From regular expressions to deterministic finite automata: $2^{\frac{n}{2}+\sqrt{n}(\log n)^{Θ(1)}}$ states are necessary and sufficient

TL;DR

The paper tackles the state-complexity of converting a regular expression of alphabetic width $n$ into a deterministic finite automaton. It introduces a two-stage approach that first forms an NFA that remembers the last symbol and then applies a refined subset construction, aided by unary-determinization via the Chrobak form and Landau's function. The main results are a tight upper bound $2^{\frac{n}{2}+ (\frac{\log_2 e}{2\sqrt{2}}+o(1))\sqrt{n\ln n}}$ and a near-matching lower bound $2^{\frac{n}{2}+ (\sqrt{2}+o(1))\sqrt{\frac{n}{\ln n}}}$ for the same class of languages, attained under a two-symbol alphabet and related constructions. This advances the theoretical understanding of regex-to-DFA transformation, sharpening the gap between best-known upper and lower bounds and highlighting the remaining open question on the precise logarithmic factor in the exponent.

Abstract

It is proved that every regular expression of alphabetic width $n$, that is, with $n$ occurrences of symbols of the alphabet, can be transformed into a deterministic finite automaton (DFA) with $2^{\frac{n}{2}+(\frac{\log_2 e}{2\sqrt{2}}+o(1))\sqrt{n\ln n}}$ states recognizing the same language (the best upper bound up to date is $2^n$). At the same time, it is also shown that this bound is close to optimal, namely, that there exist regular expressions of alphabetic width $n$ over a two-symbol alphabet, such that every DFA for the same language has at least $2^{\frac{n}{2}+(\sqrt{2} + o(1))\sqrt{\frac{n}{\ln n}}}$ states (the previously known lower bound is $\frac{5}{4}2^{\frac{n}{2}}$). The same bounds are obtained for an intermediate problem of determinizing nondetermistic finite automata (NFA) with each state having all incoming transitions by the same symbol.

Figures (4)

• Figure 1: Transformation of a regular expression to an NFA that remembers the last symbol and has a non-reenterable initial state.
• Figure 2: The Chrobak normal form of unary NFA.
• Figure 3: NFA $A_{3,5}$.
• Figure 4: Proof of Claim \ref{['state_complexity_lower_bound_lemma__reachability_claim']} in Lemma \ref{['state_complexity_lower_bound_lemma']}: for NFA $A_{3,5,7}$, applying the string $ba$ to add a a state to the cycle $P_m$, where $m=2$.

Theorems & Definitions (31)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Lemma 1
• proof
• Lemma 2: The first method for subset calculation
• proof
• Definition 5: Chrobak Chrobak
• Lemma 3