Classification of Non-redundancy of Boolean Predicates of Arity 4

TL;DR

This paper classifies the non-redundancy of Boolean predicates of arity 4, providing a near-complete classification for these predicates and identifying new asymptotic behaviors.

Contribution

It offers the first near-complete classification of non-redundancy for Boolean predicates of arity 4, solving most cases and highlighting new asymptotic phenomena.

Findings

397 out of 400 predicates classified

Two predicates reduced to extremal combinatorics problems

Identification of the first predicate with non-polynomial asymptotics

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.