On the State Complexity of Partial Derivative Automata For Regular Expressions with Intersection

Abstract

Abstract. Extended regular expressions (with complement and intersection) are used in many applications due to their succinctness. In particular, regular expressions extended with intersection only (also called semi-extended) can already be exponentially smaller than standard regular 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.

Regular Expressions and Transducers over Alphabet-Invariant and User-Defined Labels

“…Not only they are in general more succinct than other equivalent constructions but also for several operators they are easily defined (e.g. for intersection [4] or tuples [12]). The partial derivative automaton of a regular expression over Σ ⋆ was introduced independently by Mirkin [18] and Antimirov [3].…”

Position Automaton Construction for Regular Expressions with Intersection

“…For regular expressions with intersection the conversion to NFA's has exponential computational complexity [11] and both the size of A pos and A pd can be exponential in the size of the regular expression. On the average case, however, the size of these automata seem to be much smaller [2], and thus feasible for practical applications. In this scenario, algorithms for building A pd using A pos seem worthwhile to develop.…”

“…Bastos et al [2] showed also that |∂(α)| ≤ 2 |α| Σ −|α|∩−1 + 1 and asymptotically and on average an upper bound for the number of states is (1.056 + o (1)) n , where n is the size of the expression.…”

“…Bastos et al [2] presented an extension of the Antimirov construction from RE ∩ expressions. Definition 1.…”

“…However, the conversions from expressions to automata based on the notion of derivative or partial derivative can still be extended to regular expressions with intersection [5,7,2]. In this paper, we present a position automaton construction for regular expressions with intersection by generalizing the notion of position.…”