A Characterization of Context-free Languages

The developments in the theory of languages and automata in recent years have been towards defining and studying new grammars, automata and languages, abstract families of automata, languages, translations, and so on. In spite of this fact, the central position of regular sets and context-free languages (CFL's) in the theory of languages remains, and so one can still expect their further study to be useful and maybe stimulating for the general development of language theory. One of the important advantages in dealing with regular languages is the fact that each such language can be denoted explicitly by a regular expression, in contrast to the implicit representation of languages by grammars. A similar characterization of CFL's would perhaps be useful also. However, one can hardly expect to get a characterization of CFL's so elegant and so simple as the one we have developed for regular languages. In Section 2 an attempt is made to give a regular-like characterization of CFL's, using union, product, and a new operation-symbol iteration. In so doing, we are compelled to use some "auxiliary" symbols. Their role and number are discussed in Section 3. The application of the results of Section 2 to semilinear languages and their subclasses is presented in Section 4.