public override void Refine(SSA <SYMBOL> m, SSA <SYMBOL> x)
{
int inc;
if (boundInc == Config.BoundInc.One)
{
inc = 1;
}
else
{
SSA <SYMBOL> ssa;
if (boundInc == Config.BoundInc.M)
{
ssa = m;
}
else
{
ssa = x;
}
inc = ssa.States.Count();
if (halveBoundInc)
{
inc /= 2;
}
}
bound += inc;
}
/// <summary>
/// ARMC constructor.
/// </summary>
/// <param name="config">Configuration.</param>
public ARMC(Config config)
: this(SSA <SYMBOL> .Parse(config.InitFilePath),
SSA <SYMBOL> .Parse(config.BadFilePath),
config.TauFilePaths.Select(path => SST <SYMBOL> .Parse(path)).ToArray(),
config)
{
}
public FiniteLengthAbstraction(Config config, SSA <SYMBOL> init = null, SSA <SYMBOL> bad = null, SST <SYMBOL>[] tau = null)
{
if (config.InitialBound == Config.InitBound.One)
{
bound = 1;
}
else
{
SSA <SYMBOL> ssa;
if (config.InitialBound == Config.InitBound.Init)
{
ssa = init ?? SSA <SYMBOL> .Parse(config.InitFilePath);
}
else
{
ssa = bad ?? SSA <SYMBOL> .Parse(config.BadFilePath);
}
bound = ssa.States.Count();
if (config.HalveInitialBound)
{
bound /= 2;
}
}
forward = (config.LanguageDirection == Config.Direction.Forward);
trace = config.TraceLanguages;
boundInc = config.BoundIncrement;
halveBoundInc = config.HalveBoundIncrement;
}
public override bool StatesAreEquivalent(SSA <SYMBOL> m, int q1, int q2)
{
Func <int, int, SSA <SYMBOL> > boundedLang = trace ?
(forward ? (Func <int, int, SSA <SYMBOL> >)m.BoundedForwardTraceLanguage : m.BoundedBackwardTraceLanguage) :
(forward ? (Func <int, int, SSA <SYMBOL> >)m.BoundedForwardStateLanguage : m.BoundedBackwardStateLanguage);
return(boundedLang(q1, bound) == boundedLang(q2, bound));
}
/// <summary>
/// Checks if L(M1) does not equal L(M2).
/// </summary>
/// <remarks>
/// Compares references if either SSA is null.
/// </remarks>
public static bool operator !=(SSA <SYMBOL> m1, SSA <SYMBOL> m2)
{
if (object.ReferenceEquals(m1, null) || object.ReferenceEquals(m2, null))
{
return(!object.ReferenceEquals(m1, m2));
}
return(!SSA <SYMBOL> .Equivalent(m1, m2));
}
public override bool Equals(object obj)
{
SSA <SYMBOL> m = obj as SSA <SYMBOL>;
if (m != null)
{
return(this.automaton.Equals(m.automaton));
}
return(this.automaton.Equals(obj));
}
public SSA <SYMBOL> BoundedLanguage(int n)
{
var m = new SSA <SYMBOL>(
0,
Enumerable.Range(0, n + 1),
Enumerable.Range(0, n).Select(s => new Move <Predicate <SYMBOL> >(s, s + 1, Predicate <SYMBOL> .True)),
new Set <SYMBOL>(Alphabet)
);
return(Product(this, m));
}
public PredicateAbstraction(Config config, SSA <SYMBOL> init = null, SSA <SYMBOL> bad = null, SST <SYMBOL>[] taus = null)
{
var initPreds = new List <SSA <SYMBOL> >();
/* add Init? */
if (config.InitialPredicate == Config.InitPred.Init || config.InitialPredicate == Config.InitPred.Both)
{
initPreds.Add(init ?? SSA <SYMBOL> .Parse(config.InitFilePath));
}
/* add Bad? */
if (config.InitialPredicate == Config.InitPred.Bad || config.InitialPredicate == Config.InitPred.Both)
{
initPreds.Add(bad ?? SSA <SYMBOL> .Parse(config.BadFilePath));
}
/* add transducer domains and/or ranges? */
SST <SYMBOL>[] ssts = taus ?? config.TauFilePaths.Select(path => SST <SYMBOL> .Parse(path)).ToArray();
if (config.IncludeGuard)
{
foreach (SST <SYMBOL> sst in ssts)
{
