public object visitQuantifiedSentence(QuantifiedSentence sentence, object arg) { Sentence quantified = sentence.getQuantified(); ISet <Variable> universalScope = (Set <Variable>)arg; // Skolemize: Skolemization is the process of removing existential // quantifiers by elimination. This is done by introducing Skolem // functions. The general rule is that the arguments of the Skolem // function are all the universally quantified variables in whose // scope the existential quantifier appears. if (Quantifiers.isEXISTS(sentence.getQuantifier())) { IMap <Variable, Term> skolemSubst = CollectionFactory.CreateInsertionOrderedMap <Variable, Term>(); foreach (Variable eVar in sentence.getVariables()) { if (universalScope.Size() > 0) { // Replace with a Skolem Function string skolemFunctionName = parser.getFOLDomain().addSkolemFunction(); ICollection <Term> terms = CollectionFactory.CreateQueue <Term>(); foreach (Variable v in universalScope) { terms.Add(v); } skolemSubst.Put(eVar, new Function(skolemFunctionName, terms)); } else { // Replace with a Skolem Constant string skolemConstantName = parser.getFOLDomain().addSkolemConstant(); skolemSubst.Put(eVar, new Constant(skolemConstantName)); } } Sentence skolemized = substVisitor.subst(skolemSubst, quantified); return(skolemized.accept(this, arg)); } // Drop universal quantifiers. if (Quantifiers.isFORALL(sentence.getQuantifier())) { // Add to the universal scope so that // existential skolemization may be done correctly universalScope.AddAll(sentence.getVariables()); Sentence droppedUniversal = (Sentence)quantified.accept(this, arg); // Enusre my scope is removed before moving back up // the call stack when returning universalScope.RemoveAll(sentence.getVariables()); return(droppedUniversal); } // Should not reach here as have already // handled the two quantifiers. throw new IllegalStateException("Unhandled Quantifier:" + sentence.getQuantifier()); }
public object visitNotSentence(NotSentence notSentence, object arg) { // CNF requires NOT (~) to appear only in literals, so we 'move ~ // inwards' by repeated application of the following equivalences: Sentence negated = notSentence.getNegated(); // ~(~alpha) equivalent to alpha (double negation elimination) if (negated is NotSentence) { return(((NotSentence)negated).getNegated().accept(this, arg)); } if (negated is ConnectedSentence) { ConnectedSentence negConnected = (ConnectedSentence)negated; Sentence alpha = negConnected.getFirst(); Sentence beta = negConnected.getSecond(); // ~(alpha ^ beta) equivalent to (~alpha V ~beta) (De Morgan) if (Connectors.isAND(negConnected.getConnector())) { // I need to ensure the ~s are moved in deeper Sentence notAlpha = (Sentence)(new NotSentence(alpha)).accept(this, arg); Sentence notBeta = (Sentence)(new NotSentence(beta)).accept(this, arg); return(new ConnectedSentence(Connectors.OR, notAlpha, notBeta)); } // ~(alpha V beta) equivalent to (~alpha ^ ~beta) (De Morgan) if (Connectors.isOR(negConnected.getConnector())) { // I need to ensure the ~s are moved in deeper Sentence notAlpha = (Sentence)(new NotSentence(alpha)).accept(this, arg); Sentence notBeta = (Sentence)(new NotSentence(beta)).accept(this, arg); return(new ConnectedSentence(Connectors.AND, notAlpha, notBeta)); } } // in addition, rules for negated quantifiers: if (negated is QuantifiedSentence) { QuantifiedSentence negQuantified = (QuantifiedSentence)negated; // I need to ensure the ~ is moved in deeper Sentence notP = (Sentence)(new NotSentence(negQuantified.getQuantified())).accept(this, arg); // ~FORALL x p becomes EXISTS x ~p if (Quantifiers.isFORALL(negQuantified.getQuantifier())) { return(new QuantifiedSentence(Quantifiers.EXISTS, negQuantified.getVariables(), notP)); } // ~EXISTS x p becomes FORALL x ~p if (Quantifiers.isEXISTS(negQuantified.getQuantifier())) { return(new QuantifiedSentence(Quantifiers.FORALL, negQuantified.getVariables(), notP)); } } return(new NotSentence((Sentence)negated.accept(this, arg))); }