private void ProcessSubstitutionStep(UnificationContinuation currentProblem, Equation currEq, List <TrsVariable> variableNamesToPreserve)
{
// LHS will always be variable here ... preserve matched term variables
Substitution newSub = null;
if (currEq.Rhs is TrsVariable &&
variableNamesToPreserve.Contains(currEq.Lhs))
{
newSub = new Substitution
{
Variable = (TrsVariable)currEq.Rhs,
SubstitutionTerm = currEq.Lhs
};
}
else
{
newSub = new Substitution
{
Variable = (TrsVariable)currEq.Lhs,
SubstitutionTerm = currEq.Rhs
};
}
currentProblem.ComposeAndApplySubstitutions(newSub);
}
private static bool ProcessAcUnificationStep(UnificationContinuation currentProblem, Queue<UnificationContinuation> currentContinuations,
Func<UnificationContinuation, Equation, bool> processFail, Equation currEq)
{
var acLhs = currEq.Lhs as TrsAcTerm;
var acRhs = currEq.Rhs as TrsAcTerm;
bool fail = false;
if (acLhs.Name != acRhs.Name)
{
fail = !processFail(currentProblem, currEq);
}
else
{
// Reduce cardinalities to reduce number of permutations
foreach (var pair in (from lhsTerm in acLhs.OnfArguments
join rhsTerm in acRhs.OnfArguments
on lhsTerm.Term equals rhsTerm.Term
where lhsTerm.Term.IsGroundTerm
&& rhsTerm.Term.IsGroundTerm
select new {
Lhs = lhsTerm,
Rhs = rhsTerm
}).ToList())
{
var diff = Math.Min(pair.Lhs.Cardinality, pair.Rhs.Cardinality);
if (pair.Lhs.Cardinality > diff) pair.Lhs.Cardinality -= diff;
else acLhs.OnfArguments.Remove(pair.Lhs);
if (pair.Rhs.Cardinality > diff) pair.Rhs.Cardinality -= diff;
else acRhs.OnfArguments.Remove(pair.Rhs);
}
// AC partitioning can only take place from the small to the large cardinalities
List<TrsTermBase> fromArgs = null;
List<TrsTermBase> toArgs = null;
if (acLhs.TotalCardinality > acRhs.TotalCardinality)
{
fromArgs = acRhs.ExpandedArguments;
toArgs = acLhs.ExpandedArguments;
}
else
{
fromArgs = acLhs.ExpandedArguments;
toArgs = acRhs.ExpandedArguments;
}
// Generate mappings ... this is where the AC semantics are applied
var setPartitionMappings = new SetMapPartitionEnumerator<int, int>(new HashSet<int>(Enumerable.Range(0, fromArgs.Count)),
new HashSet<int>(Enumerable.Range(0, toArgs.Count)));
// The first mapping will map to the first continuation, this must be added to the
// current problem's equations in order to preserve algorithm correctness when the unification result
// contains no substitutions.
var isFirstContinuation = true;
var initialCurrentProblem = currentProblem.CreateCopy(); // currentProblem gets updated in the first iteration
foreach (var mappings in setPartitionMappings)
{
// Cater for multiple unification problems generated by the current AC problem
UnificationContinuation clone = null;
if (!isFirstContinuation) clone = initialCurrentProblem.CreateCopy();
foreach (var mapping in mappings)
{
Equation eq = new Equation();
eq.Lhs = fromArgs[mapping.FromElement];
if (mapping.ToPartition.Count == 1) eq.Rhs = toArgs[mapping.ToPartition.First()];
else eq.Rhs = new TrsAcTerm(acLhs.Name, mapping.ToPartition.Select(i => toArgs[i]));
if (!isFirstContinuation) clone.CurrentEquations.Add(eq);
else currentProblem.CurrentEquations.Add(eq);
}
if (!isFirstContinuation) currentContinuations.Enqueue(clone);
else isFirstContinuation = false;
}
}
return fail;
}
private void ProcessSubstitutionStep(UnificationContinuation currentProblem, Equation currEq, List<TrsVariable> variableNamesToPreserve)
{
// LHS will always be variable here ... preserve matched term variables
Substitution newSub = null;
if (currEq.Rhs is TrsVariable
&& variableNamesToPreserve.Contains(currEq.Lhs))
{
newSub = new Substitution
{
Variable = (TrsVariable)currEq.Rhs,
SubstitutionTerm = currEq.Lhs
};
}
else
{
newSub = new Substitution
{
Variable = (TrsVariable)currEq.Lhs,
SubstitutionTerm = currEq.Rhs
};
}
currentProblem.ComposeAndApplySubstitutions(newSub);
}
/// <summary>
/// Returns empty list if the terms in the equation cannot be unified or if Lhs = Rhs with no variables,
/// otherwise a list of substitutions defining the MGU
///
/// Based on Unification chapter from Handbook of Automated Reasoning.