initPreds.Add(sst.Domain());
}
}
if (config.IncludeAction)
{
foreach (SST <SYMBOL> sst in ssts)
{
initPreds.Add(sst.Range());
}
}
/* ensure that the automata contain no epsilon transitions and have disjunct sets of states */
predicateAutomata = new Set <SSA <SYMBOL> >();
int offset = 0;
foreach (SSA <SYMBOL> pred in initPreds)
{
SSA <SYMBOL> normPred = pred.RemoveEpsilons().Normalize(offset);
predicateAutomata.Add(normPred);
offset += normPred.States.Count();
}
forward = (config.LanguageDirection == Config.Direction.Forward);
heuristic = config.Heuristic;
ignoredLabels = new Set <int>(); // remains empty if no heuristic used
}
public override SSA <SYMBOL> Collapse(SSA <SYMBOL> m)
{
m = m.RemoveEpsilons();
Dictionary <int, Set <int> > labels = MakeLabels(m);
if (heuristic.HasValue)
{
/* remove ignored states from labels */
foreach (int state in new List <int>(labels.Keys))
{
labels[state] -= ignoredLabels;
}
}
return(m.Collapse((_, q1, q2) => labels[q1] == labels[q2]));
}
/* Collapse method ignores this in favour of more efficient alternative */
public override bool StatesAreEquivalent(SSA <SYMBOL> m, int q1, int q2)
{
Func <int, SSA <SYMBOL> > stateLang = forward ? (Func <int, SSA <SYMBOL> >)m.ForwardStateLanguage : m.BackwardStateLanguage;
SSA <SYMBOL> m1 = stateLang(q1);
SSA <SYMBOL> m2 = stateLang(q2);
foreach (SSA <SYMBOL> pa in predicateAutomata)
{
stateLang = forward ? (Func <int, SSA <SYMBOL> >)pa.ForwardStateLanguage : pa.BackwardStateLanguage;
foreach (int state in pa.States)
{
SSA <SYMBOL> p = stateLang(state);
if (SSA <SYMBOL> .ProductIsEmpty(p, m1) != SSA <SYMBOL> .ProductIsEmpty(p, m2))
{
return(false);
}
}
}
return(true);
}
/// <summary>
/// Print the counterexample (SSAs).
/// </summary>
/// <param name="counterexample">Counterexample.</param>
/// <param name="directory">Output directory path.</param>
public void PrintCounterexample(Counterexample <SYMBOL> counterexample, string directory)
{
int i = 0;
foreach (var pair in counterexample.Ms)
{
SSA <SYMBOL> m = pair.Item1;
SSA <SYMBOL> mAlpha = pair.Item2;
PrintAutomaton(m, directory, "M" + i.ToString());
if (mAlpha != null)
{
PrintAutomaton(mAlpha, directory, "M" + i.ToString() + "+");
}
i++;
}
i = 0;
foreach (SSA <SYMBOL> x in counterexample.Xs)
{
PrintAutomaton(x, directory, "X" + i.ToString());
i++;
}
}
/// <summary>
/// Checks if L(this) is a superset of L(M).
/// </summary>
public bool IsSupersetOf(SSA <SYMBOL> m)
{
return(m.IsSubsetOf(this));
}
/// <summary>
/// Checks if L(this) is a subset of L(M). If not, produces witness.
/// </summary>
public bool IsSubsetOf(SSA <SYMBOL> m, out List <Predicate <SYMBOL> > witness)
{
return(!(Automaton <Predicate <SYMBOL> > .CheckDifference(this.automaton, m.automaton, 0, out witness)));
}
/// <summary>
/// Checks if L(this) is a subset of L(M).
/// </summary>
public bool IsSubsetOf(SSA <SYMBOL> m)
{
List <Predicate <SYMBOL> > witness;
return(IsSubsetOf(m, out witness));
}
/// <summary>
/// Checks if L(M1) equals L(M2). If not, produces witness.
/// </summary>
public static bool Equivalent(SSA <SYMBOL> m1, SSA <SYMBOL> m2, out List <Predicate <SYMBOL> > witness)
{
return(m1.automaton.IsEquivalentWith(m2.automaton, out witness));
}
/// <summary>
/// ARMC constructor.
