Hard satisfiable formulas for DPLL algorithms using heuristics with small memory

Abstract: DPLL algorithm for solving the Boolean satisfiability problem (SAT) can be represented in the form of a procedure that, using heuristics $A$ and $B$, select the variable $x$ from the input formula $\varphi$ and the value $b$ and runs recursively on the formulas $\varphi[x := b]$ and $\varphi[x := 1 - b]$. Exponential lower bounds on the running time of DPLL algorithms on unsatisfiable formulas follow from the lower bounds for tree-like resolution proofs. Lower bounds on satisfiable formulas are also known for some classes of DPLL algorithms such as "myopic" and "drunken" algorithms. All lower bounds are made for the classes of DPLL algorithms that limit heuristics access to the formula. In this paper we consider DPLL algorithms with heuristics that have unlimited access to the formula but use small memory. We show that for any pair of heuristics with small memory there exists a family of satisfiable formulas $\Phi_n$ such that a DPLL algorithm that uses these heuristics runs in exponential time on the formulas $\Phi_n$.