private Dictionary <int, ReachabilityStatus> ApplyTransitionToReachabilityDictionary(Dictionary <int, ReachabilityStatus> reachabilityDictionary,
char symbol, bool includeEpsilon)
{
var newReachabilityDictionary = new Dictionary <int, ReachabilityStatus>();
foreach (var reachbilityLookup in reachabilityDictionary)
{
IEnumerable <KeyValuePair <int, ReachabilityStatus> > reachableStates = null !;
if (includeEpsilon)
{
reachableStates = _outgoingTransitionLookup[reachbilityLookup.Key].GetTransitionDictionary(symbol);
}
else
{
reachableStates = _automaton.TransitionMatrix.GetStates(reachbilityLookup.Key, symbol).Select(s =>
KeyValuePair.Create(s, ReachabilityStatus.Even()));
}
foreach (var state in reachableStates)
{
var newReachability = state.Value.Times(reachbilityLookup.Value);
if (!newReachabilityDictionary.TryGetValue(state.Key, out var newStatereachabilityStatus))
{
newStatereachabilityStatus = new ReachabilityStatus();
newReachabilityDictionary[state.Key] = newStatereachabilityStatus;
}
newStatereachabilityStatus.EvenReachable |= newReachability.EvenReachable;
newStatereachabilityStatus.OddReachable |= newReachability.OddReachable;
}
}
return(newReachabilityDictionary);
}
/// <summary>
/// Updates a reachability status lookup by adding the data from the current reachabilityStatus
/// </summary>
private void UpdateReachabilityStatus(int fromState, ReachabilityStatus reachabilityStatus,
Dictionary <int, ReachabilityStatus> reachabilityLookup, bool negated)
{
if (!reachabilityLookup.TryGetValue(fromState, out var currentReachabilityStatus))
{
currentReachabilityStatus = new ReachabilityStatus();
reachabilityLookup[fromState] = currentReachabilityStatus;
}
if (negated)
{
currentReachabilityStatus.EvenReachable |= reachabilityStatus.OddReachable;
currentReachabilityStatus.OddReachable |= reachabilityStatus.EvenReachable;
}
else
{
currentReachabilityStatus.EvenReachable |= reachabilityStatus.EvenReachable;
currentReachabilityStatus.OddReachable |= reachabilityStatus.OddReachable;
}
}
internal static Automaton PopulateDFAWithXAndEpsilonTransitionsNaive(this Automaton automaton)
{
bool changes = true;
while (changes)
{
changes = AddEpsilonStatesForXXSubPaths(automaton);
// Look for paths (RRR/SS)
foreach (var startState in automaton.States)
{
// Now look for S * X^alpha * S states
var SSreachabilityDictionary = automaton.GetStatesReachableFromStateWithSymbol(startState, Constants.RegularLanguage.S,
useEpsilonStatesFromInitial: false, useEpsilonStatesAtEnd: true)
.ToDictionary((s) => s, (s) => ReachabilityStatus.Even())
.ApplyXTransitionToReachabilityDictionary(automaton)
.ApplySOrRTransitionToReachabilityDictionary(automaton, Constants.RegularLanguage.S, false);
foreach (var(finalState, reachability) in SSreachabilityDictionary)
{
if (AddTransitionsFromReachabilityStatus(automaton, startState, finalState, reachability))
{
changes = true;
}
}
// Add R * X^alpha1 * R * X^alpha2 * R
var RRRreachabilityDictionary = automaton.GetStatesReachableFromStateWithSymbol(startState, Constants.RegularLanguage.R,
useEpsilonStatesFromInitial: false, useEpsilonStatesAtEnd: true)
.ToDictionary((s) => s, (s) => ReachabilityStatus.Even())
.ApplyXTransitionToReachabilityDictionary(automaton)
.ApplySOrRTransitionToReachabilityDictionary(automaton, Constants.RegularLanguage.R, true)
.ApplyXTransitionToReachabilityDictionary(automaton)
.ApplySOrRTransitionToReachabilityDictionary(automaton, Constants.RegularLanguage.R, false);
foreach (var(finalState, reachability) in RRRreachabilityDictionary)