/// </summary>
private List<UnificationResult> GetMgu(Equation unificationProblem, List<TrsVariable> variableNamesToPreserve)
{
if (unificationProblem == null) throw new ArgumentException();
if (unificationProblem.Lhs == null) throw new ArgumentException();
if (unificationProblem.Rhs == null) throw new ArgumentException();
if (variableNamesToPreserve == null) throw new ArgumentException();
Equation initialEquation = unificationProblem.CreateCopy();
UnificationContinuation currentProblem = new UnificationContinuation
{
CurrentEquations = new List<Equation>(),
CurrentSubstitutions = new List<Substitution>()
};
currentProblem.CurrentEquations.Add(initialEquation);
Queue<UnificationContinuation> currentContinuations = new Queue<UnificationContinuation>();
currentContinuations.Enqueue(currentProblem);
HashSet<UnificationResult> results = new HashSet<UnificationResult>();
Func<UnificationContinuation, Equation, bool> processFail = delegate (UnificationContinuation curr, Equation currEq)
{
var failResult = CustomFail(currentProblem, currEq, variableNamesToPreserve);
var succeed = true;
if (failResult.Count == 0) succeed = false;
else if (failResult.Count == 1) currentProblem = failResult.First();
else foreach (var continuation in failResult) currentContinuations.Enqueue(continuation);
return succeed;
};
while (currentContinuations.Count > 0)
{
currentProblem = currentContinuations.Dequeue();
bool fail = false;
while (currentProblem.CurrentEquations.Count > 0 && !fail)
{
var currEq = currentProblem.CurrentEquations[currentProblem.CurrentEquations.Count - 1];
currentProblem.CurrentEquations.RemoveAt(currentProblem.CurrentEquations.Count - 1);
if (currEq.Lhs.Equals(currEq.Rhs))
{
// Elimination by omission (this is a "succeed" case)
}
else if (currEq.Lhs is TrsAtom && currEq.Rhs is TrsAtom)
{
if (!currEq.Lhs.Equals(currEq.Rhs)) {
fail = !processFail(currentProblem, currEq);
}
}
else if (currEq.Lhs is TrsAtom && currEq.Rhs is TrsTerm
|| currEq.Lhs is TrsTerm && currEq.Rhs is TrsAtom
|| currEq.Lhs is TrsAtom && currEq.Rhs is TrsAcTerm
|| currEq.Lhs is TrsAcTerm && currEq.Rhs is TrsAtom
|| currEq.Lhs is TrsTerm && currEq.Rhs is TrsAcTerm
|| currEq.Lhs is TrsAcTerm && currEq.Rhs is TrsTerm)
{
fail = !processFail(currentProblem, currEq);
}
else if (currEq.Lhs is TrsTerm && currEq.Rhs is TrsTerm)
{
TrsTerm lhs = currEq.Lhs as TrsTerm;
TrsTerm rhs = currEq.Rhs as TrsTerm;
if (lhs.Name != rhs.Name || lhs.Arguments.Count != rhs.Arguments.Count)
fail = !processFail(currentProblem, currEq);
else
currentProblem.CurrentEquations.AddRange(Enumerable.Range(0, lhs.Arguments.Count).
Select(i => new Equation { Lhs = lhs.Arguments[i], Rhs = rhs.Arguments[i] }));
}
else if (currEq.Lhs is TrsAcTerm && currEq.Rhs is TrsAcTerm)
{
// Note: Failure is already processed internally in the next function call (ie. custom unifiers are called)
fail = ProcessAcUnificationStep(currentProblem, currentContinuations, processFail, currEq);
}
else if (!(currEq.Lhs is TrsVariable) && (currEq.Rhs is TrsVariable))
{
TrsTermBase lhsSwap = currEq.Lhs;
currEq.Lhs = currEq.Rhs;
currEq.Rhs = lhsSwap;
currentProblem.CurrentEquations.Add(currEq);
}
else if (currEq.Lhs is TrsVariable)
{
// Occurs check
if (currEq.Rhs.ContainsVariable((TrsVariable)currEq.Lhs))
fail = !processFail(currentProblem, currEq);
else ProcessSubstitutionStep(currentProblem, currEq, variableNamesToPreserve);
}
else throw new Exception("Invalid program state");
}
if (!fail)
results.Add(new UnificationResult
{
Succeed = true,
Unifier = currentProblem.CurrentSubstitutions
});
}
return results.ToList();
}
/// <summary>
/// Applies the custom unifiers for the current equation in case of failiure ...
