Section: New Results

Computability, Complexity and Topology

Computability and topology are closely related as computability assumptions impose topological restrictions: on a topological space, computable functions are continuous and continuous functions are computable relative to some oracle. In the same way, complexity assumptions as bounds on the computation time impose analytical restrictions, but in a way that is not understood yet. For functions from the real numbers to the real numbers, it is known that polynomial-time computable functions correspond to functions with a polynomial modulus of continuity. However for functions on other spaces no such correspondence is known. We investigate the particular case of norms on the space of continuous real functions defined on the unit interval. We introduce analytical characteristics of a norm, namely its dependency on points and the concept of relevant points, and use them to characterize the polynomial-time computable norms. This work was presented at LICS 2013 [19] . A full version including other results on non-deterministic complexity classes is currently submitted [28] .

While computability theory is well-developed and understood on large classes of topological spaces, complexity theory in analysis is still in its infancy. We argue that the usual way of representing mathematical objects by functions from finite strings to finite strings (order 1 functions) is not appropriate for general spaces. We show that as soon as the space becomes large in a topological sense, it cannot be represented by order 1 functions in a way that respects complexity notions, so we propose to represent objects using higher order functions over finite strings. However higher order complexity theory is not well-understood. The only known class to date is BFF, the class of Basic Feasible Functionals, which does not enjoy nice properties: some intuitively feasible functionals do not belong to the class. We develop a new way of carrying out complexity theory at higher order types, using an adaptation of game semantics. A preliminary version of this work was presented at CCA 2013 [26] .

As mentioned before, computable functions must be continuous. It gives a simple way of proving that some operator is not computable by showing that it is discontinuous. We recall that a function $f$ is computable if there is a single oracle Turing machine M that on each $x$ given by an oracle, computes $f\left(x\right)$. The following weaker notion is also interesting: a function $f$ preserves computability if for each computable $x$, $f\left(x\right)$ is computable. Preservation of computability no more implies continuity, so there is no topological argument to show that some operator does not preserve computability. We develop a strong notion of discontinuity and prove a general result stating that this notion of discontinuity prevents preservation of computability. We apply this result to solve an open problem about the non-computability of the ergodic decomposition. We show that many classical constructions in computability theory are instances of our result. Hence we exhibit deeper connections between computability and topology. The work has been accepted at STACS 2014 [22] . A partial result was published in [13] .