{
if (AddTransitionsFromReachabilityStatus(automaton, startState, finalState, reachability))
{
changes = true;
}
}
}
}
return(automaton);
}
private static IReadOnlyDictionary <int, ReachabilityStatus> ApplySOrRTransitionToReachabilityDictionary(
this IReadOnlyDictionary <int, ReachabilityStatus> currentReachabilityDictionary, Automaton automaton,
char symbol, bool useEpsilonStatesAtEnd)
{
var newReachabilityDictionary = new Dictionary <int, ReachabilityStatus>();
foreach (var(currentState, currentReachabilityStatus) in currentReachabilityDictionary)
{
var reachableStates = automaton.GetStatesReachableFromStateWithSymbol(currentState, symbol, false, useEpsilonStatesAtEnd);
foreach (var state in reachableStates)
{
if (!newReachabilityDictionary.TryGetValue(state, out var stateReachability))
{
stateReachability = new ReachabilityStatus();
newReachabilityDictionary[state] = stateReachability;
}
stateReachability.EvenReachable |= currentReachabilityStatus.EvenReachable;
stateReachability.OddReachable |= currentReachabilityStatus.OddReachable;
}
}
return(newReachabilityDictionary);
}
private static IReadOnlyDictionary <int, ReachabilityStatus> ApplyXTransitionToReachabilityDictionary(
this IReadOnlyDictionary <int, ReachabilityStatus> currentReachabilityDictionary, Automaton automaton
)
{
var newReachabilityDictionary = new Dictionary <int, ReachabilityStatus>(currentReachabilityDictionary);
foreach (var(currentState, currentReachabilityStatus) in currentReachabilityDictionary)
{
var reachableStates = automaton.GetStatesReachableFromStateWithSymbol(currentState, Constants.RegularLanguage.X, false);
// Now combine all those states with the current reachability dictionary
foreach (var state in reachableStates)
{
if (!newReachabilityDictionary.TryGetValue(state, out var stateReachability))
{
stateReachability = new ReachabilityStatus();
newReachabilityDictionary[state] = stateReachability;
}
// This state has reachability of the opposite of the previous state
stateReachability.OddReachable |= currentReachabilityStatus.EvenReachable;
stateReachability.EvenReachable |= currentReachabilityStatus.OddReachable;
}
}
return(newReachabilityDictionary);
}
private static bool AddTransitionsFromReachabilityStatus(Automaton automaton, int startState, int finalState, ReachabilityStatus reachabilityStatus)
{
bool changes = false;
if (reachabilityStatus.EvenReachable)
{
if (automaton.AddTransition(startState, finalState, Constants.RegularLanguage.X))
{
changes = true;
}
}
else if (reachabilityStatus.OddReachable)
{
// Odd reachable so add X transition
if (startState != finalState && automaton.AddTransition(startState, finalState, Automaton.Epsilon))
{
changes = true;
}
}
return(changes);
}
/// <summary>
/// Adds a transition to the internally stored canonical state substrings. Returns a list of all transitions that should be added directly resulting from this
/// </summary>
public IReadOnlyCollection <Transition> AddTransition(int stateFrom, char symbol, int stateTo)
{
// This process works as follows:
// - Check whether the transition already exists. If it already does, then just return.
// - Add the transition to the lookup
// - Calculate newly created transitions from the lookups
var outTransitionLookup = _outgoingTransitionLookup[stateFrom].GetTransitionDictionary(symbol);
var inTransitionLookup = _incomingTransitionLookup[stateTo].GetTransitionDictionary(symbol);
// Check whether the transition already exists, and add it if not.
