39348
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(* ========================================================================= *)
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(* RANDOM FINITE MODELS *)
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(* Copyright (c) 2003 Joe Hurd, distributed under the GNU GPL version 2 *)
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(* ========================================================================= *)
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structure Model :> Model =
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struct
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open Useful;
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(* ------------------------------------------------------------------------- *)
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(* Constants. *)
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(* ------------------------------------------------------------------------- *)
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val maxSpace = 1000;
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(* ------------------------------------------------------------------------- *)
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(* Helper functions. *)
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(* ------------------------------------------------------------------------- *)
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val multInt =
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case Int.maxInt of
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NONE => (fn x => fn y => SOME (x * y))
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| SOME m =>
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let
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val m = Real.floor (Math.sqrt (Real.fromInt m))
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in
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fn x => fn y => if x <= m andalso y <= m then SOME (x * y) else NONE
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end;
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local
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fun iexp x y acc =
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if y mod 2 = 0 then iexp' x y acc
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else
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case multInt acc x of
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SOME acc => iexp' x y acc
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| NONE => NONE
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and iexp' x y acc =
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if y = 1 then SOME acc
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else
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let
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val y = y div 2
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in
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case multInt x x of
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SOME x => iexp x y acc
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| NONE => NONE
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end;
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in
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fun expInt x y =
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if y <= 1 then
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if y = 0 then SOME 1
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else if y = 1 then SOME x
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else raise Bug "expInt: negative exponent"
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else if x <= 1 then
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if 0 <= x then SOME x
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else raise Bug "expInt: negative exponand"
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else iexp x y 1;
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end;
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fun boolToInt true = 1
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| boolToInt false = 0;
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fun intToBool 1 = true
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| intToBool 0 = false
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| intToBool _ = raise Bug "Model.intToBool";
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fun minMaxInterval i j = interval i (1 + j - i);
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(* ------------------------------------------------------------------------- *)
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(* Model size. *)
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(* ------------------------------------------------------------------------- *)
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type size = {size : int};
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(* ------------------------------------------------------------------------- *)
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(* A model of size N has integer elements 0...N-1. *)
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(* ------------------------------------------------------------------------- *)
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type element = int;
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val zeroElement = 0;
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fun incrementElement {size = N} i =
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let
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val i = i + 1
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in
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if i = N then NONE else SOME i
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end;
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fun elementListSpace {size = N} arity =
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case expInt N arity of
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NONE => NONE
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| s as SOME m => if m <= maxSpace then s else NONE;
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fun elementListIndex {size = N} =
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let
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fun f acc elts =
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case elts of
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[] => acc
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| elt :: elts => f (N * acc + elt) elts
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in
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f 0
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end;
