Abstract: 
Monadic secondorder logic on finite words (MSO) is a decidable yet
expressive logic into which many decision problems can be encoded. Since MSO
formulas correspond to regular languages, equivalence of MSO formulas can be
reduced to the equivalence of some regular structures (e.g. automata). We
verify an executable decision procedure for MSO formulas that is not based
on automata but on regular expressions.
Decision procedures for regular expression equivalence have been formalized
before, usually based on Brzozowski derivatives. Yet, for a straightforward
embedding of MSO formulas into regular expressions an extension of regular
expressions with a projection operation is required. We prove total
correctness and completeness of an equivalence checker for regular
expressions extended in that way. We also define a languagepreserving
translation of formulas into regular expressions with respect to two
different semantics of MSO.
The formalization is described in this ICFP 2013 functional pearl. 
BibTeX: 
@article{MSO_Regex_EquivalenceAFP,
author = {Dmitriy Traytel and Tobias Nipkow},
title = {Decision Procedures for MSO on Words Based on Derivatives of Regular Expressions},
journal = {Archive of Formal Proofs},
month = jun,
year = 2014,
note = {\url{http://isaafp.org/entries/MSO_Regex_Equivalence.shtml},
Formal proof development},
ISSN = {2150914x},
}
