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Prolog und Prädikatenlogik II

Prolog und Prädikatenlogik II. Prolog Grundkurs WS 99/00 Christof Rumpf rumpf@uni-duesseldorf.de. translate/1. translate(X):- implout(X,X1),nl, negin(X1,X2), skolem(X2,X3,[]), univout(X3,X4), cnf(X4,X5), clausify(X5,Clauses,[]), prologify2(Clauses,Clauses2),

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Prolog und Prädikatenlogik II

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  1. Prolog und Prädikatenlogik II Prolog Grundkurs WS 99/00 Christof Rumpf rumpf@uni-duesseldorf.de GK Prolog: Prolog und Prädikatenlogik II

  2. translate/1 • translate(X):- • implout(X,X1),nl, • negin(X1,X2), • skolem(X2,X3,[]), • univout(X3,X4), • cnf(X4,X5), • clausify(X5,Clauses,[]), • prologify2(Clauses,Clauses2), • pclauses(Clauses2). • Implikationen umformen • Negationen einwärts • Skolemisierung • Allquantoren eliminieren • Konjunktive Normalform • Klauselform • Hornklauseln • Prolog-Klauseln GK Prolog: Prolog und Prädikatenlogik II

  3. implout/2 - I • implout((P <-> Q), ((P1 + Q1) # (^P1 + ^Q1))):- !, • implout(P,P1), • implout(Q,Q1). • implout((P -> Q), (^P1 # Q1)):- !, • implout(P,P1), • implout(Q,Q1). • implout(all(X,P), all(X,P1)):- !, • implout(P,P1). • implout(exists(X,P), exists(X,P1)):- !, • implout(P,P1). GK Prolog: Prolog und Prädikatenlogik II

  4. implout/2 - II • implout((P + Q), (P1 + Q1)):- !, • implout(P,P1), • implout(Q,Q1). • implout((P # Q), (P1 # Q1)):- !, • implout(P,P1), • implout(Q,Q1). • implout((^P), (^P1)):- !, • implout(P,P1). • implout(P,P). GK Prolog: Prolog und Prädikatenlogik II

  5. negin/2 & neg/2 • negin((^P), P1):- !, neg(P, P1). • negin(all(X,P), all(X,P1)):- !, negin(P, P1). • negin(exists(X,P), exists(X,P1)):- !, negin(P, P1). • negin((P + Q), (P1 + Q1)):- !, negin(P, P1), negin(Q, Q1). • negin((P # Q), (P1 # Q1)):- !, negin(P, P1), negin(Q, Q1). • negin(P, P). • neg((^P), P1):- !, negin(P, P1). • neg(all(X,P), exists(X,P1)):- !, neg(P, P1). • neg(exists(X,P), all(X,P1)):- !, neg(P, P1). • neg((P + Q), (P1 # Q1)):- !, neg(P, P1), neg(Q, Q1). • neg((P # Q), (P1 + Q1)):- !, neg(P, P1), neg(Q, Q1). • neg(P, (^P)). GK Prolog: Prolog und Prädikatenlogik II

  6. skolem/3 • skolem(all(X,P), all(X,P1), Vars):- !, skolem(P, P1, [X|Vars]). • skolem(exists(X,P), P2, Vars):- !, • gensym(c,F), • Sk =.. [F|Vars], • subst(X, Sk, P, P1), • skolem(P1, P2, Vars). • skolem((P + Q), (P1 + Q1), Vars):- !, • skolem(P, P1, Vars), • skolem(Q, Q1, Vars). • skolem((P # Q), (P1 # Q1), Vars):- !, • skolem(P, P1, Vars), • skolem(Q, Q1, Vars). • skolem(P,P,_). GK Prolog: Prolog und Prädikatenlogik II

  7. gensym/2 • gensym(Root,Symbol):- • name(Root,L1), • symb(Root,S), • name(S,L2), • conc(L1,L2,L3), • name(Symbol,L3). • symb(R,S1):- retract(gensymsym(R,S)), !, inc(S,S1), assert(gensymsym(R,S1)). • symb(R,0):- init_gensym(R). • init_gensym(R):- retract(gensymsym(R,_)), fail. • init_gensym(R):- assert(gensymsym(R,0)), !. GK Prolog: Prolog und Prädikatenlogik II

  8. subst/4 • subst(X,S,T,S):- X == T, !. • subst(_,_,T,T):- var(T) ; atomic(T), !. • subst(X,S,P,P1):- • P =.. L, • substl(X,S,L,L1), • P1 =.. L1. • substl(_,_,[],[]):- !. • substl(X,S,[H1|T1],[H2|T2]):- • subst(X,S,H1,H2), • substl(X,S,T1,T2). GK Prolog: Prolog und Prädikatenlogik II

