Higher-order complexity in analysis

International audienceWe present ongoing work on the development of complexity theory in analysis. Kawamura and Cook recently showed how to carry out complexity theory on the space C[0,1] of continuous real functions on the unit interval. It is done, as in computable analysis, by representing objects by first-order functions (from finite words to finite words, say) and by measuring the complexity of a second-order functional in terms of second-order polynomials. We prove that this framework cannot be directly applied to spaces that are not $\sigma$-compact. However, representing objects by higher-order functions (over finite words, say) makes it possible to carry out complexity theory on such spaces: for this purpose we develop the complexity of higher-order functionals. At orders above 3, our class of polynomial-time computable functionals strictly contains the class BFF of Buss, Cook and Urquhart