Monadic logical definability of nondeterministic linear time

Abstract. We answer a question asked by J. F. Lynch by proving that existential monadic second-order logic with addition captures not only the class NTIME(n) but also the class NLIN (i.e., linear time on nondeterministic RAMs), so enlarging considerably the set of natural problems expressible in this logic, since most combinatorial NP-complete problems belong to NLIN. Moreover, our result still holds if the first-order part of the formulas is required to be $ \forall^{\ast} \exists ^{\ast} $, so improving the recent similar result by J. F. Lynch about NTIME(n).¶In addition, we explicitly state that a graph problem is recognizable in nondeterministic linear time O(n + e) (where n and e are the numbers of vertices and edges, respectively) if and only if it can be defined in existential second-order logic with unary functions and only one variable on the vertices-edges domain.

[1]  Mihalis Yannakakis,et al.  Optimization, approximation, and complexity classes , 1991, STOC '88.

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

[3]  Yuri Gurevich,et al.  Toward logic tailored for computational complexity , 1984 .

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

[5]  Neil Immerman,et al.  Relational Queries Computable in Polynomial Time , 1986, Inf. Control..

[6]  Etienne Grandjean,et al.  A Nontrivial Lower Bound for an NP Problem on Automata , 1990, SIAM J. Comput..

[7]  Etienne Grandjean.  The Spectra of First-Order Sentences and Computational Complexity , 1984, SIAM J. Comput..

[8]  Neil Immerman.  Languages which capture complexity classes , 1983, STOC '83.

[9]  Frédéric Olive,et al.  Monadic Logical Definability of NP-Complete Problems , 1994, CSL.

[10]  Saharon Shelah,et al.  Nearly Linear Time , 1989, Logic at Botik.

[11]  Etienne Grandjean.  Linear Time Algorithms and NP-Complete Problems , 1992, CSL.

[12]  Ronald Fagin.  Generalized first-order spectra, and polynomial. time recognizable sets , 1974 .

[13]  Neil Immerman,et al.  Descriptive and Computational Complexity , 1989, FCT.

[14]  Bruno Courcelle,et al.  The Expression of Graph Properties and Graph Transformations in Monadic Second-Order Logic , 1997, Handbook of Graph Grammars.

[15]  Erich Grädel.  On the Notion of Linear Time Computability , 1990, Int. J. Found. Comput. Sci..

[16]  Etienne Grandjean.  First-Order Spectra with One Variable , 1990, J. Comput. Syst. Sci..

[17]  Etienne Grandjean.  A Natural NP-Complete Problem with a Nontrivial Lower Nound , 1988, SIAM J. Comput..