/// <summary>
/// Returns a (deterministic) automaton that accepts the complement of the language of the
/// given automaton.
/// </summary>
/// <param name="a">The automaton.</param>
/// <returns>A (deterministic) automaton that accepts the complement of the language of the
/// given automaton.</returns>
/// <remarks>
/// Complexity: linear in number of states (if already deterministic).
/// </remarks>
public static Automaton Complement(Automaton a)
{
a = a.CloneExpandedIfRequired();
a.Determinize();
a.Totalize();
foreach (State p in a.GetStates())
{
p.Accept = !p.Accept;
}
a.RemoveDeadTransitions();
return(a);
}
public static string ToDot(Automaton a)
{
StringBuilder transitions = new StringBuilder(INITIAL_CAPACITY);
State finalState = null;
foreach (var state in a.GetStates())
{
transitions.Append(BuildDotTransitions(state));
finalState = state;
}
string header = BuildDotHeader(a.Initial, finalState);
sb.Clear();
sb.Append($"{header}\n");
sb.Append(transitions.ToString());
sb.Append("\n}");
return(sb.ToString());
}
/// <summary>
/// Reverses the language of the given (non-singleton) automaton while returning the set of
/// new initial states.
/// </summary>
/// <param name="a">The automaton.</param>
/// <returns></returns>
public static HashSet <State> Reverse(Automaton a)
{
// Reverse all edges.
var m = new Dictionary <State, HashSet <Transition> >();
HashSet <State> states = a.GetStates();
HashSet <State> accept = a.GetAcceptStates();
foreach (State r in states)
{
m.Add(r, new HashSet <Transition>());
r.Accept = false;
}
foreach (State r in states)
{
foreach (Transition t in r.Transitions)
{
m[t.To].Add(new Transition(t.Min, t.Max, r));
}
}
foreach (State r in states)
{
r.Transitions = m[r].ToList();
}
// Make new initial+final states.
a.Initial.Accept = true;
a.Initial = new State();
foreach (State r in accept)
{
a.Initial.AddEpsilon(r); // Ensures that all initial states are reachable.
}
a.IsDeterministic = false;
return(accept);
}
/// <summary>
/// Reverses the language of the given (non-singleton) automaton while returning the set of
/// new initial states.
/// </summary>
/// <param name="a">The automaton.</param>
/// <returns></returns>
public static HashSet<State> Reverse(Automaton a)
{
// Reverse all edges.
var m = new Dictionary<State, HashSet<Transition>>();
HashSet<State> states = a.GetStates();
HashSet<State> accept = a.GetAcceptStates();
foreach (State r in states)
{
m.Add(r, new HashSet<Transition>());
r.Accept = false;
}
foreach (State r in states)
{
foreach (Transition t in r.Transitions)
{
m[t.To].Add(new Transition(t.Min, t.Max, r));
}
}
foreach (State r in states)
{
r.Transitions = m[r].ToList();
}
// Make new initial+final states.
a.Initial.Accept = true;
a.Initial = new State();
foreach (State r in accept)
{
a.Initial.AddEpsilon(r); // Ensures that all initial states are reachable.
}
a.IsDeterministic = false;
return accept;
}
/// <summary>
/// Returns true if the given string is accepted by the automaton.
/// </summary>
/// <param name="a">The automaton.</param>
/// <param name="s">The string.</param>
/// <returns></returns>
/// <remarks>
/// Complexity: linear in the length of the string.
/// For full performance, use the RunAutomaton class.
