On the Average State Complexity of Partial Derivative Automata: An Analytic Combinatorics Approach

Abstract

The partial derivative automaton ([Formula: see text]) is usually smaller than other nondeterministic finite automata constructed from a regular expression, and it can be seen as a quotient of the Glushkov automaton ([Formula: see text]). By estimating the number of regular expressions that have ε as a partial derivative, we compute a lower bound of the average number of mergings of states in [Formula: see text] and describe its asymptotic behaviour. This depends on the alphabet size, k, and for growing k's its limit approaches half the number of states in [Formula: see text]. The lower bound corresponds to consider the [Formula: see text] automaton for the marked version of the regular expression, i.e. where all its letters are made different. Experimental results suggest that the average number of states of this automaton, and of the [Formula: see text] automaton for the unmarked regular expression, are very close to each other.

Prefix and Right-Partial Derivative Automata

“…By Proposition 3, |α R | Σ = |α| Σ and by the fact that ε ∈ π(α) if and only if ε ∈ ← − π (α), the analysis of the average size of A pd (α) presented in Broda et al [2] carries on to ← − A pd (α). Thus the average sizes of A pd and ← − A pd are asymptotically the same.…”

The Average Transition Complexity of Glushkov and Partial Derivative Automata

“…In this section we compute and study the cost generating functions E k (z) and T k (z), and their asymptotic behaviours. The other functions used herein, as well as details on how to obtain them, can be found in the above cited article and in Broda et al [BMMR11b]. A more detailed description of the below computations can be found in a companion technical report of this paper [BMMR11a].…”

“…Broda et al [BMMR11b] gave a lower bound of the number of mergings of states in π(α) with respect to Pos(α), which allowed to obtain an upper bound on the average state complexity of A pd (α). There, it was observed that the merging of states is primarily caused by sub-expressions γ of α such that ε ∈ π(γ).…”

“…In a previous paper [BMMR11b], we presented a technique for estimating some of those state mergings. This enabled us, in the framework of analytic combinatorics, to compute an upper bound for the ratio of the number of states of A pd to the number of states of A pos , which is about 1 2 for large alphabet sizes.…”

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

“…From RE to NFA a standard algorithm is the partial derivative automaton construction (A pd ) introduced by Antimirov [1], which coincides with the resolution of systems of equations by Mirkin [20]. The average complexity of these conversions was recently studied using the framework of analytic combinatorics [4,5], and also their extension to regular expressions with shuffle [7]. For these studies, Mirkin's construction is essential as it provides inductive definitions that can be used to obtain generating functions.…”