/// </summary>
/// <param name="init">SSA representing initial states.</param>
/// <param name="bad">SSA representing bad states.</param>
/// <param name="tau">SST representing transition.</param>
/// <param name="configFileName">Configuration file path.</param>
public ARMC(SSA <SYMBOL> init, SSA <SYMBOL> bad, SST <SYMBOL> tau, string configFileName)
: this(init, bad, new SST <SYMBOL>[] { tau }, new Config(configFileName))
{
}
/// <summary>
/// Checks if the intersection of L(M1) and L(M2) is an empty language. If not, produces witness.
/// </summary>
public static bool ProductIsEmpty(SSA <SYMBOL> m1, SSA <SYMBOL> m2, out List <Predicate <SYMBOL> > witness)
{
return(!Automaton <Predicate <SYMBOL> > .CheckProduct(m1.automaton, m2.automaton, 0, out witness));
}
/// <summary>
/// ARMC constructor.
/// </summary>
/// <param name="init">SSA representing initial states.</param>
/// <param name="bad">SSA representing bad states.</param>
/// <param name="taus">SSTs whose composition represents transition.</param>
/// <param name="config">Configuration.</param>
public ARMC(SSA <SYMBOL> init, SSA <SYMBOL> bad, SST <SYMBOL>[] taus, Config config)
{
/* merge alphabets */
Set <SYMBOL> alphabet = init.Alphabet + bad.Alphabet + taus.Select(tau => tau.Alphabet).Aggregate(Set <SYMBOL> .Union);
init.Alphabet = alphabet;
bad.Alphabet = alphabet;
for (int i = 0; i < taus.Length; i++)
{
taus[i].Alphabet = alphabet;
}
if (config.PredicateLanguages == config.FiniteLengthLanguages) // sanity check
{
throw ConfigException.AbstractionNotChosen();
}
this.init = init;
this.bad = bad;
this.tau = SST <SYMBOL> .Union(taus);
this.tauInv = tau.Invert();
this.abstr = config.PredicateLanguages ?
(Abstraction <SYMBOL>) new PredicateAbstraction <SYMBOL>(config, init, bad, taus) :
new FiniteLengthAbstraction <SYMBOL>(config, init, bad, taus);
this.verbose = config.Verbose;
this.printAutomata = config.PrintAutomata;
this.outputDir = config.OutputDirectory;
this.format = config.AutomataFormat;
this.imgExt = (config.ImageFormat == null) ? "" : config.ImageFormat.ToString();
this.timeout = config.Timeout.Ticks;
this.stopwatch = new Stopwatch();
this.loops = 0;
if (outputDir != "")
{
/* clear output directory */
DirectoryInfo dirInfo = Directory.CreateDirectory(outputDir);
foreach (FileInfo fi in dirInfo.EnumerateFiles())
{
fi.Delete();
}
foreach (DirectoryInfo di in dirInfo.EnumerateDirectories())
{
di.Delete(true);
}
}
if (printAutomata)
{
/* print input automata and configuration */
string dir = Path.Combine(outputDir, "armc-input");
Directory.CreateDirectory(dir);
PrintAutomaton(init, dir, "init");
PrintAutomaton(bad, dir, "bad");
PrintAutomaton(tau, dir, "tau");
if (taus.Length > 1) // no point in only printing tau1
{
for (int i = 0; i < taus.Length; i++)
{
PrintAutomaton(taus[i], dir, "tau" + (i + 1).ToString());
}
}
PrintAutomaton(tauInv, dir, "tau-inv");
config.Write(Path.Combine(dir, "armc.properties"));
}
if (config.ComputationDirection == Config.Direction.Backward)
{
/* reverse direction, i.e. check if tauInv*(bad) & init is empty */
this.init = bad;
this.bad = init;
this.tauInv = tau;
this.tau = tau.Invert();
}
if (!SSA <SYMBOL> .ProductIsEmpty(init, bad)) // no point in further verification
{
if (printAutomata)
{
string dir = Path.Combine(outputDir, "armc-counterexample");
Directory.CreateDirectory(dir);
PrintAutomaton(init & bad, dir, "initXbad");
}
throw ARMCException.InitialPropertyViolation();
}
}
/* label automaton states by predicate states whose forward/backward languages intersect */
private Dictionary <int, Set <int> > MakeLabels(SSA <SYMBOL> m)
{
var labels = new Dictionary <int, Set <int> >(m.States.Count());
foreach (int state in m.States)
{
labels[state] = new Set <int>();
}
var stack = new Stack <Tuple <int, int> >();
foreach (SSA <SYMBOL> pred in predicateAutomata)
{
if (m.Algebra.Alphabet != pred.Algebra.Alphabet)
{
throw SSAException.IncompatibleAlphabets();
}
}
Func <int, IEnumerable <Move <Predicate <SYMBOL> > > > getMoves = forward ?
