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Classification of Non-redundancy of Boolean Predicates of Arity 4

Joshua Brakensiek, Venkatesan Guruswami, Aaron Putterman

Abstract

Given a constraint satisfaction problem (CSP) predicate $P \subseteq D^r$, the non-redundancy (NRD) of $P$ is maximum-sized instance on $n$ variables such that for every clause of the instance, there is an assignment which satisfies all but that clause. The study of NRD for various CSPs is an active area of research which combines ideas from extremal combinatorics, logic, lattice theory, and other techniques. Complete classifications are known in the cases $r=2$ and $(|D|=2, r=3)$. In this paper, we give a near-complete classification of the case $(|D|=2, r=4)$. Of the 400 distinct non-trivial Boolean predicates of arity 4, we implement an algorithmic procedure which perfectly classifies 397 of them. Of the remaining three, we solve two by reducing to extremal combinatorics problems -- leaving the last one as an open question. Along the way, we identify the first Boolean predicate whose non-redundancy asymptotics are non-polynomial.

Classification of Non-redundancy of Boolean Predicates of Arity 4

Abstract

Given a constraint satisfaction problem (CSP) predicate , the non-redundancy (NRD) of is maximum-sized instance on variables such that for every clause of the instance, there is an assignment which satisfies all but that clause. The study of NRD for various CSPs is an active area of research which combines ideas from extremal combinatorics, logic, lattice theory, and other techniques. Complete classifications are known in the cases and . In this paper, we give a near-complete classification of the case . Of the 400 distinct non-trivial Boolean predicates of arity 4, we implement an algorithmic procedure which perfectly classifies 397 of them. Of the remaining three, we solve two by reducing to extremal combinatorics problems -- leaving the last one as an open question. Along the way, we identify the first Boolean predicate whose non-redundancy asymptotics are non-polynomial.
Paper Structure (21 sections, 13 theorems, 26 equations)

This paper contains 21 sections, 13 theorems, 26 equations.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Outline
  4. Preliminaries
  5. Conditional Non-redundancy
  6. Balanced Predicates and Polynomial Representations
  7. Experiment Methodology
  8. Enumeration of Boolean Predicates of Arity 4
  9. NRD Lower Bound: OR Projections
  10. NRD Upper Bound: Lattice Methods
  11. Higher Degrees
  12. Extracting a Polynomial Certificate
  13. The Three Exceptional Predicates
  14. Predicate 317
  15. Lower Bounding Conditional Non-Redundancy
  16. ...and 6 more sections

Key Result

Theorem 1.1

$\operatorname{NRD}(R_{299}, n) \in \left[\frac{n^3}{2^{O(\sqrt{\log(n)})}}, \frac{n^3}{2^{\Omega(\log^*(n))}}\right]$.

Theorems & Definitions (34)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3: Triangle Inequality brakensiek2025redundancy
  • Proposition 2.4
  • proof
  • Definition 2.5: e.g., chen2020BestCase
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • ...and 24 more