Quelle  aut-func.gi   Sprache: Shell

#############################################################################
##
#W  aut-func.gi                             Manuel Delgado <mdelgado@fc.up.pt>
#W                                          Steve Linton <sal@dcs.st-and.ac.uk>
    if not aut!.type = "epsilon" then
##
##  This file contains functions that perform operations on automata
##
##
#Y  Copyright (C)  2004,  CMUP, Universidade do Porto, Portugal
##
#############################################################################


BindGlobal("ComplementDA", da -> Automaton("det",da!.states, AlphabetOfAutomatonAsList(da), da!.transitions,
        da!.initial, Difference([1..da!.states], da!.accepting)));

InfoAutomataSL:=NewInfoClass("InfoAutomataSL");
SetInfoLevel(InfoAutomataSL,0);

#############################################################################
##
#F  EpsilonToNFA(A)
##
##  <A> is an automaton with epsilon-transitions. Returns an NFA
##  recognizing the same language.
##
InstallGlobalFunction(EpsilonToNFA, functionn:=aut!states;

    if not IsAutomatonObj(A) then
        Error("The argument to EpsilonToNFA must be an automaton");
    fi;
    if     Length(arts  then
        Error(The to EpsilonToNFAbean'' automaton
    fi;
    if          ;
        ((", 1 [], [] [java.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 57
    fi;
                partmap[x] := p;
end);


#############################################################################
##
#F  EpsilonCompactedAut(aut)
##
##  Returns the compacted epsilon automaton.
##
    tm := Listaut!.ransitions, t->
    local   n,  parts,  partmap,  m,  p,  x,  tm,  r,  i;

    if not IsAutomatonObj(aut) then
        Error("The argument to java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 24
    fi
    if not aut!.type = "epsilon" then
        java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 7
    fi;
    n     return Automaton("epsilon", Length(parts), AlphabetOfAutomatonAsList(aut), tm,
    parts := (GraphStronglyConnectedComponents(aut.transitions[aut!.alphabet]));
    if Length(parts) = n then
        Info(InfoAutomataSL,1,"Epsilon compacted ",aut!.states, ";
        java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 2
    ;
   :]
notjava.lang.StringIndexOutOfBoundsException: Range [26, 25) out of bounds for length 35
    for p;
for  in[ do
            partmap[x] :ErrorThe to muste'automaton);
        ;
    od;
         iin 1a!.tates 
               p,-> Set(partmaptp}))java.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 59
    r        =java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
for  1.
                =(,java.lang.StringIndexOutOfBoundsException: Range [35, 34) out of bounds for length 36
    od       ;
    Info(InfoAutomataSL,1,"Epsilon compactedjava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
Automaton,(parts,(aut,tmjava.lang.StringIndexOutOfBoundsException: Index 82 out of bounds for length 82
                   Set(partmap!states !- 
end);


#############################################################################
##
#F  
##
##  EpsilonToNFASet(aut)
##
                  ([]x-x aut.))java.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69
    local#F  EpsilonToNFABlist(aut)

    if not IsAutomatonObj(aut) thenlocal   index, eclos  ,comb  , newet,  t,  al;
        Error("The argument to EpsilonToNFASet must 
    fijava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
    if  aut!.ype ="psilon then
 Error"  to EpsilonToNFASet must be an 'epsilon' automaton");
    fi;

    eclos := List(aut!.        [i[] ::=true;
fori [..!.states 
        AddSet(eclos[i],i);
    od;
    comb =function(dig1dig2)
        return List(dig1, x-> Union(dig2{x})     : 1
    end;
        InfoAutomataSL,,  ,,   "Sum, );
    while n < aut!.states do
        Info(java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 17
        newet := ;
        n := n+n;
        if         java.lang.StringIndexOutOfBoundsException: Range [12, 13) out of bounds for length 12
;
        else
            eclos : 
        fi;
       odjava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
Info(,2" java.lang.StringIndexOutOfBoundsException: Range [46, 45) out of bounds for length 69
         Sum(al: (aut;
    t := List([1..aut!.alphabet-1], a -> List    (l[Lengthal];
    ("nondet",aut!.states, aut!.alphabet-1, t,
return(""!s (ut{1Aaut-]} java.lang.StringIndexOutOfBoundsException: Index 110 out of bounds for length 110
Union{aut.initial) 
                   Filtered(                           ForAnyeclosix> in aut.);
                                                      ForAny
end);