/// </remarks>
public static bool Run(Automaton a, string s)
{
if (a.IsSingleton)
{
return s.Equals(a.IsSingleton);
}
if (a.IsDeterministic)
{
State p = a.Initial;
foreach (char t in s)
{
State q = p.Step(t);
if (q == null)
{
return false;
}
p = q;
}
return p.Accept;
}
HashSet<State> states = a.GetStates();
Automaton.SetStateNumbers(states);
var pp = new LinkedList<State>();
var ppOther = new LinkedList<State>();
var bb = new BitArray(states.Count);
var bbOther = new BitArray(states.Count);
pp.AddLast(a.Initial);
var dest = new List<State>();
bool accept = a.Initial.Accept;
foreach (char c in s)
{
accept = false;
ppOther.Clear();
bbOther.SetAll(false);
foreach (State p in pp)
{
dest.Clear();
p.Step(c, dest);
foreach (State q in dest)
{
if (q.Accept)
{
accept = true;
}
if (!bbOther.Get(q.Number))
{
bbOther.Set(q.Number, true);
ppOther.AddLast(q);
}
}
}
LinkedList<State> tp = pp;
pp = ppOther;
ppOther = tp;
BitArray tb = bb;
bb = bbOther;
bbOther = tb;
}
return accept;
}
/// <summary>
/// Returns an automaton that accepts the intersection of the languages of the given automata.
/// Never modifies the input automata languages.
/// </summary>
/// <param name="a1">The a1.</param>
/// <param name="a2">The a2.</param>
/// <returns></returns>
public static Automaton Intersection(Automaton a1, Automaton a2)
{
if (a1.IsSingleton)
{
if (a2.Run(a1.Singleton))
{
return a1.CloneIfRequired();
}
return BasicAutomata.MakeEmpty();
}
if (a2.IsSingleton)
{
if (a1.Run(a2.Singleton))
{
return a2.CloneIfRequired();
}
return BasicAutomata.MakeEmpty();
}
if (a1 == a2)
{
return a1.CloneIfRequired();
}
Transition[][] transitions1 = Automaton.GetSortedTransitions(a1.GetStates());
Transition[][] transitions2 = Automaton.GetSortedTransitions(a2.GetStates());
var c = new Automaton();
var worklist = new LinkedList<StatePair>();
var newstates = new Dictionary<StatePair, StatePair>();
var p = new StatePair(c.Initial, a1.Initial, a2.Initial);
worklist.AddLast(p);
newstates.Add(p, p);
while (worklist.Count > 0)
{
p = worklist.RemoveAndReturnFirst();
p.S.Accept = p.FirstState.Accept && p.SecondState.Accept;
Transition[] t1 = transitions1[p.FirstState.Number];
Transition[] t2 = transitions2[p.SecondState.Number];
for (int n1 = 0, b2 = 0; n1 < t1.Length; n1++)
{
while (b2 < t2.Length && t2[b2].Max < t1[n1].Min)
{
b2++;
}
for (int n2 = b2; n2 < t2.Length && t1[n1].Max >= t2[n2].Min; n2++)
{
if (t2[n2].Max >= t1[n1].Min)
{
var q = new StatePair(t1[n1].To, t2[n2].To);
StatePair r;
newstates.TryGetValue(q, out r);
if (r == null)
{
q.S = new State();
worklist.AddLast(q);
newstates.Add(q, q);
r = q;
}
char min = t1[n1].Min > t2[n2].Min ? t1[n1].Min : t2[n2].Min;
char max = t1[n1].Max < t2[n2].Max ? t1[n1].Max : t2[n2].Max;
p.S.Transitions.Add(new Transition(min, max, r.S));
}
}
}
}
c.IsDeterministic = a1.IsDeterministic && a2.IsDeterministic;
c.RemoveDeadTransitions();
c.CheckMinimizeAlways();
return c;
}
/// <summary>
/// Returns a (deterministic) automaton that accepts the complement of the language of the
/// given automaton.
/// </summary>
/// <param name="a">The automaton.</param>
/// <returns>A (deterministic) automaton that accepts the complement of the language of the
/// given automaton.</returns>
/// <remarks>
/// Complexity: linear in number of states (if already deterministic).
/// </remarks>
public static Automaton Complement(Automaton a)
{
a = a.CloneExpandedIfRequired();
a.Determinize();
a.Totalize();
foreach (State p in a.GetStates())
{
p.Accept = !p.Accept;
}
a.RemoveDeadTransitions();
return a;
}
internal static void MinimizeHopcroft(Automaton a)
{
a.Determinize();
IList <Transition> tr = a.Initial.Transitions;
if (tr.Count == 1)
{
Transition t = tr[0];
if (t.To == a.Initial && t.Min == char.MinValue && t.Max == char.MaxValue)
{
return;
}
}
a.Totalize();
// Make arrays for numbered states and effective alphabet.
