Acta Informatica 37, 449–465 (2001) c  Springer-Verlag 2001 Synchronized extension systems Ferucio Laurent¸iu T ¸ iplea 1,⋆ , Erkki M ¨ akinen 2,⋆⋆ , Corina Apachite 1 1 Faculty of Computer Science, “Al. I. Cuza” University of Ias ¸i 6600 Ias ¸i, Romania (e-mail: fltiplea@infoiasi.ro) 2 Department of Computer and Information Sciences, P.O. Box 607, FIN-33014 University of Tampere, Finland (e-mail: em@cs.uta.fi) Received: 1 Feburary 2000 / 16 October 2000 Abstract. Synchronized extension systems (SE-systems, for short) are 4- tuples G =(V,L 1 ,L 2 ,S ), where V is an alphabet and L 1 , L 2 and S are languages over V . They generate languages extending L 1 by L 2 to the left or to the right, and synchronizing on words in S . Such systems appear naturally when considering stacks, queues, grammar-like generative devices, splicing systems, zigzag-codes etc. 1 Introduction There are many cases where we need to extend some words to the right (or to the left), by taking into consideration some of the last letters of the word. For example, the words in a pushdown stack are generated by such a rule: only the last letter is used for synchronization (considering the top of the stack word as its right end). We generalize this idea by introducing the concept of an SE-system and provide several examples to show that our SE-systems are poweful enough to handle various generative devices. Then we study some of their basic properties and generalize a theorem in [2] regarding the regularity of the stack language of a pushdown automaton. Encoding by a code and encoding by a zigzag-code means generation of a word by a (proper) synchronized extension to the right. This topic is discussed in Sections 4 and 5. Finally, ⋆ The work of the first author was carried out while visiting Universit¨ at Augsburg by a grant from Deutschen Akademien der Wissenschaften (Germany). ⋆⋆ The work of the second author was supported by the Academy of Finland (Project 35025).