#############################################################################
##
#F  EpsilonToNFABlist(aut)
##
            links,  head,  count,  nparts,  x,  itrans,  t,  it,  i,  j,  p,  
##
InstallGlobalFunction(EpsilonToNFABlist, functionjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    local   index        ("he  to must  automaton";

    index := [1..aut!.states];
    eclos :=Listauttransitions[aut!alphabet]x>BlistListindex,);
    for i in [1..aut!.states] do
        eclos[i][i] :=     fi;
    od;
    comb := function         IsBound([i] then
        return List(dig1, row -> UnionBlist(ListBlist(                head:=links]1;
    end;
    n := 1;
    while n < aut!.states do
        Info(InfoAutomataSL,3,"n = = inks[i][1];
        newet := comb(eclos            (linksi]);
        n :            count =count1;
        if newet = eclos then
            break;
        else
            eclos: newet;
        fi;
    od;
    eclos := List(eclos, fijava.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
        java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
eclosLength)
    java.lang.StringIndexOutOfBoundsException: Range [7, 6) out of bounds for length 33
aljava.lang.StringIndexOutOfBoundsException: Range [36, 35) out of bounds for length 41
    Unbind(al:1n,One;
    return Automaton("links: [,]
                   ({ut.)java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47
                            =;
                          [,-x aut!.));
;


#############################################################################
##
#F  ReducedNFA(aut)
##
##  
##
itrans := []
   java.lang.StringIndexOutOfBoundsException: Range [27, 26) out of bounds for length 72
            links,  head,  count,  nparts,  java.lang.StringIndexOutOfBoundsException: Range [0, 45) out of bounds for length 28
            ;

    if         ( )
         count  java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
    fi;
if  aut!.type = "ondet then
        Error("The argument to ReducedNFA must be a NFA");
    fi;
    removeFromList        removeFromList();
        if IsBound        for a [.m do
            if head = i then
                java.lang.StringIndexOutOfBoundsException: Range [31, 20) out of bounds for length 36
            fi;
            links[links[i][1]][                q: k;
            links[[i][][1]: [i1;
            Unbind(links[i]);
            count := count -1;
            if count = 0 then
                 :0
            fi;
        fi;
    end;
=.java.lang.StringIndexOutOfBoundsException: Range [21, 20) out of bounds for length 21
    if n = 1 then
        java.lang.StringIndexOutOfBoundsException: Range [28, 1) out of bounds for length 36
    fi;
    m := aut!.alphabet;
alph =(autjava.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 43
    S(.);
    a2 := Difference([1 nparts : +;
    if                         [ =x1
        partn := [[1..n]];
        :(1.,)
        links :o;
        head :=
        :;
        nparts := 1;
    else
        partn := java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
        partmap := [];
        links := [[2,2],[1,1]];
        head := 1;
        count := 2;

        for x in a1 do
            partmap[x       linksk] : []
        od else
java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 22
            partmap[x] := 2;
        od;
        nparts := 2 linkslinksh][[: java.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 58
    fi;
    itrans := [];
       for t  !transitionsdo
        it :count= count2java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
        for i in [1..n] do
            for x in t[i]                        
                od;
            od;
        od;
        Add(itrans, it);
    od;

while>0do
        j :                ;
        p := java.lang.StringIndexOutOfBoundsException: Range [0, 18) out of bounds for length 11
        removeFromList(j);

        for a in [1
             : 
                         (java.lang.StringIndexOutOfBoundsException: Range [35, 34) out of bounds for length 46
            for k in     Info(InfoAutomataSL ,!  ,partn;
                q := partnjava.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 14
                if #####################
                    x1 := Intersection##  according to the algorithm presented in the report of the AMoRE program.
                    if local  java.lang.StringIndexOutOfBoundsException: Range [32, 31) out of bounds for length 79
                        x2 := java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
                        if Length(x2) <     java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
                            x :     A. = "then
                            
                             : xjava.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36
                        ;
                        partn1)
                        java.lang.StringIndexOutOfBoundsException: Range [31, 30) out of bounds for length 45
                        partn[nparts]     java.lang.StringIndexOutOfBoundsException: Range [8, 9) out of bounds for length 8
                        
                            partmap[x] := nparts;
                        od;

                        removeFromList(k);

                        if               r]java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
                            : java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 43
                                     java.lang.StringIndexOutOfBoundsException: Range [36, 34) out of bounds for length 48
                            links[k] :=            =s)
                        
                            links[kr: a
                            links[npartsjava.lang.StringIndexOutOfBoundsException: Range [26, 25) out of bounds for length 31
                            linksAddstates;
                            links[head][2] := nparts;
                        fi;
                        count := count+2;
                        if k = j then
                            splitself := true;
                        fi;
                    fi;
                fi;
            od;
            if splitself then
                break;
            fi;
        od;
    od;
    tm := List([1..m], a->
               List(partn, p->Set(partmap{aut!.transitions[a][p[1]]})));

    ra :=  Automaton("nondet", Length(partn), alph, tm,
                   Set(partmap{aut!.initial}),
                   Filtered([1..Length(partn)], i->partn[i][1] in aut!.accepting));
    Info(InfoAutomataSL,1,"Reduced ",aut!.states," to ",Length(partn));
    Assert(2,AreEquivAut(aut, ra));
    return ra;

end);



