137. A Survey on Small Fragments of First-Order
Logic over Finite Words
Volker Diekert1, Paul Gastin2, Manfred Kufleitner1,3
1 FMI, Universit¨at Stuttgart
Universit¨atsstr. 38, D-70569 Stuttgart, Germany
{diekert,kufleitner}@fmi.uni-stuttgart.de
2 LSV, ENS Cachan, CNRS
61, Av. du Pr´esident Wilson, F-94230 Cachan, France
Paul.Gastin@lsv.ens-cachan.fr
3 LaBRI, Universit´e de Bordeaux & CNRS
351, Cours de la Lib´eration, F-33405 Talence Cedex, France
Abstract
We consider fragments of first-order logic over finite words. In particu-lar,
we deal with first-order logic with a restricted number of variables and
with the lower levels of the alternation hierarchy. We use the algebraic ap-proach
to show decidability of expressibility within these fragments. As
a byproduct, we survey several characterizations of the respective frag-ments.
We give complete proofs for all characterizations and we provide
all necessary background. Some of the proofs seem to be new and simpler
than those which can be found elsewhere. We also give a proof of Simon’s
theorem on factorization forests restricted to aperiodic monoids because
this is simpler and sucient for our purpose.
Keywords: First-order logic, monoids, factorization forests, piecewise-testable
languages
Preamble
There are many brilliant surveys on formal language theory [36, 41, 48, 85, 86].
Quite many surveys cover first-order and monadic second-order definability. But
there are also nuggets below. There are deep theorems on proper fragments of
first-order definability. The most prominent fragment is FO2; it is the class of
languages which are defined by first-order sentences which do not use more than
two names for variables. Although various characterizations are known for this
class, there seems to be little knowledge in a broad community. A reason for
this is that the proofs are spread over the literature and even in the survey [76]
many proofs are referred to the original literature which in turn is sometimes
quite dicult to read.
1
[4] Diekert先生の
サーベイ論文
FO以下の論理と言語・半
群とのvarietyの対応の
サーベイ.本資料を書く
きっかけとなった.
「言語と論理と代数の対
応」の章で紹介した全対
応について丁寧な証明が
載っている.素晴らしい
サーベイ.
なかなか難しい.
139. [6] S.Eilenberg, Automata, languages and
machines, Volume B, Academic Press, (1976).
[7] M.P.Schützenberger, On finite monoids having
only trivial subgroups, Information and control 8
(1965), 190–194.
[8] J.R. Büchi. Weak second-order arithmetic and
finite automata, Zeitschrift für, Mathematische
Logik und Grundlagen der Mathematik, 6 (1960),
66–92.
[9] R. McNaughton and S. Papert. Counter-Free
Automata. MIT Press, (1971).
[10] P. Weil, From algebra to logic: There and back
again the story of a hierarchy, Invited Talk in DLT
2014.