Reasonable Space for the λ-Calculus, Logarithmically

Multi Types and Reasonable Space (Long Version)

“…theory-independent. We invite the reader to consult Accattoli [2], Accattoli et al [9] for more details. 1 Slot and van Emde Boas speak of machines rather than computational theories because they are mainly discussing Turing and random access machines.…”

“…To study logarithmic space, indeed, one needs a separation between input space and work space, which cannot be modeled taking as space the size of -terms. Very recently, Accattoli, Dal Lago, and Vanoni have also shown that the space used by the Space KAM, a variant over the KAM, is a reasonable space cost model accounting for logarithmic space [9], solving the longstanding open problem. The Space KAM tweaks the KAM in two orthogonal ways:…”

“…Environments and stacks are instead simply unstructured strings. These changes are somewhat counter-intuitive but they are in fact mandatory for reasonable space, see [9] for extensive discussions.…”

“…Roughly, the actual space is obtained by multiplying such a number by the logarithm of the size of the initial term, which corresponds to the size of the code pointer in every closure. More precisely, the simulation of Turing machines within a linear overhead in [9] rests on some technicalities for which not all code pointers have the same size.…”

“…Such a choice of course degrades the time performance, but here we are mainly interested in space behavior. All these implementation choices, necessary to obtain the space reasonability of the Space KAM, are discussed at length in Accattoli et al [9].…”