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)));
}
