Block insertion and deletion on trajectories

In this paper, we introduce block insertion and deletion on trajectories, which provide us with a new framework to study properties of language operations. With the parallel syntactical constraint provided by trajectories, these operations properly generalize several sequential as well as parallel binary language operations such as catenation, sequential insertion, k-insertion, parallel insertion, quotient, sequential deletion, k-deletion, etc.We establish some relationships between the new operations and shuffle and deletion on trajectories, and obtain several closure properties of the families of regular and context-free languages under the new operations. Moreover, we obtain several decidability results of three types of language equation problems which involve the new operations. The first one is to answer, given languages L1,L2,L3 and a trajectory set T, whether the result of an operation between L1 and L2 on the trajectory set T is equal to L3. The second one is to answer, for three given languages L1,L2,L3, whether there exists a set of trajectories such that the block insertion or deletion between L1 and L2 on this trajectory set is equal to L3. The third problem is similar to the second one, but the language L1 is unknown while languages L2,L3 as well as a trajectory set T are given.