/// it applies the first working unifier from the list of custom unifiers.
/// </summary>
private List<UnificationContinuation> CustomFail(UnificationContinuation currentProblem, Equation currEq, List<TrsVariable> variableNamesToPreserve)
{
// First try custom unifiers ... use the first one that works.
HashSet<UnificationResult> unificationResults = new HashSet<UnificationResult>();
foreach (var unifFunc in customUnifiers)
{
// Prevent overwriting of input terms ...
var input = currEq.CreateCopy();
var unifiers = unifFunc.GetUnifier(input.Lhs, input.Rhs);
if (unifiers != null)
{
foreach (var result in unifiers)
if (result.Succeed)
{
if (result.Unifier == null) result.Unifier = new List<Substitution>();
unificationResults.Add(result);
}
}
}
unificationResults = new HashSet<UnificationResult>(unificationResults.Where(result => result.Succeed));
if (unificationResults.Count == 0)
{
return new List<UnificationContinuation>();
}
else
{
var retContinuations = new HashSet<UnificationContinuation>();
foreach (var customUnifier in unificationResults)
{
bool succeed = true;
foreach (var customSubstitution in customUnifier.Unifier)
{
// Do occurs check
if (customSubstitution.SubstitutionTerm.ContainsVariable(customSubstitution.Variable))
{
succeed = false;
break;
}
// Prevent conflicting mappings
foreach (var substitution in currentProblem.CurrentSubstitutions)
{
if (substitution.Variable.Equals(customSubstitution.Variable)
&& !substitution.SubstitutionTerm.Equals(customSubstitution.SubstitutionTerm))
{
succeed = false;
break;
}
}
}
// Apply the subtitutions
if (succeed)
{
// Create continuations here for injection back into the main algorithm loop
UnificationContinuation currentContinuation = currentProblem.CreateCopy();
foreach (var substitution in customUnifier.Unifier)
ProcessSubstitutionStep(currentContinuation, new Equation { Lhs = substitution.Variable, Rhs = substitution.SubstitutionTerm }, variableNamesToPreserve);
retContinuations.Add(currentContinuation);
}
}
return retContinuations.ToList();
}
}
/// <summary>
/// Applies the custom unifiers for the current equation in case of failiure ...
/// it applies the first working unifier from the list of custom unifiers.
/// </summary>
private List <UnificationContinuation> CustomFail(UnificationContinuation currentProblem, Equation currEq, List <TrsVariable> variableNamesToPreserve)
{
// First try custom unifiers ... use the first one that works.
HashSet <UnificationResult> unificationResults = new HashSet <UnificationResult>();
foreach (var unifFunc in customUnifiers)
{
// Prevent overwriting of input terms ...