HashSet <State> ss = a.GetStates();
var states = new State[ss.Count];
int number = 0;
foreach (State q in ss)
{
states[number] = q;
q.Number = number++;
}
char[] sigma = a.GetStartPoints();
// Initialize data structures.
var reverse = new List <List <LinkedList <State> > >();
foreach (State s in states)
{
var v = new List <LinkedList <State> >();
Initialize(ref v, sigma.Length);
reverse.Add(v);
}
var reverseNonempty = new bool[states.Length, sigma.Length];
var partition = new List <LinkedList <State> >();
Initialize(ref partition, states.Length);
var block = new int[states.Length];
var active = new StateList[states.Length, sigma.Length];
var active2 = new StateListNode[states.Length, sigma.Length];
var pending = new LinkedList <IntPair>();
var pending2 = new bool[sigma.Length, states.Length];
var split = new List <State>();
var split2 = new bool[states.Length];
var refine = new List <int>();
var refine2 = new bool[states.Length];
var splitblock = new List <List <State> >();
Initialize(ref splitblock, states.Length);
for (int q = 0; q < states.Length; q++)
{
splitblock[q] = new List <State>();
partition[q] = new LinkedList <State>();
for (int x = 0; x < sigma.Length; x++)
{
reverse[q][x] = new LinkedList <State>();
active[q, x] = new StateList();
}
}
// Find initial partition and reverse edges.
foreach (State qq in states)
{
int j = qq.Accept ? 0 : 1;
partition[j].AddLast(qq);
block[qq.Number] = j;
for (int x = 0; x < sigma.Length; x++)
{
char y = sigma[x];
State p = qq.Step(y);
reverse[p.Number][x].AddLast(qq);
reverseNonempty[p.Number, x] = true;
}
}
// Initialize active sets.
for (int j = 0; j <= 1; j++)
{
for (int x = 0; x < sigma.Length; x++)
{
foreach (State qq in partition[j])
{
if (reverseNonempty[qq.Number, x])
{
active2[qq.Number, x] = active[j, x].Add(qq);
}
}
}
}
// Initialize pending.
for (int x = 0; x < sigma.Length; x++)
{
int a0 = active[0, x].Size;
int a1 = active[1, x].Size;
int j = a0 <= a1 ? 0 : 1;
pending.AddLast(new IntPair(j, x));
pending2[x, j] = true;
}
// Process pending until fixed point.
int k = 2;
while (pending.Count > 0)
{
IntPair ip = pending.RemoveAndReturnFirst();
int p = ip.N1;
int x = ip.N2;
pending2[x, p] = false;
// Find states that need to be split off their blocks.
for (StateListNode m = active[p, x].First; m != null; m = m.Next)
{
foreach (State s in reverse[m.State.Number][x])
{
if (!split2[s.Number])
{
split2[s.Number] = true;
split.Add(s);
int j = block[s.Number];
splitblock[j].Add(s);
if (!refine2[j])
{
refine2[j] = true;
refine.Add(j);
}
}
}
}
// Refine blocks.
foreach (int j in refine)
{
if (splitblock[j].Count < partition[j].Count)
{
LinkedList <State> b1 = partition[j];
LinkedList <State> b2 = partition[k];
foreach (State s in splitblock[j])
{
b1.Remove(s);
b2.AddLast(s);
block[s.Number] = k;
for (int c = 0; c < sigma.Length; c++)
{
StateListNode sn = active2[s.Number, c];
if (sn != null && sn.StateList == active[j, c])
{
sn.Remove();
active2[s.Number, c] = active[k, c].Add(s);
}
}
}
// Update pending.
