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On guarded extensions of MMSNP

Alexey Barsukov, Florent R. Madelaine

TL;DR

The paper analyzes guarded extensions of MMSNP, revisiting the fundamental MMSNP–CSP dichotomy and examining three gaps created by relaxing one property at a time: Guarded Monotone SNP (GMSNP) for monotonicity, MMSNP with guarded inequality for no inequality, and MPART as a Matrix-Partition–style extension bridging MMSNP and Monadic SNP without inequality. It proves that MMSNP with guarded inequality is no more expressive than MMSNP in terms of dichotomy status, and shows that GMSNP's dichotomy is reducible to signatures with a single input relation, while introducing MPART as a broader framework between MMSNP and Matrix Partitions. A central result reaffirms Feder and Vardi's embedding of NP into MMSNP with inequality by providing a detailed, constructive proof using oblivious Turing machines and an explicit space-time diagram encoding, establishing that the right-hand sides of the gaps do not admit a dichotomy unless $P=NP$. Altogether, the work delineates the boundaries of dichotomy phenomena for guarded extensions and showcases concrete mechanisms (enrichment, single-relations reductions, and space-time encodings) to relate these logics to known complexity classes.

Abstract

Feder and Vardi showed that the class Monotone Monadic SNP without inequality (MMSNP) has a P vs NP-complete dichotomy if and only if such a dichotomy holds for finite-domain Constraint Satisfaction Problems (CSPs). Moreover, they showed that none of the three classes obtained by removing one of the defining properties of MMSNP (monotonicity, monadicity, no inequality) has a dichotomy. The overall objective of this paper is to study the gaps between MMSNP and each of these three superclasses, where the existence of a dichotomy remains unknown. For the gap between MMSNP and Monotone SNP without inequality, we study the class Guarded Monotone SNP without inequality (GMSNP) introduced by Bienvenu, ten Cate, Lutz, and Wolter, and prove that GMSNP has a dichotomy if and only if a dichotomy holds for GMSNP problems over signatures consisting of a unique relation symbol. For the gap between MMSNP and MMSNP with inequality, we introduce a new class MMSNP with guarded inequality, that lies between MMSNP and MMSNP with inequality and that is strictly more expressive than the former and still has a dichotomy. For the gap between MMSNP and Monadic SNP without inequality, we introduce a logic that extends the class of Matrix Partitions in a similar way how MMSNP extends finite-domain CSP, and pose an open question about the existence of a dichotomy for this class. Finally, we revisit the theorem of Feder and Vardi, which claims that the class NP embeds into MMSNP with inequality. We give a detailed proof of this theorem as it ensures no dichotomy for the right-hand side class of each of the three gaps.

On guarded extensions of MMSNP

TL;DR

The paper analyzes guarded extensions of MMSNP, revisiting the fundamental MMSNP–CSP dichotomy and examining three gaps created by relaxing one property at a time: Guarded Monotone SNP (GMSNP) for monotonicity, MMSNP with guarded inequality for no inequality, and MPART as a Matrix-Partition–style extension bridging MMSNP and Monadic SNP without inequality. It proves that MMSNP with guarded inequality is no more expressive than MMSNP in terms of dichotomy status, and shows that GMSNP's dichotomy is reducible to signatures with a single input relation, while introducing MPART as a broader framework between MMSNP and Matrix Partitions. A central result reaffirms Feder and Vardi's embedding of NP into MMSNP with inequality by providing a detailed, constructive proof using oblivious Turing machines and an explicit space-time diagram encoding, establishing that the right-hand sides of the gaps do not admit a dichotomy unless . Altogether, the work delineates the boundaries of dichotomy phenomena for guarded extensions and showcases concrete mechanisms (enrichment, single-relations reductions, and space-time encodings) to relate these logics to known complexity classes.