var input = currEq.CreateCopy();
var unifiers = unifFunc.GetUnifier(input.Lhs, input.Rhs);
if (unifiers != null)
{
foreach (var result in unifiers)
{
if (result.Succeed)
{
if (result.Unifier == null)
{
result.Unifier = new List <Substitution>();
}
unificationResults.Add(result);
}
}
}
}
unificationResults = new HashSet <UnificationResult>(unificationResults.Where(result => result.Succeed));
if (unificationResults.Count == 0)
{
return(new List <UnificationContinuation>());
}
else
{
var retContinuations = new HashSet <UnificationContinuation>();
foreach (var customUnifier in unificationResults)
{
bool succeed = true;
foreach (var customSubstitution in customUnifier.Unifier)
{
// Do occurs check
if (customSubstitution.SubstitutionTerm.ContainsVariable(customSubstitution.Variable))
{
succeed = false;
break;
}
// Prevent conflicting mappings
foreach (var substitution in currentProblem.CurrentSubstitutions)
{
if (substitution.Variable.Equals(customSubstitution.Variable) &&
!substitution.SubstitutionTerm.Equals(customSubstitution.SubstitutionTerm))
{
succeed = false;
break;
}
}
}
// Apply the subtitutions
if (succeed)
{
// Create continuations here for injection back into the main algorithm loop
UnificationContinuation currentContinuation = currentProblem.CreateCopy();
foreach (var substitution in customUnifier.Unifier)
{
ProcessSubstitutionStep(currentContinuation, new Equation {
Lhs = substitution.Variable, Rhs = substitution.SubstitutionTerm
}, variableNamesToPreserve);
}
retContinuations.Add(currentContinuation);
}
}
return(retContinuations.ToList());
}
}
private static bool ProcessAcUnificationStep(UnificationContinuation currentProblem, Queue <UnificationContinuation> currentContinuations,
Func <UnificationContinuation, Equation, bool> processFail, Equation currEq)
{
var acLhs = currEq.Lhs as TrsAcTerm;
var acRhs = currEq.Rhs as TrsAcTerm;
bool fail = false;
if (acLhs.Name != acRhs.Name)
{
fail = !processFail(currentProblem, currEq);
}
else
{
// Reduce cardinalities to reduce number of permutations
foreach (var pair in (from lhsTerm in acLhs.OnfArguments
join rhsTerm in acRhs.OnfArguments
on lhsTerm.Term equals rhsTerm.Term
where lhsTerm.Term.IsGroundTerm &&
rhsTerm.Term.IsGroundTerm
select new {
Lhs = lhsTerm,
Rhs = rhsTerm
}).ToList())
{
var diff = Math.Min(pair.Lhs.Cardinality, pair.Rhs.Cardinality);
if (pair.Lhs.Cardinality > diff)
{
pair.Lhs.Cardinality -= diff;
}
else
{
acLhs.OnfArguments.Remove(pair.Lhs);
}
if (pair.Rhs.Cardinality > diff)
{
pair.Rhs.Cardinality -= diff;
}
else
{
acRhs.OnfArguments.Remove(pair.Rhs);
}
}
// AC partitioning can only take place from the small to the large cardinalities
List <TrsTermBase> fromArgs = null;
List <TrsTermBase> toArgs = null;
if (acLhs.TotalCardinality > acRhs.TotalCardinality)
{
fromArgs = acRhs.ExpandedArguments;
toArgs = acLhs.ExpandedArguments;
}
else
{
fromArgs = acLhs.ExpandedArguments;
toArgs = acRhs.ExpandedArguments;
}
// Generate mappings ... this is where the AC semantics are applied
var setPartitionMappings = new SetMapPartitionEnumerator <int, int>(new HashSet <int>(Enumerable.Range(0, fromArgs.Count)),
new HashSet <int>(Enumerable.Range(0, toArgs.Count)));
// The first mapping will map to the first continuation, this must be added to the
// current problem's equations in order to preserve algorithm correctness when the unification result
// contains no substitutions.
var isFirstContinuation = true;
var initialCurrentProblem = currentProblem.CreateCopy(); // currentProblem gets updated in the first iteration
foreach (var mappings in setPartitionMappings)
{
// Cater for multiple unification problems generated by the current AC problem
UnificationContinuation clone = null;
if (!isFirstContinuation)
{
clone = initialCurrentProblem.CreateCopy();
}
foreach (var mapping in mappings)
{
Equation eq = new Equation();
eq.Lhs = fromArgs[mapping.FromElement];
if (mapping.ToPartition.Count == 1)
{
eq.Rhs = toArgs[mapping.ToPartition.First()];
}
else
{
eq.Rhs = new TrsAcTerm(acLhs.Name, mapping.ToPartition.Select(i => toArgs[i]));
}
if (!isFirstContinuation)
{
clone.CurrentEquations.Add(eq);
}
else
{
currentProblem.CurrentEquations.Add(eq);
}
}
if (!isFirstContinuation)
{
currentContinuations.Enqueue(clone);
}
else
{
isFirstContinuation = false;
}
}
}
return(fail);
}
/// <summary>
/// Returns empty list if the terms in the equation cannot be unified or if Lhs = Rhs with no variables,
/// otherwise a list of substitutions defining the MGU
///
/// Based on Unification chapter from Handbook of Automated Reasoning.
