Further Closure Properties of Input-Driven Pushdown Automata

Abstract

The paper investigates the closure of the language family defined by input-driven pushdown automata (IDPDA) under the following operations: insertion $ins(L, K)=\{xyz \mid xz \in L, \, y \in K\}$, deletion $del(L, K)=\{xz \mid xyz \in L, \, y \in K\}$, square root $\sqrt{L}=\{w \mid ww \in L\}$, and the first half $\frac{1}{2}L=\{u \mid \exists v: |u|=|v|, \, uv \in L\}$. For K and L recognized by nondeterministic IDPDA, with m and with n states, respectively, insertion requires $mn+2m$ states, as long as K is well-nested; deletion is representable with 2n states, for well-nested K; square root requires $n^3-O(n^2)$ states, for well-nested L; the well-nested subset of the first half is representable with $2^{O(n^2)}$ states. Without the well-nestedness constraints, non-closure is established in each case.

A. Okhotin—Supported by Russian Science Foundation, project 18-11-00100.