# Resolution Simulates Ordered Binary Decision Diagrams for Formulas in Conjunctive Normal Form

## TL;DR

This paper demonstrates that resolution can polynomially simulate Ordered Binary Decision Diagrams (OBDDs) when both are limited to formulas in Conjunctive Normal Form, resolving an open question about their relative efficiency.

## Contribution

It proves that resolution and OBDDs are polynomially comparable on CNF formulas, challenging previous beliefs about their exponential separation.

## Key findings

- Resolution simulates OBDDs polynomially on CNFs
- Negates the existence of polynomial OBDD refutations with exponential resolution
- Provides insights into the relative efficiency of proof systems in propositional logic

## Abstract

A classical question of propositional logic is one of the shortest proof of a tautology. A related fundamental problem is to determine the relative efficiency of standard proof systems, where the relative complexity is measured using the notion of polynomial simulation. Presently, the state-of-the-art satisfiability algorithms are based on resolution in combination with search. An Ordered Binary Decision Diagram (OBDD) is a data structure that is used to represent Boolean functions. Groote and Zantema have proved that there is exponential separation between resolution and a proof system based on limited OBDD derivations. However, formal comparison of these methods is not straightforward because OBDDs work on arbitrary formulas, whereas resolution can only be applied to formulas in Conjunctive Normal Form (CNFs). Contrary to popular belief, we argue that resolution simulates OBDDs polynomially if we limit both to CNFs and thus answer negatively the open question of Groote and Zantema whether there exist unsatisfiable CNFs having polynomial OBDD refutations and requiring exponentially long resolution refutations.