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appendixB.pl
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %
% Prolog programs from Appendix B of the book %
% SIMPLY LOGICAL: Intelligent reasoning by example %
% (c) Peter A. Flach/John Wiley & Sons, 1994. %
% %
% Predicates: transform/2 %
% complete/2 %
% %
% NB. This file needs predicates defined in %
% the file 'library'. %
% %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
:-consult(library).
%%% B.1 From predicate logic to clausal logic %%%
:-op(900,xfx,'=>'). % implication
:-op(800,xfy,&). % conjunction
:-op(800,xfy,v). % disjunction
:-op(400,fy,-). % negation
transform(Formula,Clauses):-
rewrite_implications(Formula,F1),
negations_inside(F1,F2),
prenex_normal_form(F2,F3),
skolemise(F3,F4),
conjunctive_normal_form(F4,F5),
clausal_form(F5,Clauses).
% rewrite_implications(F1,F2) <- F2 is a predicate logic formula
% without implications, logically equivalent to F1
rewrite_implications(A,A):- % base case
logical_atom(A).
rewrite_implications(A => B, -C v D):- % implication
rewrite_implications(A,C),
rewrite_implications(B,D).
rewrite_implications(A & B, C & D):- % no change;
rewrite_implications(A,C), % try rest of formula
rewrite_implications(B,D).
rewrite_implications(A v B, C v D):-
rewrite_implications(A,C),
rewrite_implications(B,D).
rewrite_implications(-A,-C):-
rewrite_implications(A,C).
rewrite_implications(forall(X,B), forall(X,D)):-
rewrite_implications(B,D).
rewrite_implications(exists(X,B), exists(X,D)):-
rewrite_implications(B,D).
% negations_inside(F1,F2) <- F2 is a predicate logic formula with negations
% only preceding literals, logically equivalent to F1
negations_inside(A,A):- % base case
literal(A).
negations_inside(-(A & B), C v D):- % De Morgan (1)
negations_inside(-A,C),
negations_inside(-B,D).
negations_inside(-(A v B), C & D):- % De Morgan (2)
negations_inside(-A,C),
negations_inside(-B,D).
negations_inside(-(-A),B):- % double negation
negations_inside(A,B).
negations_inside(-exists(X,A),forall(X,B)):- % quantifiers
negations_inside(-A,B).
negations_inside(-forall(X,A),exists(X,B)):-
negations_inside(-A,B).
negations_inside(A & B, C & D):- % no change;
negations_inside(A,C), % try rest of formula
negations_inside(B,D).
negations_inside(A v B, C v D):-
negations_inside(A,C),
negations_inside(B,D).
negations_inside(exists(X,A),exists(X,B)):-
negations_inside(A,B).
negations_inside(forall(X,A),forall(X,B)):-
negations_inside(A,B).
% prenex_normal_form(F1,F2) <- F2 is a predicate logic formula with all
% quantifiers in front, logically equivalent to F1
prenex_normal_form(F,PNF):-
pnf(F,PNF,B,B).
pnf(A,V,V,A):- % base case
literal(A).
pnf(forall(X,F),forall(X,Quants),V,Body):-
pnf(F,Quants,V,Body).
pnf(exists(X,F),exists(X,Quants),V,Body):-
pnf(F,Quants,V,Body).
pnf(A & B,Quants,V,BodyA & BodyB):-
pnf(A,Quants,QB,BodyA),
pnf(B,QB,V,BodyB).
pnf(A v B,Quants,V,BodyA v BodyB):-
pnf(A,Quants,QB,BodyA),
pnf(B,QB,V,BodyB).
% skolemise(F1,F2) <- F2 is obtained from F1 by replacing
% all existentially quantified variables by Skolem terms
skolemise(F1,F2):-
skolemise(F1,[],1,F2).
skolemise(forall(X,F1),VarList,N,F2):-!, % remove univ. quantifier,
skolemise(F1,[X|VarList],N,F2). % memorise variable
skolemise(exists(X,F1),VarList,N,F2):-!,
skolem_term(X,VarList,N), % unify with Skolem term
N1 is N+1,
skolemise(F1,VarList,N1,F2).
skolemise(F,_,_,F). % copy rest of formula
skolem_term(X,VarList,N):-
C is N+48, % number -> character
name(Functor,[115,107,C]), % Skolem functor skN
X =.. [Functor|VarList].
conjunctive_normal_form(A,A):- % base case
disjunction_of_literals(A),!.
conjunctive_normal_form((A & B) v C, D & E ):-!, % distribution
conjunctive_normal_form(A v C,D),
conjunctive_normal_form(B v C,E).
conjunctive_normal_form(A v (B & C), D & E ):- !, % distribution
conjunctive_normal_form(A v B,D),
conjunctive_normal_form(A v C,E).
conjunctive_normal_form(A & B,C & D):- % conjuction
conjunctive_normal_form(A,C),
conjunctive_normal_form(B,D).
conjunctive_normal_form(A v B,E):- % other cases
conjunctive_normal_form(A,C),
conjunctive_normal_form(B,D),
conjunctive_normal_form(C v D,E).
clausal_form(A,[Clause]):-
disjunction_of_literals(A),
make_clause(A,Clause).
clausal_form(A & B,Clauses):-
clausal_form(A,ClausesA),
clausal_form(B,ClausesB),
append(ClausesA,ClausesB,Clauses).
make_clause(P,([P]:-[])):-
logical_atom(P).
make_clause(-N,([]:-[N])):-
logical_atom(N).
make_clause(A v B,(HeadAB:-BodyAB)):-
make_clause(A,(HeadA:-BodyA)),
make_clause(B,(HeadB:-BodyB)),
append(HeadA,HeadB,HeadAB),
append(BodyA,BodyB,BodyAB).