// We will check the outgoing transition from the stateFrom, but we expect it to be consistent between the outgoing and incoming transition
// I.e. if s0 ----S----> s1, then we have S outgoing from s0 to s1, and S incoming on s1 from s0
// Therefore, if something breaks here then we know that our code has produced an inconsistent state
ReachabilityStatus fromReachabilityStatus;
if (outTransitionLookup.TryGetValue(stateTo, out var toReachabilityStatus))
{
fromReachabilityStatus = inTransitionLookup[stateFrom];
}
else
{
toReachabilityStatus = new ReachabilityStatus();
outTransitionLookup[stateTo] = toReachabilityStatus;
fromReachabilityStatus = new ReachabilityStatus();
inTransitionLookup[stateFrom] = fromReachabilityStatus;
}
if (symbol != Constants.RegularLanguage.X)
{
if (toReachabilityStatus.EvenReachable)
{
// Transition already exists
return(Array.Empty <Transition>());
}
else
{
fromReachabilityStatus.EvenReachable = true;
toReachabilityStatus.EvenReachable = true;
}
}
else
{
if (toReachabilityStatus.OddReachable)
{
// Transition already exists
return(Array.Empty <Transition>());
}
else
{
toReachabilityStatus.OddReachable = true;
fromReachabilityStatus.OddReachable = true;
}
}
var toAddTransitions = new List <Transition>();
Action <int, int, char> addTransition = (fromState, toState, symbol) =>
{
if (fromState == toState && symbol == Automaton.Epsilon)
{
return;
}
if (!_automaton.AddTransition(fromState, toState, symbol))
{
return;
}
toAddTransitions.Add(new Transition(fromState, toState, symbol));
};
// Adds an X or epsilon transition from a given state to another, based on an existing reachability status
Action <int, int, ReachabilityStatus, bool> addTransitionFromReachabilityStatus = (fromState, toState, reachabilityStatus, negated) =>
{
if ((reachabilityStatus.EvenReachable && negated) || (reachabilityStatus.OddReachable && !negated))
{
addTransition(fromState, toState, Constants.RegularLanguage.X);
}
if ((reachabilityStatus.OddReachable && negated) || (reachabilityStatus.EvenReachable && !negated))
{
addTransition(fromState, toState, Automaton.Epsilon);
}
};
Func <int, Dictionary <int, ReachabilityStatus> > getEpsilonReachabilityDictionary = (state) =>
{
var rightReachabilityDictionary = _outgoingTransitionLookup[state].EpsilonTransitions.ToDictionary(kv => kv.Key, kv => kv.Value);
ReachabilityStatus?reachabilityStatus = null;
if (!rightReachabilityDictionary.TryGetValue(stateTo, out reachabilityStatus))
{
reachabilityStatus = new ReachabilityStatus();
rightReachabilityDictionary[stateTo] = reachabilityStatus;
}
reachabilityStatus.EvenReachable |= true;
return(rightReachabilityDictionary);
};
// Adds transitions for each side combined together with the middle. Should only be called on combinations that make sense.
