A note on synchronized extension systems

The concept of a synchronized extension system (SE-system, for short) has been introduced in by Ţiplea et al. [Acta Inform., to appear] as a 4-tuple G = (V ,L1,L2, S), where V is an alphabet and L1, L2 and S are languages over V . Such systems generate languages extending L1 by L2 to the left or to the right, and synchronizing on words in S. Ţiplea et al. showed that the language of type r− generated by an SE-system of type (reg, reg, f ) is regular. As a particular case, the stack language of a pushdown automaton is regular. In this note we show that for any SE-system of type (reg, reg, f ), it is possible to construct a pushdown automaton A whose stack language is {z0}Lr(G), for some symbol z0. Therefore, assuming that the stack language of a pushdown automaton is regular, we obtain that the language of type r− generated by an SE-system of type (reg, reg, f ) is regular. From this point of view, our result acts as a complete relationship between languages of type r− generated by SE-systems of type (reg, reg, f ) and stack languages of pushdown automata.  2001 Elsevier Science B.V. All rights reserved.