Reasonable Space for the $\lambda$-Calculus, Logarithmically

Can the $\lambda$-calculus be considered as a reasonable computational model? Can we use it for measuring the time and space consumption of algorithms? While the literature contains positive answers about time, much less is known about space. This paper presents a new reasonable space cost model for the $\lambda$-calculus, based on a variant over the Krivine's abstract machine. For the first time, this cost model is able to account for logarithmic space. Moreover, we study the time behavior of our machine and show how to transport our results to the call-by-value $\lambda$-calculus