Action <Dictionary <int, ReachabilityStatus>, char, int, bool> combineWithBothSides =
(incomingReachabilityLookup, symbol, rightSymbolCount, negated) =>
{
var rightReachabilityDictionary = getEpsilonReachabilityDictionary(stateTo);
for (int i = 1; i <= rightSymbolCount; i++)
{
rightReachabilityDictionary = ApplyTransitionToReachabilityDictionary(rightReachabilityDictionary, symbol, i != rightSymbolCount);
}
foreach (var leftReachabilityStatus in incomingReachabilityLookup)
{
foreach (var rightReachabilityStatus in rightReachabilityDictionary)
{
var stateReachabilityStatus = leftReachabilityStatus.Value.Times(rightReachabilityStatus.Value);
addTransitionFromReachabilityStatus(leftReachabilityStatus.Key, rightReachabilityStatus.Key, stateReachabilityStatus, negated);
}
}
};
Action <Dictionary <int, ReachabilityStatus>, bool> combineLeft =
(incomingReachabilityLookup, negated) =>
{
foreach (var stateReachabilityLookup in incomingReachabilityLookup)
{
addTransitionFromReachabilityStatus(stateReachabilityLookup.Key, stateTo, stateReachabilityLookup.Value, negated);
}
};
Action <char, int, bool> combineRight =
(symbol, rightSymbolCount, negated) =>
{
var rightReachabilityDictionary = getEpsilonReachabilityDictionary(stateTo);
for (int i = 1; i <= rightSymbolCount; i++)
{
rightReachabilityDictionary = ApplyTransitionToReachabilityDictionary(rightReachabilityDictionary, symbol, i != rightSymbolCount);
}
foreach (var rightReachabilityLookup in rightReachabilityDictionary)
{
addTransitionFromReachabilityStatus(stateFrom, rightReachabilityLookup.Key, rightReachabilityLookup.Value, negated);
}
};
// Update a given transition set, by iterating through all possibilities specified in the iterate lookup
Action <int, Dictionary <int, ReachabilityStatus>, Func <StateTransitions, Dictionary <int, ReachabilityStatus> >, bool> updateTransitionSetEpsilon =
(inState, iterateReachabilityLookup, getToReachabilitySet, negated) =>
{
foreach (var stateReachabilityLookup in iterateReachabilityLookup)
{
var currentOutState = stateReachabilityLookup.Key;
var currentReachabilityStatus = stateReachabilityLookup.Value;
UpdateReachabilityStatus(currentOutState, currentReachabilityStatus, getToReachabilitySet(_incomingTransitionLookup[inState]), negated);
UpdateReachabilityStatus(inState, currentReachabilityStatus, getToReachabilitySet(_outgoingTransitionLookup[currentOutState]), negated);
}
};
Action <Dictionary <int, ReachabilityStatus>, Func <StateTransitions, Dictionary <int, ReachabilityStatus> >, bool> updateTransitionSetRight =
(iterateReachabilityLookup, getToReachabilitySet, negated) =>
{
foreach (var stateReachabilityLookup in iterateReachabilityLookup)
{
var currentReachabilityStatus = stateReachabilityLookup.Value;
UpdateReachabilityStatus(stateFrom, currentReachabilityStatus, getToReachabilitySet(_incomingTransitionLookup[stateReachabilityLookup.Key]), negated);
UpdateReachabilityStatus(stateReachabilityLookup.Key, currentReachabilityStatus, getToReachabilitySet(_outgoingTransitionLookup[stateFrom]), negated);
}
};
Action <Dictionary <int, ReachabilityStatus>, Func <StateTransitions, Dictionary <int, ReachabilityStatus> >, bool> updateTransitionSetLeft =
(iterateReachabilityLookup, getToReachabilitySet, negated) =>
{
foreach (var stateReachabilityLookup in iterateReachabilityLookup)
{
foreach (var epsilonReachability in getEpsilonReachabilityDictionary(stateTo))
{
var combinedReachability = stateReachabilityLookup.Value.Times(epsilonReachability.Value);
UpdateReachabilityStatus(epsilonReachability.Key, combinedReachability, getToReachabilitySet(_outgoingTransitionLookup[stateReachabilityLookup.Key]), false);
UpdateReachabilityStatus(stateReachabilityLookup.Key, combinedReachability, getToReachabilitySet(_incomingTransitionLookup[epsilonReachability.Key]), false);
}
}
};
Action <Dictionary <int, ReachabilityStatus>, Dictionary <int, ReachabilityStatus>, Func <StateTransitions, Dictionary <int, ReachabilityStatus> >, bool> updateTransitionSetBothSides =
(leftIterateSet, rightIterateSet, getToReachabilitySet, negated) =>
{
foreach (var lefStateReachabilityLookup in leftIterateSet)
{
foreach (var rightStateReachabilityLookup in rightIterateSet)
{
var leftState = lefStateReachabilityLookup.Key;
var rightState = rightStateReachabilityLookup.Key;
var currentReachabilityStatus = lefStateReachabilityLookup.Value.Times(rightStateReachabilityLookup.Value);
UpdateReachabilityStatus(leftState, currentReachabilityStatus, getToReachabilitySet(_incomingTransitionLookup[rightState]), negated);
UpdateReachabilityStatus(rightState, currentReachabilityStatus, getToReachabilitySet(_outgoingTransitionLookup[leftState]), negated);
}
}
};
// We have sk ---symbol---> sj.
