Hard Satisfiable Formulas for Splittings by Linear Combinations

Itsykson and Sokolov in 2014 introduced the class of DPLL(⊕) algorithms that solve Boolean satisfiability problem using the splitting by linear combinations of variables modulo 2. This class extends the class of DPLL algorithms that split by variables. DPLL(⊕) algorithms solve in polynomial time systems of linear equations modulo 2 that are hard for DPLL, PPSZ and CDCL algorithms. Itsykson and Sokolov have proved first exponential lower bounds for DPLL(⊕) algorithms on unsatisfiable formulas. In this paper we consider a subclass of DPLL(⊕) algorithms that arbitrary choose a linear form for splitting and randomly (with equal probabilities) choose a value to investigate first; we call such algorithms drunken DPLL(⊕). We give a construction of a family of satisfiable CNF formulas Ψn of size poly(n) such that any drunken DPLL(⊕) algorithm with probability at least 1 − 2−Ω(n) runs at least 2 steps on Ψn; thus we solve an open question stated in the paper [12]. This lower bound extends the result of Alekhnovich, Hirsch and Itsykson [1] from drunken DPLL to drunken DPLL(⊕).