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(* ------------------------------------------------------------------------- *)
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(* The parts of the model that are fixed. *)
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(* ------------------------------------------------------------------------- *)
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type fixedFunction = size -> element list -> element option;
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type fixedRelation = size -> element list -> bool option;
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datatype fixed =
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Fixed of
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{functions : fixedFunction NameArityMap.map,
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relations : fixedRelation NameArityMap.map};
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val uselessFixedFunction : fixedFunction = K (K NONE);
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val uselessFixedRelation : fixedRelation = K (K NONE);
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val emptyFunctions : fixedFunction NameArityMap.map = NameArityMap.new ();
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val emptyRelations : fixedRelation NameArityMap.map = NameArityMap.new ();
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fun fixed0 f sz elts =
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case elts of
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[] => f sz
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| _ => raise Bug "Model.fixed0: wrong arity";
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fun fixed1 f sz elts =
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case elts of
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[x] => f sz x
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| _ => raise Bug "Model.fixed1: wrong arity";
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fun fixed2 f sz elts =
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case elts of
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[x,y] => f sz x y
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| _ => raise Bug "Model.fixed2: wrong arity";
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val emptyFixed =
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let
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val fns = emptyFunctions
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and rels = emptyRelations
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in
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Fixed
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{functions = fns,
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relations = rels}
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end;
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fun peekFunctionFixed fix name_arity =
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let
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val Fixed {functions = fns, ...} = fix
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in
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NameArityMap.peek fns name_arity
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end;
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fun peekRelationFixed fix name_arity =
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let
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val Fixed {relations = rels, ...} = fix
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in
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NameArityMap.peek rels name_arity
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end;
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fun getFunctionFixed fix name_arity =
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case peekFunctionFixed fix name_arity of
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SOME f => f
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| NONE => uselessFixedFunction;
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fun getRelationFixed fix name_arity =
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case peekRelationFixed fix name_arity of
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SOME rel => rel
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| NONE => uselessFixedRelation;
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fun insertFunctionFixed fix name_arity_fn =
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let
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val Fixed {functions = fns, relations = rels} = fix
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val fns = NameArityMap.insert fns name_arity_fn
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in
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Fixed
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{functions = fns,
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relations = rels}
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end;
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fun insertRelationFixed fix name_arity_rel =
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let
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val Fixed {functions = fns, relations = rels} = fix
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val rels = NameArityMap.insert rels name_arity_rel
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in
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Fixed
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{functions = fns,
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relations = rels}
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end;
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local
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fun union _ = raise Bug "Model.unionFixed: nameArity clash";
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in
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fun unionFixed fix1 fix2 =
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let
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val Fixed {functions = fns1, relations = rels1} = fix1
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and Fixed {functions = fns2, relations = rels2} = fix2
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val fns = NameArityMap.union union fns1 fns2
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val rels = NameArityMap.union union rels1 rels2
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in
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Fixed
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{functions = fns,
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relations = rels}
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end;
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end;
