On monadic NP vs. monadic co-NP

It is proved that connectivity of finite graphs is not in monadic NP, even in the presence of arbitrary built-in relations of moderate degree (that is, degree (log n) /sup o(1)/). This results in a strong separation between monadic NP and monadic co-NP. The proof uses a combination of three techniques: (1) a technique of W. Hanf (1965) for showing that two (infinite) structures agree on all first-order sentences, under certain conditions; (2) a recent approach to second-order Ehrenfeucht-Fraisse games by M. Ajtai and R. Fagin (1990); and (3) playing Ehrenfeucht-Fraisse games over random structures. Regarding (1), a version of Hanf's result that is better suited for use as a tool in inexpressibility proofs for classes of finite structures is given. The power of these techniques is further demonstrated by using the first two techniques to give a very simple proof of the separation of monadic NP from monadic co-NP without the presence of built-in relations.<<ETX>>

[1]  Neil Immerman,et al.  Languages that Capture Complexity Classes , 1987, SIAM J. Comput..

[2]  Thomas Schwentick,et al.  Graph connectivity and monadic NP , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

[3]  Herbert B. Enderton,et al.  A mathematical introduction to logic , 1972 .

[4]  Neil D. Jones,et al.  Turing machines and the spectra of first-order formulas with equality , 1972, STOC.

[5]  John M. Lewis,et al.  The Node-Deletion Problem for Hereditary Properties is NP-Complete , 1980, J. Comput. Syst. Sci..

[6]  David Harel,et al.  Structure and complexity of relational queries , 1980, 21st Annual Symposium on Foundations of Computer Science (sfcs 1980).

[7]  Bruno Courcelle,et al.  The Monadic Second-Order Logic of Graphs. I. Recognizable Sets of Finite Graphs , 1990, Inf. Comput..

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

[9]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[10]  Ronald Fagin,et al.  A spectrum hierarchy , 1975, Math. Log. Q..

[11]  Neil D. Jones,et al.  Turing machines and the spectra of first-order formulas , 1974, Journal of Symbolic Logic.

[12]  György Turán,et al.  On the definability of properties of finite graphs , 1984, Discret. Math..

[13]  Stavros S. Cosmadakis,et al.  Logical reducibility and monadic NP , 1993, Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science.

[14]  Michel de Rougemont.  Second-order and Inductive Definability on Finite Structures , 1987, Math. Log. Q..

[15]  Miklós Ajtai,et al.  ∑11-Formulae on finite structures , 1983, Ann. Pure Appl. Log..

[16]  Joel H. Spencer,et al.  Threshold spectra via the Ehrenfeucht game , 1991, Discret. Appl. Math..

[17]  Petr Hájek.  On Logics of Discovery , 1975, MFCS.

[18]  A. Ehrenfeucht.  An application of games to the completeness problem for formalized theories , 1961 .

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

[20]  Joseph R. Shoenfield,et al.  Mathematical logic , 1967 .

[21]  Ronald Fagin,et al.  Monadic generalized spectra , 1975, Math. Log. Q..

[22]  Ashok K. Chandra.  Theory of database queries , 1988, PODS '88.

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

[24]  Ronald Fagin,et al.  Finite-Model Theory - A Personal Perspective , 1990, Theor. Comput. Sci..

[25]  Ronald Fagin,et al.  Reachability is harder for directed than for undirected finite graphs , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

[26]  Moshe Y. Vardi.  The complexity of relational query languages (Extended Abstract) , 1982, STOC '82.

[27]  Wolfgang Thomas.  On Logics, Tilings, and Automata , 1991, ICALP.

[28]  Alfred V. Aho,et al.  Universality of data retrieval languages , 1979, POPL.