static void fOL_Demodulation()
{
System.Console.WriteLine("-----------------");
System.Console.WriteLine("Demodulation Demo");
System.Console.WriteLine("-----------------");
FOLDomain domain = new FOLDomain();
domain.addConstant("A");
domain.addConstant("B");
domain.addConstant("C");
domain.addConstant("D");
domain.addConstant("E");
domain.addPredicate("P");
domain.addFunction("F");
domain.addFunction("G");
domain.addFunction("H");
domain.addFunction("J");
FOLParser parser = new FOLParser(domain);
Predicate expression = (Predicate)parser.parse("P(A,F(B,G(A,H(B)),C),D)");
TermEquality assertion = (TermEquality)parser.parse("B = E");
Demodulation demodulation = new Demodulation();
Predicate altExpression = (Predicate)demodulation.apply(assertion, expression);
System.Console.WriteLine("Demodulate '" + expression + "' with '" + assertion + "' to give");
System.Console.WriteLine(altExpression.ToString());
System.Console.WriteLine("and again to give");
System.Console.WriteLine(demodulation.apply(assertion, altExpression).ToString());
System.Console.WriteLine("");
}
public void testSimpleAtomicNonExample()
{
FOLDomain domain = new FOLDomain();
domain.addConstant("A");
domain.addConstant("B");
domain.addConstant("C");
domain.addConstant("D");
domain.addConstant("E");
domain.addPredicate("P");
domain.addFunction("F");
domain.addFunction("G");
domain.addFunction("H");
domain.addFunction("J");
FOLParser parser = new FOLParser(domain);
Predicate expression = (Predicate)parser.parse("P(A,G(x,B),C)");
TermEquality assertion = (TermEquality)parser.parse("G(A,y) = J(y)");
Predicate altExpression = (Predicate)demodulation.apply(assertion,
expression);
Assert.IsNull(altExpression);
}
public void testDefaultClauseSimplifier()
{
FOLDomain domain = new FOLDomain();
domain.addConstant("ZERO");
domain.addConstant("ONE");
domain.addPredicate("P");
domain.addFunction("Plus");
domain.addFunction("Power");
FOLParser parser = new FOLParser(domain);
ICollection <TermEquality> rewrites = CollectionFactory.CreateQueue <TermEquality>();
rewrites.Add((TermEquality)parser.parse("Plus(x, ZERO) = x"));
rewrites.Add((TermEquality)parser.parse("Plus(ZERO, x) = x"));
rewrites.Add((TermEquality)parser.parse("Power(x, ONE) = x"));
rewrites.Add((TermEquality)parser.parse("Power(x, ZERO) = ONE"));
DefaultClauseSimplifier simplifier = new DefaultClauseSimplifier(rewrites);
Sentence s1 = parser.parse("((P(Plus(y,ZERO),Plus(ZERO,y)) OR P(Power(y, ONE),Power(y,ZERO))) OR P(Power(y,ZERO),Plus(y,ZERO)))");
CNFConverter cnfConverter = new CNFConverter(parser);
CNF cnf = cnfConverter.convertToCNF(s1);
Assert.AreEqual(1, cnf.getNumberOfClauses());
Clause simplified = simplifier.simplify(cnf.getConjunctionOfClauses().Get(0));
Assert.AreEqual("[P(y,y), P(y,ONE), P(ONE,y)]", simplified.ToString());
}
public void testCascadedOccursCheck()
{
FOLDomain domain = new FOLDomain();
domain.addPredicate("P");
domain.addFunction("F");
domain.addFunction("SF0");
domain.addFunction("SF1");
FOLParser parser = new FOLParser(domain);
Sentence s1 = parser.parse("P(SF1(v2),v2)");
Sentence s2 = parser.parse("P(v3,SF0(v3))");
IMap <Variable, Term> result = unifier.unify(s1, s2);
Assert.IsNull(result);
s1 = parser.parse("P(v1,SF0(v1),SF0(v1),SF0(v1),SF0(v1))");
s2 = parser.parse("P(v2,SF0(v2),v2, v3, v2)");
result = unifier.unify(s1, s2);
Assert.IsNull(result);
s1 = parser
.parse("P(v1, F(v2),F(v2),F(v2),v1, F(F(v1)),F(F(F(v1))),v2)");
s2 = parser
.parse("P(F(v3),v4, v5, v6, F(F(v5)),v4, F(v3), F(F(v5)))");
result = unifier.unify(s1, s2);
Assert.IsNull(result);
}
public void testAdditionalVariableMixtures()
{
FOLDomain domain = new FOLDomain();
domain.addConstant("A");
domain.addConstant("B");
domain.addFunction("F");
domain.addFunction("G");
domain.addFunction("H");
domain.addPredicate("P");
FOLParser parser = new FOLParser(domain);
// Test Cascade Substitutions handled correctly
Sentence s1 = parser.parse("P(z, x)");
Sentence s2 = parser.parse("P(x, a)");
IMap <Variable, Term> result = unifier.unify(s1, s2);
Assert.AreEqual("[[z, a], [x, a]]", result.ToString());
s1 = parser.parse("P(x, z)");
s2 = parser.parse("P(a, x)");
result = unifier.unify(s1, s2);
Assert.AreEqual("[[x, a], [z, a]]", result.ToString());
s1 = parser.parse("P(w, w, w)");
s2 = parser.parse("P(x, y, z)");
result = unifier.unify(s1, s2);
Assert.AreEqual("[[w, z], [x, z], [y, z]]", result.ToString());
s1 = parser.parse("P(x, y, z)");
s2 = parser.parse("P(w, w, w)");
result = unifier.unify(s1, s2);
Assert.AreEqual("[[x, w], [y, w], [z, w]]", result.ToString());
s1 = parser.parse("P(x, B, F(y))");
s2 = parser.parse("P(A, y, F(z))");
result = unifier.unify(s1, s2);
Assert.AreEqual("[[x, A], [y, B], [z, B]]", result.ToString());
s1 = parser.parse("P(F(x,B), G(y), F(z,A))");
s2 = parser.parse("P(y, G(F(G(w),w)), F(w,z))");
