Complexité d'ordre supérieur et analyse récursive

Hugo Férée 1
1 CARTE - Theoretical adverse computations, and safety
Inria Nancy - Grand Est, LORIA - FM - Department of Formal Methods
Abstract : While first order complexity is well defined and studied, higher order lacks a satisfactory notion of complexity. Such a theory already exists at order 2 and provides a complexity class analogue to usual polynomial time computable functions. This is already especially interesting in the domain of computable analysis, where real numbers or real functions for example can be represented by first order functions. In particular, there is a clear link between computability and continuity, and we also illustrate in the case of real operators that complexity can be related to some analytical properties. However, we prove that, from a complexity point of view, some mathematical spaces can not be faithfully represented by order 1 functions and require higher order ones. This result underlines that there is a need to develop a notion of complexity at higher types which will be, in particular but not only, useful to computable analysis. We have developed a computational model for higher type sequential computations based on a game semantics approach, where the application of two functions is represented by the game opposing two strategies. By defining the size of such strategies, we are able to define a robust and meaningful notion of complexity at all types, together with a class of polynomial time computable higher order functionals which seems to be a good candidate for a class of feasible functionals at higher types.
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Hugo Férée. Complexité d'ordre supérieur et analyse récursive. Complexité [cs.CC]. Université de Lorraine, 2014. Français. ⟨NNT : 2014LORR0173⟩. ⟨tel-01751160v2⟩

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