// Get a propositional formula and return its (raw) CNF form
// addedVars: the names of the new variables introduced by the transformation
private static ISet <ISet <string> > TseytinTransformation(FormulaNode formula, out ISet <string> addedVars)
{
ISet <ISet <string> > res = new HashSet <ISet <string> >();
ISet <TseytinBlock> blockSet = new HashSet <TseytinBlock>();
ticketMachine.Reset();
TseytinBlock top = GetTseytinBlocks(formula, blockSet);
if (top == null)
{
// The formula is simply a variable - no Tseitin blocks required
res.Add(new HashSet <string> {
formula.GetName()
});
}
else
{
// Create a conjunction of all Tseitin blocks
foreach (TseytinBlock block in blockSet)
{
res.UnionWith(TseytinBlockToCNF(block));
}
// Add the requirement that the topmost formula is true
ISet <string> topClause = new HashSet <string>();
topClause.Add(top.ticket.ToString());
res.Add(topClause);
}
addedVars = new HashSet <string>();
foreach (TseytinBlock block in blockSet)
{
addedVars.Add(block.ticket.ToString());
}
// Force the TRUE variable to true
res.Add(new HashSet <string> {
FormulaNode.TRUE_LITERAL
});
return(res);
}
/* Replace the variables describing the state <from> with the ones that describe
* state <to>
*/
public static FormulaNode Substitute(FormulaNode formula, SymbolicState from, SymbolicState to)
{
if (formula.GetLogicOperator() == LogicOperator.VAR)
{
if (from.nameToElementary.Keys.Contains(formula.GetName()))
{
// this variable represents an elementary formula
var elementary = from.nameToElementary[formula.GetName()];
return(new FormulaNode(to.elementaryNames[elementary]));
}
else
{
return(new FormulaNode(formula.GetName()));
}
}
if (formula.GetLogicOperator() == LogicOperator.ALL ||
formula.GetLogicOperator() == LogicOperator.EXISTS)
{
FormulaNode result = new FormulaNode(formula.GetLogicOperator());
if (from.nameToElementary.Keys.Contains(formula.GetName()))
{
// this variable represents an elementary formula
var elementary = from.nameToElementary[formula.GetName()];
result.SetName(to.elementaryNames[elementary]);
}
else
{
result.SetName(formula.GetName());
}
result.SetChildren(Substitute(formula[0], from, to), null);
return(result);
}
FormulaNode res = new FormulaNode(formula.GetLogicOperator());
FormulaNode left = Substitute(formula[0], from, to);
FormulaNode right = null;
if (formula[1] != null)
{
right = Substitute(formula[1], from, to);
}
res.SetChildren(left, right);
return(res);
}
// Get a PNF formula and return its QBCNF
public static QBCNFormula ConvertToCNF(FormulaNode formula)
{
QBCNFormula res = new QBCNFormula();
FormulaNode node = formula;
ISet <string> addedVars;
res.quantifiers = new List <string>();
while (node.GetLogicOperator() == LogicOperator.EXISTS ||
node.GetLogicOperator() == LogicOperator.ALL)
{
if (node.GetLogicOperator() == LogicOperator.EXISTS)
{
res.quantifiers.Add("e" + node.GetName());
}
else
{
res.quantifiers.Add("a" + node.GetName());
}
node = node[0];
}
if (res.quantifiers.Count != formula.GetVariables().Count)
{
throw new Exception("Non-quantified variables in formula!");
}
res.propositional = TseytinTransformation(node, out addedVars);
//Replace strings with numbers
ISet <string> literals = res.GetLiterals();
Dictionary <string, string> changes = new Dictionary <string, string>();
foreach (string literal in literals)
{
int n;
bool isNum = int.TryParse(literal, out n);
if (!changes.ContainsKey(literal) & !isNum)
{
int ticket = ticketMachine.GetTicket();
if (literal[0] != '-')
{
changes.Add(literal, ticket.ToString());
changes.Add("-" + literal, (-ticket).ToString());
}
else
{
changes.Add(literal, (-ticket).ToString());
changes.Add(literal.Substring(1), ticket.ToString());
}
}
}
res.ReplaceLiterals(changes);
// Quantify the Tseytin variables with EXISTS
foreach (string var in addedVars)
{
res.quantifiers.Add("e" + var);
}
return(res);
}
/* Assumes the formula is in NNF form */
public static ISet <FormulaNode> elementaryFormulas(FormulaNode formula)
{
HashSet <FormulaNode> result = new HashSet <FormulaNode>();
switch (formula.GetLogicOperator())
{
case LogicOperator.AU:
case LogicOperator.AR:
// AX(f) is elementary
