public object VisitQuantifiedSentence(QuantifiedSentence sentence, object arg) { ISentence quantified = sentence.Quantified; var universalScope = (ISet <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.Quantifier)) { IDictionary <Variable, ITerm> skolemSubst = new Dictionary <Variable, ITerm>(); foreach (Variable eVar in sentence.Variables) { if (universalScope.Count > 0) { // Replace with a Skolem Function var skolemFunctionName = parser.GetFOLDomain() .AddSkolemFunction(); skolemSubst[eVar] = new Function(skolemFunctionName, new List <ITerm>((IEnumerable <ITerm>)universalScope)); } else { // Replace with a Skolem Constant String skolemConstantName = parser.GetFOLDomain() .AddSkolemConstant(); skolemSubst[eVar] = new Constant(skolemConstantName); } } ISentence skolemized = substVisitor.Subst(skolemSubst, quantified); return(skolemized.Accept(this, arg)); } // Drop universal quantifiers. if (Quantifiers.IsForall(sentence.Quantifier)) { // Add to the universal scope so that // existential skolemization may be done correctly universalScope.UnionWith(sentence.Variables); ISentence droppedUniversal = (ISentence)quantified.Accept(this, arg); // Enusre my scope is removed before moving back up // the call stack when returning foreach (var v in sentence.Variables) { universalScope.Remove(v); } return(droppedUniversal); } // Should not reach here as have already // handled the two quantifiers. throw new InvalidOperationException("Unhandled Quantifier:" + sentence.Quantifier); }
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: ISentence negated = notSentence.Negated; // ~(~alpha) equivalent to alpha (double negation elimination) if (negated is NotSentence) { return(((NotSentence)negated).Negated.Accept(this, arg)); } if (negated is ConnectedSentence) { ConnectedSentence negConnected = (ConnectedSentence)negated; ISentence alpha = negConnected.First; ISentence beta = negConnected.Second; // ~(alpha ^ beta) equivalent to (~alpha V ~beta) (De Morgan) if (Connectors.IsAnd(negConnected.Connector)) { // I need to ensure the ~s are moved in deeper ISentence notAlpha = (ISentence)(new NotSentence(alpha)).Accept( this, arg); ISentence notBeta = (ISentence)(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.Connector)) { // I need to ensure the ~s are moved in deeper ISentence notAlpha = (ISentence)(new NotSentence(alpha)).Accept( this, arg); ISentence notBeta = (ISentence)(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 ISentence notP = (ISentence)(new NotSentence(negQuantified .Quantified)).Accept(this, arg); // ~ForAll x p becomes Exists x ~p if (Quantifiers.IsForall(negQuantified.Quantifier)) { return(new QuantifiedSentence(Quantifiers.Exists, negQuantified .Variables, notP)); } // ~Exists x p becomes ForAll x ~p if (Quantifiers.IsExists(negQuantified.Quantifier)) { return(new QuantifiedSentence(Quantifiers.ForAll, negQuantified .Variables, notP)); } } return(new NotSentence((ISentence)negated.Accept(this, arg))); }