On Inverse Operations and Their Descriptional Complexity  Maria Paola Bianchi 1 , Markus Holzer 2 , Sebastian Jakobi 2 , and Giovanni Pighizzini 1 1 Dipartimento di Informatica, Universit`a degli Studi di Milano, Via Comelico 39, 20135 Milano, Italy {maria.bianchi,giovanni.pighizzini}@unimi.it 2 Institut f¨ ur Informatik, Universit¨ at Giessen, Arndtstr. 2, 35392 Giessen, Germany {holzer,jakobi}@informatik.uni-giessen.de Abstract. We investigate the descriptional complexity of some inverse language operations applied to languages accepted by finite automata. For instance, the inverse Kleene star operation for a language L asks for the smallest language S such that S ∗ is equal to L, if it exists [J. Brzo- zowski. Roots of star events. J. ACM 14, 1967]. Other inverse operations based on the chop operation or on insertion/deletion operations can be defined appropriately. We present a general framework, that allows us to give an easy characterization of inverse operations, whenever simple conditions on the originally considered language operation are fulfilled. It turns out, that in most cases we obtain exponential upper and lower bounds that are asymptotically close, for the investigated inverse lan- guage operation problems. 1 Introduction The study of the descriptional complexity of language operations is a vivid area of research. After its decline in the mid 1970’s, a renewal initiated by the late Sheng Yu in his influential paper [17] brought descriptional complexity issues, not only for finite automata, back to life. Since then many aspects of descriptional complexity of deterministic and nondeterministic finite automata, pushdown au- tomata, and other language-accepting or -generating devices were studied. For a recent survey on descriptional complexity issues on finite automata we refer to [10] and [18]. In truth there is much more to regular languages, deterministic finite automata, nondeterministic finite automata, etc., than one can summarize in these surveys. The operation problem on languages is well studied in the literature, and is defined as follows: let ◦ be a fixed binary operation on languages that preserves regularity; then given an n-state finite automaton A and an m-state finite au- tomaton B, how many states are sufficient and necessary in the worst case (in  This paper is partially supported by CRUI/DAAD under the project “Programma Vigoni: Descriptional Complexity of Non-Classical Computational Models.” M. Kutrib, N. Moreira, and R. Reis (Eds.): DCFS 2012, LNCS 7386, pp. 89–102, 2012. c  Springer-Verlag Berlin Heidelberg 2012