Separating the Classes of Recursively Enumerable Languages Based on Machine Size

Jan van Leeuwen, Jiří Wiedermann
2015 International Journal of Foundations of Computer Science  
Many years ago it was observed that the r.e. languages form an infinite proper hierarchy RE1 ⊂ RE2 ⊂ · · · based on the size of the Turing machines that accept them. Aside from some basic facts, little seems known about it in general. We examine the position of the finite languages and their complements in the hierarchy. We show that for every finite language L one has L,L ∈ REn for some n ≤ p·(m− log 2 p +1)+1 where m is the length of the longest word in L, c is the cardinality of L, and p =
more » ... n(c, 2 m−1 ). If L ∈ REn, thenL ∈ REs for some s = O(n + m). We also prove that for every n, there is a finite language Ln with m = O(n log 2 n) such that Ln ∈ REn but Ln,Ln ∈ REs for some s = O(n log 2 n). Extending this, we show that there exist families {Fn} n≥1 of finite languages such that F1 ⊂ F2 ⊂ · · · where for every n, Fn ∈ REn but Fn ∈ REs for an s with s = O(n log 2 n). The proofs make use of several auxiliary results for Turing machines with advice over a fixed alphabet.
doi:10.1142/s0129054115500380 fatcat:a3egpgjiwjbn7jvnqhgawogno4