/// </summary>
private List <UnificationResult> GetMgu(Equation unificationProblem, List <TrsVariable> variableNamesToPreserve)
{
if (unificationProblem == null)
{
throw new ArgumentException();
}
if (unificationProblem.Lhs == null)
{
throw new ArgumentException();
}
if (unificationProblem.Rhs == null)
{
throw new ArgumentException();
}
if (variableNamesToPreserve == null)
{
throw new ArgumentException();
}
Equation initialEquation = unificationProblem.CreateCopy();
UnificationContinuation currentProblem = new UnificationContinuation
{
CurrentEquations = new List <Equation>(),
CurrentSubstitutions = new List <Substitution>()
};
currentProblem.CurrentEquations.Add(initialEquation);
Queue <UnificationContinuation> currentContinuations = new Queue <UnificationContinuation>();
currentContinuations.Enqueue(currentProblem);
HashSet <UnificationResult> results = new HashSet <UnificationResult>();
Func <UnificationContinuation, Equation, bool> processFail = delegate(UnificationContinuation curr, Equation currEq)
{
var failResult = CustomFail(currentProblem, currEq, variableNamesToPreserve);
var succeed = true;
if (failResult.Count == 0)
{
succeed = false;
}
else if (failResult.Count == 1)
{
currentProblem = failResult.First();
}
else
{
foreach (var continuation in failResult)
{
currentContinuations.Enqueue(continuation);
}
}
return(succeed);
};
while (currentContinuations.Count > 0)
{
currentProblem = currentContinuations.Dequeue();
bool fail = false;
while (currentProblem.CurrentEquations.Count > 0 && !fail)
{
var currEq = currentProblem.CurrentEquations[currentProblem.CurrentEquations.Count - 1];
currentProblem.CurrentEquations.RemoveAt(currentProblem.CurrentEquations.Count - 1);
if (currEq.Lhs.Equals(currEq.Rhs))
{
// Elimination by omission (this is a "succeed" case)
}
else if (currEq.Lhs is TrsAtom && currEq.Rhs is TrsAtom)
{
if (!currEq.Lhs.Equals(currEq.Rhs))
{
fail = !processFail(currentProblem, currEq);
}
}
else if (currEq.Lhs is TrsAtom && currEq.Rhs is TrsTerm ||
currEq.Lhs is TrsTerm && currEq.Rhs is TrsAtom ||
currEq.Lhs is TrsAtom && currEq.Rhs is TrsAcTerm ||
currEq.Lhs is TrsAcTerm && currEq.Rhs is TrsAtom ||
currEq.Lhs is TrsTerm && currEq.Rhs is TrsAcTerm ||
currEq.Lhs is TrsAcTerm && currEq.Rhs is TrsTerm)
{
fail = !processFail(currentProblem, currEq);
}
else if (currEq.Lhs is TrsTerm && currEq.Rhs is TrsTerm)
{
TrsTerm lhs = currEq.Lhs as TrsTerm;
TrsTerm rhs = currEq.Rhs as TrsTerm;
if (lhs.Name != rhs.Name || lhs.Arguments.Count != rhs.Arguments.Count)
{
fail = !processFail(currentProblem, currEq);
}
else
{
currentProblem.CurrentEquations.AddRange(Enumerable.Range(0, lhs.Arguments.Count).
Select(i => new Equation {
Lhs = lhs.Arguments[i], Rhs = rhs.Arguments[i]
}));
}
}
else if (currEq.Lhs is TrsAcTerm && currEq.Rhs is TrsAcTerm)
{
// Note: Failure is already processed internally in the next function call (ie. custom unifiers are called)
fail = ProcessAcUnificationStep(currentProblem, currentContinuations, processFail, currEq);
}
else if (!(currEq.Lhs is TrsVariable) && (currEq.Rhs is TrsVariable))
{
TrsTermBase lhsSwap = currEq.Lhs;
currEq.Lhs = currEq.Rhs;
currEq.Rhs = lhsSwap;
currentProblem.CurrentEquations.Add(currEq);
}
else if (currEq.Lhs is TrsVariable)
{
// Occurs check
if (currEq.Rhs.ContainsVariable((TrsVariable)currEq.Lhs))
{
fail = !processFail(currentProblem, currEq);
}
else
{
ProcessSubstitutionStep(currentProblem, currEq, variableNamesToPreserve);
}
}
else
{
throw new Exception("Invalid program state");
}
}
if (!fail)
{
results.Add(new UnificationResult
{
Succeed = true,
Unifier = currentProblem.CurrentSubstitutions
});
}
}
return(results.ToList());
}
