On the Average Size of Glushkov and Partial Derivative Automata

Abstract

In this paper, the relation between the Glushkov automaton [Formula: see text] and the partial derivative automaton [Formula: see text] of a given regular expression, in terms of transition complexity, is studied. The average transition complexity of [Formula: see text] was proved by Nicaud to be linear in the size of the corresponding expression. This result was obtained using an upper bound of the number of transitions of [Formula: see text]. Here we present a new quadratic construction of [Formula: see text] that leads to a more elegant and straightforward implementation, and that allows the exact counting of the number of transitions. Based on that, a better estimation of the average size is presented. Asymptotically, and as the alphabet size grows, the number of transitions per state is on average 2. Broda et al. computed an upper bound for the ratio of the number of states of [Formula: see text] to the number of states of [Formula: see text] which is about ½ for large alphabet sizes. Here we show how to obtain an upper bound for the number of transitions in [Formula: see text], which we then use to get an average case approximation. In conclusion, assymptotically, and for large alphabets, the size of [Formula: see text] is half the size of the [Formula: see text]. This is corroborated by some experiments, even for small alphabets and small regular expressions.

Deciding Synchronous Kleene Algebra with Derivatives

“…For regular expressions it is known that after eliminating ε-transitions from the Thompson automaton one obtains the Glushkov automaton [13], of which the partial derivative automaton is a quotient. Asymptotically and on average the size of the partial derivative automaton is half the size of the Glushkov automaton [8], which on the other hand is linear on the size of the expression. As noticed before, for the synchronous product the Thompson construction considers a product automaton and thus a quadratic number of transitions is expected.…”

Prefix and Right-Partial Derivative Automata

“…However, ← − A pd (α) has only one final state and its number of initial states is the number of final states of A pd (α R ). As studied by Nicaud [14], the size of last(α) tends asymptotically to a constant depending on k and |λ(α)| is half that size [3]. Thus, that constant value will be also the number of initial states of ← − A pd .…”

“…As noted by Broda et al [3] and Maia et al [12], following Mirkin's construction, the partial derivative automaton of α can be inductively constructed. A (right) support for α is a set of regular expressions {α 1 , .…”

“…In particular, for REs under special conditions, A pd is an optimal conversion method [12]. Moreover, asymptotically and on average, the size of A pd is half the size of A pos [3]. Reductions on the size of NFAs using left-relations was studied recently by Ko and Han [11].…”

The Average Transition Complexity of Glushkov and Partial Derivative Automata

“…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].…”