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val unionListFixed =
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let
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fun union (fix,acc) = unionFixed acc fix
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in
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List.foldl union emptyFixed
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end;
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local
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fun hasTypeFn _ elts =
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case elts of
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[x,_] => SOME x
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| _ => raise Bug "Model.hasTypeFn: wrong arity";
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fun eqRel _ elts =
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case elts of
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[x,y] => SOME (x = y)
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| _ => raise Bug "Model.eqRel: wrong arity";
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in
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val basicFixed =
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let
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val fns = NameArityMap.singleton (Term.hasTypeFunction,hasTypeFn)
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val rels = NameArityMap.singleton (Atom.eqRelation,eqRel)
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in
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Fixed
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{functions = fns,
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relations = rels}
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end;
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end;
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(* ------------------------------------------------------------------------- *)
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(* Renaming fixed model parts. *)
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(* ------------------------------------------------------------------------- *)
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type fixedMap =
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{functionMap : Name.name NameArityMap.map,
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relationMap : Name.name NameArityMap.map};
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fun mapFixed fixMap fix =
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let
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val {functionMap = fnMap, relationMap = relMap} = fixMap
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and Fixed {functions = fns, relations = rels} = fix
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val fns = NameArityMap.compose fnMap fns
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val rels = NameArityMap.compose relMap rels
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in
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Fixed
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{functions = fns,
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relations = rels}
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end;
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local
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fun mkEntry tag (na,n) = (tag,na,n);
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fun mkList tag m = map (mkEntry tag) (NameArityMap.toList m);
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fun ppEntry (tag,source_arity,target) =
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Print.blockProgram Print.Inconsistent 2
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[Print.addString tag,
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Print.addBreak 1,
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NameArity.pp source_arity,
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Print.addString " ->",
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Print.addBreak 1,
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Name.pp target];
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in
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fun ppFixedMap fixMap =
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let
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val {functionMap = fnMap, relationMap = relMap} = fixMap
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in
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case mkList "function" fnMap @ mkList "relation" relMap of
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[] => Print.skip
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| entry :: entries =>
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Print.blockProgram Print.Consistent 0
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(ppEntry entry ::
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map (Print.sequence Print.addNewline o ppEntry) entries)
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end;
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end;
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(* ------------------------------------------------------------------------- *)
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(* Standard fixed model parts. *)
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(* ------------------------------------------------------------------------- *)
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(* Projections *)
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val projectionMin = 1
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and projectionMax = 9;
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val projectionList = minMaxInterval projectionMin projectionMax;
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fun projectionName i =
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let
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val _ = projectionMin <= i orelse
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raise Bug "Model.projectionName: less than projectionMin"
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val _ = i <= projectionMax orelse
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raise Bug "Model.projectionName: greater than projectionMax"
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in
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Name.fromString ("project" ^ Int.toString i)
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end;
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fun projectionFn i _ elts = SOME (List.nth (elts, i - 1));
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fun arityProjectionFixed arity =
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let
