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 ) = { x y z | x z ∈ L , y ∈ K } , deletion del ( L , K ) = { x z | x y z ∈ L , y ∈ K } , square root L = { w | w w ∈ L } , the first half 1 2 L = { u | ∃ v : | u | = | v | , u v ∈ L } and cyclic shift . For K and L recognized by nondeterministic IDPDA, with m and with n states, respectively, insertion requires exactly m n + 2 m states, as long as K is well-nested; deletion requires exactly 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; the well-nested subset of the cyclic shift requires exactly 2 n 2 states. Without the well-nestedness constraints, non-closure is established in each case.