result = unifier.unify(s1, s2);
Assert.IsNull(result);
s1 = parser.parse("P(F(G(A)), x, F(H(z,z)), H(y, G(w)))");
s2 = parser.parse("P(y, G(z), F(v ), H(F(w), x ))");
result = unifier.unify(s1, s2);
Assert.AreEqual(
"[[y, F(G(A))], [x, G(G(A))], [v, H(G(A),G(A))], [w, G(A)], [z, G(A)]]",
result.ToString());
}
public void testAdditionalVariableMixtures()
{
FOLDomain domain = new FOLDomain();
domain.addConstant("A");
domain.addConstant("B");
domain.addFunction("F");
domain.addFunction("G");
domain.addFunction("H");
domain.addPredicate("P");
FOLParser parser = new FOLParser(domain);
// Test Cascade Substitutions handled correctly
Sentence s1 = parser.parse("P(z, x)");
Sentence s2 = parser.parse("P(x, a)");
Dictionary <Variable, Term> result = unifier.unify(s1, s2);
Assert.AreEqual("{z=a, x=a}", result.ToString());
s1 = parser.parse("P(x, z)");
s2 = parser.parse("P(a, x)");
result = unifier.unify(s1, s2);
Assert.AreEqual("{x=a, z=a}", result.ToString());
s1 = parser.parse("P(w, w, w)");
s2 = parser.parse("P(x, y, z)");
result = unifier.unify(s1, s2);
Assert.AreEqual("{w=z, x=z, y=z}", result.ToString());
s1 = parser.parse("P(x, y, z)");
s2 = parser.parse("P(w, w, w)");
result = unifier.unify(s1, s2);
Assert.AreEqual("{x=w, y=w, z=w}", result.ToString());
s1 = parser.parse("P(x, B, F(y))");
s2 = parser.parse("P(A, y, F(z))");
result = unifier.unify(s1, s2);
Assert.AreEqual("{x=A, y=B, z=B}", result.ToString());
s1 = parser.parse("P(F(x,B), G(y), F(z,A))");
s2 = parser.parse("P(y, G(F(G(w),w)), F(w,z))");
result = unifier.unify(s1, s2);
Assert.IsNull(result);
s1 = parser.parse("P(F(G(A)), x, F(H(z,z)), H(y, G(w)))");
s2 = parser.parse("P(y, G(z), F(v ), H(F(w), x ))");
result = unifier.unify(s1, s2);
Assert.AreEqual(
"{y=F(G(A)), x=G(G(A)), v=H(G(A),G(A)), w=G(A), z=G(A)}",
result.ToString());
}
public void setUp()
{
FOLDomain domain = new FOLDomain();
domain.addConstant("P");
domain.addConstant("John");
domain.addConstant("Saladin");
domain.addFunction("LeftLeg");
domain.addFunction("BrotherOf");
domain.addFunction("EnemyOf");
domain.addPredicate("HasColor");
domain.addPredicate("King");
lexer = new FOLLexer(domain);
}
public void testNegativeTermEquality()
{
FOLDomain domain = new FOLDomain();
domain.addConstant("A");
domain.addConstant("B");
domain.addConstant("C");
domain.addPredicate("P");
domain.addFunction("F");
FOLParser parser = new FOLParser(domain);
ICollection <Literal> lits = CollectionFactory.CreateQueue <Literal>();
AtomicSentence a1 = (AtomicSentence)parser.parse("P(y, F(A,y))");
lits.Add(new Literal(a1));
Clause c1 = new Clause(lits);
lits.Clear();
a1 = (AtomicSentence)parser.parse("F(x,B) = x");
lits.Add(new Literal(a1, true));
Clause c2 = new Clause(lits);
ISet <Clause> paras = paramodulation.apply(c1, c2);
Assert.AreEqual(0, paras.Size());
}
public void testSimpleClauseNonExample()
{
FOLDomain domain = new FOLDomain();
domain.addConstant("A");
domain.addConstant("B");
domain.addConstant("C");
domain.addPredicate("P");
domain.addFunction("F");
FOLParser parser = new FOLParser(domain);
ICollection <Literal> lits = CollectionFactory.CreateQueue <Literal>();
Predicate p1 = (Predicate)parser.parse("P(y, F(A,y))");
lits.Add(new Literal(p1));
Clause clExpression = new Clause(lits);
TermEquality assertion = (TermEquality)parser.parse("F(x,B) = C");
Clause altClExpression = demodulation.apply(assertion, clExpression);
Assert.IsNull(altClExpression);
}
public void testBypassReflexivityAxiom()
{
FOLDomain domain = new FOLDomain();
domain.addConstant("A");
domain.addConstant("B");
domain.addConstant("C");
domain.addPredicate("P");
domain.addFunction("F");
FOLParser parser = new FOLParser(domain);
List <Literal> lits = new List <Literal>();
Predicate p1 = (Predicate)parser.parse("P(y, F(A,y))");
lits.Add(new Literal(p1));
Clause clExpression = new Clause(lits);
TermEquality assertion = (TermEquality)parser.parse("x = x");
Clause altClExpression = demodulation.apply(assertion, clExpression);
Assert.IsNull(altClExpression);
}
public void testBypassReflexivityAxiom()
{
FOLDomain domain = new FOLDomain();
domain.addConstant("A");
domain.addConstant("B");
domain.addConstant("C");
domain.addPredicate("P");
domain.addFunction("F");
FOLParser parser = new FOLParser(domain);
List <Literal> lits = new List <Literal>();
AtomicSentence a1 = (AtomicSentence)parser.parse("P(y, F(A,y))");
lits.Add(new Literal(a1));
Clause c1 = new Clause(lits);
lits.Clear();
a1 = (AtomicSentence)parser.parse("x = x");
lits.Add(new Literal(a1));
Clause c2 = new Clause(lits);
List <Clause> paras = paramodulation.apply(c1, c2);
Assert.AreEqual(0, paras.Count);
}
// Note: see -
// http://logic.stanford.edu/classes/cs157/2008/lectures/lecture15.pdf
// slide 16,17, and 18 for where this test example was taken from.