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fun mkProj i = ((projectionName i, arity), projectionFn i)
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fun addProj i acc =
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if i > arity then acc
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else addProj (i + 1) (NameArityMap.insert acc (mkProj i))
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val fns = addProj projectionMin emptyFunctions
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val rels = emptyRelations
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in
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Fixed
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{functions = fns,
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relations = rels}
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end;
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val projectionFixed =
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unionListFixed (map arityProjectionFixed projectionList);
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(* Arithmetic *)
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val numeralMin = ~100
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and numeralMax = 100;
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val numeralList = minMaxInterval numeralMin numeralMax;
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fun numeralName i =
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let
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val _ = numeralMin <= i orelse
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raise Bug "Model.numeralName: less than numeralMin"
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val _ = i <= numeralMax orelse
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raise Bug "Model.numeralName: greater than numeralMax"
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val s = if i < 0 then "negative" ^ Int.toString (~i) else Int.toString i
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in
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Name.fromString s
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end;
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val addName = Name.fromString "+"
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and divName = Name.fromString "div"
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and dividesName = Name.fromString "divides"
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and evenName = Name.fromString "even"
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and expName = Name.fromString "exp"
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and geName = Name.fromString ">="
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and gtName = Name.fromString ">"
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and isZeroName = Name.fromString "isZero"
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and leName = Name.fromString "<="
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and ltName = Name.fromString "<"
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and modName = Name.fromString "mod"
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and multName = Name.fromString "*"
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and negName = Name.fromString "~"
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and oddName = Name.fromString "odd"
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and preName = Name.fromString "pre"
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and subName = Name.fromString "-"
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and sucName = Name.fromString "suc";
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local
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(* Support *)
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fun modN {size = N} x = x mod N;
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fun oneN sz = modN sz 1;
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fun multN sz (x,y) = modN sz (x * y);
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(* Functions *)
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fun numeralFn i sz = SOME (modN sz i);
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fun addFn sz x y = SOME (modN sz (x + y));
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fun divFn {size = N} x y =
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let
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val y = if y = 0 then N else y
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in
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SOME (x div y)
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end;
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fun expFn sz x y = SOME (exp (multN sz) x y (oneN sz));
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fun modFn {size = N} x y =
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let
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val y = if y = 0 then N else y
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in
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SOME (x mod y)
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end;
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fun multFn sz x y = SOME (multN sz (x,y));
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fun negFn {size = N} x = SOME (if x = 0 then 0 else N - x);
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fun preFn {size = N} x = SOME (if x = 0 then N - 1 else x - 1);
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fun subFn {size = N} x y = SOME (if x < y then N + x - y else x - y);
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fun sucFn {size = N} x = SOME (if x = N - 1 then 0 else x + 1);
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(* Relations *)
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fun dividesRel _ x y = SOME (divides x y);
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fun evenRel _ x = SOME (x mod 2 = 0);
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fun geRel _ x y = SOME (x >= y);
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fun gtRel _ x y = SOME (x > y);
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fun isZeroRel _ x = SOME (x = 0);
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fun leRel _ x y = SOME (x <= y);
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fun ltRel _ x y = SOME (x < y);
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fun oddRel _ x = SOME (x mod 2 = 1);
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in
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val modularFixed =
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let
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val fns =
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NameArityMap.fromList
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(map (fn i => ((numeralName i,0), fixed0 (numeralFn i)))
