// NOTE: Maybe should REMOVE this method
// because the formulas that are And
// will eventually be expanded by ExpandToAnd(Symbol).
// It will also make the Tableau more easy to look.
/// <summary>
/// Expand all the Ands formula
/// </summary>
/// <param name="formulas"></param>
/// <returns>A set of formulas with all the And expanded</returns>
public static HashSet <Symbol> ExpandAllAnds(
TableauNode curNode, bool alphaRules, bool deltaRules)
{
List <Symbol> delList = new List <Symbol>();
List <Symbol> addList = new List <Symbol>();
HashSet <Symbol> formulas = curNode.SetFormulas;
// Subformula that is expanded to And
foreach (Symbol subformula in formulas)
{
List <Symbol> expansion = new List <Symbol>();
if (alphaRules)
{
expansion = ExpandToAnd(subformula);
}
// Delta rules - Expand !x and ~@x
if (deltaRules)
{
expansion = DeltaRules(subformula, curNode);
}
if (!(expansion is null))
{
foreach (Symbol term in expansion)
{
addList.Add(term);
}
delList.Add(subformula);
}
}
foreach (Symbol subformula in delList)
{
formulas.Remove(subformula);
}
foreach (Symbol subformula in addList)
{
formulas.Add(subformula);
}
delList.Clear();
addList.Clear();
return(formulas);
}
public void Build()
{
// Root of Tableau
Root = new TableauNode(new HashSet <Symbol>(new SymbolComparer())
{
new Not(Formula)
});
Root.RuleApplied = TableauRule.NONE;
bool unsatis = BuildUtil(Root);
State = 0;
if (unsatis)
{
State = 1;
}
}
public static List <Symbol> DeltaRules(Symbol formula, TableauNode currentNode)
{
char newVar = '#';
for (char c = 'a'; c <= 'z'; c++)
{
if (!currentNode.CurrentVariables[c])
{
newVar = c;
break;
}
}
if (formula is Existential)
{
char boundedVar = ((Existential)formula).BoundVariables[0]; // extend later
formula = formula.ChangeVariableName(boundedVar, newVar, true).Childs[0];
currentNode.CurrentVariables[newVar] = true;
}
else if (formula is Not && formula.Childs[0] is Universal)
{
formula = formula.Childs[0];
char boundedVar = ((Universal)formula).BoundVariables[0]; // extend later
formula = new Not(formula.ChangeVariableName(boundedVar, newVar, true).Childs[0]);
currentNode.CurrentVariables[newVar] = true;
}
else
{
return(null);
}
return(new List <Symbol>()
{
formula
});
}
public static List <Symbol> GammaRules(Symbol formula, TableauNode currentNode)
{
List <Symbol> returnList = new List <Symbol>();
if (formula is Universal)
{
foreach (char c in currentNode.ListVariables)
{
if (formula.GammaApplied[c])
{
continue;
}
char boundVariable = ((Universal)formula).BoundVariables[0];
returnList.Add(formula.ChangeVariableName(boundVariable, c, true).Childs[0]);
formula.GammaApplied[c] = true;
}
}
else if (formula is Not && formula.Childs[0] is Existential)
{
char boundVariable = ((Existential)formula.Childs[0]).BoundVariables[0];
foreach (char c in currentNode.ListVariables)
{
if (formula.GammaApplied[c])
{
continue;
}
returnList.Add(new Not(formula.ChangeVariableName(boundVariable, c, true).Childs[0].Childs[0]));
formula.GammaApplied[c] = true;
}
}
return(returnList.Count == 0 ? null : returnList);
}
/*
* 1) Check contradiction
* 2) Alpha rules
* 3) Delta rules
* 4) Beta rules
* 5) Gamma rules
*/
/// <summary>
/// Build the Semantic Tableau
/// </summary>
/// <returns>True if the formula is Unsatifiable (has contradiction)</returns>
public bool BuildUtil(TableauNode curNode) // return true if contradiction
{
TableauNode newNode = new TableauNode(curNode.SetFormulas, curNode.CurrentVariables);
if (CheckContradiction(newNode.SetFormulas))
{
newNode.SetFormulas = new HashSet <Symbol>()
{
new Constant('0')
};
curNode.Childs.Add(newNode);
return(true);
}
HashSet <Symbol> formulas = newNode.SetFormulas;
//////////////////////////////////////////////////////////////
// Alpha Rules
//////////////////////////////////////////////////////////////
formulas = ExpandAllAnds(newNode, true, false);
if (!TableauNode.EqualSet(formulas, curNode.SetFormulas)) // successful
{
newNode.RuleApplied = TableauRule.ALPHA;
curNode.Childs.Add(newNode);
return(BuildUtil(newNode));
}
// Haven't applied Alpha rules
// Continue with
//////////////////////////////////////////////////////////////
// Delta rules
//////////////////////////////////////////////////////////////
formulas = ExpandAllAnds(newNode, false, true);
if (!TableauNode.EqualSet(formulas, curNode.SetFormulas)) // successful
{
newNode.RuleApplied = TableauRule.DELTA;
curNode.Childs.Add(newNode);
return(BuildUtil(newNode));
}
//////////////////////////////////////////////////////////////
// Beta Rules
//////////////////////////////////////////////////////////////
bool CurrentlyContradicted = true;
Symbol chosenFormula = null; // The formula that can be applied beta rules
List <Symbol> expansion = new List <Symbol>(); // expanded version of the chosen formula
// Find a subformula that can be applied beta rules
foreach (Symbol subformula in formulas)
{
expansion = ExpandToOr(subformula);
if (expansion is null)
{
continue;
}
chosenFormula = subformula;
break;
}
// There exist a formula expanded.
