public static FA Single(Integers terminal)
{
var start = new State();
var end = new State();
start.Add(terminal, end);
return(FA.From(start, end));
}
public static FA Plus(FA fa)
{
var clone = fa.ToNfa(true);
Debug.Assert(clone.Final != null);
clone.Final.Add(clone.Start);
return(clone);
}
public static FA None()
{
var start = new State();
var illegal = new State();
start.Add(Integers.Any, illegal);
illegal.Add(Integers.Any, illegal);
return(FA.From(start, start));
}
public static FA Negate(FA dfa)
{
EnsureDfa(dfa);
for (var i = 0; i < dfa.States.Count; ++i)
{
dfa.States[i].IsFinal = !dfa.States[i].IsFinal;
}
return(dfa);
}
public static FA Or(FA first, params FA[] nexts)
{
var fa = first;
foreach (var next in nexts)
{
fa = fa.Or(next);
}
return(fa);
}
public static FA Concat(FA fa, FA other)
{
var first = fa.ToNfa(true);
Debug.Assert(first.Final != null);
other = other.ToNfa(true);
Debug.Assert(other.Final != null);
first.Final.Add(other.Start);
return(FA.From(first.Start, other.Final));
}
public FA Minimize(FA dfa)
{
EnsureDfa(dfa);
dfa = MergeStates(dfa);
dfa = MergeTransitions(dfa);
var result = From(dfa.Start);
EnsureDfa(result);
return(result);
}
private static void EnsureDfa(FA dfa)
{
if (dfa.Final != null)
{
throw new Exception("DFA: dfa.Final must be null");
}
if (!dfa.Finals.Any())
{
throw new Exception("DFA: no final state");
}
if (dfa.States.Count < 1)
{
throw new Exception("DFA: at least on state");
}
EnsureDfaTransitions(dfa);
}
public static FA From(string sequence)
{
Debug.Assert(!string.IsNullOrEmpty(sequence));
var start = new State();
var current = start;
var next = (State?)null;
foreach (var ch in sequence)
{
next = new State();
current.Add(Integers.From(ch), next);
current = next;
}
Debug.Assert(next != null);
return(FA.From(start, next));
}
public static FA Or(FA fa, FA other)
{
var first = fa.ToNfa(true);
Debug.Assert(first.Final != null);
var second = other.ToNfa(true);
Debug.Assert(second.Final != null);
var newEnd = new State();
first.Start.Add(second.Start);
first.Final.Add(newEnd);
second.Final.Add(newEnd);
return(FA.From(first.Start, newEnd));
}
public static FA Build(FA dfa1, FA dfa2, Func <State, State, bool> isFinal)
{
EnsureDfa(dfa1);
EnsureDfa(dfa2);
dfa1 = Complete(dfa1);
dfa2 = Complete(dfa2);
EnsureComplete(dfa1);
EnsureComplete(dfa2);
var cross = new State[dfa1.States.Count, dfa2.States.Count];
var t1 = dfa1.States.Select(s => GetTransitions(s)).ToList();
var t2 = dfa2.States.Select(s => GetTransitions(s)).ToList();
for (var n1 = 0; n1 < dfa1.States.Count; n1 += 1)
{
for (var n2 = 0; n2 < dfa2.States.Count; n2 += 1)
{
cross[n1, n2] = new State(isFinal(dfa1.States[n1], dfa2.States[n2]));
}
}
for (var n1 = 0; n1 < dfa1.States.Count; n1 += 1)
{
for (var n2 = 0; n2 < dfa2.States.Count; n2 += 1)
{
var state = cross[n1, n2];
var ctranses = GetMerge(t1[n1], t2[n2]);
foreach (var ctrans in ctranses)
{
state.Add(Codepoints.From(ctrans.Range.Min, ctrans.Range.Max), cross[ctrans.Target1, ctrans.Target2]);
}
}
}
var dfa = From(cross[0, 0]);
EnsureComplete(dfa);
return(dfa.RemoveDead().Minimize());
}
public static FA ToNfa(FA dfa)
{
if (dfa.Final != null)
{
return(dfa);
}
dfa.Final = new State();
foreach (var final in dfa.Finals)
{
final.Add(dfa.Final);
}
dfa.Final.Id = dfa.States.Count;
dfa.States.Add(dfa.Final);
foreach (var state in dfa.States)
{
state.IsFinal = false;
}
return(dfa);
}
