# Dichotomy for symmetric Boolean PCSPs

## TL;DR

This paper proves a complete complexity classification for symmetric Boolean PCSPs, removing previous restrictions and establishing a clear dichotomy between polynomial-time solvable and NP-complete cases.

## Contribution

It extends the dichotomy theorem for Boolean PCSPs by removing the negation assumption, providing a comprehensive classification of their computational complexity.

## Key findings

- Dichotomy established for all symmetric Boolean PCSPs.
- Identifies new reasons for computational hardness.
- Shows that tractability conditions are similar to previous results.

## Abstract

A PCSP is a combination of two CSPs defined by two similar templates; the computational question is to distinguish a YES instance of the first one from a NO instance of the second. The computational complexity of many PCSPs remains unknown. Even the case of Boolean templates (solved for CSP by Schaefer [STOC'78]) remains wide open. The main result of Brakensiek and Guruswami [SODA'18] shows that Boolean PCSPs exhibit a dichotomy (PTIME vs. NPC) when "all the clauses are symmetric and allow for negation of variables''. In this paper we remove the "allow for negation of variables'' assumption from the theorem. The "symmetric" assumption means that changing the order of variables in a constraint does not change its satisfiability. The "negation of variables" means that both of the templates share a relation which can be used to effectively negate Boolean variables. The main result of this paper establishes dichotomy for all the symmetric boolean templates. The tractability case of our theorem and the theorem of Brakensiek and Guruswami are almost identical. The main difference, and the main contribution of this work, is the new reason for hardness and the reasoning proving the split.