Journal of Logic, Language and Information

, Volume 17, Issue 3, pp 323–344 | Cite as

Partially Ordered Connectives and Monadic Monotone Strict NP

Article

Abstract

Motivated by constraint satisfaction problems, Feder and Vardi (SIAM Journal of Computing, 28, 57–104, 1998) set out to search for fragments \({\mathcal{L}}\) of \(\Sigma_1^1\) satisfying the dichotomy property: every problem definable in \({\mathcal{L}}\) is either in P or else NP-complete. Feder and Vardi considered in this connection two logics, strict NP (or SNP) and monadic, monotone, strict NP without inequalities (or MMSNP). The former consists of formulas of the form \(\exists \vec{X}\forall \vec{x} \phi\), where \(\phi\) is a quantifier-free formula in a relational vocabulary; and the latter is the fragment of SNP whose formulas involve only negative occurrences of relation symbols, only monadic second-order quantifiers, and no occurrences of the equality symbol. It remains an open problem whether MMSNP enjoys the dichotomy property. In the present paper, SNP and MMSNP are characterized in terms of partially ordered connectives. More specifically, SNP is characterized using the logic D of partially ordered connectives introduced in Blass and Gurevich (Annals of Pure and Applied Logic, 32, 1–16, 1986), Sandu and Väänänen (Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 38, 361–372 1992), and MMSNP employing a generalization C of D introduced in the present paper.

Keywords

Constraint satisfaction problems Generalized quantifiers Henkin quantifiers MMSNP Partially ordered connectives SNP 

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Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  • Lauri Hella
    • 1
  • Merlijn Sevenster
    • 2
  • Tero Tulenheimo
    • 3
  1. 1.Department of Mathematics and StatisticsUniversity of TampereTampereFinland
  2. 2.Philips Research, EindhovenEindhovenThe Netherlands
  3. 3.Department of PhilosophyUniversity of HelsinkiHelsinkiFinland

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