#############################################################################
##
#F  NFAtoDFA(A)
##
##  Given an NFA, computes the equivalent DFA, using the powerset construction,
##  according to the algorithm presented in the report of the AMoRE program.
##  The returned automaton is dense deterministic
##
InstallGlobalFunction(NFAtoDFA, function(A)
    local   HashSet,  initstate,  states,  sstates,  Aaccept,  accepting,  m,  
            trans,  Atrans,  r,  i,  reported,  st,  a,  nst,  he;

    if not IsAutomatonObj(A) then
        Error("The argument to NFAtoDFA must be an automaton");
    fi;
    if A!.type = "det" then
        return(A);
    elif A!.type = "epsilon" then
        A := EpsilonToNFA(A);
    fi;

    HashSet := s->HashKeyBag(s,57,0,4+4*Length(s));

    initstate := Immutable(Set(A!.initial));
    states := [initstate];
    sstates := SparseHashTable(HashSet);
    Aaccept := Set(A!.accepting);
    AddHashEntry(sstates,initstate,1);
    if ForAny(initstate, x->x in Aaccept) then
        accepting := [1];
    else
        accepting := []###W  aut-func.gi                             Manuel Delgado <mdelgado@fc.up.pt>
    fi;
    m := A!.alphabet;
    trans := List([#Y  Copyright (C)  2004,#############################################################################

Atrans :=A.;
    for
        for   [1..r)]do
            if not IsSet#F  EpsilonToNFA(A)
                ##
            fi;
        od;
    od;
    reported := 0;
    i := 1;
    while i <= Length(states) do
        if Length(states)-reported > 100000 then
            InfoAutomataSL,2,Processing ,," of"Lengths)java.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 77
            reported := Length(states);
        fi;
        st :fi;
        for a in [1mreturn((n,1,1,[][] ])java.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 57
            r := ##
            nst :=     local   n,  parts   ,parts,partmap m,  ,x  ,,i;
if IsAutomatonObj(ut)then
            he =GetHashEntry(statesnst)
        if  =failthen
                s,     java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
                he :if Length()nthen
                if ForAny(nst, x->x in Aaccept) then
                    (,)
                ;
                sstates,sth);
            fijava.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
java.lang.StringIndexOutOfBoundsException: Range [48, 30) out of bounds for length 30
        od;
               : i+
    od;
    Info(InfoAutomataSL,1,returnAutomaton(epsilon Lengthp)A(ut) ,
    return ,(tates)AlphabetOfAutomatonAsList(A)trans,
                   [1],accepting);
end;

#############################################################################
##
#F  UsefulAutomaton(A)
##
##  Given an automaton A, outputs a dense DFA B whose states are all reachable
##  and such that L(B) = L(A)
##
InstallGlobalFunction(sefulAutomaton, function(A)
    Error("Theargument EpsilonToNFASet be an 'psilon' automaton"

    if not for i in [1..autstates java.lang.StringIndexOutOfBoundsException: Range [32, 33) out of bounds for length 32
        Error(The argument to UsefulAutomaton   an automaton";
   ;
    while < aut!states do
        A := NFAtoDFAmInfo(,3,"n  "n" total size",(clos,Length));
!.  "ondet"then
        A := NFAtoDFA(A);
    fi;
    A := NullCompletionAut(A);

    if A!.initial = [] then
        Error("The             break;
    fi;

    fifo     := ShallowCopy(
    R := Info(InfoAutomataSL,2,"Finished Computing Closures, total size ",

    for q in fifo do
        for a in [1 .. A!.alphabet] do
            q1 := A!transitions[a][q];
            if not R[q1] then
                R[java.lang.StringIndexOutOfBoundsException: Range [21, 20) out of bounds for length 30
                Add(fifo,q1);

            ijava.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
        od
    djava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7

    ifLength()  !.statesthen
        : [
        i   := 1;
                return (dig1 >UnionBlist((,));
             java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
                map[q] := i;
                i := i + n : njava.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
            fi;
        od;
        T [java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16
        for a  [ . A! alphabet]do
            T[a] := [];
            forq in[1. A!.states] 
                q 
                    T[a[[] = A!.ransitions[a][q]];
                fi;
            od;
        od;
        wacc:= Filtered(A!accepting, q->R[q]);
        return(Automaton(                   Union(eclos{aut!.initial}),java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47
                      AlphabetOfAutomatonAsList) T,[apA!i1],mapn);  
    else        return(A);
java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
end)#F  ReducedNFA(aut)##  

