On the extension of Gladkij's Theorem and the Hierarchies of languages

• Authors: Yoshihide Igarashi;Namio Honda
• Venue: Journal of Computer and System Sciences
• Year: 1973

Let T be a computable, monotonic increasing function from non-negative integers to positive integers. Then it is said that @a@?@S^* is in a class L"T if there exists a phrase structure grammar G, which generates all words of length n in @a within length T(n) of derivations for each n. A main result of this paper is an extension of the A. V. Gladkij's Nonlinear Theorem on Context-Sensitive Grammars. Our extended theorem is as follows: Let be a function defined on the set of all strings on an alphabet @S = {a"1, a"2}, taking as values non-void subsets of @S^*. Let be a language , where b is a symbol not in @S. Let be a function from @S^* to positive integres defined by , and let be a function from non-negative integers to positive integers defined by , . If T is a time function such that , then L"T does not contain . From this result, an open problem proposed by R. V. Book are solved. Moreover from this result, it is shown that there exist infinitely long chains of distinct complexity classes betwwen certain two distinct complexity classes.