Context-Free Languages of Sub-exponential Growth

There do not exist context–free languages of intermediate growth. The function γ whose value at each non–negative integer n is the number of words on length n in a fixed formal language L is called the growth function of L. Flajolet [3] asked if there are context–free languages of intermediate growth; that is, such that γ is not bounded above by a polynomial, but lim sup γ(n)/r = 0 for all r > 1. The answer to this question is a corollary to the following theorem. Theorem 1. If L is a context–free language with growth function γ, then either there is a number r > 1 and integer n0 such that γ(n) ≥ r n for all n ≥ n0, or else L is a bounded language. A bounded language is one which is a subset of w 1 · · ·w n for some words {w1, . . . , wn}. Since it is clear that the growth of a bounded language is bounded above by a polynomial, we have the desired corollary. Corollary 2. There do not exist context–free languages of intermediate growth. We note that by a recent result of Grigorchuk and Mach̀i there are indexed languages of intermediate growth [4]. Corollary 2 was obtained independently by Roberto Incitti [6]. Theorem 1 occurs in our previous work [1] as a remark that the proof given there of the weaker result [1, Proposition 1.3] suffices for Theorem 1. In this note we give a quicker proof of Theorem 1 based on work of Ginsburg and Spanier [5], who also obtain a corresponding decidability result. Theorem 3 ([5, Theorem 5.2]). It is decidable whether or not the language L generated by a given context–free grammar is bounded; and if L is bounded, one can effectively find words {w1, . . . , wn} such that L ⊂ w 1 · · ·w n. 1991 Mathematics Subject Classification. 68Q45.