Complexity of operation problems

Abstract

The operation problem for several classes of automata and other language descriptors is addressed: Fix an operation on formal languages. Given a class of automata (or other language descriptors), is the application of this operation to the given class still a language represented by a device of that class? In particular, several aspects of complexity in connection with these problems are considered. Is the problem decidable or not? What is the computational complexity of the decision procedure, or what is its precise level in the arithmetic hierarchy? What is the blow-up of the size of the resulting device, if it exists, in terms of the sizes of the given ones? Otherwise, is there a so-called non-recursive trade-off between the representation by devices combined with the operation and the representation by just one device? We present some selected results on the computational and descriptional complexity of operation problems and draw attention to the overall picture and some of the main ideas involved. © 2014 Springer International Publishing.