// Now combine this with the incoming and outgoing transitions with sk and sjs
switch (symbol)
{
// For S Combine with S transitions on each side (S + S = X)
case Constants.RegularLanguage.S:
updateTransitionSetRight(_outgoingTransitionLookup[stateTo].EpsilonTransitions, (s) => s.STransitions, false);
// Combine new S transition with any incoming and outgoing S transitions
combineRight('S', 1, true);
combineLeft(_incomingTransitionLookup[stateFrom].STransitions, true);
break;
// For R Combine left with R and RR transitions on each side.(R + R = RR, RR + R = R + RR = X, R + R + R = X)
// In other words:
// For R + RR = X or RR + R = X add a transition
// For R + R + R = X add a transition
// For R + R = RR update the dictionaries.
case Constants.RegularLanguage.R:
// Update R + X/epsilon
updateTransitionSetRight(_outgoingTransitionLookup[stateTo].EpsilonTransitions, (s) => s.RTransitions, false);
// Update R + R transitions either side
updateTransitionSetLeft(_incomingTransitionLookup[stateFrom].RTransitions, (s) => s.RRTransitions, false);
updateTransitionSetRight(_outgoingTransitionLookup[stateTo].RTransitions, (s) => s.RRTransitions, false);
// Combine new R transition with any incoming RR transitions
combineRight('R', 2, true);
combineLeft(_incomingTransitionLookup[stateFrom].RRTransitions, true);
// Update incoming and outgoing transitions for R + R + R
combineWithBothSides(_incomingTransitionLookup[stateFrom].RTransitions, 'R', 1, true);
// Combine new R transition with any incoming R transitions
break;
case Constants.RegularLanguage.X:
case Automaton.Epsilon:
// For X and Epsilon Combine with X, S, R and RR transitions on each side
var even = true;
if (symbol == Constants.RegularLanguage.X)
{
// Update chains set
foreach (var chainEnd in _outgoingTransitionLookup[stateTo].EpsilonChains.Append(stateTo))
{
_outgoingTransitionLookup[stateFrom].XEpsilonChains.Add(chainEnd);
_incomingTransitionLookup[chainEnd].XEpsilonChains.Add(stateFrom);
}
even = false;
// Add epsilon transitions for X + X
// TODO: Potential optimisation:
// We shouldn't need to add the state to the queue here as the state should already be consistent at this point,
// so we should just add the state to the automaton, so long as it doesn't go to itself.
foreach (var xChainStart in _incomingTransitionLookup[stateFrom].XEpsilonChains)
{
addTransition(xChainStart, stateTo, Automaton.Epsilon);
}
var xStates = _automaton.GetStatesReachableFromStateWithSymbol(stateTo, 'X', true, false);
foreach (var xState in xStates)
{
addTransition(stateFrom, xState, Automaton.Epsilon);
}
}
else
{
// Update chains set
_outgoingTransitionLookup[stateFrom].EpsilonChains.Add(stateTo);
_incomingTransitionLookup[stateTo].EpsilonChains.Add(stateFrom);
foreach (var chainEnd in _outgoingTransitionLookup[stateTo].EpsilonChains.Append(stateTo))
{
foreach (var chainStart in _incomingTransitionLookup[stateFrom].XEpsilonChains)
{
_outgoingTransitionLookup[chainStart].XEpsilonChains.Add(chainEnd);
_incomingTransitionLookup[chainEnd].XEpsilonChains.Add(chainStart);
}
foreach (var chainStart in _incomingTransitionLookup[stateFrom].EpsilonChains)
{
_outgoingTransitionLookup[chainStart].EpsilonChains.Add(chainEnd);
_incomingTransitionLookup[chainEnd].EpsilonChains.Add(chainStart);
}
}
// Combine with X on each side X
var incomingXEpsilonChains = _incomingTransitionLookup[stateFrom].XEpsilonChains;
var xStates = _automaton.GetStatesReachableFromStateWithSymbol(stateTo, 'X', true, false);
var newTransitions = new List <(int from, int to)>();
foreach (var startXState in incomingXEpsilonChains)
{
foreach (var outgoingX in xStates)
{
// Both X. Combine.
