What is an Efficient Implementation of the λ-calculus ?

We propose to measure the efficiency of any implementation of the λcalculus as a function of a new parameter ν, that is itself a function of any λ-expression. Complexity is expressed here as a function of ν just as runtime is expressed as a function of the input size n in ordinary analysis of algorithms. This enables implementations to be compared for worst case efficiency. We argue that any implementation must have complexity Ω(ν), i.e. a linear lower bound. Furthermore, we show that implementations based upon Turner Combinators or Hughes Super-combinators have complexities 2, i.e. an exponential lower bound. It is open whether any implementation of polynomial complexity, ν, exists, although some implementations have been implicitly claimed to have this complexity.