FormulaNode ax_f = new FormulaNode(LogicOperator.AX);
ax_f.SetChildren(formula, null);
result.Add(ax_f);
result.Add(nnfNegate(ax_f));
result.UnionWith(elementaryFormulas(formula[0]));
result.UnionWith(elementaryFormulas(formula[1]));
break;
case LogicOperator.EU:
case LogicOperator.ER:
// EX(f) is elementary
FormulaNode ex_f = new FormulaNode(LogicOperator.EX);
ex_f.SetChildren(formula, null);
result.Add(ex_f);
result.Add(nnfNegate(ex_f));
result.UnionWith(elementaryFormulas(formula[0]));
result.UnionWith(elementaryFormulas(formula[1]));
break;
case LogicOperator.EX:
case LogicOperator.AX:
// the formula itself is elementary
result.Add(formula);
result.Add(nnfNegate(formula));
result.UnionWith(elementaryFormulas(formula[0]));
break;
case LogicOperator.OR:
case LogicOperator.AND:
result.UnionWith(elementaryFormulas(formula[0]));
result.UnionWith(elementaryFormulas(formula[1]));
break;
case LogicOperator.NOT:
Debug.Assert(formula[0].GetLogicOperator() == LogicOperator.VAR,
"Input to elementaryFormulas must be in NNF form");
result.UnionWith(elementaryFormulas(formula[0]));
break;
case LogicOperator.VAR:
if (formula.GetName() != FormulaNode.TRUE_LITERAL)
{
result.Add(formula);
result.Add(nnfNegate(formula));
}
break;
default:
throw new Exception("Not implemented");
}
return(result);
}
/* Build a propositional formula representing the value of the given CTL
* formula in this state
*/
public FormulaNode ValueOf(FormulaNode formula)
{
FormulaNode result, er;
switch (formula.GetLogicOperator())
{
case LogicOperator.VAR:
case LogicOperator.EX:
// literals and EX(...) formulas are considered positive elementary,
// and so should correspond directly to a propositional variable
if (formula.GetLogicOperator() == LogicOperator.VAR &&
formula.GetName() == FormulaNode.TRUE_LITERAL)
{
return(new FormulaNode(FormulaNode.TRUE_LITERAL));
}
return(new FormulaNode(elementaryNames[formula]));
case LogicOperator.AX:
FormulaNode notBody = CTLUtils.nnfNegate(formula[0]);
FormulaNode ex = new FormulaNode(LogicOperator.EX, notBody, null);
FormulaNode exValue = ValueOf(ex);
return(new FormulaNode(LogicOperator.NOT, exValue, null));
case LogicOperator.EU:
result = ValueOf(new FormulaNode(LogicOperator.EX, formula, null));
result = new FormulaNode(LogicOperator.AND, ValueOf(formula[0]), result);
result = new FormulaNode(LogicOperator.OR, ValueOf(formula[1]), result);
return(result);
case LogicOperator.AU:
er = new FormulaNode(LogicOperator.ER,
CTLUtils.nnfNegate(formula[0]),
CTLUtils.nnfNegate(formula[1]));
result = ValueOf(new FormulaNode(LogicOperator.EX, er, null));
result = new FormulaNode(LogicOperator.NOT, result, null);
result = new FormulaNode(LogicOperator.AND, ValueOf(formula[0]), result);
result = new FormulaNode(LogicOperator.OR, ValueOf(formula[1]), result);
return(result);
case LogicOperator.ER:
result = ValueOf(new FormulaNode(LogicOperator.EX, formula, null));
result = new FormulaNode(LogicOperator.OR, ValueOf(formula[0]), result);
result = new FormulaNode(LogicOperator.AND, ValueOf(formula[1]), result);
return(result);
case LogicOperator.AR:
er = new FormulaNode(LogicOperator.EU,
CTLUtils.nnfNegate(formula[0]),
CTLUtils.nnfNegate(formula[1]));
result = ValueOf(new FormulaNode(LogicOperator.EX, er, null));
result = new FormulaNode(LogicOperator.NOT, result, null);
result = new FormulaNode(LogicOperator.OR, ValueOf(formula[0]), result);
result = new FormulaNode(LogicOperator.AND, ValueOf(formula[1]), result);
return(result);
case LogicOperator.AND:
case LogicOperator.OR:
return(new FormulaNode(formula.GetLogicOperator(), ValueOf(formula[0]), ValueOf(formula[1])));
case LogicOperator.NOT:
FormulaNode bodyVar;
if (formula[0].GetLogicOperator() == LogicOperator.VAR &&
formula[0].GetName() == FormulaNode.TRUE_LITERAL)
{
bodyVar = new FormulaNode(FormulaNode.TRUE_LITERAL);
}
else if (!elementaryNames.ContainsKey(formula[0]))
{
throw new Exception("Argument to SymbolicState.valueOf must be contained in the closure, and in NNF form.");
}
else
{
bodyVar = new FormulaNode(elementaryNames[formula[0]]);
}
return(new FormulaNode(LogicOperator.NOT, bodyVar, null));
default:
throw new NotImplementedException();
}
}