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numeralList @
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[((addName,2), fixed2 addFn),
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((divName,2), fixed2 divFn),
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((expName,2), fixed2 expFn),
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((modName,2), fixed2 modFn),
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((multName,2), fixed2 multFn),
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((negName,1), fixed1 negFn),
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((preName,1), fixed1 preFn),
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((subName,2), fixed2 subFn),
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((sucName,1), fixed1 sucFn)])
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val rels =
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NameArityMap.fromList
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[((dividesName,2), fixed2 dividesRel),
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((evenName,1), fixed1 evenRel),
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((geName,2), fixed2 geRel),
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((gtName,2), fixed2 gtRel),
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((isZeroName,1), fixed1 isZeroRel),
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((leName,2), fixed2 leRel),
|
|
460 |
((ltName,2), fixed2 ltRel),
|
|
461 |
((oddName,1), fixed1 oddRel)]
|
|
462 |
in
|
|
463 |
Fixed
|
|
464 |
{functions = fns,
|
|
465 |
relations = rels}
|
|
466 |
end;
|
|
467 |
end;
|
|
468 |
|
|
469 |
local
|
|
470 |
(* Support *)
|
|
471 |
|
|
472 |
fun cutN {size = N} x = if x >= N then N - 1 else x;
|
|
473 |
|
|
474 |
fun oneN sz = cutN sz 1;
|
|
475 |
|
|
476 |
fun multN sz (x,y) = cutN sz (x * y);
|
|
477 |
|
|
478 |
(* Functions *)
|
|
479 |
|
|
480 |
fun numeralFn i sz = if i < 0 then NONE else SOME (cutN sz i);
|
|
481 |
|
|
482 |
fun addFn sz x y = SOME (cutN sz (x + y));
|
|
483 |
|
|
484 |
fun divFn _ x y = if y = 0 then NONE else SOME (x div y);
|
|
485 |
|
|
486 |
fun expFn sz x y = SOME (exp (multN sz) x y (oneN sz));
|
|
487 |
|
|
488 |
fun modFn {size = N} x y =
|
|
489 |
if y = 0 orelse x = N - 1 then NONE else SOME (x mod y);
|
|
490 |
|
|
491 |
fun multFn sz x y = SOME (multN sz (x,y));
|
|
492 |
|
|
493 |
fun negFn _ x = if x = 0 then SOME 0 else NONE;
|
|
494 |
|
|
495 |
fun preFn _ x = if x = 0 then NONE else SOME (x - 1);
|
|
496 |
|
|
497 |
fun subFn {size = N} x y =
|
|
498 |
if y = 0 then SOME x
|
|
499 |
else if x = N - 1 orelse x < y then NONE
|
|
500 |
else SOME (x - y);
|
|
501 |
|
|
502 |
fun sucFn sz x = SOME (cutN sz (x + 1));
|
|
503 |
|
|
504 |
(* Relations *)
|
|
505 |
|
|
506 |
fun dividesRel {size = N} x y =
|
|
507 |
if x = 1 orelse y = 0 then SOME true
|
|
508 |
else if x = 0 then SOME false
|
|
509 |
else if y = N - 1 then NONE
|
|
510 |
else SOME (divides x y);
|
|
511 |
|
|
512 |
fun evenRel {size = N} x =
|
|
513 |
if x = N - 1 then NONE else SOME (x mod 2 = 0);
|
|
514 |
|
|
515 |
fun geRel {size = N} y x =
|
|
516 |
if x = N - 1 then if y = N - 1 then NONE else SOME false
|
|
517 |
else if y = N - 1 then SOME true else SOME (x <= y);
|
|
518 |
|
|
519 |
fun gtRel {size = N} y x =
|
|
520 |
if x = N - 1 then if y = N - 1 then NONE else SOME false
|
|
521 |
else if y = N - 1 then SOME true else SOME (x < y);
|
|
522 |
|
|
523 |
fun isZeroRel _ x = SOME (x = 0);
|
|
524 |
|
|
525 |
fun leRel {size = N} x y =
|
|
526 |
if x = N - 1 then if y = N - 1 then NONE else SOME false
|
|
527 |
else if y = N - 1 then SOME true else SOME (x <= y);
|
|
528 |
|
|
529 |
fun ltRel {size = N} x y =
|
|
530 |
if x = N - 1 then if y = N - 1 then NONE else SOME false
|
|
531 |
else if y = N - 1 then SOME true else SOME (x < y);
|
|
532 |
|
|
533 |
fun oddRel {size = N} x =
|
|
534 |
if x = N - 1 then NONE else SOME (x mod 2 = 1);
|
|
535 |
in
|
|
536 |
val overflowFixed =
|
|
537 |
let
|
|
538 |
val fns =
|
|
539 |
NameArityMap.fromList
|
|
540 |
(map (fn i => ((numeralName i,0), fixed0 (numeralFn i)))
|
|
541 |
numeralList @
|
|
542 |
[((addName,2), fixed2 addFn),
|
|
543 |
((divName,2), fixed2 divFn),
|
|
544 |
((expName,2), fixed2 expFn),
|
|
545 |
((modName,2), fixed2 modFn),
|
|
546 |
((multName,2), fixed2 multFn),
|
|
547 |
((negName,1), fixed1 negFn),
|
|
548 |
((preName,1), fixed1 preFn),
|
|
549 |
((subName,2), fixed2 subFn),
|
|
550 |
((sucName,1), fixed1 sucFn)])
|
|
551 |
|
|
552 |
val rels =
|
|
553 |
NameArityMap.fromList
|
|
554 |
[((dividesName,2), fixed2 dividesRel),
|
|
555 |
((evenName,1), fixed1 evenRel),
|
|
556 |
((geName,2), fixed2 geRel),
|
|
557 |
((gtName,2), fixed2 gtRel),
|
|
558 |
((isZeroName,1), fixed1 isZeroRel),
|
|
559 |
((leName,2), fixed2 leRel),
|
|
560 |
((ltName,2), fixed2 ltRel),
|
|
561 |
((oddName,1), fixed1 oddRel)]
|
|
562 |
in
|
|
563 |
Fixed
|
|
564 |
{functions = fns,
|
|
565 |
relations = rels}
|
|
566 |
end;
|
|
567 |
end;
|
|
568 |
|
|
569 |
(* Sets *)
|
|
570 |
|
|
571 |
val cardName = Name.fromString "card"
|
|
572 |
and complementName = Name.fromString "complement"
|
|
573 |
and differenceName = Name.fromString "difference"
|
|
574 |
and emptyName = Name.fromString "empty"
|
|
575 |
and memberName = Name.fromString "member"
|
|
576 |
and insertName = Name.fromString "insert"
|
|
577 |
and intersectName = Name.fromString "intersect"
|
|
578 |
and singletonName = Name.fromString "singleton"
|
|
579 |
and subsetName = Name.fromString "subset"
|
|
580 |
and symmetricDifferenceName = Name.fromString "symmetricDifference"
|
|
581 |
and unionName = Name.fromString "union"
|
|
582 |
and universeName = Name.fromString "universe";
|
|
583 |
|
|
584 |
local
|
|
585 |
(* Support *)
|
|
586 |
|
|
587 |
fun eltN {size = N} =
|
|
588 |
let
|
|
589 |
fun f 0 acc = acc
|
|
590 |
| f x acc = f (x div 2) (acc + 1)
|
|
591 |
in
|
|
592 |
f N ~1
|
|
593 |
end;
|
|
594 |
|
|
595 |
fun posN i = Word.<< (0w1, Word.fromInt i);
|
|
596 |
|
|
597 |
fun univN sz = Word.- (posN (eltN sz), 0w1);
|
|
598 |
|
|
599 |
fun setN sz x = Word.andb (Word.fromInt x, univN sz);
|
|
600 |
|
|
601 |
(* Functions *)
|
|
602 |
|
|
603 |
fun cardFn sz x =
|
|
604 |
let
|
|
605 |
fun f 0w0 acc = acc
|
|
606 |
| f s acc =
|
|
607 |
let
|
|
608 |
val acc = if Word.andb (s,0w1) = 0w0 then acc else acc + 1
|
|
609 |
in
|
|
610 |
f (Word.>> (s,0w1)) acc
|
|
611 |
end
|
|
612 |
in
|
|
613 |
SOME (f (setN sz x) 0)
|
|
614 |
end;
|
|
615 |
|
|
616 |
fun complementFn sz x = SOME (Word.toInt (Word.xorb (univN sz, setN sz x)));
|
|
617 |
|
|
618 |
fun differenceFn sz x y =
|
|
619 |
let
|
|
620 |
val x = setN sz x
|
|
621 |
and y = setN sz y
|
|
622 |
in
|
|
623 |
SOME (Word.toInt (Word.andb (x, Word.notb y)))
|
|
624 |
end;
|
|
625 |
|
|
626 |
fun emptyFn _ = SOME 0;
|
|
627 |
|
|
628 |
fun insertFn sz x y =
|
|
629 |
let
|
|