public static FOLKnowledgeBase createABCDEqualityAndSubstitutionKnowledgeBase(
InferenceProcedure infp, bool includeEqualityAxioms)
{
FOLDomain domain = new FOLDomain();
domain.addConstant("A");
domain.addConstant("B");
domain.addConstant("C");
domain.addConstant("D");
domain.addPredicate("P");
domain.addFunction("F");
FOLKnowledgeBase kb = new FOLKnowledgeBase(domain, infp);
kb.tell("F(A) = B");
kb.tell("F(B) = A");
kb.tell("C = D");
kb.tell("P(A)");
kb.tell("P(C)");
if (includeEqualityAxioms)
{
// Reflexivity Axiom
kb.tell("x = x");
// Symmetry Axiom
kb.tell("(x = y => y = x)");
// Transitivity Axiom
kb.tell("((x = y AND y = z) => x = z)");
// Function F Substitution Axiom
kb.tell("((x = y AND F(y) = z) => F(x) = z)");
// Predicate P Substitution Axiom
kb.tell("((x = y AND P(y)) => P(x))");
}
return(kb);
}
public static FOLDomain crusadesDomain()
{
FOLDomain domain = new FOLDomain();
domain.addConstant("John");
domain.addConstant("Richard");
domain.addConstant("England");
domain.addConstant("Saladin");
domain.addConstant("Crown");
domain.addFunction("LeftLegOf");
domain.addFunction("BrotherOf");
domain.addFunction("EnemyOf");
domain.addFunction("LegsOf");
domain.addPredicate("King");
return(domain);
}
public void testSimpleClauseExamples()
{
FOLDomain domain = new FOLDomain();
domain.addConstant("A");
domain.addConstant("B");
domain.addConstant("C");
domain.addConstant("D");
domain.addConstant("E");
domain.addPredicate("P");
domain.addPredicate("Q");
domain.addPredicate("W");
domain.addFunction("F");
domain.addFunction("G");
domain.addFunction("H");
domain.addFunction("J");
FOLParser parser = new FOLParser(domain);
List <Literal> lits = new List <Literal>();
Predicate p1 = (Predicate)parser.parse("Q(z, G(D,B))");
Predicate p2 = (Predicate)parser.parse("P(x, G(A,C))");
Predicate p3 = (Predicate)parser.parse("W(z,x,u,w,y)");
lits.Add(new Literal(p1));
lits.Add(new Literal(p2));
lits.Add(new Literal(p3));
Clause clExpression = new Clause(lits);
TermEquality assertion = (TermEquality)parser.parse("G(x,y) = x");
Clause altClExpression = demodulation.apply(assertion, clExpression);
Assert.AreEqual("[P(x,G(A,C)), Q(z,D), W(z,x,u,w,y)]",
altClExpression.ToString());
altClExpression = demodulation.apply(assertion, altClExpression);
Assert.AreEqual("[P(x,A), Q(z,D), W(z,x,u,w,y)]", altClExpression
.ToString());
}
public void testSimpleAtomicExamples()
{
FOLDomain domain = new FOLDomain();
domain.addConstant("A");
domain.addConstant("B");
domain.addConstant("C");
domain.addConstant("D");
domain.addConstant("E");
domain.addPredicate("P");
domain.addFunction("F");
domain.addFunction("G");
domain.addFunction("H");
domain.addFunction("J");
FOLParser parser = new FOLParser(domain);
Predicate expression = (Predicate)parser
.parse("P(A,F(B,G(A,H(B)),C),D)");
TermEquality assertion = (TermEquality)parser.parse("B = E");
Predicate altExpression = (Predicate)demodulation.apply(assertion,
expression);
Assert.IsFalse(expression.Equals(altExpression));
Assert
.AreEqual("P(A,F(E,G(A,H(B)),C),D)", altExpression
.ToString());
altExpression = (Predicate)demodulation
.apply(assertion, altExpression);
Assert
.AreEqual("P(A,F(E,G(A,H(E)),C),D)", altExpression
.ToString());
assertion = (TermEquality)parser.parse("G(x,y) = J(x)");
altExpression = (Predicate)demodulation.apply(assertion, expression);
Assert.AreEqual("P(A,F(B,J(A),C),D)", altExpression.ToString());
}
public static FOLDomain knowsDomain()
{
FOLDomain domain = new FOLDomain();
domain.addConstant("John");
domain.addConstant("Jane");
domain.addConstant("Bill");
domain.addConstant("Elizabeth");
domain.addFunction("Mother");
domain.addPredicate("Knows");
return(domain);
}
public void testInductionAxiomSchema()
{
FOLDomain domain = new FOLDomain();
domain.addPredicate("Equal");
domain.addFunction("Plus");
domain.addConstant("A");
domain.addConstant("B");
domain.addConstant("N");
domain.addConstant("ONE");
domain.addConstant("ZERO");
FOLParser parser = new FOLParser(domain);
CNFConverter cnfConv = new CNFConverter(parser);
// Base Case:
Sentence sent = parser
.parse("NOT(FORALL x (FORALL y (Equal(Plus(Plus(x,y),ZERO), Plus(x,Plus(y,ZERO))))))");
CNF cnf = cnfConv.convertToCNF(sent);
Assert.AreEqual(
"[~Equal(Plus(Plus(SC0,SC1),ZERO),Plus(SC0,Plus(SC1,ZERO)))]",
cnf.ToString());
// Instance of Induction Axion Scmema
sent = parser
.parse("(("
+ "Equal(Plus(Plus(A,B),ZERO), Plus(A,Plus(B,ZERO)))"
+ " AND "
+ "(FORALL x (FORALL y (FORALL z("
+ "Equal(Plus(Plus(x,y),z), Plus(x,Plus(y,z)))"
+ " => "
+ "Equal(Plus(Plus(x,y),Plus(z,ONE)), Plus(x,Plus(y,Plus(z,ONE))))"
+ "))))" + ")" + " => "
+ "FORALL x (FORALL y (FORALL z("
+ "Equal(Plus(Plus(x,y),z), Plus(x,Plus(y,z)))"
+ "))))");
cnf = cnfConv.convertToCNF(sent);
Assert
.AreEqual(
