Calibrating Computational Feasibility by Abstraction Rank

We characterize computationally the functions provable in second order logic with set existence restricted to natural classes of rst order formulas. A classi cation of rst-order set-existence by implicational rank yields a natural hierarchy of complexity classes within the class of Kalmar-elementary functions: The functions over f0; 1g constructively provable using set existence for formulas of implicational rank k are precisely the functions computable in deterministic time O(expk(n)), where exp0 = [k( n:n ), and expk+1 = 2 exp k . In particular, set-existence for positive formulas yields exactly PTime. We thus obtain lean and natural formalisms for codifying feasible mathematics, which are expressive both in allowing second order de nitions and reasoning, and in incorporating equational programming and reasoning about program convergence in a direct and uncoded style. Through a formula-as-type morphism, we also obtain a link with lambda de nability, which we exhibit in the full paper: The functions over f0; 1g de nable in the polymorphic lambda calculus F2 over a base of type of words, using rst-order type-arguments of rank k, are precisely the functions computable in deterministic time O(expk(n)). 2 The poly-time case was proved (directly) in [15].