#############################################################################
##
#F  MinimalizedAut(A)
##
##  Given an automaton A = (Q, sigma, delta, q0, F), this function computes the
##  minimal DFA B = (Q', sigma, delta', q0', F') such that L(B) = L(A).
##  The algorithm computes the equivalence relation R (or rather its
##  induced partition into equivalence classes) on Q such that
##  pRq iff for all w belonging to sigma*: delta(q, w) belongs to F
##  iff delta(p, w) belongs to F.
##
##  Q'  = Q/R
##  q0' = [q0]
##  F'  = {[f]| f belongs to F}
##  delta'([q], a) = [delta(q, a)] for all q belonging to Q and
##
##
InstallGlobalFunction(MinimalizedAut, function(aut)
local  ,  m  a1,a2,    partmap x  ,t  iti 
            ,  qstarts,qcount, ,  ,  ,,  p, x1,x2,  tm,  
            mma;

    fi
 Error("  toMinimalizedAut  be an ";
    fi;
    aut := UsefulAutomaton(aut);
    Info(InfoAutomataSL 3, "A: ",aut,"\")
    n := aut!.states;
    if  n = 1 then
         ;
    fi;
    m := aut!.alphabet;
    a1: aut.ccepting);
    a2 : Difference.n,a1;
    if Length(a1) = 0 or Length(a2) = 0 then
        partn :=         java.lang.StringIndexOutOfBoundsException: Range [14, 13) out of bounds for length 26
        partmap =List1.n]O)
    else

        ) a1)then
             :a1java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
            a1 := a2;
            a2 := x;
        fi;
        partn := [a1,a2];
        partmap := [];
         in a1 do
            partmap[x] := 1;
        java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7

        for x in a2 do
            partmap[x] := 2;
        od;

        itrans := [];
        for t in aut!.transitions do
            it := [];
            for i  1..] do
                x := t[i];
                if IsBound(Add(, it);
                    Add(it[t[i]],i);
                
                    itjava.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 18
                fi;
            od;
            
                 iti)then
                    Sort(it[i]);
                fi;
            od;
            Add(itrans fi
        

        qlinks: count: count+;
        qstarts := java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
        qcount := m;

        while qcount > 0 do
            Info(break
            fora [1m]od;
                if qstarts[a] <> 0 then
                    break;
                ra : fi
            od;
            i := java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 0
            qstarts[a] := qlinks[a] : [;
            qcount  : qcount -java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32
            Unbindqlinks[]]);
            c : partn[];
            ci := [];
             : itrans[a];
            for x in , the equivalentDFA usingthe construction#  according to the algorithm presented in the report of the AMoRE program.
ifjava.lang.StringIndexOutOfBoundsException: Range [29, 26) out of bounds for length 38
                    java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
fi
            .  "java.lang.StringIndexOutOfBoundsException: Range [21, 15) out of bounds for length 40
            (ci)java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
            if Size(ci) = 0 or
               Size(ci) = nfi;
                continue
            fi;
            (InfoAutomataSL,":"ci,"njava.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 49
            for Set(artmapci}}djava.lang.StringIndexOutOfBoundsException: Range [40, 41) out of bounds for length 40
                p := states := [initstate];
                    sstates: end);
                    x1 = java.lang.StringIndexOutOfBoundsException: Range [39, 38) out of bounds for length 45
                     Length ##  Given an automaton A = (Q, sigma, delta, q0, F), this function computes the
=   equivalence Qjava.lang.StringIndexOutOfBoundsException: Range [62, 57) out of bounds for length 62
                         injava.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 13
                                        notIsSet(r[i]) java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 2
                             x2
                            x2 :=od;
                        
java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 39
            (artn,x1);
                         inx1 do
                            [ :=(partn)
                        od;
                        java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 42
                            qlinks[a][LengthInfo(, 3,reported : Length(states);
                            : Length(partnjava.lang.StringIndexOutOfBoundsException: Index 56 out of bounds for length 56
                        
                        qcount := qcount +                MakeImmutable(st;
                    ifhe =  then
                fi;
                            he:=Length((states)java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
        od;
    fi;
    tm := List([1..m], a->
               [aut!.java.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 36

    mma :=  ##
                    [partmap[aut!.initial[1]]],
                    Filtered([1..Length(partn)], i->partn[i][1] in aut!.accepting));
    Info(InfoAutomataSL,1,"Minimized ",aut!.states," to ",Length(partn));
    #Assert(2,AreEquivAut(aut, mma));
    #    Assert(2,mma!.states = MinimalAutomaton(aut)!.states);
    return NullCompletionAut(mma);
end);

