Tree Automata and Tree Grammars

Preface I wrote these lecture notes during my stay in Aarhus in the academic year 1974/75. As a young researcher I had a wonderful time at DAIMI, and I have always been happy to have had that early experience. I wish to thank Heiko Vogler for his noble plan to move these notes into the digital world, and I am grateful to Florian Starke and Markus Napierkowski (and Heiko) for the excellent transformation of my handwritten manuscript into L A T E X. Apart from the reparation of errors and some cosmetical changes, the text of the lecture notes has not been changed. Of course, many things have happened in tree language theory since 1975. In particular, most of the problems mentioned in these notes have been solved. The developments until 1984 are described in the book " Tree Automata " by Ferenc Gécseg and Magnus Steinby, and for recent developments I recommend the Appendix of the reissue of that book at arXiv.org/abs/1509.06233. Tree automata and tree grammars To appreciate the theory of tree automata and tree grammars one should already be motivated by the goals and results of formal language theory. In particular one should be interested in " derivation trees ". A derivation tree models the grammatical structure of a sentence in a (context-free) language. By considering only the bottom of the tree the sentence may be recovered from the tree. The first idea in tree language theory is to generalize the notion of a finite automaton working on strings to that of a finite automaton operating on trees. It turns out that a large part of the theory of regular languages can rather easily be generalized to a theory of regular tree languages. Moreover, since a regular tree language is (almost) the same as the set of derivation trees of some context-free language, one obtains results about context-free languages by " taking the bottom " of results about regular tree languages. The second idea in tree language theory is to generalize the notion of a generalized sequential machine (that is, finite automaton with output) to that of a finite state tree transducer. Tree transducers are more complicated than string transducers since they are equipped with the basic capabilities of copying, deleting and reordering (of subtrees). The part of (tree) language theory that is concerned with translation of languages is mainly motivated by compiler writing (and, to a …