"[~Equal(Plus(Plus(A,B),ZERO),Plus(A,Plus(B,ZERO))), Equal(Plus(Plus(q0,q1),q2),Plus(q0,Plus(q1,q2))), Equal(Plus(Plus(SC2,SC3),SC4),Plus(SC2,Plus(SC3,SC4)))],[~Equal(Plus(Plus(A,B),ZERO),Plus(A,Plus(B,ZERO))), ~Equal(Plus(Plus(SC2,SC3),Plus(SC4,ONE)),Plus(SC2,Plus(SC3,Plus(SC4,ONE)))), Equal(Plus(Plus(q0,q1),q2),Plus(q0,Plus(q1,q2)))]",
cnf.ToString());
// Goal
sent = parser
.parse("NOT(FORALL x (FORALL y (FORALL z (Equal(Plus(Plus(x,y),z), Plus(x,Plus(y,z)))))))");
cnf = cnfConv.convertToCNF(sent);
Assert.AreEqual(
"[~Equal(Plus(Plus(SC5,SC6),SC7),Plus(SC5,Plus(SC6,SC7)))]",
cnf.ToString());
}
public void testBadCascadeSubstitution_LCL418_1()
{
FOLDomain domain = new FOLDomain();
domain.addPredicate("ISATHEOREM");
domain.addFunction("EQUIVALENT");
FOLParser parser = new FOLParser(domain);
Sentence s1 = parser
.parse("ISATHEOREM(EQUIVALENT(EQUIVALENT(c1744,c1743),EQUIVALENT(c1742,c1743)))");
Sentence s2 = parser
.parse("ISATHEOREM(EQUIVALENT(EQUIVALENT(c1752,c1751),c1752))");
IMap <Variable, Term> result = unifier.unify(s1, s2);
Assert.AreEqual(
"[[c1744, EQUIVALENT(c1742,c1751)], [c1743, c1751], [c1752, EQUIVALENT(c1742,c1751)]]",
result.ToString());
}
public void testSimpleExample()
{
FOLDomain domain = new FOLDomain();
domain.addConstant("A");
domain.addConstant("B");
domain.addPredicate("P");
domain.addPredicate("Q");
domain.addPredicate("R");
domain.addFunction("F");
FOLParser parser = new FOLParser(domain);
List <Literal> lits = new List <Literal>();
AtomicSentence a1 = (AtomicSentence)parser.parse("P(F(x,B),x)");
AtomicSentence a2 = (AtomicSentence)parser.parse("Q(x)");
lits.Add(new Literal(a1));
lits.Add(new Literal(a2));
Clause c1 = new Clause(lits);
lits.Clear();
a1 = (AtomicSentence)parser.parse("F(A,y) = y");
a2 = (AtomicSentence)parser.parse("R(y)");
lits.Add(new Literal(a1));
lits.Add(new Literal(a2));
Clause c2 = new Clause(lits);
List <Clause> paras = paramodulation.apply(c1, c2);
Assert.AreEqual(2, paras.Count);
foreach (Clause c in paras)
{
Assert.AreEqual("[P(B,A), Q(A), R(B)]", c.ToString());
Assert.AreEqual("[P(F(A,F(x,B)),x), Q(x), R(F(x,B))]", c
.ToString());
}
}
static void fOL_Paramodulation()
{
System.Console.WriteLine("-------------------");
System.Console.WriteLine("Paramodulation Demo");
System.Console.WriteLine("-------------------");
FOLDomain domain = new FOLDomain();
domain.addConstant("A");
domain.addConstant("B");
domain.addPredicate("P");
domain.addPredicate("Q");
domain.addPredicate("R");
domain.addFunction("F");
FOLParser parser = new FOLParser(domain);
ICollection <Literal> lits = CollectionFactory.CreateQueue <Literal>();
AtomicSentence a1 = (AtomicSentence)parser.parse("P(F(x,B),x)");
AtomicSentence a2 = (AtomicSentence)parser.parse("Q(x)");
lits.Add(new Literal(a1));
lits.Add(new Literal(a2));
Clause c1 = new Clause(lits);
lits.Clear();
a1 = (AtomicSentence)parser.parse("F(A,y) = y");
a2 = (AtomicSentence)parser.parse("R(y)");
lits.Add(new Literal(a1));
lits.Add(new Literal(a2));
Clause c2 = new Clause(lits);
Paramodulation paramodulation = new Paramodulation();
ISet <Clause> paras = paramodulation.apply(c1, c2);
System.Console.WriteLine("Paramodulate '" + c1 + "' with '" + c2 + "' to give");
System.Console.WriteLine(paras.ToString());
System.Console.WriteLine("");
}
public void testIsTautology()
{
FOLDomain domain = new FOLDomain();
domain.addConstant("A");
domain.addPredicate("P");
domain.addPredicate("Q");
domain.addPredicate("R");
domain.addFunction("F");
FOLParser parser = new FOLParser(domain);
// {p(f(a)),~p(f(a))}
Clause c = new Clause();
c.addPositiveLiteral((Predicate)parser.parse("P(F(A))"));
Assert.IsFalse(c.isTautology());
c.addNegativeLiteral((Predicate)parser.parse("P(F(A))"));
Assert.IsTrue(c.isTautology());
// {p(x),q(y),~q(y),r(z)}
c = new Clause();
c.addPositiveLiteral((Predicate)parser.parse("P(x)"));
Assert.IsFalse(c.isTautology());
c.addPositiveLiteral((Predicate)parser.parse("Q(y)"));
Assert.IsFalse(c.isTautology());
c.addNegativeLiteral((Predicate)parser.parse("Q(y)"));
Assert.IsTrue(c.isTautology());
c.addPositiveLiteral((Predicate)parser.parse("R(z)"));
Assert.IsTrue(c.isTautology());
// {~p(a),p(x)}
c = new Clause();
c.addNegativeLiteral((Predicate)parser.parse("P(A)"));
Assert.IsFalse(c.isTautology());
c.addPositiveLiteral((Predicate)parser.parse("P(x)"));
Assert.IsFalse(c.isTautology());
}
public void testSimpleExample()
{
FOLDomain domain = new FOLDomain();
domain.addConstant("A");
domain.addConstant("B");
domain.addPredicate("P");
domain.addPredicate("Q");
domain.addPredicate("R");
domain.addFunction("F");
FOLParser parser = new FOLParser(domain);
ICollection <Literal> lits = CollectionFactory.CreateQueue <Literal>();
AtomicSentence a1 = (AtomicSentence)parser.parse("P(F(x,B),x)");
AtomicSentence a2 = (AtomicSentence)parser.parse("Q(x)");
lits.Add(new Literal(a1));
lits.Add(new Literal(a2));
Clause c1 = new Clause(lits);