  9. univout/2 • univout(all(X,P), P1):- !, univout(P,P1). • univout((P + Q), (P1 + Q1)):- !, • univout(P, P1), • univout(Q, Q1). • univout((P # Q), (P1 # Q1)):- !, • univout(P, P1), • univout(Q, Q1). • univout(P, P). GK Prolog: Prolog und Prädikatenlogik II

  10. cnf/2 • cnf((P # Q), R):- !, • cnf(P, P1), • cnf(Q, Q1), • cnf1((P1 # Q1), R). • cnf((P + Q), (P1 + Q1)):- !, • cnf(P, P1), • cnf(Q, Q1). • cnf(P, P). GK Prolog: Prolog und Prädikatenlogik II

  11. cnf1/2 • cnf1(((P + Q) # R), (P1 + Q1)):- !, • cnf((P # R), P1), • cnf((Q # R), Q1). • cnf1((R # (P + Q)), (P1 + Q1)):- !, • cnf((R # P), P1), • cnf((R # Q), Q1). • cnf1(P, P). GK Prolog: Prolog und Prädikatenlogik II

  12. clausify/3 • clausify((P + Q), C1, C2):- !, • clausify(P, C1, C3), • clausify(Q, C3, C2). • clausify(P, [cl(A, B)|Cs], Cs):- • inclause(P, A, [], B, []), !. • clausify(_, C, C). GK Prolog: Prolog und Prädikatenlogik II

  13. inclause/5 • inclause((P # Q), A, A1, B, B1):- !, • inclause(P, A2, A1, B2, B1), • inclause(Q, A, A2, B, B2). • inclause((^P), A, A, B1, B):- !, • notin(P, A), • putin(P, B, B1). • inclause(P, A1, A, B, B):- !, • notin(P, B), • putin(P, A, A1). GK Prolog: Prolog und Prädikatenlogik II

  14. notin/2 & putin/3 • notin(X, [X|_]):- !, fail. • notin(X, [_|L]):- !, notin(X, L). • notin(X, []). • putin(X, [], [X]):- !. • putin(X, [X|L], L):- !. • putin(X, [Y|L], [Y|L1]):- putin(X, L, L1). GK Prolog: Prolog und Prädikatenlogik II

  15. pclauses/1 & pclause/2 • pclauses([]):- !, nl. • pclauses([cl(A,B)|Cs]):- • pclause(A, B), nl, tab(4), pclauses(Cs). • pclause(L, []):- !, • pdisj(L), write('.'). • pclause([], L):- !, • write(':- '), pconj(L), write('.'). • pclause(L1,L2):- • pdisj(L1), write(':- '), • pconj(L2), write('.'). GK Prolog: Prolog und Prädikatenlogik II

  16. pdisj/1 & pconj/1 • pdisj([L]):- !, write(L). • pdisj([H|T]):- • write(H), write('; '), pdisj(T). • pconj([L]):- !, write(L). • pconj([H|T]):- • write(H), write(', '), pconj(T). GK Prolog: Prolog und Prädikatenlogik II

  17. prologify1/2 • % Solution 1: produces infinite loops with indirect recursions. • % Clauses with n disjuncts in the head are transformed to n clauses • % with single heads. Every disjunct gets its own clause where it • % represents the single head of the clause while all other disjuncts • % become negated conjuncts in the body of that clause. • prologify1([],[]):- !. • prologify1([cl(H,B)|Cs],Clauses):- • prologify_clause1(H,H,B,Clauses1), • prologify1(Cs,Clauses2), • append(Clauses1,Clauses2,Clauses). GK Prolog: Prolog und Prädikatenlogik II

  18. prologify_clause1/2 • prologify_clause1([],_,_,[]):- !. • prologify_clause1([H|T],Disj,B,[cl([H],B2)|Cs]):- • delete(H,Disj,NegConj), !, • prologify_disj1(NegConj,B1), • append(B,B1,B2), • prologify_clause1(T,Disj,B,Cs). • prologify_disj1([],[]):- !. • prologify_disj1([D|Ds],[not(D)|Cs]):- • prologify_disj1(Ds,Cs). GK Prolog: Prolog und Prädikatenlogik II

  19. prologify2/2 • % Solution 2: selects the first disjunct as the head of the clause. • % The first disjunct in the complex head of a clause is arbitrarily • % choosen to represent the single head of the clause, while all other • % disjuncts become negated conjuncts in the body of that clause. • prologify2([],[]):- !. • prologify2([cl(H,B)|Cs],[Clause|Clauses]):- • prologify_clause2(H,H,B,Clause), • prologify2(Cs,Clauses). • prologify_clause2([],_,_,[]):- !. • prologify_clause2([H|T],Disj,B,cl([H],B2)):- • delete(H,Disj,NegConj), !, • prologify_disj(NegConj,B1), % as for the 1st solution • append(B,B1,B2). GK Prolog: Prolog und Prädikatenlogik II