630 |
val x = x mod eltN sz
|
|
631 |
and y = setN sz y
|
|
632 |
in
|
|
633 |
SOME (Word.toInt (Word.orb (posN x, y)))
|
|
634 |
end;
|
|
635 |
|
|
636 |
fun intersectFn sz x y =
|
|
637 |
SOME (Word.toInt (Word.andb (setN sz x, setN sz y)));
|
|
638 |
|
|
639 |
fun singletonFn sz x =
|
|
640 |
let
|
|
641 |
val x = x mod eltN sz
|
|
642 |
in
|
|
643 |
SOME (Word.toInt (posN x))
|
|
644 |
end;
|
|
645 |
|
|
646 |
fun symmetricDifferenceFn sz x y =
|
|
647 |
let
|
|
648 |
val x = setN sz x
|
|
649 |
and y = setN sz y
|
|
650 |
in
|
|
651 |
SOME (Word.toInt (Word.xorb (x,y)))
|
|
652 |
end;
|
|
653 |
|
|
654 |
fun unionFn sz x y =
|
|
655 |
SOME (Word.toInt (Word.orb (setN sz x, setN sz y)));
|
|
656 |
|
|
657 |
fun universeFn sz = SOME (Word.toInt (univN sz));
|
|
658 |
|
|
659 |
(* Relations *)
|
|
660 |
|
|
661 |
fun memberRel sz x y =
|
|
662 |
let
|
|
663 |
val x = x mod eltN sz
|
|
664 |
and y = setN sz y
|
|
665 |
in
|
|
666 |
SOME (Word.andb (posN x, y) <> 0w0)
|
|
667 |
end;
|
|
668 |
|
|
669 |
fun subsetRel sz x y =
|
|
670 |
let
|
|
671 |
val x = setN sz x
|
|
672 |
and y = setN sz y
|
|
673 |
in
|
|
674 |
SOME (Word.andb (x, Word.notb y) = 0w0)
|
|
675 |
end;
|
|
676 |
in
|
|
677 |
val setFixed =
|
|
678 |
let
|
|
679 |
val fns =
|
|
680 |
NameArityMap.fromList
|
|
681 |
[((cardName,1), fixed1 cardFn),
|
|
682 |
((complementName,1), fixed1 complementFn),
|
|
683 |
((differenceName,2), fixed2 differenceFn),
|
|
684 |
((emptyName,0), fixed0 emptyFn),
|
|
685 |
((insertName,2), fixed2 insertFn),
|
|
686 |
((intersectName,2), fixed2 intersectFn),
|
|
687 |
((singletonName,1), fixed1 singletonFn),
|
|
688 |
((symmetricDifferenceName,2), fixed2 symmetricDifferenceFn),
|
|
689 |
((unionName,2), fixed2 unionFn),
|
|
690 |
((universeName,0), fixed0 universeFn)]
|
|
691 |
|
|
692 |
val rels =
|
|
693 |
NameArityMap.fromList
|
|
694 |
[((memberName,2), fixed2 memberRel),
|
|
695 |
((subsetName,2), fixed2 subsetRel)]
|
|
696 |
in
|
|
697 |
Fixed
|
|
698 |
{functions = fns,
|
|
699 |
relations = rels}
|
|
700 |
end;
|
|
701 |
end;
|
|
702 |
|
|
703 |
(* Lists *)
|
|
704 |
|
|
705 |
val appendName = Name.fromString "@"
|
|
706 |
and consName = Name.fromString "::"
|
|
707 |
and lengthName = Name.fromString "length"
|
|
708 |
and nilName = Name.fromString "nil"
|
|
709 |
and nullName = Name.fromString "null"
|
|
710 |
and tailName = Name.fromString "tail";
|
|
711 |
|
|
712 |
local
|
|
713 |
val baseFix =
|
|
714 |
let
|
|
715 |
val fix = unionFixed projectionFixed overflowFixed
|
|
716 |
|
|
717 |
val sucFn = getFunctionFixed fix (sucName,1)
|
|
718 |
|
|
719 |
fun suc2Fn sz _ x = sucFn sz [x]
|
|
720 |
in
|
|
721 |
insertFunctionFixed fix ((sucName,2), fixed2 suc2Fn)
|
|
722 |
end;
|
|
723 |
|
|
724 |
val fixMap =
|
|
725 |
{functionMap = NameArityMap.fromList
|
|
726 |
[((appendName,2),addName),
|
|
727 |
((consName,2),sucName),
|
|
728 |
((lengthName,1), projectionName 1),
|
|
729 |
((nilName,0), numeralName 0),
|
|
730 |
((tailName,1),preName)],
|
|
731 |
relationMap = NameArityMap.fromList
|
|
732 |
[((nullName,1),isZeroName)]};
|
|
733 |
|
|
734 |
in
|
|
735 |
val listFixed = mapFixed fixMap baseFix;
|
|
736 |
end;
|
|
737 |
|
|
738 |
(* ------------------------------------------------------------------------- *)
|
|
739 |
(* Valuations. *)
|
|
740 |
(* ------------------------------------------------------------------------- *)
|
|
741 |
|
|
742 |
datatype valuation = Valuation of element NameMap.map;
|
|
743 |
|
|
744 |
val emptyValuation = Valuation (NameMap.new ());
|
|
745 |
|
|
746 |
fun insertValuation (Valuation m) v_i = Valuation (NameMap.insert m v_i);
|
|
747 |
|
|
748 |
fun peekValuation (Valuation m) v = NameMap.peek m v;
|
|
749 |
|
|
750 |
fun constantValuation i =
|
|
751 |
let
|
|
752 |
fun add (v,V) = insertValuation V (v,i)
|
|
753 |
in
|
|
754 |
NameSet.foldl add emptyValuation
|
|
755 |
end;
|
|
756 |
|
|
757 |
val zeroValuation = constantValuation zeroElement;
|
|
758 |
|
|
759 |
fun getValuation V v =
|
|
760 |
case peekValuation V v of
|
|
761 |
SOME i => i
|
|
762 |
| NONE => raise Error "Model.getValuation: incomplete valuation";
|
|
763 |
|
|
764 |
fun randomValuation {size = N} vs =
|
|
765 |
let
|
|
766 |
fun f (v,V) = insertValuation V (v, Portable.randomInt N)
|
|
767 |
in
|
|
768 |
NameSet.foldl f emptyValuation vs
|
|
769 |
end;
|
|
770 |
|
|
771 |
fun incrementValuation N vars =
|
|
772 |
let
|
|
773 |
fun inc vs V =
|
|
774 |
case vs of
|
|
775 |
[] => NONE
|
|
776 |
| v :: vs =>
|
|
777 |
let
|
|
778 |
val (carry,i) =
|
|
779 |
case incrementElement N (getValuation V v) of
|
|
780 |
SOME i => (false,i)
|
|
781 |
| NONE => (true,zeroElement)
|
|
782 |
|
|
783 |
val V = insertValuation V (v,i)
|
|
784 |
in
|
|
785 |
if carry then inc vs V else SOME V
|
|
786 |
end
|
|
787 |
in
|
|
788 |
inc (NameSet.toList vars)
|
|
789 |
end;
|
|
790 |
|
|
791 |
fun foldValuation N vars f =
|
|
792 |
let
|
|
793 |
val inc = incrementValuation N vars
|
|
794 |
|
|
795 |
fun fold V acc =
|
|
796 |
let
|
|
797 |
val acc = f (V,acc)
|
|
798 |
in
|
|
799 |
case inc V of
|
|
800 |
NONE => acc
|
|
801 |
| SOME V => fold V acc
|
|
802 |
end
|
|
803 |
|
|
804 |
val zero = zeroValuation vars
|
|
805 |
in
|
|
806 |
fold zero
|
|
807 |
end;
|
|
808 |
|
|
809 |
(* ------------------------------------------------------------------------- *)
|
|
810 |
(* A type of random finite mapping Z^n -> Z. *)
|
|
811 |
(* ------------------------------------------------------------------------- *)
|
|
812 |
|
|
813 |
val UNKNOWN = ~1;
|
|
814 |
|
|
815 |
datatype table =
|
|
816 |
ForgetfulTable
|
|
817 |
| ArrayTable of int Array.array;
|
|
818 |
|
|
819 |
fun newTable N arity =
|
|
820 |
case elementListSpace {size = N} arity of
|
|
821 |
NONE => ForgetfulTable
|
|
822 |
| SOME space => ArrayTable (Array.array (space,UNKNOWN));
|
|
823 |
|
|
824 |
local
|
|
825 |
fun randomResult R = Portable.randomInt R;
|
|
826 |
in
|
|
827 |
fun lookupTable N R table elts =
|
|
828 |
case table of
|
|
829 |
ForgetfulTable => randomResult R
|
|
830 |
| ArrayTable a =>
|
|
831 |
let
|
|
832 |
val i = elementListIndex {size = N} elts
|
|
833 |
|
|
834 |
val r = Array.sub (a,i)
|
|
835 |
in
|
|
836 |
if r <> UNKNOWN then r
|
|
837 |
else
|
|
838 |
let
|
|
839 |
val r = randomResult R
|
|
840 |
|
|
841 |
val () = Array.update (a,i,r)
|
|
842 |
in
|
|
843 |
r
|
|
844 |
end
|
|
845 |
end;
|
|
846 |
end;
|
|
847 |
|
|
848 |
fun updateTable N table (elts,r) =
|
|
849 |
case table of
|
|
850 |
ForgetfulTable => ()
|
|
851 |
| ArrayTable a =>
|
|
852 |
let
|
|
853 |
val i = elementListIndex {size = N} elts
|
|
854 |
|
|
855 |
val () = Array.update (a,i,r)
|
|
856 |
in
|
|
857 |
()
|
|
858 |
end;
|
|
859 |
|
|
860 |
(* ------------------------------------------------------------------------- *)
|
|
861 |
(* A type of random finite mappings name * arity -> Z^arity -> Z. *)
|
|
862 |
(* ------------------------------------------------------------------------- *)
