On the expressive power of monadic least fixed point logic

Monadic least fixed point logic MLFP is a natural logic whose expressiveness lies between that of first-order logic FO and monadic second-order logic MSO. In this paper, we take a closer look at the expressive power of MLFP. Our results are: (1) MLFP can describe graph properties beyond any fixed level of the monadic second-order quantifier alternation hierarchy. (2) On strings with built-in addition, MLFP can describe at least all languages that belong to the linear time complexity class DLIN. (3) Settling the question whether addition-invariant MLFP = addition-invariant MSO on finite strings or, equivalently, settling the question whether MLFP = MSO on finite strings with addition would solve open problems in complexity theory: "=" would imply that PH = PTIME whereas "≠" would imply that DLIN ≠ LINH. Apart from this we give a self-contained proof of the previously known result that MLFP is strictly less expressive than MSO on the class of finite graphs.

[1]  Jörg Flum,et al.  Query evaluation via tree-decompositions , 2001, JACM.

[2]  Ludwig Staiger,et al.  Ω-languages , 1997 .

[3]  Benjamin Rossman.  Successor-invariance in the finite , 2003, 18th Annual IEEE Symposium of Logic in Computer Science, 2003. Proceedings..

[4]  Frédéric Olive,et al.  Rudimentary Languages and Second Order Logic , 1997, Math. Log. Q..

[5]  Wolfgang Thomas,et al.  Languages, Automata, and Logic , 1997, Handbook of Formal Languages.

[6]  Julian C. Bradfield.  The Modal µ-Calculus Alternation Hierarchy is Strict , 1998, Theor. Comput. Sci..

[7]  Thomas Schwentick,et al.  Machine-Independent Characterizations and Complete Problems for Deterministic Linear Time , 2002, SIAM J. Comput..

[8]  Erich Grädel.  Model Checking Games , 2002, Electron. Notes Theor. Comput. Sci..

[9]  Frédéric Olive,et al.  Monadic logical definability of nondeterministic linear time , 1998, computational complexity.

[10]  Jörg Flum,et al.  Finite model theory , 1995, Perspectives in Mathematical Logic.

[11]  Franz W. Peren.  Arithmetic , 1903, Nature.

[12]  Alex K. Simpson,et al.  Computational Adequacy in an Elementary Topos , 1998, CSL.

[13]  Ronald Fagin,et al.  The Closure of Monadic NP , 2000, J. Comput. Syst. Sci..

[14]  Nicole Schweikardt,et al.  Comparing the Succinctness of Monadic Query Languages over Finite Trees , 2003, CSL.

[15]  Neil Immerman,et al.  Descriptive Complexity , 1999, Graduate Texts in Computer Science.

[16]  Etienne Grandjean.  Invariance properties of RAMS and linear time , 2005, computational complexity.

[17]  Arnaud Durand,et al.  Nonerasing, Counting, and Majority over the Linear Time Hierarchy , 2002, Inf. Comput..

[18]  Nicole Schweikardt,et al.  Arithmetic, first-order logic, and counting quantifiers , 2002, TOCL.

[19]  Anuj Dawar.  A Restricted Second Order Logic for Finite Structures , 1998, Inf. Comput..

[20]  Grzegorz Rozenberg,et al.  Handbook of Formal Languages , 1997, Springer Berlin Heidelberg.

[21]  Thomas Schwentick,et al.  Descriptive Complexity, Lower Bounds and Linear Time , 1998, CSL.

[22]  Nicole Schweikardt,et al.  The Monadic Quantifier Alternation Hierarchy over Grids and Graphs , 2002, Inf. Comput..

[23]  James F. Lynch.  The quantifier structure of sentences that characterize nondeterministic time complexity , 2005, computational complexity.

[24]  Arnaud Durand,et al.  Counting Results in Weak Formalisms , 2006, Circuits, Logic, and Games.

[25]  Larry J. Stockmeyer,et al.  The Polynomial-Time Hierarchy , 1976, Theor. Comput. Sci..

[26]  Nicole Schweikardt.  On the Expressive Power of Monadic Least Fixed Point Logic , 2004, ICALP.

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

[28]  Thomas Schwentick,et al.  Locality of Order-Invariant First-Order Formulas , 1998, MFCS.

[29]  Etienne Grandjean,et al.  Sorting, linear time and the satisfiability problem , 1996, Annals of Mathematics and Artificial Intelligence.

[30]  Richard M. Karp,et al.  Reducibility among combinatorial problems" in complexity of computer computations , 1972 .

[31]  James F. Lynch,et al.  Complexity classes and theories of finite models , 1981, Mathematical systems theory.