lits.Clear();
a1 = (AtomicSentence)parser.parse("F(A,y) = y");
a2 = (AtomicSentence)parser.parse("R(y)");
lits.Add(new Literal(a1));
lits.Add(new Literal(a2));
Clause c2 = new Clause(lits);
ISet <Clause> paras = paramodulation.apply(c1, c2);
Assert.AreEqual(2, paras.Size());
Assert.AreEqual("[P(B,A), Q(A), R(B)]", Util.first(paras).ToString());
paras.Remove(Util.first(paras));
Assert.AreEqual("[P(F(A,F(x,B)),x), Q(x), R(F(x,B))]", Util.first(paras).ToString());
}
public void testNOTSentence()
{
FOLDomain domain = new FOLDomain();
domain.addConstant("A");
domain.addConstant("B");
domain.addConstant("C");
domain.addFunction("Plus");
domain.addPredicate("P");
FOLParser parser = new FOLParser(domain);
Sentence s1 = parser.parse("NOT(P(A))");
Sentence s2 = parser.parse("NOT(P(A))");
IMap <Variable, Term> result = unifier.unify(s1, s2);
Assert.IsNotNull(result);
Assert.AreEqual(0, result.Size());
s1 = parser.parse("NOT(P(A))");
s2 = parser.parse("NOT(P(B))");
result = unifier.unify(s1, s2);
Assert.IsNull(result);
s1 = parser.parse("NOT(P(A))");
s2 = parser.parse("NOT(P(x))");
result = unifier.unify(s1, s2);
Assert.IsNotNull(result);
Assert.AreEqual(1, result.Size());
Assert.AreEqual(new Constant("A"), result.Get(new Variable("x")));
}
public void testMultipleTermEqualitiesInBothClausesExample()
{
FOLDomain domain = new FOLDomain();
domain.addConstant("A");
domain.addConstant("B");
domain.addConstant("C");
domain.addConstant("D");
domain.addPredicate("P");
domain.addPredicate("Q");
domain.addPredicate("R");
domain.addFunction("F");
FOLParser parser = new FOLParser(domain);
ICollection <Literal> lits = CollectionFactory.CreateQueue <Literal>();
AtomicSentence a1 = (AtomicSentence)parser.parse("F(C,x) = D");
AtomicSentence a2 = (AtomicSentence)parser.parse("A = D");
AtomicSentence a3 = (AtomicSentence)parser.parse("P(F(x,B),x)");
AtomicSentence a4 = (AtomicSentence)parser.parse("Q(x)");
AtomicSentence a5 = (AtomicSentence)parser.parse("R(C)");
lits.Add(new Literal(a1));
lits.Add(new Literal(a2));
lits.Add(new Literal(a3));
lits.Add(new Literal(a4));
lits.Add(new Literal(a5));
Clause c1 = new Clause(lits);
lits.Clear();
a1 = (AtomicSentence)parser.parse("F(A,y) = y");
a2 = (AtomicSentence)parser.parse("F(B,y) = C");
a3 = (AtomicSentence)parser.parse("R(y)");
a4 = (AtomicSentence)parser.parse("R(A)");
lits.Add(new Literal(a1));
lits.Add(new Literal(a2));
lits.Add(new Literal(a3));
lits.Add(new Literal(a4));
Clause c2 = new Clause(lits);
ISet <Clause> paras = paramodulation.apply(c1, c2);
Assert.AreEqual(5, paras.Size());
Assert.AreEqual(
"[F(B,B) = C, F(C,A) = D, A = D, P(B,A), Q(A), R(A), R(B), R(C)]",
Util.first(paras).ToString());
paras.Remove(Util.first(paras));
Assert.AreEqual(
"[F(A,F(C,x)) = D, F(B,F(C,x)) = C, A = D, P(F(x,B),x), Q(x), R(F(C,x)), R(A), R(C)]",
Util.first(paras).ToString());
paras.Remove(Util.first(paras));
Assert.AreEqual(
"[F(A,B) = B, F(C,B) = D, A = D, P(C,B), Q(B), R(A), R(B), R(C)]",
Util.first(paras).ToString());
paras.Remove(Util.first(paras));
Assert.AreEqual(
"[F(F(B,y),x) = D, F(A,y) = y, A = D, P(F(x,B),x), Q(x), R(y), R(A), R(C)]",
Util.first(paras).ToString());
paras.Remove(Util.first(paras));
Assert.AreEqual(
"[F(B,y) = C, F(C,x) = D, F(D,y) = y, P(F(x,B),x), Q(x), R(y), R(A), R(C)]",
Util.first(paras).ToString());
}
public void testTermEquality()
{
FOLDomain domain = new FOLDomain();
domain.addConstant("A");
domain.addConstant("B");
domain.addFunction("Plus");
FOLParser parser = new FOLParser(domain);
TermEquality te1 = (TermEquality)parser.parse("x = x");
TermEquality te2 = (TermEquality)parser.parse("x = x");
// Both term equalities the same,
// should unify but no substitutions.
IMap <Variable, Term> result = unifier.unify(te1, te2);
Assert.IsNotNull(result);
Assert.AreEqual(0, result.Size());
// Different variable names but should unify.
te1 = (TermEquality)parser.parse("x1 = x1");
te2 = (TermEquality)parser.parse("x2 = x2");
result = unifier.unify(te1, te2);
Assert.IsNotNull(result);
Assert.AreEqual(1, result.Size());
Assert.AreEqual("[[x1, x2]]", result.ToString());
// Test simple unification with reflexivity axiom
te1 = (TermEquality)parser.parse("x1 = x1");
te2 = (TermEquality)parser.parse("Plus(A,B) = Plus(A,B)");
result = unifier.unify(te1, te2);
Assert.IsNotNull(result);
Assert.AreEqual(1, result.Size());
Assert.AreEqual("[[x1, Plus(A,B)]]", result.ToString());
// Test more complex unification with reflexivity axiom
te1 = (TermEquality)parser.parse("x1 = x1");
te2 = (TermEquality)parser.parse("Plus(A,B) = Plus(A,z1)");
result = unifier.unify(te1, te2);
Assert.IsNotNull(result);
Assert.AreEqual(2, result.Size());
Assert.AreEqual("[[x1, Plus(A,B)], [z1, B]]", result.ToString());
// Test reverse of previous unification with reflexivity axiom
// Should still be the same.