for (int c = 0; c < sigma.Length; c++)
{
int aj = active[j, c].Size;
int ak = active[k, c].Size;
if (!pending2[c, j] && 0 < aj && aj <= ak)
{
pending2[c, j] = true;
pending.AddLast(new IntPair(j, c));
}
else
{
pending2[c, k] = true;
pending.AddLast(new IntPair(k, c));
}
}
k++;
}
foreach (State s in splitblock[j])
{
split2[s.Number] = false;
}
refine2[j] = false;
splitblock[j].Clear();
}
split.Clear();
refine.Clear();
}
// Make a new state for each equivalence class, set initial state.
var newstates = new State[k];
for (int n = 0; n < newstates.Length; n++)
{
var s = new State();
newstates[n] = s;
foreach (State q in partition[n])
{
if (q == a.Initial)
{
a.Initial = s;
}
s.Accept = q.Accept;
s.Number = q.Number; // Select representative.
q.Number = n;
}
}
// Build transitions and set acceptance.
foreach (State s in newstates)
{
s.Accept = states[s.Number].Accept;
foreach (Transition t in states[s.Number].Transitions)
{
s.Transitions.Add(new Transition(t.Min, t.Max, newstates[t.To.Number]));
}
}
a.RemoveDeadTransitions();
}
/// <summary>
/// Minimizes the given automaton using Huffman's algorithm.
/// </summary>
/// <param name="a">The automaton.</param>
internal static void MinimizeHuffman(Automaton a)
{
a.Determinize();
a.Totalize();
HashSet <State> ss = a.GetStates();
var transitions = new Transition[ss.Count][];
State[] states = ss.ToArray();
var mark = new List <List <bool> >();
var triggers = new List <List <HashSet <IntPair> > >();
foreach (State t in states)
{
var v = new List <HashSet <IntPair> >();
Initialize(ref v, states.Length);
triggers.Add(v);
}
// Initialize marks based on acceptance status and find transition arrays.
for (int n1 = 0; n1 < states.Length; n1++)
{
states[n1].Number = n1;
transitions[n1] = states[n1].GetSortedTransitions(false).ToArray();
for (int n2 = n1 + 1; n2 < states.Length; n2++)
{
if (states[n1].Accept != states[n2].Accept)
{
mark[n1][n2] = true;
}
}
}
// For all pairs, see if states agree.
for (int n1 = 0; n1 < states.Length; n1++)
{
for (int n2 = n1 + 1; n2 < states.Length; n2++)
{
if (!mark[n1][n2])
{
if (MinimizationOperations.StatesAgree(transitions, mark, n1, n2))
{
MinimizationOperations.AddTriggers(transitions, triggers, n1, n2);
}
else
{
MinimizationOperations.MarkPair(mark, triggers, n1, n2);
}
}
}
}
// Assign equivalence class numbers to states.
int numclasses = 0;
foreach (State t in states)
{
t.Number = -1;
}
for (int n1 = 0; n1 < states.Length; n1++)
{
if (states[n1].Number == -1)
{
states[n1].Number = numclasses;
for (int n2 = n1 + 1; n2 < states.Length; n2++)
{
if (!mark[n1][n2])
{
states[n2].Number = numclasses;
}
}
numclasses++;
}
}
// Make a new state for each equivalence class.
var newstates = new State[numclasses];
for (int n = 0; n < numclasses; n++)
{
newstates[n] = new State();
}
// Select a class representative for each class and find the new initial state.
for (int n = 0; n < states.Length; n++)
{
newstates[states[n].Number].Number = n;
if (states[n] == a.Initial)
{
a.Initial = newstates[states[n].Number];
}
}
// Build transitions and set acceptance.
for (int n = 0; n < numclasses; n++)
{
State s = newstates[n];
s.Accept = states[s.Number].Accept;
foreach (Transition t in states[s.Number].Transitions)
{
s.Transitions.Add(new Transition(t.Min, t.Max, newstates[t.To.Number]));
}
}
a.RemoveDeadTransitions();
}
/// <summary>
/// Returns true if the given string is accepted by the automaton.
/// </summary>
/// <param name="a">The automaton.</param>
/// <param name="s">The string.</param>
/// <returns></returns>
/// <remarks>
/// Complexity: linear in the length of the string.
