On the State Complexity of Partial Derivative Automata For Regular Expressions with Intersection Rafaela Bastos, Sabine Broda, Ant´ onio Machiavelo, Nelma Moreira (B ) , and Rog´ erio Reis CMUP and Faculdade de Ciˆ encias da Universidade do Porto, Porto, Portugal {rrbastos,sbb,nam,rvr}@dcc.fc.up.pt, ajmachia@fc.up.pt Abstract. Extended regular expressions (with complement and inter- section) are used in many applications due to their succinctness. In par- ticular, regular expressions extended with intersection only (also called semi-extended) can already be exponentially smaller than standard reg- ular expressions or equivalent nondeterministic finite automata (NFA). For practical purposes it is important to study the average behaviour of conversions between these models. In this paper, we focus on the conversion of regular expressions with intersection to nondeterministic finite automata, using partial derivatives and the notion of support. First, we give a tight upper bound of 2 O(n) for the worst-case number of states of the resulting partial derivative automaton, where n is the size of the expression. Using the framework of analytic combinatorics, we then establish an upper bound of (1.056 + o(1)) n for its asymptotic average-state complexity, which is significantly smaller than the one for the worst case. 1 Introduction Regular expressions with additional operators are used in applications such as pro- gramming languages [12], XML processing [23], or runtime verification [22]. Most of these operators do not increase their language expressive power but lead to gains in the succinctness of the representation. This is the case for intersection. For regu- lar expressions with intersection (RE ∩ ) (or semi-extended), several computational complexity decision problems, such as membership, equivalence and emptiness, were studied by various authors. Petersen [21] has shown that the membership problem is LOGCFL-complete, while for standard regular expressions (RE) it is NL-complete [19]. F¨ urer [14] has proved that inequivalence and non-empty comple- ment are EXPSPACE-complete, which contrasts with the PSPACE-completeness This work was partially supported by CMUP (UID/MAT/00144/2013), which is funded by FCT (Portugal) with national (MEC) and european structural funds through the programs FEDER, under the partnership agreement PT2020. c  IFIP International Federation for Information Processing 2016 Published by Springer International Publishing Switzerland 2016. All Rights Reserved C. Cˆampeanu et al. (Eds.): DCFS 2016, LNCS 9777, pp. 45–59, 2016. DOI: 10.1007/978-3-319-41114-9 4