te1 = (TermEquality)parser.parse("x1 = x1");
te2 = (TermEquality)parser.parse("Plus(A,z1) = Plus(A,B)");
result = unifier.unify(te1, te2);
Assert.IsNotNull(result);
Assert.AreEqual(2, result.Size());
Assert.AreEqual("[[x1, Plus(A,B)], [z1, B]]", result.ToString());
// Test with nested terms
te1 = (TermEquality)parser.parse("Plus(Plus(Plus(A,B),B, A)) = Plus(Plus(Plus(A,B),B, A))");
te2 = (TermEquality)parser.parse("Plus(Plus(Plus(A,B),B, A)) = Plus(Plus(Plus(A,B),B, A))");
result = unifier.unify(te1, te2);
Assert.IsNotNull(result);
Assert.AreEqual(0, result.Size());
// Simple term equality unification fails
te1 = (TermEquality)parser.parse("Plus(A,B) = Plus(B,A)");
te2 = (TermEquality)parser.parse("Plus(A,B) = Plus(A,B)");
result = unifier.unify(te1, te2);
Assert.IsNull(result);
}
public void testQuantifiedSentence()
{
FOLDomain domain = new FOLDomain();
domain.addConstant("A");
domain.addConstant("B");
domain.addConstant("C");
domain.addFunction("Plus");
domain.addPredicate("P");
FOLParser parser = new FOLParser(domain);
Sentence s1 = parser
.parse("FORALL x,y ((P(x) AND P(A)) OR (P(A) => P(y)))");
Sentence s2 = parser
.parse("FORALL x,y ((P(x) AND P(A)) OR (P(A) => P(y)))");
IMap <Variable, Term> result = unifier.unify(s1, s2);
Assert.IsNotNull(result);
Assert.AreEqual(0, result.Size());
s1 = parser.parse("FORALL x,y ((P(x) AND P(A)) OR (P(A) => P(y)))");
s2 = parser.parse("FORALL x ((P(x) AND P(A)) OR (P(A) => P(y)))");
result = unifier.unify(s1, s2);
Assert.IsNull(result);
s1 = parser.parse("FORALL x,y ((P(x) AND P(A)) OR (P(A) => P(y)))");
s2 = parser.parse("FORALL x,y ((P(x) AND P(A)) OR (P(B) => P(y)))");
result = unifier.unify(s1, s2);
Assert.IsNull(result);
s1 = parser.parse("FORALL x,y ((P(x) AND P(A)) OR (P(A) => P(y)))");
s2 = parser.parse("FORALL x,y ((P(A) AND P(A)) OR (P(A) => P(y)))");
result = unifier.unify(s1, s2);
Assert.IsNotNull(result);
Assert.AreEqual(1, result.Size());
Assert.AreEqual(new Constant("A"), result.Get(new Variable("x")));
//
s1 = parser.parse("EXISTS x,y ((P(x) AND P(A)) OR (P(A) => P(y)))");
s2 = parser.parse("EXISTS x,y ((P(x) AND P(A)) OR (P(A) => P(y)))");
result = unifier.unify(s1, s2);
Assert.IsNotNull(result);
Assert.AreEqual(0, result.Size());
s1 = parser.parse("EXISTS x,y ((P(x) AND P(A)) OR (P(A) => P(y)))");
s2 = parser.parse("EXISTS x ((P(x) AND P(A)) OR (P(A) => P(y)))");
result = unifier.unify(s1, s2);
Assert.IsNull(result);
s1 = parser.parse("EXISTS x,y ((P(x) AND P(A)) OR (P(A) => P(y)))");
s2 = parser.parse("EXISTS x,y ((P(x) AND P(A)) OR (P(B) => P(y)))");
result = unifier.unify(s1, s2);
Assert.IsNull(result);
s1 = parser.parse("EXISTS x,y ((P(x) AND P(A)) OR (P(A) => P(y)))");
s2 = parser.parse("EXISTS x,y ((P(A) AND P(A)) OR (P(A) => P(y)))");
result = unifier.unify(s1, s2);
Assert.IsNotNull(result);
Assert.AreEqual(1, result.Size());
Assert.AreEqual(new Constant("A"), result.Get(new Variable("x")));
}
public void testComplexEquals()
{
FOLDomain domain = new FOLDomain();
domain.addConstant("A");
domain.addConstant("B");
domain.addConstant("C");
domain.addConstant("D");
domain.addPredicate("P");
domain.addPredicate("Animal");
domain.addPredicate("Kills");
domain.addFunction("F");
domain.addFunction("SF0");
FOLParser parser = new FOLParser(domain);
CNFConverter cnfConverter = new CNFConverter(parser);
Sentence s1 = parser.parse("((x1 = y1 AND y1 = z1) => x1 = z1)");
Sentence s2 = parser.parse("((x2 = y2 AND F(y2) = z2) => F(x2) = z2)");
CNF cnf1 = cnfConverter.convertToCNF(s1);
CNF cnf2 = cnfConverter.convertToCNF(s2);
Clause c1 = cnf1.getConjunctionOfClauses().Get(0);
Clause c2 = cnf2.getConjunctionOfClauses().Get(0);
Assert.IsFalse(c1.Equals(c2));
s1 = parser.parse("((x1 = y1 AND y1 = z1) => x1 = z1)");
s2 = parser.parse("((x2 = y2 AND y2 = z2) => x2 = z2)");
cnf1 = cnfConverter.convertToCNF(s1);
cnf2 = cnfConverter.convertToCNF(s2);
c1 = cnf1.getConjunctionOfClauses().Get(0);
c2 = cnf2.getConjunctionOfClauses().Get(0);
Assert.IsTrue(c1.Equals(c2));
s1 = parser.parse("((x1 = y1 AND y1 = z1) => x1 = z1)");
s2 = parser.parse("((y2 = z2 AND x2 = y2) => x2 = z2)");
cnf1 = cnfConverter.convertToCNF(s1);
cnf2 = cnfConverter.convertToCNF(s2);
c1 = cnf1.getConjunctionOfClauses().Get(0);
c2 = cnf2.getConjunctionOfClauses().Get(0);
Assert.IsTrue(c1.Equals(c2));
s1 = parser.parse("(((x1 = y1 AND y1 = z1) AND z1 = r1) => x1 = r1)");
s2 = parser.parse("(((x2 = y2 AND y2 = z2) AND z2 = r2) => x2 = r2)");