if (!(chosenFormula == null))
{
formulas.Remove(chosenFormula); // remove the formula
foreach (Symbol term in expansion)
{
formulas.Add(term); // Add an expanded subformula
var newSet = new HashSet <Symbol>(new SymbolComparer());
foreach (var item in formulas)
{
newSet.Add(ObjectExtensions.Copy(item));
}
TableauNode tmpNode = new TableauNode(newSet, curNode.CurrentVariables);
tmpNode.RuleApplied = TableauRule.BETA;
curNode.Childs.Add(tmpNode);
if (!BuildUtil(tmpNode))
{
CurrentlyContradicted = false;
}
formulas.Remove(term);
}
return(CurrentlyContradicted); // both childs must be contradition
}
//////////////////////////////////////////////////////////////
// Gamma rules
//////////////////////////////////////////////////////////////
var expansionStorage = new HashSet <Symbol>(new SymbolComparer());
expansion = new List <Symbol>();
foreach (var subformula in formulas)
{
expansion = GammaRules(subformula, newNode);
if (expansion is null)
{
continue;
}
foreach (var expansionTerm in expansion)
{
expansionStorage.Add(expansionTerm);
}
}
if (expansionStorage.Count > 0)
{
foreach (var expansionTerm in expansionStorage)
{
formulas.Add(expansionTerm);
}
curNode.Childs.Add(newNode);
newNode.RuleApplied = TableauRule.GAMMA;
return(BuildUtil(newNode));
}
return(false);
}
public int GenerateGraphUtil(StreamWriter sw, TableauNode u)
{
string content = ""; // text of the Tableau Node (the set of formulas at the current node)
// Add all formulas to the content
foreach (var formula in u.SetFormulas)
{
content += formula.ToString() + "\n";
}
//content = content.Remove(content.Length - 1); // remove last ','
//content = content.Remove(1, 1); // remove first space
//content += "]";
// List active variables
content += " [";
foreach (char c in u.ListVariables)
{
content += c + ",";
}
if (content[content.Length - 1] == ',')
{
content = content.Remove(content.Length - 1); // remove last ','
}
content += "]";
int uNum = Counter; // ID of current node
string fillColor = (u.RuleApplied == TableauRule.ALPHA ? "yellow" :
u.RuleApplied == TableauRule.BETA ? "palegreen" :
u.RuleApplied == TableauRule.DELTA ? "skyblue" :
u.RuleApplied == TableauRule.GAMMA ? "brown1" :
"gray88");
// content of the node
string graphvizContent = $"\tnode{uNum} [shape=rectangle; style = filled;" +
$"label = \"{content}\"; color = black; fillcolor={fillColor}];";
// if this node is a contradiction (a node containing only false)
// then color it red
if (TableauNode.EqualSet(u.SetFormulas, new HashSet <Symbol>()
{
new Constant('0')
}))
{
graphvizContent = $"\tnode{uNum} [shape=rectangle; style=filled;" +
$"label = \"{content}\"; color=crimson fillcolor={fillColor}];";
}
// draw current node
sw.WriteLine(graphvizContent);
foreach (TableauNode v in u.Childs)
{
Counter++;
int vNum = GenerateGraphUtil(sw, v);
sw.WriteLine("\tnode{0} -- node{1};", uNum, vNum); // draw edge to child node
}
return(uNum); // return ID of current node
}