Two Lower Bounds on Computational Complexity of Infinite Words

The most of the previous work on the complexity of in nite words has measured the complexity as descriptional one, i. e. an in nite word w had a \small" complexity if it was generated by a morphism or another simple machinery, and w has been considered to be \complex" if one needs to use more complex devices (gsm's) to generate it. In [J. Hromkovi c, J. Karhum aki, A. Lepist o: \Comparing descriptive and computational complexity of in nite words" In: Results and Trends in Theoretical Computer Science , Lecture Notes in Theoretical Computer Science 812, SpringerVerlag 1994, pp. 169-182] the study of the computational complexity of in nite word generation and of its relation to the descriptional characterizations mentioned above was started. The complexity classes GSPACE(f) = f in nite words generated in space f(n)g are de ned there, and some fundamental mechanisms for in nite word generation are related to them. It is also proved there, that there is no hierarchy between GSPACE(O(1)) and GSPACE(log2 n). Here, GSPACE(f) GSPACE(g) for g(n) f(n) log2 n; f(n) = o(g(n)) is proved. The main result of this paper is a new lower bound on the computational complexity of in nite word generation: real-time, binary working alphabet, and o(n=(log n)2) space is insu cient to generate a concrete in nite word over two-letter alphabet.