Partially Ordered Connectives and Monadic Monotone Strict NP

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 Vaananen (Zeitschrift fur Mathematische Logik und Grundlagen der Mathematik, 38, 361---372 1992), and MMSNP employing a generalization C of D introduced in the present paper.