Functional languages can be enriched with logic variables to provide new computational features such as incremental construction of data structures. In this paper, we present a novel application for logic variables that highlights their importance: we argue that they are essential for efficient implementations of pure functional languages. This point is made by demonstrating that logic variables are required for explicating the process of demand propagation in lazy evaluation of functional programs. There are two applications of this result. For dataflow researchers, it offers a simple and efficient implementation of laziness on dataflow machines. For researchers investigating lazy graph reduction, it suggests new strictness analysis algorithms in which logic variables play an important role.