|
|
863 |
|
|
864 |
datatype tables =
|
|
865 |
Tables of
|
|
866 |
{domainSize : int,
|
|
867 |
rangeSize : int,
|
|
868 |
tableMap : table NameArityMap.map ref};
|
|
869 |
|
|
870 |
fun newTables N R =
|
|
871 |
Tables
|
|
872 |
{domainSize = N,
|
|
873 |
rangeSize = R,
|
|
874 |
tableMap = ref (NameArityMap.new ())};
|
|
875 |
|
|
876 |
fun getTables tables n_a =
|
|
877 |
let
|
|
878 |
val Tables {domainSize = N, rangeSize = _, tableMap = tm} = tables
|
|
879 |
|
|
880 |
val ref m = tm
|
|
881 |
in
|
|
882 |
case NameArityMap.peek m n_a of
|
|
883 |
SOME t => t
|
|
884 |
| NONE =>
|
|
885 |
let
|
|
886 |
val (_,a) = n_a
|
|
887 |
|
|
888 |
val t = newTable N a
|
|
889 |
|
|
890 |
val m = NameArityMap.insert m (n_a,t)
|
|
891 |
|
|
892 |
val () = tm := m
|
|
893 |
in
|
|
894 |
t
|
|
895 |
end
|
|
896 |
end;
|
|
897 |
|
|
898 |
fun lookupTables tables (n,elts) =
|
|
899 |
let
|
|
900 |
val Tables {domainSize = N, rangeSize = R, ...} = tables
|
|
901 |
|
|
902 |
val a = length elts
|
|
903 |
|
|
904 |
val table = getTables tables (n,a)
|
|
905 |
in
|
|
906 |
lookupTable N R table elts
|
|
907 |
end;
|
|
908 |
|
|
909 |
fun updateTables tables ((n,elts),r) =
|
|
910 |
let
|
|
911 |
val Tables {domainSize = N, ...} = tables
|
|
912 |
|
|
913 |
val a = length elts
|
|
914 |
|
|
915 |
val table = getTables tables (n,a)
|
|
916 |
in
|
|
917 |
updateTable N table (elts,r)
|
|
918 |
end;
|
|
919 |
|
|
920 |
(* ------------------------------------------------------------------------- *)
|
|
921 |
(* A type of random finite models. *)
|
|
922 |
(* ------------------------------------------------------------------------- *)
|
|
923 |
|
|
924 |
type parameters = {size : int, fixed : fixed};
|
|
925 |
|
|
926 |
datatype model =
|
|
927 |
Model of
|
|
928 |
{size : int,
|
|
929 |
fixedFunctions : (element list -> element option) NameArityMap.map,
|
|
930 |
fixedRelations : (element list -> bool option) NameArityMap.map,
|
|
931 |
randomFunctions : tables,
|
|
932 |
randomRelations : tables};
|
|
933 |
|
|
934 |
fun new {size = N, fixed} =
|
|
935 |
let
|
|
936 |
val Fixed {functions = fns, relations = rels} = fixed
|
|
937 |
|
|
938 |
val fixFns = NameArityMap.transform (fn f => f {size = N}) fns
|
|
939 |
and fixRels = NameArityMap.transform (fn r => r {size = N}) rels
|
|
940 |
|
|
941 |
val rndFns = newTables N N
|
|
942 |
and rndRels = newTables N 2
|
|
943 |
in
|
|
944 |
Model
|
|
945 |
{size = N,
|
|
946 |
fixedFunctions = fixFns,
|
|
947 |
fixedRelations = fixRels,
|
|
948 |
randomFunctions = rndFns,
|
|
949 |
randomRelations = rndRels}
|
|
950 |
end;
|
|
951 |
|
|
952 |
fun size (Model {size = N, ...}) = N;
|
|
953 |
|
|
954 |
fun peekFixedFunction M (n,elts) =
|
|
955 |
let
|
|
956 |
val Model {fixedFunctions = fixFns, ...} = M
|
|
957 |
in
|
|
958 |
case NameArityMap.peek fixFns (n, length elts) of
|
|
959 |
NONE => NONE
|
|
960 |
| SOME fixFn => fixFn elts
|
|
961 |
end;
|
|
962 |
|
|
963 |
fun isFixedFunction M n_elts = Option.isSome (peekFixedFunction M n_elts);
|
|
964 |
|
|
965 |
fun peekFixedRelation M (n,elts) =
|
|
966 |
let
|
|
967 |
val Model {fixedRelations = fixRels, ...} = M
|
|
968 |
in
|
|
969 |
case NameArityMap.peek fixRels (n, length elts) of
|
|
970 |
NONE => NONE
|
|
971 |
| SOME fixRel => fixRel elts
|
|
972 |
end;
|
|
973 |
|
|
974 |
fun isFixedRelation M n_elts = Option.isSome (peekFixedRelation M n_elts);
|
|
975 |
|
|
976 |
(* A default model *)
|
|
977 |
|
|
978 |
val defaultSize = 8;
|
|
979 |
|
|
980 |
val defaultFixed =
|
|
981 |
unionListFixed
|
|
982 |
[basicFixed,
|
|
983 |
projectionFixed,
|
|
984 |
modularFixed,
|
|
985 |
setFixed,
|
|
986 |
listFixed];
|
|
987 |
|
|
988 |
val default = {size = defaultSize, fixed = defaultFixed};
|
|
989 |
|
|
990 |
(* ------------------------------------------------------------------------- *)
|
|
991 |
(* Taking apart terms to interpret them. *)
|
|
992 |
(* ------------------------------------------------------------------------- *)
|
|
993 |
|
|
994 |
fun destTerm tm =
|
|
995 |
case tm of
|
|
996 |
Term.Var _ => tm
|
|
997 |
| Term.Fn f_tms =>
|
|
998 |
case Term.stripApp tm of
|
|
999 |
(_,[]) => tm
|
|
1000 |
| (v as Term.Var _, tms) => Term.Fn (Term.appName, v :: tms)
|
|
1001 |
| (Term.Fn (f,tms), tms') => Term.Fn (f, tms @ tms');
|
|
1002 |
|
|
1003 |
(* ------------------------------------------------------------------------- *)
|
|
1004 |
(* Interpreting terms and formulas in the model. *)
|
|
1005 |
(* ------------------------------------------------------------------------- *)
|
|
1006 |
|
|
1007 |
fun interpretFunction M n_elts =
|
|
1008 |
case peekFixedFunction M n_elts of
|
|
1009 |
SOME r => r
|
|
1010 |
| NONE =>
|
|
1011 |
let
|
|
1012 |
val Model {randomFunctions = rndFns, ...} = M
|
|
1013 |
in
|
|
1014 |
lookupTables rndFns n_elts
|
|
1015 |
end;
|
|
1016 |
|
|
1017 |
fun interpretRelation M n_elts =
|
|
1018 |
case peekFixedRelation M n_elts of
|
|
1019 |
SOME r => r
|
|
1020 |
| NONE =>
|
|
1021 |
let
|
|
1022 |
val Model {randomRelations = rndRels, ...} = M
|
|
1023 |
in
|
|
1024 |
intToBool (lookupTables rndRels n_elts)
|
|
1025 |
end;
|
|
1026 |
|
|
1027 |
fun interpretTerm M V =
|
|
1028 |
let
|
|
1029 |
fun interpret tm =
|
|
1030 |
case destTerm tm of
|
|
1031 |
Term.Var v => getValuation V v
|
|
1032 |
| Term.Fn (f,tms) => interpretFunction M (f, map interpret tms)
|
|
1033 |
in
|
|
1034 |
interpret
|
|
1035 |
end;
|
|
1036 |
|
|
1037 |
fun interpretAtom M V (r,tms) =
|
|
1038 |
interpretRelation M (r, map (interpretTerm M V) tms);
|
|
1039 |
|
|
1040 |
fun interpretFormula M =
|
|
1041 |
let
|
|
1042 |
val N = size M
|
|
1043 |
|
|
1044 |
fun interpret V fm =
|
|
1045 |
case fm of
|
|
1046 |
Formula.True => true
|
|
1047 |
| Formula.False => false
|
|
1048 |
| Formula.Atom atm => interpretAtom M V atm
|
|
1049 |
| Formula.Not p => not (interpret V p)
|
|
1050 |
| Formula.Or (p,q) => interpret V p orelse interpret V q
|
|
1051 |
| Formula.And (p,q) => interpret V p andalso interpret V q
|
|
1052 |
| Formula.Imp (p,q) => interpret V (Formula.Or (Formula.Not p, q))
|
|
1053 |
| Formula.Iff (p,q) => interpret V p = interpret V q
|
|
1054 |
| Formula.Forall (v,p) => interpret' V p v N
|
|
1055 |
| Formula.Exists (v,p) =>
|
|
1056 |
interpret V (Formula.Not (Formula.Forall (v, Formula.Not p)))
|
|
1057 |
|
|
1058 |
and interpret' V fm v i =
|
|
1059 |
i = 0 orelse
|
|
1060 |
let
|
|
1061 |
val i = i - 1
|
|
1062 |
val V' = insertValuation V (v,i)
|
|
1063 |
in
|
|
1064 |
interpret V' fm andalso interpret' V fm v i
|
|
1065 |
end
|
|
1066 |
in
|
|
1067 |
interpret
|
|
1068 |
end;
|
|
1069 |
|
|
1070 |
fun interpretLiteral M V (pol,atm) =
|
|
1071 |
let
|
|
1072 |
val b = interpretAtom M V atm
|
|
1073 |
in
|
|
1074 |
if pol then b else not b
|
|
1075 |
end;
|
|
1076 |
|
|
1077 |
fun interpretClause M V cl = LiteralSet.exists (interpretLiteral M V) cl;
|
|
1078 |
|
|
1079 |
(* ------------------------------------------------------------------------- *)