private static void EnsureComplete(FA dfa)
{
EnsureDfa(dfa);
foreach (var state in dfa.States)
{
var all = new Integers();
var sum = 0;
foreach (var transition in state.Transitions)
{
sum += transition.Set.Cardinality;
all.Add(transition.Set);
}
if (!UnicodeSets.IsAny(all))
{
throw new Exception("DFA: incomplete transition set (DFA is not `complete´)");
}
if (sum != all.Cardinality)
{
throw new Exception("DFA: overlapping transitions (DFA is not `sane´)");
}
}
}
private static void EnsureDfa(FA dfa)
{
if (dfa.Final != null)
{
throw new Exception("DFA: dfa.Final must be null");
}
if (!dfa.Finals.Any())
{
throw new Exception("DFA: no final state");
}
if (dfa.States.Count < 1)
{
throw new Exception("DFA: at least on state");
}
for (var i = 0; i < dfa.States.Count; ++i)
{
if (dfa.States[i].Id != i)
{
throw new Exception("DFA: not properly numbered");
}
}
EnsureDfaTransitions(dfa);
}
public static FA Complete(FA dfa)
{
EnsureDfa(dfa);
State?sink = null;
foreach (var state in dfa.States)
{
var rest = UnicodeSets.Any();
foreach (var transition in state.Transitions)
{
rest = rest.ExceptWith(transition.Set);
}
if (!rest.IsEmpty)
{
if (sink == null)
{
sink = new State();
sink.Add(Integers.Any, sink);
}
state.Add(Integers.From(rest), sink);
}
}
if (sink != null)
{
sink.Id = dfa.States.Count;
dfa.States.Add(sink);
}
EnsureComplete(dfa);
return(dfa);
}
public static FA ToDfa(FA nfa)
{
if (nfa.Final == null)
{
EnsureDfa(nfa);
return(nfa);
}
var once = new UniqueQueue <Closure>();
var start = new Closure(nfa.Start, nfa.Final);
once.Enqueue(start);
while (once.Count > 0)
{
var closure = once.Dequeue();
var transitions = closure.UnambiguateTransitions();
foreach (var transition in transitions)
{
var terminal = transition.Key;
var targets = transition.Value;
var targetClosure = new Closure(targets, nfa.Final);
once.Enqueue(targetClosure, out targetClosure);
var target = targetClosure.DfaState;
closure.DfaState.Add(Integers.From(terminal), target);
}
}
var dfa = From(start.DfaState);
EnsureDfa(dfa);
return(dfa);
}
public FA Difference(FA other, bool cloned = false)
{
return(Operations.Difference(CloneIf(cloned), other.CloneIf(cloned)));
}
public FA Substract(FA other, bool cloned = false)
{
return(Operations.Substract(CloneIf(cloned), other.CloneIf(cloned)));
}
public static FA Build(FA dfa1, FA dfa2, Func <State, State, bool> isFinal)
{
EnsureDfa(dfa1);
EnsureDfa(dfa2);
dfa1 = Complete(dfa1);
dfa2 = Complete(dfa2);
EnsureComplete(dfa1);
EnsureComplete(dfa2);
var cross = new State[dfa1.States.Count, dfa2.States.Count];
var t1 = dfa1.States.Select(s => GetTransitions(s)).ToList();
var t2 = dfa2.States.Select(s => GetTransitions(s)).ToList();
for (var n1 = 0; n1 < dfa1.States.Count; n1 += 1)
{
var state1 = dfa1.States[n1];
for (var n2 = 0; n2 < dfa2.States.Count; n2 += 1)
{
var state2 = dfa2.States[n2];
var newState = new State(isFinal(state1, state2));
if (state1.IsPayload && !state2.IsPayload)
{
newState.AddPayload(state1);
}
else if (!state1.IsPayload && state2.IsPayload)
{
newState.AddPayload(state2);
}
else if (state1.IsPayload && state2.IsPayload)
{
Debug.Assert(true);
#if true
if (state1.Transitions.Count == 1 && state1.Transitions[0].Set.IsAny)
{
newState.AddPayload(state1);
}
else if (state2.Transitions.Count == 1 && state2.Transitions[0].Set.IsAny)
{
newState.AddPayload(state2);
}
else
{
if (state1.Payload > state2.Payload)
{