cnf1 = cnfConverter.convertToCNF(s1);
cnf2 = cnfConverter.convertToCNF(s2);
c1 = cnf1.getConjunctionOfClauses().Get(0);
c2 = cnf2.getConjunctionOfClauses().Get(0);
Assert.IsTrue(c1.Equals(c2));
s1 = parser.parse("(((x1 = y1 AND y1 = z1) AND z1 = r1) => x1 = r1)");
s2 = parser.parse("(((z2 = r2 AND y2 = z2) AND x2 = y2) => x2 = r2)");
cnf1 = cnfConverter.convertToCNF(s1);
cnf2 = cnfConverter.convertToCNF(s2);
c1 = cnf1.getConjunctionOfClauses().Get(0);
c2 = cnf2.getConjunctionOfClauses().Get(0);
Assert.IsTrue(c1.Equals(c2));
s1 = parser.parse("(((x1 = y1 AND y1 = z1) AND z1 = r1) => x1 = r1)");
s2 = parser.parse("(((x2 = y2 AND y2 = z2) AND z2 = y2) => x2 = r2)");
cnf1 = cnfConverter.convertToCNF(s1);
cnf2 = cnfConverter.convertToCNF(s2);
c1 = cnf1.getConjunctionOfClauses().Get(0);
c2 = cnf2.getConjunctionOfClauses().Get(0);
Assert.IsFalse(c1.Equals(c2));
s1 = parser
.parse("(((((x1 = y1 AND y1 = z1) AND z1 = r1) AND r1 = q1) AND q1 = s1) => x1 = r1)");
s2 = parser
.parse("(((((x2 = y2 AND y2 = z2) AND z2 = r2) AND r2 = q2) AND q2 = s2) => x2 = r2)");
cnf1 = cnfConverter.convertToCNF(s1);
cnf2 = cnfConverter.convertToCNF(s2);
c1 = cnf1.getConjunctionOfClauses().Get(0);
c2 = cnf2.getConjunctionOfClauses().Get(0);
Assert.IsTrue(c1.Equals(c2));
s1 = parser
.parse("((((NOT(Animal(c1920)) OR NOT(Animal(c1921))) OR NOT(Kills(c1922,c1920))) OR NOT(Kills(c1919,c1921))) OR NOT(Kills(SF0(c1922),SF0(c1919))))");
s2 = parser
.parse("((((NOT(Animal(c1929)) OR NOT(Animal(c1928))) OR NOT(Kills(c1927,c1929))) OR NOT(Kills(c1930,c1928))) OR NOT(Kills(SF0(c1930),SF0(c1927))))");
cnf1 = cnfConverter.convertToCNF(s1);
cnf2 = cnfConverter.convertToCNF(s2);
c1 = cnf1.getConjunctionOfClauses().Get(0);
c2 = cnf2.getConjunctionOfClauses().Get(0);
Assert.IsTrue(c1.Equals(c2));
}
public void testNonTrivialFactors()
{
FOLDomain domain = new FOLDomain();
domain.addConstant("A");
domain.addConstant("B");
domain.addFunction("F");
domain.addFunction("G");
domain.addFunction("H");
domain.addPredicate("P");
domain.addPredicate("Q");
FOLParser parser = new FOLParser(domain);
// p(x,y), q(a,b), p(b,a), q(y,x)
Clause c = new Clause();
c.addPositiveLiteral((Predicate)parser.parse("P(x,y)"));
c.addPositiveLiteral((Predicate)parser.parse("Q(A,B)"));
c.addNegativeLiteral((Predicate)parser.parse("P(B,A)"));
c.addPositiveLiteral((Predicate)parser.parse("Q(y,x)"));
Assert.AreEqual("[[~P(B,A), P(B,A), Q(A,B)]]", c.getNonTrivialFactors().ToString());
// p(x,y), q(a,b), p(b,a), q(y,x)
c = new Clause();
c.addPositiveLiteral((Predicate)parser.parse("P(x,y)"));
c.addPositiveLiteral((Predicate)parser.parse("Q(A,B)"));
c.addNegativeLiteral((Predicate)parser.parse("P(B,A)"));
c.addNegativeLiteral((Predicate)parser.parse("Q(y,x)"));
Assert.AreEqual("[]", c.getNonTrivialFactors().ToString());
// p(x,f(y)), p(g(u),x), p(f(y),u)
c = new Clause();
c.addPositiveLiteral((Predicate)parser.parse("P(x,F(y))"));
c.addPositiveLiteral((Predicate)parser.parse("P(G(u),x)"));
c.addPositiveLiteral((Predicate)parser.parse("P(F(y),u)"));
// Should be: [{P(F(c#),F(c#)),P(G(F(c#)),F(c#))}]
c = Util.first(c.getNonTrivialFactors());
Literal p = c.getPositiveLiterals().Get(0);
Assert.AreEqual("P", p.getAtomicSentence().getSymbolicName());
Function f = (Function)p.getAtomicSentence().getArgs().Get(0);
Assert.AreEqual("F", f.getFunctionName());
Variable v = (Variable)f.getTerms().Get(0);
f = (Function)p.getAtomicSentence().getArgs().Get(1);
Assert.AreEqual("F", f.getFunctionName());
Assert.AreEqual(v, f.getTerms().Get(0));
//
p = c.getPositiveLiterals().Get(1);
f = (Function)p.getAtomicSentence().getArgs().Get(0);
Assert.AreEqual("G", f.getFunctionName());
f = (Function)f.getTerms().Get(0);
Assert.AreEqual("F", f.getFunctionName());
Assert.AreEqual(v, f.getTerms().Get(0));
f = (Function)p.getAtomicSentence().getArgs().Get(1);
Assert.AreEqual("F", f.getFunctionName());
Assert.AreEqual(v, f.getTerms().Get(0));
// p(g(x)), q(x), p(f(a)), p(x), p(g(f(x))), q(f(a))
c = new Clause();
c.addPositiveLiteral((Predicate)parser.parse("P(G(x))"));
c.addPositiveLiteral((Predicate)parser.parse("Q(x)"));
c.addPositiveLiteral((Predicate)parser.parse("P(F(A))"));
c.addPositiveLiteral((Predicate)parser.parse("P(x)"));
c.addPositiveLiteral((Predicate)parser.parse("P(G(F(x)))"));
c.addPositiveLiteral((Predicate)parser.parse("Q(F(A))"));
Assert.AreEqual("[[P(F(A)), P(G(F(F(A)))), P(G(F(A))), Q(F(A))]]", c.getNonTrivialFactors().ToString());
}
public void testTermEquality()