########################################################################
##
#F  MinimalAutomaton(A)
##
##  Minimalizes the automaton A
##
InstallMethod(MinimalAutomaton,"for finite automata", true,
        [IsAutomatonObj and IsAutomatonRep], 0,
        function(A)
    return MinimalizedAut(A);
end);
#############################################################################
#F  AreEquivAut(A1,A2)
##
##  Tests if the automata <A1> and <A2> are equivalent. This means that the
##  corresponding minimal automata are isomorphic.
##
InstallGlobalFunction(AreEquivAut, function(A1, A2)
    local   I1,  I2,  bijection,  dom,  range,  see_this_time,  
            see_next_time,  q,  a,  p2,  p1;

    if IsAutomaton(A1) then
        A1 := MinimalAutomaton(A1);
    elif IsRationalExpression(A1) then
        A1 := RatExpToAut(A1);
    else
        Error("The arguments must be rational expressions or automata");
    fi;
    if IsAutomaton(A2) then
        A2 := MinimalAutomaton(A2);
    elif IsRationalExpression(A2) then
        A2 := RatExpToAut(A2);
    else
        Error("The arguments must be rational expressions or automata");
    fi;
    if not AlphabetOfAutomatonAsList(A1) = AlphabetOfAutomatonAsList(A2) then
        Print("The given languages are not over the same alphabet","\n");
        return(false);
    fi;
    if A1!.states <> A2!.states then
        return(false);
    fi;
    if not Length(A1!.accepting) = Length(A2!.accepting) then
        return(false);
    fi;

    I1 := A1!.initial[1];
    I2 := A2!.initial[1];
    bijection     := [];
    dom           := List([1 .. A1!.states], s -> false);
    range         := List([1 .. A2!.states], s -> false);
    bijection[I1] := I2;
    dom[I1]       := true;
    range[I2]     := true;
    see_this_time := [I1];
    see_next_time := [];
    while IsBound(see_this_time[1]) do
        for q in see_this_time do
            for a in [1 .. A1!.alphabet] do
                p2 := A2!.transitions[a][bijection[q]];
                p1 := A1!.transitions[a][q];
                if dom[p1] then
                    if bijection[p1] <> p2 then
                        return(false);
                    fi;
                else
                    if range[p2] then
                        return(false);
                    fi;
                    bijection[p1] := p2;
                    dom[p1]  := true;
                    range[p2]:= true;
                    Add(see_next_time, p1);
                fi;
            od;
        od;
        see_this_time := List(see_next_time, s -> s);
        see_next_time := [];
    od;
    return(Set(List(A1!.accepting, x -> bijection[x])) = Set(A2!.accepting));
end);

#############################################################################
##
#F  AccessibleStates(aut[,p])
##
##  Computes the list of states of  the automaton aut 
##  which are accessible from state p. When p is not given, returns the 
##  states  which are accessible from any initial state.
##
InstallGlobalFunction(AccessibleStates, function(arg)
    local   aut,  newacc,  acc,  N,  a,  q;

    aut := arg[1];
    if not IsAutomaton(aut) then
        Error(" aut must be an automaton");
    fi;

    if not(aut!.type = "det" or aut!.type = "nondet") then
        Error(" aut must be a deterministic or nondeterministic automaton");
    fi;

    if IsBound(arg[2]) then
        if IsPosInt(arg[2]) then
            newacc:= [arg[2]];
        else
            Error("p must be a positive integer");
        fi;
    else
        newacc:= aut!.initial;
    fi;

    acc := [];           #list of accessible states

    while newacc <> [] do  
        N := [];
        acc := Union(acc, newacc);    
        for a in [1 ..                      Jose Morais <josejoao@fc.up.pt>
            newaccdo
                 : (N (aut![a]]))java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
            
        
        =N(,0)java.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 48
    od fi;

    return acc;
end);


#############################################################################
##
#F  AccessibleAutomaton(aut)
##
##  If "aut" is a deterministic automaton, not necessarily dense, an 
##  equivalent dense deterministic accessible automaton is returned. 
##
##  If "aut" is not deterministic with a single initial state, an equivalent 
##  accessible automaton is returned.
##
(AccessibleAutomatonfunction(java.lang.StringIndexOutOfBoundsException: Index 56 out of bounds for length 56
    local   ,L,      , TR,  nt  java.lang.StringIndexOutOfBoundsException: Range [72, 71) out of bounds for length 79
            n1]=java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28

    if not ( 
        Error("The java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
    fi;
    if java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 30
        return UsefulAutomaton(aut);
    java.lang.StringIndexOutOfBoundsException: Range [0, 8) out of bounds for length 2
        A :
    else
        Error(" aut must be a deterministic or nondeterministic automaton");
    fiError"   java.lang.StringIndexOutOfBoundsException: Range [47, 46) out of bounds for length 70

