A pumping lemma for regular closure of prefix-free languages

Abstract

Let Σ be an infinite set of distinct symbols. Let PFL⊆Power(Σ⁎) be the set of all prefix-free languages. For N⊆Power(Σ⁎), let ΓREG(N) be the regular closure of N, i.e., let ΓREG(N) be the smallest set such that N⊆ΓREG(N) and L1∪L2, L1L2, L1⁎∈ΓREG(N) for any L1,L2∈ΓREG(N). In this paper, we provide a pumping lemma for ΓREG(PFL). This lemma enables us to prove that some languages do not belong to ΓREG(PFL). It is notable that we immediately obtain a pumping lemma for ΓREG(DCFL) as a corollary, due to the known fact that ΓREG(DCFL)=ΓREG(DCFL∩PFL), where we define DCFL as DCFL:=∪Σ0DCFL(Σ0) with the union taken over all non-empty finite set Σ0⊆Σ, and DCFL(Σ0) is the family of all L⊆Σ0⁎ such that L is accepted by a deterministic pushdown automaton over Σ0 by final state.