A note on synchronized extension systems

Abstract 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,L 1 ,L 2 ,S) , where V is an alphabet and L 1 , L 2 and S are languages over V . Such systems generate languages extending L 1 by L 2 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 {z 0 }L r − (G) , for some symbol z 0 . 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.