Infinitely-Often Universal Languages and Diagonalization

Diagonalization is a powerful technique in recursion theory and in computational complexity [2]. The limits of this technique are not clear. On the one hand, many people argue that conflicting relativizations mean a complexity question cannot be resolved using only diagonalization. On the other hand, it is not clear that diagonalization arguments necessarily relativize. In [5], the authors proposed a definition of “separation by strong diagonalization” in which to separate class from a proof is required that contains a universal language for . However, in this paper we show that such an argument does not capture every separation that could be considered to be by diagonalization. Therefore, we consider various weakenings of the notion of universal language and corresponding formalizations of separation by diagonalization. We introduce four notions of infinitely-often universal language. For each notion, we give answers or partial answers to the following questions: 1. Under what conditions does the existence of a variant of a universal language for in show ? More precisely, what closure properties are needed on and ? 2. Can any separation be reformulated as this kind of diagonalization argument? More precisely, are there complexity classes with nice closure properties, so that has no such variant of a universal language for ? 3. Are these variants of universal language different from the other notions we have defined? The main examples of a separation by diagonalization are the time and space hierarchy theorems. We explore the following question: is any separation of a from where is closed under polynomial-time Turing reducibility essentially a separation by the time hiearchy theorem?