(Func <int, IEnumerable <Move <Predicate <SYMBOL> > > >)m.GetMovesTo : m.GetMovesFrom;
foreach (SSA <SYMBOL> pred in predicateAutomata)
{
if (forward)
{
foreach (int mState in m.FinalStates)
{
foreach (int pState in pred.FinalStates)
{
stack.Push(new Tuple <int, int>(mState, pState));
}
}
}
else
{
stack.Push(new Tuple <int, int>(m.InitialState, pred.InitialState));
}
while (stack.Count > 0)
{
var pair = stack.Pop();
int mState = pair.Item1;
int pState = pair.Item2;
labels[mState].Add(pair.Item2);
foreach (var mMove in getMoves(mState))
{
int state = forward ? mMove.SourceState : mMove.TargetState;
foreach (var pMove in (forward ? pred.GetMovesTo(pState) : pred.GetMovesFrom(pState)))
{
if (!m.Algebra.IsSatisfiable(mMove.Label & pMove.Label))
{
continue;
}
int stateLabel = forward ? pMove.SourceState : pMove.TargetState;
if (!labels[state].Contains(stateLabel))
{
stack.Push(new Tuple <int, int>(state, stateLabel));
}
}
}
}
}
return(labels);
}
public override void Refine(SSA <SYMBOL> m, SSA <SYMBOL> x)
{
int offset = predicateAutomata.Sum(pred => pred.States.Count());
x = x.RemoveEpsilons().Normalize(offset);
predicateAutomata.Add(x);
if (heuristic.HasValue)
{
var xStates = new Set <int>(x.States);
/* find important states (appear in labels) */
Dictionary <int, Set <int> > labels = MakeLabels(m);
var importantStates = new Set <int>();
foreach (int mState in m.States)
{
foreach (int xState in labels[mState])
{
if (xState < offset || xState >= xStates.Count + offset)
{
continue;
}
importantStates.Add(xState);
}
}
if (((Config.PredHeuristic)heuristic) == Config.PredHeuristic.KeyStates)
{
/* try to find one key state among important states */
foreach (int state in importantStates)
{
/* try ignoring all but one state */
ignoredLabels += xStates;
ignoredLabels.Remove(state);
/* check if the collapsed automaton still intersects */
if (SSA <SYMBOL> .ProductIsEmpty(Collapse(m), x))
{
return;
}
/* failed, restore temporarily ignored states */
ignoredLabels -= xStates;
}
/* couldn't find just one key state, try to find two */
foreach (int state1 in importantStates)
{
foreach (int state2 in importantStates.Where(s => s != state1))
{
ignoredLabels += xStates;
ignoredLabels.Remove(state1);
ignoredLabels.Remove(state2);
if (SSA <SYMBOL> .ProductIsEmpty(Collapse(m), x))
{
return;
}
ignoredLabels -= xStates;
}
}
/* fall back on important states heuristic */
}
/* ignore all unimportant states */
ignoredLabels += xStates - importantStates;
}
}
/// <summary>
/// ARMC constructor.
/// </summary>
/// <param name="init">SSA representing initial states.</param>
/// <param name="bad">SSA representing bad states.</param>
/// <param name="tau">SST representing transition.</param>
/// <param name="config">Configuration.</param>
public ARMC(SSA <SYMBOL> init, SSA <SYMBOL> bad, SST <SYMBOL> tau, Config config)
: this(init, bad, new SST <SYMBOL>[] { tau }, config)
{
}
/// <summary>
/// Checks if L(this) is a superset of L(M). If not, produces witness.
/// </summary>
public bool IsSupersetOf(SSA <SYMBOL> m, out List <Predicate <SYMBOL> > witness)
{
return(m.IsSubsetOf(this, out witness));
}
/// <summary>
/// Checks if L(this) is a proper superset of L(M).