|
|
1080 |
(* Check whether random groundings of a formula are true in the model. *)
|
|
1081 |
(* Note: if it's cheaper, a systematic check will be performed instead. *)
|
|
1082 |
(* ------------------------------------------------------------------------- *)
|
|
1083 |
|
|
1084 |
fun check interpret {maxChecks} M fv x =
|
|
1085 |
let
|
|
1086 |
val N = size M
|
|
1087 |
|
|
1088 |
fun score (V,{T,F}) =
|
|
1089 |
if interpret M V x then {T = T + 1, F = F} else {T = T, F = F + 1}
|
|
1090 |
|
|
1091 |
fun randomCheck acc = score (randomValuation {size = N} fv, acc)
|
|
1092 |
|
|
1093 |
val maxChecks =
|
|
1094 |
case maxChecks of
|
|
1095 |
NONE => maxChecks
|
|
1096 |
| SOME m =>
|
|
1097 |
case expInt N (NameSet.size fv) of
|
|
1098 |
SOME n => if n <= m then NONE else maxChecks
|
|
1099 |
| NONE => maxChecks
|
|
1100 |
in
|
|
1101 |
case maxChecks of
|
|
1102 |
SOME m => funpow m randomCheck {T = 0, F = 0}
|
|
1103 |
| NONE => foldValuation {size = N} fv score {T = 0, F = 0}
|
|
1104 |
end;
|
|
1105 |
|
|
1106 |
fun checkAtom maxChecks M atm =
|
|
1107 |
check interpretAtom maxChecks M (Atom.freeVars atm) atm;
|
|
1108 |
|
|
1109 |
fun checkFormula maxChecks M fm =
|
|
1110 |
check interpretFormula maxChecks M (Formula.freeVars fm) fm;
|
|
1111 |
|
|
1112 |
fun checkLiteral maxChecks M lit =
|
|
1113 |
check interpretLiteral maxChecks M (Literal.freeVars lit) lit;
|
|
1114 |
|
|
1115 |
fun checkClause maxChecks M cl =
|
|
1116 |
check interpretClause maxChecks M (LiteralSet.freeVars cl) cl;
|
|
1117 |
|
|
1118 |
(* ------------------------------------------------------------------------- *)
|
|
1119 |
(* Updating the model. *)
|
|
1120 |
(* ------------------------------------------------------------------------- *)
|
|
1121 |
|
|
1122 |
fun updateFunction M func_elts_elt =
|
|
1123 |
let
|
|
1124 |
val Model {randomFunctions = rndFns, ...} = M
|
|
1125 |
|
|
1126 |
val () = updateTables rndFns func_elts_elt
|
|
1127 |
in
|
|
1128 |
()
|
|
1129 |
end;
|
|
1130 |
|
|
1131 |
fun updateRelation M (rel_elts,pol) =
|
|
1132 |
let
|
|
1133 |
val Model {randomRelations = rndRels, ...} = M
|
|
1134 |
|
|
1135 |
val () = updateTables rndRels (rel_elts, boolToInt pol)
|
|
1136 |
in
|
|
1137 |
()
|
|
1138 |
end;
|
|
1139 |
|
|
1140 |
(* ------------------------------------------------------------------------- *)
|
|
1141 |
(* A type of terms with interpretations embedded in the subterms. *)
|
|
1142 |
(* ------------------------------------------------------------------------- *)
|
|
1143 |
|
|
1144 |
datatype modelTerm =
|
|
1145 |
ModelVar
|
|
1146 |
| ModelFn of Term.functionName * modelTerm list * int list;
|
|
1147 |
|
|
1148 |
fun modelTerm M V =
|
|
1149 |
let
|
|
1150 |
fun modelTm tm =
|
|
1151 |
case destTerm tm of
|
|
1152 |
Term.Var v => (ModelVar, getValuation V v)
|
|
1153 |
| Term.Fn (f,tms) =>
|
|
1154 |
let
|
|
1155 |
val (tms,xs) = unzip (map modelTm tms)
|
|
1156 |
in
|
|
1157 |
(ModelFn (f,tms,xs), interpretFunction M (f,xs))
|
|
1158 |
end
|
|
1159 |
in
|
|
1160 |
modelTm
|
|
1161 |
end;
|
|
1162 |
|
|
1163 |
(* ------------------------------------------------------------------------- *)
|
|
1164 |
(* Perturbing the model. *)
|
|
1165 |
(* ------------------------------------------------------------------------- *)
|
|
1166 |
|
|
1167 |
datatype perturbation =
|
|
1168 |
FunctionPerturbation of (Term.functionName * element list) * element
|
|
1169 |
| RelationPerturbation of (Atom.relationName * element list) * bool;
|
|
1170 |
|
|
1171 |
fun perturb M pert =
|
|
1172 |
case pert of
|
|
1173 |
FunctionPerturbation func_elts_elt => updateFunction M func_elts_elt
|
|
1174 |
| RelationPerturbation rel_elts_pol => updateRelation M rel_elts_pol;
|
|
1175 |
|
|
1176 |
local
|
|
1177 |
fun pertTerm _ [] _ acc = acc
|
|
1178 |
| pertTerm M target tm acc =
|
|
1179 |
case tm of
|
|
1180 |
ModelVar => acc
|
|
1181 |
| ModelFn (func,tms,xs) =>
|
|
1182 |
let
|
|
1183 |
fun onTarget ys = mem (interpretFunction M (func,ys)) target
|
|
1184 |
|
|
1185 |
val func_xs = (func,xs)
|
|
1186 |
|
|
1187 |
val acc =
|
|
1188 |
if isFixedFunction M func_xs then acc
|
|
1189 |
else
|
|
1190 |
let
|
|
1191 |
fun add (y,acc) = FunctionPerturbation (func_xs,y) :: acc
|
|
1192 |
in
|
|
1193 |
foldl add acc target
|
|
1194 |
end
|
|
1195 |
in
|
|
1196 |
pertTerms M onTarget tms xs acc
|
|
1197 |
end
|
|
1198 |
|
|
1199 |
and pertTerms M onTarget =
|
|
1200 |
let
|
|
1201 |
val N = size M
|
|
1202 |
|
|
1203 |
fun filterElements pred =
|
|
1204 |
let
|
|
1205 |
fun filt 0 acc = acc
|
|
1206 |
| filt i acc =
|
|
1207 |
let
|
|
1208 |
val i = i - 1
|
|
1209 |
val acc = if pred i then i :: acc else acc
|
|
1210 |
in
|
|
1211 |
filt i acc
|
|
1212 |
end
|
|
1213 |
in
|
|
1214 |
filt N []
|
|
1215 |
end
|
|
1216 |
|
|
1217 |
fun pert _ [] [] acc = acc
|
|
1218 |
| pert ys (tm :: tms) (x :: xs) acc =
|
|
1219 |
let
|
|
1220 |
fun pred y =
|
|
1221 |
y <> x andalso onTarget (List.revAppend (ys, y :: xs))
|
|
1222 |
|
|
1223 |
val target = filterElements pred
|
|
1224 |
|
|
1225 |
val acc = pertTerm M target tm acc
|
|
1226 |
in
|
|
1227 |
pert (x :: ys) tms xs acc
|
|
1228 |
end
|
|
1229 |
| pert _ _ _ _ = raise Bug "Model.pertTerms.pert"
|
|
1230 |
in
|
|
1231 |
pert []
|
|
1232 |
end;
|
|
1233 |
|
|
1234 |
fun pertAtom M V target (rel,tms) acc =
|
|
1235 |
let
|
|
1236 |
fun onTarget ys = interpretRelation M (rel,ys) = target
|
|
1237 |
|
|
1238 |
val (tms,xs) = unzip (map (modelTerm M V) tms)
|
|
1239 |
|
|
1240 |
val rel_xs = (rel,xs)
|
|
1241 |
|
|
1242 |
val acc =
|
|
1243 |
if isFixedRelation M rel_xs then acc
|
|
1244 |
else RelationPerturbation (rel_xs,target) :: acc
|
|
1245 |
in
|
|
1246 |
pertTerms M onTarget tms xs acc
|
|
1247 |
end;
|
|
1248 |
|
|
1249 |
fun pertLiteral M V ((pol,atm),acc) = pertAtom M V pol atm acc;
|
|
1250 |
|
|
1251 |
fun pertClause M V cl acc = LiteralSet.foldl (pertLiteral M V) acc cl;
|
|
1252 |
|
|
1253 |
fun pickPerturb M perts =
|
|
1254 |
if null perts then ()
|
|
1255 |
else perturb M (List.nth (perts, Portable.randomInt (length perts)));
|
|
1256 |
in
|
|
1257 |
fun perturbTerm M V (tm,target) =
|
|
1258 |
pickPerturb M (pertTerm M target (fst (modelTerm M V tm)) []);
|
|
1259 |
|
|
1260 |
fun perturbAtom M V (atm,target) =
|
|
1261 |
pickPerturb M (pertAtom M V target atm []);
|
|
1262 |
|
|
1263 |
fun perturbLiteral M V lit = pickPerturb M (pertLiteral M V (lit,[]));
|
|
1264 |
|
|
1265 |
fun perturbClause M V cl = pickPerturb M (pertClause M V cl []);
|
|
1266 |
end;
|
|
1267 |
|
|
1268 |
(* ------------------------------------------------------------------------- *)
|
|
1269 |
(* Pretty printing. *)
|
|
1270 |
(* ------------------------------------------------------------------------- *)
|
|
1271 |
|
|
1272 |
fun pp M =
|
|
1273 |
Print.program
|
|
1274 |
[Print.addString "Model{",
|
|
1275 |
Print.ppInt (size M),
|
|
1276 |
Print.addString "}"];
|
|
1277 |
|
|
1278 |
end
|