newState.AddPayload(state1);
}
else
{
newState.AddPayload(state2);
}
//throw new InternalException("missing tie break");
}
#endif
}
cross[n1, n2] = newState;
}
}
for (var n1 = 0; n1 < dfa1.States.Count; n1 += 1)
{
for (var n2 = 0; n2 < dfa2.States.Count; n2 += 1)
{
var state = cross[n1, n2];
var ctranses = GetMerge(t1[n1], t2[n2]);
foreach (var ctrans in ctranses)
{
state.Add(Integers.From(ctrans.Range.Min, ctrans.Range.Max), cross[ctrans.Target1, ctrans.Target2]);
}
}
}
var dfa = From(cross[0, 0]);
EnsureComplete(dfa);
dfa = dfa.RemoveDead();
dfa = dfa.Minimize();
dfa = dfa.RemoveDead();
return(dfa);
}
public static FA Difference(FA dfa1, FA dfa2)
{
return(CrossBuilder.Build(dfa1, dfa2, (s1, s2) => s1.Final && !s2.Final));
}
public static FA Intersect(FA dfa1, FA dfa2)
{
return(CrossBuilder.Build(dfa1, dfa2, (s1, s2) => s1.Final && s2.Final));
}
public static FA Union(FA dfa1, FA dfa2)
{
return(CrossBuilder.Build(dfa1, dfa2, (s1, s2) => s1.Final || s2.Final));
}
public static FA RemoveDead(FA dfa)
{
EnsureDfa(dfa);
var closures = new Dictionary <State, HashSet <State> >();
void Initial(State state)
{
var set = new HashSet <State>()
{
state
};
foreach (var transition in state.Transitions)
{
set.Add(transition.Target);
}
closures.Add(state, set);
}
bool Close(State state)
{
var set = closures[state];
var before = set.Count;
foreach (var buddy in set.Where(s => s != state).ToList())
{
var buddySet = closures[buddy];
set.UnionWith(buddySet);
}
var after = set.Count;
return(after > before);
}
foreach (var state in dfa.States)
{
Initial(state);
}
var added = true;
while (added)
{
added = false;
foreach (var state in dfa.States)
{
added = added || Close(state);
}
}
var dead = new HashSet <State>();
foreach (var state in dfa.States)
{
var set = closures[state];
if (set.All(s => !s.Final))
{
dead.UnionWith(set);
}
}
return(RemoveDead(dfa, dead));
}
public FA And(FA other) => Builder.Concat(this, other);
public static FA Substract(FA dfa1, FA dfa2)
{
return(CrossBuilder.Build(dfa1, dfa2, (s1, s2) => s1.IsFinal && !s2.IsFinal));
}
public static FA Minimize(FA dfa)
{
return(new Minimizer().Minimize(dfa));
}
public FA Minimize(FA dfa)
{
EnsureDfa(dfa);
dfa = Complete(dfa);
EnsureComplete(dfa);
var s = dfa.States;
var n = s.Count;
var table = new bool[n, n];
for (var i = 0; i < n; ++i)
{
for (var j = 0; j < n; ++j)
{
if (i >= j)
{
continue;
}
table[i, j] = s[i].Final && !s[j].Final ||
!s[i].Final && s[j].Final ||
s[i].Final && s[j].Final && s[i].Payload != s[j].Payload;
}
}
bool more = true;
while (more)
{
more = false;
for (var i = 0; i < n; ++i)
{
for (var j = 0; j < n; ++j)
{
if (i >= j)
{
continue;
}
if (!table[i, j])
{
foreach (var ti in s[i].Transitions)
{
foreach (var tj in s[j].Transitions)
{
if (ti.Set.Overlaps(tj.Set))
{
var si = ti.Target.Id;
var sj = tj.Target.Id;
if (si == sj)
{
continue;
}
if (si > sj)
{
var tmp = si;
si = sj;
sj = tmp;
Debug.Assert(si < sj);
}
var before = more;
if (table[si, sj])
{
var isij = table[i, j];
more = more || !table[i, j];
if (!before && more)
{
Debug.Assert(true);
}
table[i, j] = true;
}
}
}
}
}
}
}
}
var combines = new List <StateSet>();
for (var i = 0; i < n; ++i)
{
for (var j = 0; j < n; ++j)
{
if (i >= j)
{
continue;
}
if (!table[i, j])
{
combines.Add(new StateSet {
s[i], s[j]
});
}
}
}
var current = 0;
while (current < combines.Count)
{
for (var i = current + 1; i < combines.Count;)
{
