qtm'] (Multiple (NONE, NameArityMap.new ())))
| add a (Var :: qtms) (Multiple (vs,fs)) =
Multiple (SOME (oadd a qtms vs), fs)
| add a (Fn (f,l) :: qtms) (Multiple (vs,fs)) =
let
val n = NameArityMap.peek fs f
in
Multiple (vs, NameArityMap.insert fs (f, oadd a (l @ qtms) n))
end
| add _ _ _ = raise Bug "TermNet.insert: Match"
and oadd a qtms NONE = singles qtms a
| oadd a qtms (SOME n) = add a qtms n;
fun ins a qtm (i,n) = SOME (i + 1, oadd (Result [a]) [qtm] n);
in
fun insert (Net (p,k,n)) (tm,a) =
Net (p, k + 1, ins (k,a) (termToQterm tm) (pre n))
handle Error _ => raise Bug "TermNet.insert: should never fail";
end;
fun fromList parm l = List.foldl (fn (tm_a,n) => insert n tm_a) (new parm) l;
fun filter pred =
let
fun filt (Result l) =
(case List.filter (fn (_,a) => pred a) l of
[] => NONE
| l => SOME (Result l))
| filt (Single (qtm,n)) =
(case filt n of
NONE => NONE
| SOME n => SOME (Single (qtm,n)))
| filt (Multiple (vs,fs)) =
let
val vs = Option.mapPartial filt vs
val fs = NameArityMap.mapPartial (fn (_,n) => filt n) fs
in
if not (Option.isSome vs) andalso NameArityMap.null fs then NONE
else SOME (Multiple (vs,fs))
end
in
fn net as Net (_,_,NONE) => net
| Net (p, k, SOME (_,n)) => Net (p, k, netSize (filt n))
end
handle Error _ => raise Bug "TermNet.filter: should never fail";
fun toString net = "TermNet[" ^ Int.toString (size net) ^ "]";
(* ------------------------------------------------------------------------- *)
(* Specialized fold operations to support matching and unification. *)
(* ------------------------------------------------------------------------- *)
local
fun norm (0 :: ks, (f as (_,n)) :: fs, qtms) =
let
val (a,qtms) = revDivide qtms n
in
addQterm (Fn (f,a)) (ks,fs,qtms)
end
| norm stack = stack
and addQterm qtm (ks,fs,qtms) =
let
val ks = case ks of [] => [] | k :: ks => (k - 1) :: ks
in
norm (ks, fs, qtm :: qtms)
end
and addFn (f as (_,n)) (ks,fs,qtms) = norm (n :: ks, f :: fs, qtms);
in
val stackEmpty = ([],[],[]);
val stackAddQterm = addQterm;
val stackAddFn = addFn;
fun stackValue ([],[],[qtm]) = qtm
| stackValue _ = raise Bug "TermNet.stackValue";
end;
local
fun fold _ acc [] = acc
| fold inc acc ((0,stack,net) :: rest) =
fold inc (inc (stackValue stack, net, acc)) rest
| fold inc acc ((n, stack, Single (qtm,net)) :: rest) =
fold inc acc ((n - 1, stackAddQterm qtm stack, net) :: rest)
| fold inc acc ((n, stack, Multiple (v,fns)) :: rest) =
let
val n = n - 1
val rest =
case v of
NONE => rest
| SOME net => (n, stackAddQterm Var stack, net) :: rest
fun getFns (f as (_,k), net, x) =
(k + n, stackAddFn f stack, net) :: x
in
fold inc acc (NameArityMap.foldr getFns rest fns)
end
| fold _ _ _ = raise Bug "TermNet.foldTerms.fold";
in
fun foldTerms inc acc net = fold inc acc [(1,stackEmpty,net)];
end;
fun foldEqualTerms pat inc acc =
let
fun fold ([],net) = inc (pat,net,acc)
| fold (pat :: pats, Single (qtm,net)) =
if equalQterm pat qtm then fold (pats,net) else acc
| fold (Var :: pats, Multiple (v,_)) =
(case v of NONE => acc | SOME net => fold (pats,net))
| fold (Fn (f,a) :: pats, Multiple (_,fns)) =
(case NameArityMap.peek fns f of
NONE => acc
| SOME net => fold (a @ pats, net))
| fold _ = raise Bug "TermNet.foldEqualTerms.fold";
in
fn net => fold ([pat],net)
end;
local
fun fold _ acc [] = acc
| fold inc acc (([],stack,net) :: rest) =
fold inc (inc (stackValue stack, net, acc)) rest
| fold inc acc ((Var :: pats, stack, net) :: rest) =
let
fun harvest (qtm,n,l) = (pats, stackAddQterm qtm stack, n) :: l
in
fold inc acc (foldTerms harvest rest net)
end
| fold inc acc ((pat :: pats, stack, Single (qtm,net)) :: rest) =
(case unifyQtermQterm pat qtm of
NONE => fold inc acc rest
| SOME qtm =>
fold inc acc ((pats, stackAddQterm qtm stack, net) :: rest))
| fold
inc acc
(((pat as Fn (f,a)) :: pats, stack, Multiple (v,fns)) :: rest) =
let
val rest =
case v of
NONE => rest
| SOME net => (pats, stackAddQterm pat stack, net) :: rest
val rest =
case NameArityMap.peek fns f of
NONE => rest
| SOME net => (a @ pats, stackAddFn f stack, net) :: rest
in
fold inc acc rest
end