Abstract

Feder and Vardi showed that the class Monotone Monadic SNP without inequality (MMSNP) has a P vs NP-complete dichotomy if and only if such a dichotomy holds for finite-domain Constraint Satisfaction Problems (CSPs). Moreover, they showed that none of the three classes obtained by removing one of the defining properties of MMSNP (monotonicity, monadicity, no inequality) has a dichotomy. The overall objective of this paper is to study the gaps between MMSNP and each of these three superclasses, where the existence of a dichotomy remains unknown. For the gap between MMSNP and Monotone SNP without inequality, we study the class Guarded Monotone SNP without inequality (GMSNP) introduced by Bienvenu, ten Cate, Lutz, and Wolter, and prove that GMSNP has a dichotomy if and only if a dichotomy holds for GMSNP problems over signatures consisting of a unique relation symbol. For the gap between MMSNP and MMSNP with inequality, we introduce a new class MMSNP with guarded inequality, that lies between MMSNP and MMSNP with inequality and that is strictly more expressive than the former and still has a dichotomy. For the gap between MMSNP and Monadic SNP without inequality, we introduce a logic that extends the class of Matrix Partitions in a similar way how MMSNP extends finite-domain CSP, and pose an open question about the existence of a dichotomy for this class. Finally, we revisit the theorem of Feder and Vardi, which claims that the class NP embeds into MMSNP with inequality. We give a detailed proof of this theorem as it ensures no dichotomy for the right-hand side class of each of the three gaps.
Paper Structure (16 sections, 19 theorems, 45 equations, 10 figures)

This paper contains 16 sections, 19 theorems, 45 equations, 10 figures.

Table of Contents

  1. Introduction
  2. The dichotomy question
  3. Three gaps above MMSNP
  4. Outline
  5. Preliminaries
  6. Studying the three gaps
  7. MMSNP with guarded inequality
  8. Guarded Monotone SNP over one-element signatures
  9. Matrix Partitions
  10. NP is polynomial-time equivalent to MMSNP with inequality
  11. Obliviousness
  12. Construction of the space-time diagram.
  13. Construction of the sentence.
  14. Reduction from $M$ to $\Phi$
  15. Reduction from $\Phi$ to $M$
  16. ...and 1 more sections

Key Result

Theorem 2.1

For every sentence $\Phi$ in $\textrm{SNP}$, the class $\mathrm{fm}(\Phi)$ is closed under inverse homomorphisms only if $\Phi$ is logically equivalent to a sentence in monotone SNP without inequality.

Figures (10)

  • Figure 1: Classes at the bottom exhibit a $\textrm{P}$vs$\textrm{NP-complete}$ dichotomy, while those at the top do not, and it remains open for classes in the middle. Undirected edges stand for straightforward inclusions and directed edges stand for inclusions under polynomial-time reductions. The classes and inclusions discussed in this paper are highlighted with red. The sign denotes classes extended with guarded inequality.
  • Figure 2: All the 5 equivalence relations on a $3$-element set.
  • Figure 3: The classes introduced in Subsection \ref{['section:mp']} and their relation to other classes from this article. Dashed lines divide the figure in 3 parts: "dichotomy", "unknown", "no dichotomy". Undirected edges stand for inclusions and directed edges stand for inclusions under polynomial-time reductions.
  • Figure 4: The domain and the relations for a structure $\mathfrak{A}$ from Example \ref{['ex:construction']}.
  • Figure 5: The rule (left) initiating the relations $\EuRoman{I}$ and $\EuRoman{H}$ (denoted by $\Box$ and by a circle, respectively) and the rule (right) that forces the relation $\EuRoman{I}$ to spread. The $\EuRoman{I}$- and $\EuRoman{H}$-atoms in the head of each rule are coloured in green.
  • ...and 5 more figures

Theorems & Definitions (28)

  • Example 1.1
  • Theorem 2.1: federV03
  • Definition 2.2
  • Remark 2.3
  • Theorem 3.1
  • Example 3.2
  • Lemma 3.3
  • Definition 3.4
  • Example 3.5
  • Proposition 3.6
  • ...and 18 more