newTransitions.Add((startXState, outgoingX));
}
}
foreach (var transition in newTransitions)
{
addTransition(transition.from, transition.to, Automaton.Epsilon);
}
}
// First update the Transition lookup.
// Update epsilon transitions
updateTransitionSetRight(_outgoingTransitionLookup[stateTo].EpsilonTransitions, (s) => s.EpsilonTransitions, !even);
// updateTransitionSetLeft(_incomingTransitionLookup[stateFrom].EpsilonTransitions, (s) => s.EpsilonTransitions, !even);
updateTransitionSetEpsilon(stateTo, _incomingTransitionLookup[stateFrom].EpsilonTransitions, (s) => s.EpsilonTransitions, !even);
// Combine with X,S,R,RR from front side. We only do the front side as the rules we are checking are XX, SXS, RXRXR.
foreach (var reachabilityLookup in _outgoingTransitionLookup[stateFrom].EpsilonTransitions)
{
if (reachabilityLookup.Value.EvenReachable)
{
updateTransitionSetEpsilon(reachabilityLookup.Key, _incomingTransitionLookup[stateFrom].STransitions, (s) => s.STransitions, false);
updateTransitionSetEpsilon(reachabilityLookup.Key, _incomingTransitionLookup[stateFrom].RTransitions, (s) => s.RTransitions, false);
updateTransitionSetEpsilon(reachabilityLookup.Key, _incomingTransitionLookup[stateFrom].RRTransitions, (s) => s.RRTransitions, false);
}
if (reachabilityLookup.Value.OddReachable)
{
updateTransitionSetEpsilon(reachabilityLookup.Key, _incomingTransitionLookup[stateFrom].STransitions, (s) => s.STransitions, true);
updateTransitionSetEpsilon(reachabilityLookup.Key, _incomingTransitionLookup[stateFrom].RTransitions, (s) => s.RTransitions, true);
updateTransitionSetEpsilon(reachabilityLookup.Key, _incomingTransitionLookup[stateFrom].RRTransitions, (s) => s.RRTransitions, true);
}
}
// Combine R + R
var rightReachability = getEpsilonReachabilityDictionary(stateTo);
rightReachability = ApplyTransitionToReachabilityDictionary(rightReachability, 'R', false);
foreach (var incomingRReachability in _incomingTransitionLookup[stateFrom].RTransitions)
{
foreach (var epsilonReachability in rightReachability)
{
var combinedReachability = incomingRReachability.Value.Times(epsilonReachability.Value);
UpdateReachabilityStatus(epsilonReachability.Key, combinedReachability, _outgoingTransitionLookup[incomingRReachability.Key].RRTransitions, !even);
UpdateReachabilityStatus(incomingRReachability.Key, combinedReachability, _incomingTransitionLookup[epsilonReachability.Key].RRTransitions, !even);
}
}
// Combine S + S
combineWithBothSides(_incomingTransitionLookup[stateFrom].STransitions, 'S', 1, even);
// Combine R + RR
combineWithBothSides(_incomingTransitionLookup[stateFrom].RTransitions, 'R', 2, even);
// Combine RR + R
combineWithBothSides(_incomingTransitionLookup[stateFrom].RRTransitions, 'R', 1, even);
break;
default:
throw new ArgumentException($"Character {symbol} has no understood implementation for canonicalisation");
}
return(toAddTransitions);
}