     =!java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
     ]#java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 51
cc=A!initial

    while newacc <> [java.lang.StringIndexOutOfBoundsException: Range [8, 7) out of bounds for length 8
         =[]java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16
         := nion ;java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
        :=newetjava.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27
            for q in newacc do
                 isused java.lang.StringIndexOutOfBoundsException: Index 75 out of bounds for length 75
                    
                    
                    N=Union(,A.transitions[a][)
                fi;

            
        java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
        ForAny(clos-> inaut!.))
    od;
    java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 2

    ###delete the columns corresponding to inaccessible states
    TR := TransposedMat(A!.a,  splitself,  pi,  k,  qpi,  k,  q,  x1,  x2,  tm,  ra;
    nt := TR{acc};
    java.lang.StringIndexOutOfBoundsException: Range [29, 12) out of bounds for length 43

    ### unbind the entries corresponding to non accessible states
    for   newtablejava.lang.StringIndexOutOfBoundsException: Range [26, 27) out of bounds for length 26
        for qa in qqa do
            if not qa = 0 then
                for q in qa do
                    if not (q in    if aut!.type n"then
                        Unbind(qa[q]);
                        #qa[q] := 0;
                    ;
                od;
            ;
            Set(qalinks[[2]]i[1]java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 49
        od;
    odend
n1 :(;
    :java.lang.StringIndexOutOfBoundsException: Range [17, 16) out of bounds for length 30
    newnewtable := NullMat(n1,n2);
    for r in [1 .. n1] do
        for s in [1 .. n2] do
            #            newnewtable[r][s] := [0];
            if Length(a1) = 0 or Length(a2) = 0 then
        odjava.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
    od;
    for r in [1 .. n1    java.lang.StringIndexOutOfBoundsException: Range [8, 9) out of bounds for length 8
 [ .n2 do
            for i partmap[x] := 2;
                ;
                    [][]i newtable[r][];
                ;

            od;
        od;
    od;

    =IntersectionA!initial,acc);
    init := [];
    for icount> do
        java.lang.StringIndexOutOfBoundsException: Range [8, 1) out of bounds for length 26
    od;
    q :=  for Set({pi})do
    f := [];
          if()> then
        Add(f, Position(acc i);
    ;

     java.lang.StringIndexOutOfBoundsException: Range [21, 20) out of bounds for length 59
                   init, f);
endx1 = ;

#############################################################################
##
#F  ProductAutomaton(A1,A2)
##
##  Note: (p,q)->(p-1)m+q is a bijection from n*m to mn.
##  A1 and A2 are deterministic autamata
##
partmap=nparts
od;
            qi;

                         =0 java.lang.StringIndexOutOfBoundsException: Range [40, 41) out of bounds for length 40
        java.lang.StringIndexOutOfBoundsException: Range [28, 12) out of bounds for length 56
    fi;
    if not IsAutomatonObj(A2) then
        Error("The second argument must be a deterministic automatonlinks[]:= head,]
    fi;
    if not A1!.type = "det" then
        Error("The first argument must befi;
    fi;
    if not A2!.type = "det" then
        argument  a deterministicjava.lang.StringIndexOutOfBoundsException: Range [69, 68) out of bounds for length 71
    fi;
    a := A1!.alphabet;
   not java.lang.StringIndexOutOfBoundsException: Range [37, 36) out of bounds for length 77
        Errorjava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
    ;
     :=  (nondet" Length(partn), alph,,
    n2 := A2!.states;
    n := n1 * n2;
    T1             ([1.(partn)] i-i][1 inaut!accepting);
    nsjava.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
 =a)java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
    fin := [];
    init := ##
    for s in [1##  according to the algorithm presented in the report of the AMoRE program.
        java.lang.StringIndexOutOfBoundsException: Range [19, 17) out of bounds for length 33
            p := QuoInt(s,n2)java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
            q: RemIntn2)java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30
        elifreturn(A);
            p := QuoInt(s,n2);
             = ;
        fi;## s corresponds to (p,q) via the bijection above
               i [1.]do
            if IsBound(T1[i][p]) and T1[i]java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
and[]> 0then
                = [ii]]
                = T2[[q;
                T[i][s] := (pi -  ccepting: [1]
            fi;
        trans=List[..i->[])
    od;
    if A1!  in 1..(do
        for p inri =Set([i]);
            for q in A2!.accepting        ;
                Add((statesdo
            od;
                 (tates)reported>100000  then
    fi;
    if =states)java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39
        for p in A1!ans[a];
            for              =Union(r{st});
                Add( he : snst);
            od;
        od;
    fi;

    return Automaton("det",n,(1)Tinitfin;
end)AddHashEntrysstatesnst,;


