A Unified Construction of the Glushkov, Follow, and Antimirov Automata

Abstract

Abstract. Many techniques have been introduced in the last few decades to create -free automata representing regular expressions: Glushkov automata, the so-called follow automata, and Antimirov automata. This paper presents a simple and unified view of all these -free automata both in the case of unweighted and weighted regular expressions. It describes simple and general algorithms with running time complexities at least as good as that of the best previously known techniques, and provides concise proofs. The construction methods are all based on two standard automata algorithms: epsilon-removal and minimization. This contrasts with the multitude of complicated and special-purpose techniques and proofs put forward by others to construct these automata. Our analysis provides a better understanding of -free automata representing regular expressions: they are all the results of the application of some combinations of epsilon-removal and minimization to the classical Thompson automata. This makes it straightforward to generalize these algorithms to the weighted case, which also results in much simpler algorithms than existing ones. For weighted regular expressions over a closed semiring, we extend the notion of follow automata to the weighted case. We also present the first algorithm to compute the Antimirov automata in the weighted case. IntroductionThe construction of finite automata representing regular expressions has been widely studied due to its multiple applications to pattern-matching and many Report Documentation PageForm Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number.

Significance Score of Motifs in Biological Sequences

“…We admit that the language of a NFA is always regular (see Hopcroft et al, 2001, for the formal proof) but we will prove the reciprocal with the Glushkov's construction (Allauzen & Mohri, 2006). This construction provides a simple way to build the NFA directly from the regular expression of the language.…”

Derived-Term Automata of Multitape Expressions with Composition

“…Alas, we are then exposed to the same problems as the techniques that start from the Thompson automaton to compute different types of automata [1]: for some valid expressions, we generate invalid automata. For instance in Q, the expression (a * + −1 1) * is valid, as a * + −1 1 is proper, yet its Thompson automaton is invalid, as it contains a spontaneous cycle whose weight, 1, is not starrable:…”

Adding Pebbles to Weighted Automata

“…Clearly, the number of states of the automaton A(E) is the litteral-length (E) of expression E. The time complexity is cubic in the length of E. It should be possible to get a quadratic algorithm by generalizing the notion of star normal form introduced in [9] for word languages or the algorithm presented in [1] for classical weighted expressions and automata.…”