/// </summary>
public bool IsProperSupersetOf(SSA <SYMBOL> m)
{
return(m.IsSubsetOf(this) && !this.IsSubsetOf(m));
}
/// <summary>
/// Performs one step (inner loop) of ARMC.
/// </summary>
/// <returns>Verification result, or <code>null</code> if undecided.</returns>
/// <param name="counterexample">Counterexample (if encountered, i.e. <code>false</code> returned).</param>
public bool?VerifyStep(out Counterexample <SYMBOL> counterexample)
{
var ssas = new Stack <Tuple <SSA <SYMBOL>, SSA <SYMBOL> > >();
SSA <SYMBOL> m;
SSA <SYMBOL> ml;
SSA <SYMBOL> mAlpha = null;
SSA <SYMBOL> x;
var xs = new Stack <SSA <SYMBOL> >();
int i = 0;
int l = 0;
string ext = "." + format.ToString().ToLower();
string dir = Path.Combine(outputDir, "armc-loop-" + loops.ToString());
stopwatch.Start();
if (printAutomata)
{
Directory.CreateDirectory(dir);
abstr.Print(dir, this);
}
counterexample = null;
m = init;
if (printAutomata)
{
PrintAutomaton(m, dir, "M0");
}
while (true)
{
if (verbose)
{
Log("\r" + loops.ToString() + "." + i.ToString());
}
if (i > 0 && !SSA <SYMBOL> .ProductIsEmpty(m, bad))
{
if (verbose)
{
LogLine(": counterexample encountered");
}
l = i;
x = m & bad;
x.Name = "<i>X</i><sub>" + l.ToString() + "</sub>";
if (printAutomata)
{
PrintAutomaton(x, dir, "X" + l.ToString());
}
ml = m;
xs.Push(x);
break;
}
mAlpha = abstr.Collapse(m).Determinize().Minimize();
mAlpha.Name = "<i>M</i><sub>" + i.ToString() + "</sub><sup>α</sup>";
if (printAutomata)
{
PrintAutomaton(mAlpha, dir, "M" + i.ToString() + "+");
}
if (i > 0 && mAlpha == ssas.Peek().Item2)
{
stopwatch.Stop();
if (verbose)
{
LogLine(": fixpoint reached");
LogLine("time = " + stopwatch.Elapsed.ToString());
}
return(true);
}
if (timeout > 0 && stopwatch.ElapsedTicks > timeout)
{
stopwatch.Stop();
if (verbose)
{
LogLine(": timeout (" + stopwatch.Elapsed.ToString() + ")");
}
throw ARMCException.Timeout();
}
ssas.Push(Tuple.Create(m, mAlpha));
i++;
m = tau.Apply(mAlpha).Determinize().Minimize();
m.Name = "<i>M</i><sub>" + i.ToString() + "</sub>";
if (printAutomata)
{
PrintAutomaton(m, dir, "M" + i.ToString());
}
}
bool spurious = false;
foreach (var pair in ssas)
{
m = pair.Item1;
mAlpha = pair.Item2;
i--;
if (verbose)
{
Log("\r" + loops.ToString() + "." + l.ToString() + "-" + i.ToString());
}
x = (tauInv.Apply(x) & mAlpha).Determinize().Minimize();
x.Name = "<i>X</i><sub>" + i.ToString() + "</sub>";
xs.Push(x);
if (printAutomata)
{
PrintAutomaton(x, dir, "X" + i.ToString());
}
if (SSA <SYMBOL> .ProductIsEmpty(x, m))
{
spurious = true;
break;
}
}
stopwatch.Stop();
if (spurious)
{
if (verbose)
{
LogLine(": counterexample is spurious");
}
abstr.Refine(m, x);
return(null);
}
else
{
if (verbose)
{
LogLine(": counterexample is real");
LogLine("time = " + stopwatch.Elapsed.ToString());
}
List <Tuple <SSA <SYMBOL>, SSA <SYMBOL> > > ms = ssas.ToList();
ms.Reverse();
ms.Add(new Tuple <SSA <SYMBOL>, SSA <SYMBOL> >(ml, null));
counterexample = new Counterexample <SYMBOL> {
Ms = ms,
Xs = xs.ToList()
};
return(false);
}
}
/// <summary>
/// Checks if the intersection of L(M1) and L(M2) is an empty language.