  20. prologify3/2 • % Solution 3: perform an intelligent search on the clauses to find out • % which predicates are already defined or have to be defined to • % allow decuctions from the database. Facts might give rise to the • % decision that their predicated should not appear as rule heads • % unless the rule includes recusion on that predicate. • % ...to be implemented... GK Prolog: Prolog und Prädikatenlogik II

  21. Alle Menschen sind sterblich. • PL1 all(A,mensch(A) -> sterblich(A)) • IMP all(A,^ mensch(A) # sterblich(A)) • NEG all(A,^ mensch(A) # sterblich(A)) • SKO all(A,^ mensch(A) # sterblich(A)) • UNI ^ mensch(A) # sterblich(A) • CNF ^ mensch(A) # sterblich(A) • CL1 [cl([sterblich(A)],[mensch(A)])] • CL2 [cl([sterblich(A)],[mensch(A)])] • PRO sterblich(A):- mensch(A). GK Prolog: Prolog und Prädikatenlogik II

  22. Alle Männer lieben irgendeine Frau. • PL1 all(A,man(A) -> exists(B,woman(B) + loves(A,B))) • IMP all(A,^ man(A) # exists(B,woman(B) + loves(A,B))) • NEG all(A,^ man(A) # exists(B,woman(B) + loves(A,B))) • SKO all(A,^ man(A) # woman(c4(A)) + loves(A,c4(A))) • UNI ^ man(A) # woman(c4(A)) + loves(A,c4(A)) • CNF (^ man(A) # woman(c4(A))) + ^ man(A) # loves(A,c4(A)) • CL1 [cl([woman(c4(A))],[man(A)]),cl([loves(A,c4(A))],[man(A)])] • CL2 [cl([woman(c4(A))],[man(A)]),cl([loves(A,c4(A))],[man(A)])] • PRO woman(c4(A)):- man(A). • loves(A,c4(A)):- man(A). GK Prolog: Prolog und Prädikatenlogik II

  23. Irgendein Mann liebt irgendeine Frau. • PL1 exists(A,man(A) + exists(B,woman(B) + loves(A,B))) • IMP exists(A,man(A) + exists(B,woman(B) + loves(A,B))) • NEG exists(A,man(A) + exists(B,woman(B) + loves(A,B))) • SKO man(c2) + woman(c3) + loves(c2,c3) • UNI man(c2) + woman(c3) + loves(c2,c3) • CNF man(c2) + woman(c3) + loves(c2,c3) • CL1 [cl([man(c2)],[]),cl([woman(c3)],[]),cl([loves(c2,c3)],[])] • CL2 [cl([man(c2)],[]),cl([woman(c3)],[]),cl([loves(c2,c3)],[])] • PRO man(c2). • woman(c3). • loves(c2,c3). GK Prolog: Prolog und Prädikatenlogik II

  24. vorfahr/2 • NLS Elternteile von Individuen sind Vorfahren dieser Individuen und • Elternteile von Vorfahren von Individuen sind Vorfahren dieser Individuen. • PL1 all(A,all(B,all(C,(et(A,C) + v(C,B) -> v(A,B)) + (et(A,B) -> v(A,B))))) • IMP all(A,all(B,all(C,(^ (et(A,C) + v(C,B)) # v(A,B)) + ^ et(A,B) # v(A,B)))) • NEG all(A,all(B,all(C,((^ et(A,C) # ^ v(C,B)) # v(A,B)) + ^ et(A,B) # v(A,B)))) • SKO all(A,all(B,all(C,((^ et(A,C) # ^ v(C,B)) # v(A,B)) + ^ et(A,B) # v(A,B)))) • UNI ((^ et(A,B) # ^ v(B,C)) # v(A,C)) + ^ et(A,C) # v(A,C) • CNF ((^ et(A,B) # ^ v(B,C)) # v(A,C)) + ^ et(A,C) # v(A,C) • CL1 [cl([v(A,C)],[et(A,B),v(B,C)]),cl([v(A,C)],[et(A,C)])] • CL2 [cl([v(A,C)],[et(A,B),v(B,C)]),cl([v(A,C)],[et(A,C)])] • PRO v(A,C):- et(A,B), v(B,C). • v(A,C):- et(A,C). GK Prolog: Prolog und Prädikatenlogik II

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