%%%%%%%%%%%%%% basics %%%%%%%%%%%%%%%%%%%
disjunction_of_literals(A):-
literal(A).
disjunction_of_literals(C v D):-
disjunction_of_literals(C),
disjunction_of_literals(D).
literal(A):-
logical_atom(A).
literal(-A):-
logical_atom(A).
logical_atom(A):-
functor(A,P,N),
not logical_symbol(P).
logical_symbol(=>).
logical_symbol(-).
logical_symbol(&).
logical_symbol(v).
logical_symbol(exists).
logical_symbol(forall).
%%% B.2 Predicate Completion %%%
complete(Program,Comp):-
separate_definitions(Program,Definitions),
complete_definitions(Definitions,CompDefs,Heads),
handle_undefined(Program,Heads,CompDefs,Comp).
separate_definitions([],[]).
separate_definitions([([Head]:-Body)|Clauses],[[([Head]:-Body)|Def]|Defs]):-
get_definition(Clauses,Head,Def,Rest),
separate_definitions(Rest,Defs).
get_definition([],Head,[],[]).
get_definition([([H]:-B)|Clauses],Head,[([H]:-B)|Def],Rest):-
same_predicate(H,Head),
get_definition(Clauses,Head,Def,Rest).
get_definition([([H]:-B)|Clauses],Head,Def,[([H]:-B)|Rest]):-
not same_predicate(H,Head),
get_definition(Clauses,Head,Def,Rest).
handle_undefined(Program,Heads,CompDefs,Comp):-
setof0(L,
H^B^(element((H:-B),Program),
((element(L,B),not L=not(X))
;element(not L,B)),
not element(L,Heads)),
Undefs),
undef_formulas(Undefs,CompDefs,Comp).
undef_formulas([],Comp,Comp).
undef_formulas([L|Ls],Comp0,Comp):-
quantify(L,F),
undef_formulas(Ls,F & Comp0,Comp).
quantify(L,F):-
L =.. [P|As],
variablise(As,Vs,F,-NewL),
NewL =.. [P|Vs].
variablise([],[],L,L).
variablise([A|As],[V|Vs],forall(V,F),L):-
variablise(As,Vs,F,L).
complete_definitions([Def],Comp,[Head]):-!,
complete_definition(Def,Comp,Head).
complete_definitions([Def|Defs],Comp & Comps,[Head|Heads]):-
complete_definition(Def,Comp,Head),
complete_definitions(Defs,Comps,Heads).
complete_definition(Definition,Comp,Head):-
unifications_and_quantifiers(Definition,F),
complete_formula(F,Comp,Head).
unifications_and_quantifiers([],[]).
unifications_and_quantifiers([Clause|Clauses],[C|Cs]):-
unifs_and_quants(Clause,C),
unifications_and_quantifiers(Clauses,Cs).
unifs_and_quants(([Head] :- Body),([NewHead]:-NewBody)):-
Head=..[Pred|Args],
explicit_unifications(Args,NewArgs,Body,TmpBody),
existential_quantifiers(TmpBody,NewArgs,NewBody),
NewHead=..[Pred|NewArgs].
explicit_unifications([],[],Body,Body).
explicit_unifications([Term|Args],[NewVar|NewArgs],Body,[NewVar = Term|NewBody]):-
nonvar(Term),
explicit_unifications(Args,NewArgs,Body,NewBody).
explicit_unifications([Var|Args],[Var|NewArgs],Body,NewBody):-
var(Var),
explicit_unifications(Args,NewArgs,Body,NewBody).
existential_quantifiers(Body,HeadVars,NewBody):-
varsin(Body,BodyVars),
body_form(Body,Conj),
body_quants(BodyVars,HeadVars,Conj,NewBody).
body_form([not Lit],-Lit):-!.
body_form([Lit],Lit):-!.
body_form([not Lit|List],-Lit & Conj):-!,
body_form(List,Conj).
body_form([Lit|List],Lit & Conj):-
body_form(List,Conj).
body_quants([],HeadVars,Conj,Conj).
body_quants([BVar|BVars],HeadVars,Conj,exists(BVar,F)):-
not var_element(BVar,HeadVars),
body_quants(BVars,HeadVars,Conj,F).
body_quants([BVar|BVars],HeadVars,Conj,F):-
var_element(BVar,HeadVars),
body_quants(BVars,HeadVars,Conj,F).
complete_formula(F,Formula,Head):-
combine_clauses(F,Head,Body),
varsin(Head,HeadVars),
head_quants(HeadVars,Head => Body,Formula).
combine_clauses([([Head]:-Body)],Head,Body):- !.
combine_clauses([([Head]:-Body)|Rest],Head,Body v RestBody):-
combine_clauses(Rest,Head,RestBody).
head_quants([],Formula,Formula).
head_quants([HVar|HVars],Formula,forall(HVar,F)):-
head_quants(HVars,Formula,F).
%%% Queries %%%
query1(PL,CL):-
pl(PL),
transform(PL,CL).
pl( forall(Y,exists(X,mother_of(X,Y))) &
-forall(Z,exists(W,woman(Z) => mother_of(Z,W)))).
pl( forall(X,exists(Y,mouse(X) => tail_of(Y,X)))).
pl( forall(X,exists(Y,loves(X,Y)) &
forall(Z,loves(Y,Z)))).
pl( forall(X,forall(Y,exists(Z,number(X) & number(Y) => maximum(Z,X,Y))))).
query2(P,F,CP):-
program(P),
complete(P,F),
transform(F,CP).
program([ ([bird(tweety)]:-[]),
([flies(X)]:-[bird(X),not abnormal(X)]) ]).
program([ ([likes(peter,S)]:-[student_of(S,peter)]),
([student_of(paul,peter)]:-[]) ]).