/// </summary>
public static bool ProductIsEmpty(SSA <SYMBOL> m1, SSA <SYMBOL> m2)
{
List <Predicate <SYMBOL> > witness;
return(SSA <SYMBOL> .ProductIsEmpty(m1, m2, out witness));
}
/// <summary>
/// Constructs SSA accepting L(this) minus L(M2).
/// </summary>
public SSA <SYMBOL> Minus(SSA <SYMBOL> m)
{
return(new SSA <SYMBOL>(this.automaton.Minus(m.automaton)));
}
/// <summary>
/// Checks if L(M1) equals L(M2).
/// </summary>
public static bool Equivalent(SSA <SYMBOL> m1, SSA <SYMBOL> m2)
{
return(m1.automaton.IsEquivalentWith(m2.automaton));
}
/// <summary>
/// ARMC constructor.
/// </summary>
/// <param name="init">SSA representing initial states.</param>
/// <param name="bad">SSA representing bad states.</param>
/// <param name="tau">SST representing transition.</param>
public ARMC(SSA <SYMBOL> init, SSA <SYMBOL> bad, SST <SYMBOL> tau)
: this(init, bad, new SST <SYMBOL>[] { tau }, new Config())
{
}
/// <summary>
/// Applies this transducer to automaton M in order to create a new automaton
/// that accepts the translations of all words in L(M).
/// </summary>
public SSA <SYMBOL> Apply(SSA <SYMBOL> m)
{
if (this.Algebra.Alphabet != m.Algebra.Alphabet)
{
throw SSTException.IncompatibleAlphabets();
}
m = m.RemoveEpsilons();
Automaton <Label <SYMBOL> > tau = this.automaton;
var stack = new Stack <Tuple <int, int> >();
var finalStates = new List <int>();
var moves = new List <Move <Predicate <SYMBOL> > >();
var stateDict = new Dictionary <Tuple <int, int>, int>();
int id = 0;
var init = new Tuple <int, int>(tau.InitialState, m.InitialState);
stateDict[init] = id++; // initial state will be 0
stack.Push(init);
Action <Tuple <int, int>, Tuple <int, int>, Predicate <SYMBOL> > addMove = (sourcePair, targetPair, label) => {
int targetState;
if (!stateDict.TryGetValue(targetPair, out targetState))
{
stateDict[targetPair] = targetState = id++;
stack.Push(targetPair);
}
moves.Add(new Move <Predicate <SYMBOL> >(stateDict[sourcePair], targetState, label));
};
while (stack.Count > 0)
{
Tuple <int, int> sourcePair = stack.Pop();
int tauState = sourcePair.Item1;
int mState = sourcePair.Item2;
foreach (Move <Label <SYMBOL> > tauMove in tau.GetMovesFrom(tauState))
{
if (tauMove.Label.Input == null)
{
addMove(
sourcePair,
new Tuple <int, int>(tauMove.TargetState, mState),
tauMove.Label.IsIdentity ? null : tauMove.Label.Output
);
continue;
}
foreach (Move <Predicate <SYMBOL> > mMove in m.GetMovesFrom(mState))
{
if (!m.Algebra.IsSatisfiable(tauMove.Label.Input & mMove.Label))
{
continue;
}
Predicate <SYMBOL> newLabel;
if (tauMove.Label.IsIdentity)
{
newLabel = tauMove.Label.Input & mMove.Label;
}
else
{
newLabel = tauMove.Label.Output;
if (newLabel != null && !m.Algebra.IsSatisfiable(newLabel))
{
continue;
}
}
addMove(
sourcePair,
new Tuple <int, int>(tauMove.TargetState, mMove.TargetState),
newLabel
);
}
}
}
foreach (var pair in stateDict)
{
if (tau.IsFinalState(pair.Key.Item1) && m.IsFinalState(pair.Key.Item2))
{
finalStates.Add(pair.Value);
}
}
return(new SSA <SYMBOL>(0, finalStates, moves, m.Algebra.Alphabet));
}
/// <summary>
/// Constructs SSA accepting union of L(M1) and L(M2).
/// </summary>
public static SSA <SYMBOL> Sum(SSA <SYMBOL> m1, SSA <SYMBOL> m2)
{
return(new SSA <SYMBOL>(Automaton <Predicate <SYMBOL> > .MkSum(m1.automaton, m2.automaton)));
}