{
FOLDomain domain = new FOLDomain();
domain.addPredicate("P");
domain.addPredicate("Q");
domain.addPredicate("R");
domain.addConstant("A");
domain.addConstant("B");
domain.addConstant("C");
domain.addConstant("D");
domain.addFunction("Plus");
domain.addConstant("ONE");
domain.addConstant("ZERO");
FOLParser parser = new FOLParser(domain);
CNFConverter cnfConv = new CNFConverter(parser);
// x=y
Sentence sent = parser.parse("x = y");
CNF cnf = cnfConv.convertToCNF(sent);
Assert.AreEqual("[x = y]", cnf.ToString());
// x!=y
sent = parser.parse("NOT(x = y)");
cnf = cnfConv.convertToCNF(sent);
Assert.AreEqual("[~x = y]", cnf.ToString());
// A=B
sent = parser.parse("A = B");
cnf = cnfConv.convertToCNF(sent);
Assert.AreEqual("[A = B]", cnf.ToString());
// A!=B
sent = parser.parse("NOT(A = B)");
cnf = cnfConv.convertToCNF(sent);
Assert.AreEqual("[~A = B]", cnf.ToString());
// ~(((~A=B or ~D=C) => ~(A=B or D=C)) => A=D)
sent = parser
.parse("NOT(((((NOT(A = B) OR NOT(D = C))) => NOT((A = B OR D = C))) => A = D))");
cnf = cnfConv.convertToCNF(sent);
Assert.AreEqual(
"[~A = B, A = B],[~A = B, D = C],[~D = C, A = B],[~D = C, D = C],[~A = D]",
cnf.ToString());
//
// Induction Axiom Schema using Term Equality
// Base Case:
sent = parser
.parse("NOT(FORALL x (FORALL y (Plus(Plus(x,y),ZERO) = Plus(x,Plus(y,ZERO)))))");
cnf = cnfConv.convertToCNF(sent);
Assert.AreEqual(
"[~Plus(Plus(SC0,SC1),ZERO) = Plus(SC0,Plus(SC1,ZERO))]",
cnf.ToString());
// Instance of Induction Axion Scmema
sent = parser.parse("(("
+ "Plus(Plus(A,B),ZERO) = Plus(A,Plus(B,ZERO))" + " AND "
+ "(FORALL x (FORALL y (FORALL z("
+ "Plus(Plus(x,y),z) = Plus(x,Plus(y,z))" + " => "
+ "Plus(Plus(x,y),Plus(z,ONE)) = Plus(x,Plus(y,Plus(z,ONE)))"
+ "))))" + ")" + " => " + "FORALL x (FORALL y (FORALL z("
+ "Plus(Plus(x,y),z) = Plus(x,Plus(y,z))" + "))))");
cnf = cnfConv.convertToCNF(sent);
Assert.AreEqual(
"[~Plus(Plus(A,B),ZERO) = Plus(A,Plus(B,ZERO)), Plus(Plus(q0,q1),q2) = Plus(q0,Plus(q1,q2)), Plus(Plus(SC2,SC3),SC4) = Plus(SC2,Plus(SC3,SC4))],[~Plus(Plus(A,B),ZERO) = Plus(A,Plus(B,ZERO)), ~Plus(Plus(SC2,SC3),Plus(SC4,ONE)) = Plus(SC2,Plus(SC3,Plus(SC4,ONE))), Plus(Plus(q0,q1),q2) = Plus(q0,Plus(q1,q2))]",
cnf.ToString());
// Goal
sent = parser
.parse("NOT(FORALL x (FORALL y (FORALL z (Plus(Plus(x,y),z) = Plus(x,Plus(y,z))))))");
cnf = cnfConv.convertToCNF(sent);
Assert.AreEqual(
"[~Plus(Plus(SC5,SC6),SC7) = Plus(SC5,Plus(SC6,SC7))]",
cnf.ToString());
}
public void testMultipleTermEqualitiesInBothClausesExample()
{
FOLDomain domain = new FOLDomain();
domain.addConstant("A");
domain.addConstant("B");
domain.addConstant("C");
domain.addConstant("D");
domain.addPredicate("P");
domain.addPredicate("Q");
domain.addPredicate("R");
domain.addFunction("F");
FOLParser parser = new FOLParser(domain);
List <Literal> lits = new List <Literal>();
AtomicSentence a1 = (AtomicSentence)parser.parse("F(C,x) = D");
AtomicSentence a2 = (AtomicSentence)parser.parse("A = D");
AtomicSentence a3 = (AtomicSentence)parser.parse("P(F(x,B),x)");
AtomicSentence a4 = (AtomicSentence)parser.parse("Q(x)");
AtomicSentence a5 = (AtomicSentence)parser.parse("R(C)");
lits.Add(new Literal(a1));
lits.Add(new Literal(a2));
lits.Add(new Literal(a3));
lits.Add(new Literal(a4));
lits.Add(new Literal(a5));
Clause c1 = new Clause(lits);
lits.Clear();
a1 = (AtomicSentence)parser.parse("F(A,y) = y");
a2 = (AtomicSentence)parser.parse("F(B,y) = C");
a3 = (AtomicSentence)parser.parse("R(y)");
a4 = (AtomicSentence)parser.parse("R(A)");
lits.Add(new Literal(a1));
lits.Add(new Literal(a2));
lits.Add(new Literal(a3));
lits.Add(new Literal(a4));
Clause c2 = new Clause(lits);
List <Clause> paras = paramodulation.apply(c1, c2);
Assert.AreEqual(5, paras.Count);
List <Clause> .Enumerator it = paras.GetEnumerator();
it.MoveNext();
Assert
.AreEqual(
"[F(B,B) = C, F(C,A) = D, A = D, P(B,A), Q(A), R(A), R(B), R(C)]",
it.Current.ToString());
it.MoveNext();
Assert
.AreEqual(
"[F(A,F(C,x)) = D, F(B,F(C,x)) = C, A = D, P(F(x,B),x), Q(x), R(F(C,x)), R(A), R(C)]",
it.Current.ToString());
it.MoveNext();
Assert
.AreEqual(
"[F(A,B) = B, F(C,B) = D, A = D, P(C,B), Q(B), R(A), R(B), R(C)]",
it.Current.ToString());
it.MoveNext();
Assert
.AreEqual(
"[F(F(B,y),x) = D, F(A,y) = y, A = D, P(F(x,B),x), Q(x), R(y), R(A), R(C)]",
it.Current.ToString());
it.MoveNext();
Assert
.AreEqual(
"[F(B,y) = C, F(C,x) = D, F(D,y) = y, P(F(x,B),x), Q(x), R(y), R(A), R(C)]",
it.Current.ToString());
}