#############################################################################
##
#F  IntersectionLanguage(A1,A2)
##
##  The same as IntersectionAutomaton; accepts both automata or rational 
##  expressions as arguments
##
InstallGlobalFunction(IntersectionLanguage, java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 29
    local  HashPair       ,  ,q,q1map, ,,  java.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 57

    IsAutomaton( then
 
    ( a1)  then
(java.lang.StringIndexOutOfBoundsException: Range [28, 26) out of bounds for length 29
    else
        ;
    fi;
     then
     ;
    elif IsRationalExpression( a2 )  then
     := RatExpToAut );
     :BlistList(.!.tates],.)java.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 46
   Error("The must   automaton or a  expression ;
    fi;

    if a1!.            ifR[]then
     iffifostates then
               : 1java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
     m1 := AlphabetOfAutomaton                 :i;
    fi;
    if a2!.type =T =[]java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16
     m2 := for q in [1 A.java.lang.StringIndexOutOfBoundsException: Range [37, 36) out of bounds for length 40
    elsejava.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
     m2 := AlphabetOfAutomaton(a2);
    fi;
     m1>m2then
     Error( "The arguments must be two automata over the java.lang.StringIndexOutOfBoundsException: Range [0, 61) out of bounds for length 45
java.lang.StringIndexOutOfBoundsException: Index 80 out of bounds for length 7

    if a1!end;
     #############################################################################
    fi;
    if a2!.type <> "epsilon" then
     a2 := Automaton("epsilon##  induced partition into equivalence classes) on Q such that
    fi;

    HashPair##  F'  = {[f]| f belongs to F}
     return java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 2
    end;
    ht := SparseHashTable( HashPair );
    init := [ ];
    s1  .java.lang.StringIndexOutOfBoundsException: Range [26, 25) out of bounds for length 28
     for s2 in java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
   Add(  ,s2]java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
      (,[s1,s2 , ength(   );
     od;
    od;

    numofinitst:=Length(init);

    states := initm= ut.java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23
    m := a1!.alphabet;
         partmap[.],One;
    t1 := a1!.java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 8
    =.java.lang.StringIndexOutOfBoundsException: Range [26, 25) out of bounds for length 26
    t := List( [ 1 .. m;

    java.lang.StringIndexOutOfBoundsException: Range [8, 6) out of bounds for length 22
     st := java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 11
     for a in [ 1 ..             partmap[x];
            t[a][i] :=
      if a = m then
     in[a]st1]do
     :  ,st2]]
                      x] then
        he                
        if he = fail  then
         Add(             for i injava.lang.StringIndexOutOfBoundsException: Range [16, 15) out of bounds for length 30
   he );
         AddHashEntry( ht, nst, he );
        fi;
        Add(java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
      java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10
       x []st[] do
        nst := [ st[1], x ];
        MakeImmutablefor in[.m java.lang.StringIndexOutOfBoundsException: Range [30, 31) out of bounds for length 30
      =( htjava.lang.StringIndexOutOfBoundsException: Range [36, 35) out of bounds for length 38
     if  = fail  then
            i :qstarts[a];
         he := Length( states );
         AddHashEntry(:=-java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32
        fi;
        Add(t[a][i], he);
       od;
      else
             for x in t1[a][st[1]] if ([x)
                 ;
                     nst := [ java.lang.StringIndexOutOfBoundsException: Range [15, 14) out of bounds for length 30
                     MakeImmutable( nst );
         he := GetHashEntry( ht, nst );
                        if he = fail  then
                         Add( states, nst );
                         he := Length( states );
                         AddHashEntry( ht, nst, he );
                        fi;
                     Add(t[a][i], he);
                 od;
                od;
      fi;
     od;
     i := i + 1;
    od;
    finals := [  ];
    for p in a1!.accepting  do
     for q in a2!.accepting  do
            he := GetHashEntry( ht, [ p, q ] );
            if he <> fail then
                AddSet( finals, he );
            fi;
     od;
    od;

    ];
    for s1 in a1!.initial do
                (InfoAutomataSL,,C ,,"\)java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 49
      he= GetHashEntry( ht  s1  ];
      if he <> fail then
     Add(, )java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23
      fi;
 java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8
    od;

    return x2 =xjava.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36

end)