if (combines[current].Overlaps(combines[i]))
{
combines[current].Add(combines[i]);
combines.RemoveAt(i);
}
else
{
i += 1;
}
}
current += 1;
}
foreach (var set in combines)
{
var premium = set.First();
var remove = set.Skip(1).ToList();
foreach (var state in s)
{
foreach (var transition in state.Transitions)
{
if (remove.Contains(transition.Target))
{
transition.Retarget(premium);
}
}
}
EnsureDistinctTransitions(premium);
}
var mini = From(dfa.Start);
mini = MergeTransitions(mini);
return(mini);
}
public FA Union(FA other, bool cloned = false)
{
return(Operations.Union(CloneIf(cloned), other.CloneIf(cloned)));
}
private FA MergeStates(FA dfa)
{
//
// https://en.wikipedia.org/wiki/DFA_minimization
// Hopcroft's algorithm
//
#if true
var nons = new StateSet(dfa.Nons.ToList());
var all = dfa.States.ToList();
var partitions = new List <StateSet> {
nons
};
var working = new List <StateSet> {
nons
};
var fgroups = from state in dfa.Finals group state by state.Payload;
foreach (var fgroup in fgroups)
{
partitions.Add(new StateSet(fgroup));
}
#else
var finals = new StateSet(dfa.Finals.ToList());
var nons = new StateSet(dfa.Nons.ToList());
var all = dfa.States.ToList();
var partitions = new List <StateSet> {
finals, nons
};
var working = new List <StateSet> {
finals
};
#endif
var terminals = new HashSet <Integers>(all.SelectMany(state => state.Transitions).Select(transition => transition.Set));
while (working.Count > 0)
{
var a = working[0];
working.RemoveAt(0);
foreach (var terminal in terminals)
{
var x = new StateSet();
foreach (var state in all)
{
foreach (var transition in state.Transitions)
{
if (transition.Set.Overlaps(terminal))
{
if (a.Contains(transition.Target))
{
x.Add(state);
break;
}
}
}
}
var i = 0;
while (i < partitions.Count)
{
var y = partitions[i];
var x_intersect_y = x.IntersectWith(y);
var y_without_x = y.ExceptWith(x);
if (x_intersect_y.IsEmpty || y_without_x.IsEmpty)
{
i += 1;
continue;
}
partitions[i] = x_intersect_y;
i += 1;
partitions.Insert(i, y_without_x);
i += 1;
var j = 0;
while (j < working.Count)
{
if (working[j] == y)
{
working.RemoveAt(j);
working.Add(x_intersect_y);
working.Add(y_without_x);
break;
}
j += 1;
}
if (j == working.Count)
{
// ReSharper disable once ConvertIfStatementToConditionalTernaryExpression
if (x_intersect_y.Count <= y_without_x.Count)
{
working.Add(x_intersect_y);
}
else
{
working.Add(y_without_x);
}
}
}
}
}
foreach (var partition in partitions)
{
if (partition.Count >= 1)
{
#if true
var groups = from s in partition group s by s.Payload;
foreach (var group in groups)
{
List <State> dfaStates = group.ToList();
var premium = dfaStates.First();
var remove = dfaStates.Skip(1).ToList();
foreach (var state in all)
{
foreach (var transition in state.Transitions)
{
if (remove.Contains(transition.Target))
{
transition.Retarget(premium);
}
}
}
EnsureDistinctTransitions(premium);
}
#else
List <State> dfaStates = partition.ToList();
var premium = dfaStates.First();
var remove = dfaStates.Skip(1).ToList();
foreach (var state in all)
{
foreach (var transition in state.Transitions)
{
if (remove.Contains(transition.Target))
{
transition.Retarget(premium);
}
}
}
EnsureDistinctTransitions(premium);
#endif
}
}
var result = FA.From(dfa.Start);
EnsureDfa(result);
return(result);
}
public FA Intersect(FA other, bool cloned = false)
{
return(Operations.Intersect(CloneIf(cloned), other.CloneIf(cloned)));
}