Lattices of Logical Fragments over Words - (Extended Abstract)

This paper introduces an abstract notion of fragments of monadic second-order logic. This concept is based on purely syntactic closure properties. We show that over finite words, every logical fragment defines a lattice of languages with certain closure properties. Among these closure properties are residuals and inverse $\mathcal C$-morphisms. Here, depending on certain closure properties of the fragment, $\mathcal C$ is the family of arbitrary, non-erasing, length-preserving, length-multiplying, or lengthreducing morphisms. In particular, definability in a certain fragment can often be characterized in terms of the syntactic morphism. This work extends a result of Straubing in which he investigated certain restrictions of first-order formulae. As motivating examples, we present (1) a fragment which captures the stutter-invariant part of piecewise-testable languages and (2) an acyclic fragment of Σ2. As it turns out, the latter has the same expressive power as two-variable first-order logic FO2.

[1]  Denis Thérien,et al.  DIAMONDS ARE FOREVER: THE VARIETY DA , 2002 .

[2]  J. Büchi.  Weak Second‐Order Arithmetic and Finite Automata , 1960 .

[3]  Wolfgang Thomas,et al.  Classifying Regular Events in Symbolic Logic , 1982, J. Comput. Syst. Sci..

[4]  Serge Grigorieff,et al.  Duality and Equational Theory of Regular Languages , 2008, ICALP.

[5]  C. C. Elgot.  Decision problems of finite automata design and related arithmetics , 1961 .

[6]  Howard Straubing,et al.  Actions, wreath products of C-varieties and concatenation product , 2006, Theor. Comput. Sci..

[7]  Howard Straubing,et al.  Regular Languages Defined with Generalized Quanifiers , 1995, Inf. Comput..

[8]  Howard Straubing,et al.  On Logical Descriptions of Regular Languages , 2002, LATIN.

[9]  Paul Gastin,et al.  A Survey on Small Fragments of First-Order Logic over Finite Words , 2008, Int. J. Found. Comput. Sci..

[10]  Denis Thérien,et al.  Logic Meets Algebra: the Case of Regular Languages , 2007, Log. Methods Comput. Sci..

[11]  Howard Straubing,et al.  First Order Formulas with Modular Ppredicates , 2006, 21st Annual IEEE Symposium on Logic in Computer Science (LICS'06).

[12]  R. McNaughton,et al.  Counter-Free Automata , 1971 .

[13]  Jean-Éric Pin.  Expressive power of existential first-order sentences of Bu"chi's sequential calculus , 2005, Discret. Math..

[14]  Howard Straubing,et al.  Some results on $\mathcal {C}$-varieties , 2005 .

[15]  Michal Kunc.  Equational description of pseudovarieties of homomorphisms , 2003, RAIRO Theor. Informatics Appl..

[16]  Paul Gastin,et al.  First-order definable languages , 2008, Logic and Automata.

[17]  Imre Simon,et al.  Piecewise testable events , 1975, Automata Theory and Formal Languages.

[18]  Johan Anthory Willem Kamp,et al.  Tense logic and the theory of linear order , 1968 .

[19]  Martin Lange,et al.  The Complexity of Model Checking Higher-Order Fixpoint Logic , 2007, Log. Methods Comput. Sci..

[20]  Marcel Paul Schützenberger,et al.  On Finite Monoids Having Only Trivial Subgroups , 1965, Inf. Control..

[21]  Howard Straubing,et al.  Some results on $\cal C$-varieties , 2005, RAIRO Theor. Informatics Appl..

[22]  Thomas Wilke,et al.  Over words, two variables are as powerful as one quantifier alternation , 1998, STOC '98.