/// For full performance, use the RunAutomaton class.
/// </remarks>
public static bool Run(Automaton a, string s)
{
if (a.IsSingleton)
{
return(s.Equals(a.Singleton, System.StringComparison.CurrentCulture));
}
if (a.IsDeterministic)
{
State p = a.Initial;
foreach (char t in s)
{
State q = p.Step(t);
if (q == null)
{
return(false);
}
p = q;
}
return(p.Accept);
}
HashSet <State> states = a.GetStates();
Automaton.SetStateNumbers(states);
var pp = new LinkedList <State>();
var ppOther = new LinkedList <State>();
var bb = new BitArray(states.Count);
var bbOther = new BitArray(states.Count);
pp.AddLast(a.Initial);
var dest = new List <State>();
bool accept = a.Initial.Accept;
foreach (char c in s)
{
accept = false;
ppOther.Clear();
bbOther.SetAll(false);
foreach (State p in pp)
{
dest.Clear();
p.Step(c, dest);
foreach (State q in dest)
{
if (q.Accept)
{
accept = true;
}
if (!bbOther.Get(q.Number))
{
bbOther.Set(q.Number, true);
ppOther.AddLast(q);
}
}
}
LinkedList <State> tp = pp;
pp = ppOther;
ppOther = tp;
BitArray tb = bb;
bb = bbOther;
bbOther = tb;
}
return(accept);
}
/// <summary>
/// Returns an automaton that accepts the intersection of the languages of the given automata.
/// Never modifies the input automata languages.
/// </summary>
/// <param name="a1">The a1.</param>
/// <param name="a2">The a2.</param>
/// <returns></returns>
public static Automaton Intersection(Automaton a1, Automaton a2)
{
if (a1.IsSingleton)
{
if (a2.Run(a1.Singleton))
{
return(a1.CloneIfRequired());
}
return(BasicAutomata.MakeEmpty());
}
if (a2.IsSingleton)
{
if (a1.Run(a2.Singleton))
{
return(a2.CloneIfRequired());
}
return(BasicAutomata.MakeEmpty());
}
if (a1 == a2)
{
return(a1.CloneIfRequired());
}
Transition[][] transitions1 = Automaton.GetSortedTransitions(a1.GetStates());
Transition[][] transitions2 = Automaton.GetSortedTransitions(a2.GetStates());
var c = new Automaton();
var worklist = new LinkedList <StatePair>();
var newstates = new Dictionary <StatePair, StatePair>();
var p = new StatePair(c.Initial, a1.Initial, a2.Initial);
worklist.AddLast(p);
newstates.Add(p, p);
while (worklist.Count > 0)
{
p = worklist.RemoveAndReturnFirst();
p.S.Accept = p.FirstState.Accept && p.SecondState.Accept;
Transition[] t1 = transitions1[p.FirstState.Number];
Transition[] t2 = transitions2[p.SecondState.Number];
for (int n1 = 0, b2 = 0; n1 < t1.Length; n1++)
{
while (b2 < t2.Length && t2[b2].Max < t1[n1].Min)
{
b2++;
}
for (int n2 = b2; n2 < t2.Length && t1[n1].Max >= t2[n2].Min; n2++)
{
if (t2[n2].Max >= t1[n1].Min)
{
var q = new StatePair(t1[n1].To, t2[n2].To);
StatePair r;
newstates.TryGetValue(q, out r);
if (r == null)
{
q.S = new State();
worklist.AddLast(q);
newstates.Add(q, q);
r = q;
}
char min = t1[n1].Min > t2[n2].Min ? t1[n1].Min : t2[n2].Min;
char max = t1[n1].Max < t2[n2].Max ? t1[n1].Max : t2[n2].Max;
p.S.Transitions.Add(new Transition(min, max, r.S));
}
}
}
}
c.IsDeterministic = a1.IsDeterministic && a2.IsDeterministic;
c.RemoveDeadTransitions();
c.CheckMinimizeAlways();
return(c);
}
