Some results in tree automata
The following three results concerning tree automata are presented in this paper. (1) Rounds has presented the following open problem: For every recognizable setR, can we construct a deterministic finite-state transformation recognizingR? We show that this is not possible, in fact, even for a local set. However, the following is true: For every recognizable setR there is an inverse projectionR′ effectively obtained such thatR′ is recognized by a deterministic finite-state transformation. (2) Martin and Vere in their study of tree automata leave open the question of whether Generalized Syntax Directed Transductions (GSDT's) are closed under Arden's transformation or Greibach's transformation, and conjecture that they are not. We prove that this conjecture is true. It is also shown that GSDT's are not closed under transformation to LR(k) grammars. (3) Peters and Ritchie have shown that if, in a grammar where the generative rules are context-free, there are “recognition” rules which are context-sensitive, the language recognized is still context-free. A tree-automata-oriented proof is given by Rounds. We show that a similar result holds also for right linear grammars, i.e., if the generative rules are right linear, then using context-sensitive rules for “recognition”, one can still recognize only regular languages. Some other related results concerning context-sensitive extensions of subclasses of context-free languages are also presented.
KeywordsComputational Mathematic Open Problem Generative Rule Related Result Regular Language
Unable to display preview. Download preview PDF.
- W. C. Rounds, Mappings and grammars on trees,Math. Systems Theory 4 (1970), 257–287.Google Scholar
- J. W. Thatcher, Characterizing derivation trees of context-free grammars through a generalization of finite automata theory,J. Comput. Systems Sci. 1 (1967), 317–322.Google Scholar
- D. F. Martin andS. A. Vere, On syntax-directed transduction and tree transducers,Second Annual Symposium on Theory of Computing, 1970, pp. 129–135.Google Scholar
- P. S. Peters andR. W. Ritchie, Context-sensitive immediate constituent analysis: Context-free languages revisited,Math. Systems Theory 6 (1972), 324–333.Google Scholar
- W. C. Rounds, Tree-oriented proofs of some theorems on context-free and indexed languages, Second Annual ACM Symposium on Theory of Computing, 1970, pp. 109–116.Google Scholar
- J. W. Thatcher andJ. B. Wright, Generalized finite automata theory with an application to a decision problem of second-order logic,Math. Systems Theory 2 (1968), 57–81.Google Scholar
- A. V. Aho andJ. D. Ullman, Translations on a context-free grammar, 1969 ACM Symposium on Theory of Computing, pp. 93–112.Google Scholar
- J. W. Thatcher, There's a lot more to finite automata theory than you would have thought, Proceedings of the Fourth Annual Princeton Conference on Information Sciences and Systems, 1970, pp. 263–276.Google Scholar
- L. S. Levy, Generalized local adjunction and replacement in adjunct languages, Moore School Report No. 70-29, 1970.Google Scholar
- J. E. Hopcroft andJ. D. Ullman,Formal Languages and Their Relations to Automata, Addison Wesley, Reading, Mass., 1969.Google Scholar