| fold _ _ _ = raise Bug "TermNet.foldUnifiableTerms.fold";
in
fun foldUnifiableTerms pat inc acc net =
fold inc acc [([pat],stackEmpty,net)];
end;
(* ------------------------------------------------------------------------- *)
(* Matching and unification queries. *)
(* *)
(* These function return OVER-APPROXIMATIONS! *)
(* Filter afterwards to get the precise set of satisfying values. *)
(* ------------------------------------------------------------------------- *)
local
fun idwise ((m,_),(n,_)) = Int.compare (m,n);
fun fifoize ({fifo, ...} : parameters) l = if fifo then sort idwise l else l;
in
fun finally parm l = List.map snd (fifoize parm l);
end;
local
fun mat acc [] = acc
| mat acc ((Result l, []) :: rest) = mat (l @ acc) rest
| mat acc ((Single (qtm,n), tm :: tms) :: rest) =
mat acc (if matchQtermTerm qtm tm then (n,tms) :: rest else rest)
| mat acc ((Multiple (vs,fs), tm :: tms) :: rest) =
let
val rest = case vs of NONE => rest | SOME n => (n,tms) :: rest
val rest =
case tm of
Term.Var _ => rest
| Term.Fn (f,l) =>
case NameArityMap.peek fs (f, length l) of
NONE => rest
| SOME n => (n, l @ tms) :: rest
in
mat acc rest
end
| mat _ _ = raise Bug "TermNet.match: Match";
in
fun match (Net (_,_,NONE)) _ = []
| match (Net (p, _, SOME (_,n))) tm =
finally p (mat [] [(n,[tm])])
handle Error _ => raise Bug "TermNet.match: should never fail";
end;
local
fun unseenInc qsub v tms (qtm,net,rest) =
(NameMap.insert qsub (v,qtm), net, tms) :: rest;
fun seenInc qsub tms (_,net,rest) = (qsub,net,tms) :: rest;
fun mat acc [] = acc
| mat acc ((_, Result l, []) :: rest) = mat (l @ acc) rest
| mat acc ((qsub, Single (qtm,net), tm :: tms) :: rest) =
(case matchTermQterm qsub tm qtm of
NONE => mat acc rest
| SOME qsub => mat acc ((qsub,net,tms) :: rest))
| mat acc ((qsub, net as Multiple _, Term.Var v :: tms) :: rest) =
(case NameMap.peek qsub v of
NONE => mat acc (foldTerms (unseenInc qsub v tms) rest net)
| SOME qtm => mat acc (foldEqualTerms qtm (seenInc qsub tms) rest net))
| mat acc ((qsub, Multiple (_,fns), Term.Fn (f,a) :: tms) :: rest) =
let
val rest =
case NameArityMap.peek fns (f, length a) of
NONE => rest
| SOME net => (qsub, net, a @ tms) :: rest
in
mat acc rest
end
| mat _ _ = raise Bug "TermNet.matched.mat";
in
fun matched (Net (_,_,NONE)) _ = []
| matched (Net (parm, _, SOME (_,net))) tm =
finally parm (mat [] [(NameMap.new (), net, [tm])])
handle Error _ => raise Bug "TermNet.matched: should never fail";
end;
local
fun inc qsub v tms (qtm,net,rest) =
(NameMap.insert qsub (v,qtm), net, tms) :: rest;
fun mat acc [] = acc
| mat acc ((_, Result l, []) :: rest) = mat (l @ acc) rest
| mat acc ((qsub, Single (qtm,net), tm :: tms) :: rest) =
(case unifyQtermTerm qsub qtm tm of
NONE => mat acc rest
| SOME qsub => mat acc ((qsub,net,tms) :: rest))
| mat acc ((qsub, net as Multiple _, Term.Var v :: tms) :: rest) =
(case NameMap.peek qsub v of
NONE => mat acc (foldTerms (inc qsub v tms) rest net)
| SOME qtm => mat acc (foldUnifiableTerms qtm (inc qsub v tms) rest net))
| mat acc ((qsub, Multiple (v,fns), Term.Fn (f,a) :: tms) :: rest) =
let
val rest = case v of NONE => rest | SOME net => (qsub,net,tms) :: rest
val rest =
case NameArityMap.peek fns (f, length a) of
NONE => rest
| SOME net => (qsub, net, a @ tms) :: rest
in
mat acc rest
end
| mat _ _ = raise Bug "TermNet.unify.mat";
in
fun unify (Net (_,_,NONE)) _ = []
| unify (Net (parm, _, SOME (_,net))) tm =
finally parm (mat [] [(NameMap.new (), net, [tm])])
handle Error _ => raise Bug "TermNet.unify: should never fail";
end;
(* ------------------------------------------------------------------------- *)
(* Pretty printing. *)
(* ------------------------------------------------------------------------- *)
local
fun inc (qtm, Result l, acc) =
List.foldl (fn ((n,a),acc) => (n,(qtm,a)) :: acc) acc l
| inc _ = raise Bug "TermNet.pp.inc";
fun toList (Net (_,_,NONE)) = []
| toList (Net (parm, _, SOME (_,net))) =
finally parm (foldTerms inc [] net);
in
fun pp ppA =
Print.ppMap toList (Print.ppList (Print.ppOp2 " |->" ppQterm ppA));
end;
end