#############################################################################
##
#F  FuseSymbolsAut(aut, n1, n2)
##
##  
##
InstallGlobalFunction(FuseSymbolsAut, function(aut, n1, n2)
    local   tm,  ntm,  i,  row,  a,  odjava.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15

    if not IsAutomatonObj(aut) then
        Error("The first argument to FuseSymbolsAut must be an automaton"
    fi;
    =.java.lang.StringIndexOutOfBoundsException: Range [27, 26) out of bounds for length 27
    ntm: [java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
    for i in [1..Length(tm)] do
        if i  java.lang.StringIndexOutOfBoundsException: Range [22, 17) out of bounds for length 22
 java.lang.StringIndexOutOfBoundsException: Range [29, 28) out of bounds for length 34
            Add(#######################################################################
        elif##
            row := List(InstallMethod(java.lang.StringIndexOutOfBoundsException: Range [31, 30) out of bounds for length 59
            (,row
        fi;
    od;
    ##
        for i in [1..aut!.states##  corresponding minimal automata are isomorphic.
            row =Flata[i];
            for j in [1..java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 0
     java.lang.StringIndexOutOfBoundsException: Range [30, 29) out of bounds for length 38
                    Unbind        (Therationalor
                fi;
           ;
            ntm[a][i] := Set(Compacted(row));
        od;
    od;
    alph :=         Error("Themust berational expressions automata";
    (alphn2];
    alph := if not AlphabetOfAutomat) () then
#    Error(",,");
     java.lang.StringIndexOutOfBoundsException: Range [22, 20) out of bounds for length 53
                   aut!.
;


#############################################################################
##
#F AutomatonAllPairsPaths(A)
##
## Given an automaton A, with n states, outputs a n x n matrix P,
## such that P[i][j] is the list of simple paths from state i to
## state j in A.
java.lang.StringIndexOutOfBoundsException: Range [22, 21) out of bounds for length 57
    local   if dom[p1then

    pathClosure := return()
                            ;

        list2 := [];
        i := 1;
        while IsBound(list[i]) bijection[p1: p2
            if i = 1 then
            else
                                ;
            fiodjava.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
             : i +1;
        od;
len : ;
        len2 := i-2;
        list3 := ShallowCopy(#############################################################################
#F  AccessibleStates(aut[,p])
        AddSet(paths[list[1]][list[len]], ##  which are accessible from state p. When p is not given, returns the 
        for iInstallGlobalFunctionAccessibleStates, function(arg)
            for   newacc,,acc  N,a  ;
                if not p = not (t
                    if p[1] in list3 then
                        (autbea deterministicor nondeterministic";
                    fi;
java.lang.StringIndexOutOfBoundsException: Range [25, 24) out of bounds for length 33
                    j := 2;
                    ;
                        if         !initial;
                             : ]           list accessible 
                            break;
                        ;
                        ja [ .autalphabet] do
                    od;
                    flag andnot [1 =p[j-1 then
                        listx := java.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 11
                        if not     odjava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
                            java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 0
                        fi#F  AccessibleAutomaton(aut)

                    fi;
                fi;
            od;
        odjava.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
        : Concatenation([list[1]], list3);
        for i in [1..V] do
            for p in paths[list[len]][i] do
                if not p = [] then
                    lenx := Length(p);
                    flag := true;
                    for j in [1.. (aut);
                        if p[j] in          : java.lang.StringIndexOutOfBoundsException: Range [17, 16) out of bounds for length 17
                            flagL: !alphabetjava.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
                            break;
                        fi;
                    od;
                    if flag and              q inin newacc do
 lenx list3 then
                            continue;
                        fi;
                        listx := Concatenation(list2, p);
                        ;
java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
                        
                    fi =Filtered,  ->IsPosIntn 
                fi;
    ;
        
    end;
    re  -


    BFS := function(v)
        local   w;

        if not visited[v] then
            visited[v] := true;
            for Unbiqaq]);
                pathClosure([v,w]);
            od;
 G[] 
                BFS(od;
            od;
        fi;
    end;
    ##  End of BFS()  --
    #==============================================n2 = (newtable[1];


    if not IsAutomaton(A) then
        Error("The argument must be an automaton");
    newnewtabler[ =java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36

    G := UnderlyingGraphOfAutomaton  1 .n2do
    V := Length(G);
    paths := [];
    visited := [];
    for i in [1..V] do
 ]java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23
        visited[]:= false;
        for j in [1..V] do
            java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
         od
    od;
    BFS(A!.initial[1] = [;

    return(paths);
end);


        Add(f, Position(acc, i));
##