/** 
     * Finds regular expression in table or computes it if not there yet. (Used by {@link #toRegExp}.) 
     */
    private RegExp.Node tableLookup(State p, State q, int k, Map<StatePair,RegExp.Node[]> table, Map<Integer,State> statemap) {
        RegExp.Node x = table.get(new StatePair(p, q))[k];
        if (x==null) {
            if (k==0) {
                x = new RegExp.EmptyLanguageNode();
                for (char c : alphabet.symbols)
                    if (delta(p,c)==q)
                        x = new RegExp.UnionNode(x, new RegExp.SymbolNode(c));
                if (p==q)
                    x = new RegExp.UnionNode(x, new RegExp.EmptyStringNode());
            } else {
                State r = statemap.get(k);
                RegExp.Node x1 = tableLookup(p, q, k-1, table, statemap);
                RegExp.Node x2 = tableLookup(p, r, k-1, table, statemap);
                RegExp.Node x3 = tableLookup(r, r, k-1, table, statemap);
                RegExp.Node x4 = tableLookup(r, q, k-1, table, statemap);
                x = new RegExp.UnionNode(x1, new RegExp.ConcatNode(x2, new RegExp.ConcatNode(new RegExp.StarNode(x3), x4)));
                
            }
            table.get(new StatePair(p, q))[k] = x;
        }
        return x;
    }

    /** 
     * Converts this automaton into an equivalent {@link RegExp} regular expression. [Martin, Th. 3.30] 
     */
    public RegExp toRegExp() {
        Map<Integer,State> statemap = new HashMap<Integer,State>(); 
        Map<StatePair,RegExp.Node[]> table = new HashMap<StatePair,RegExp.Node[]>(); // fill out this table lazily
        // initialize
        int n = 0;
        for (State p : states) {
            statemap.put(++n, p); // states are numbered from 1 to states.size()
            for (State q : states)
                table.put(new StatePair(p, q), new RegExp.Node[states.size()+1]);
        }
        // construct the abstract syntax tree for the regular expression
        RegExp.Node x = new RegExp.EmptyLanguageNode();
        for (State p : accept) 
            x = new RegExp.UnionNode(x, tableLookup(initial, p, states.size(), table, statemap));
        return new RegExp(x, alphabet);
    }