The Complexity of Composition: New Approaches to Depth and Space

The __composition__ of two given functions f and g is a fixed way of combining them into a single new function f ◦ g. A __composition theorem__ for a complexity measure s(·) states that s(f ◦ g) ≈ s(f) + s(g); in other words, computing the combined function f ◦ g is no easier (with respect to s) than computing f and g individually. If true, then we would gain a natural approach towards proving lower bounds on s(F) for an explicit F by repeatedly composing smaller hard functions in such a way that their complexities are additive by the composition theorem. We study the composition problem for two measures: __formula depth__ and __space complexity.__ The KRW conjecture [KRW95] states that the formula depth required to compute f ◦g is approximately depth(f)+depth(g), where f ◦g is the function given by replacing every input variable of f with a disjoint copy of g. This conjecture is known to imply NC1 ⊊ P. We work towards proving this conjecture by way of proving new __lifting theorems__ from query complexity to communication complexity. Our new proof of the classic result of Raz and McKenzie [RM99] allows us to intimately connect lifting to combinatorics, and in doing so we provide a novel improvement to a key parameter called the gadget size. This result also allows us to prove conditional hardness for __automating__ the __Cutting Planes__ proof system. Cook et al. [CMW+12] introduced the __tree evaluation problem__ as a way of showing L ⊊ P; their central conjecture partially relies on showing that the space to compute a function f while remembering the output of another function g is approximately the space to compute f plus the size of g’s output. This conjecture, which we call the __z-f conjecture__, was challenged by Buhrman et al. [BCK+14], who defined a new type of space computation called __catalytic computing__ and used it to show that composition does not hold for space-bounded computation in some settings. We give further evidence against composition by using the catalytic computing framework to give the first upper bounds on tree evaluation since the problem’s definition in [CMW+12], refuting their central conjecture. Using these techniques we also prove new results on __amortized__ computation by improving constructions for __catalytic branching programs__.

Récupéré en direct depuis OpenAlex et désinversé. Les résumés ne sont pas conservés dans cette base de données : les index inversés représentent 8,6 Go des 9,3 Go de texte de la base, et le serveur dispose de 13 Go libres.