Tailoring Recursion to Characterize Non-Deterministic Complexity Classes Over Arbitrary Structures

TL;DR: This work provides new completeness results for geometrical problems when this structure is the set of real numbers with addition and order, and provides several machine independent characterizations of complexity classes over arbitrary structures.

TL;DR: In this article, the authors present a panorama of quelques-uns of persons who have expressed their opinions about the polynomiality of a certain notion of concurrence entre agents.

Abstract: ALAN COBHAM. The intrinsic computational difficulty of functions. Logic, methodology and philosophy of science, Proceedings of the 1964 International Congress, edited by Yehoshua BarHillel, Studies in logic and the foundations of mathematics, North-Holland Publishing Company, Amsterdam 1965, pp. 24-30. This is perhaps the best general discussion in print of the subject of the title. Although the interesting work of Blum (see the following review) demonstrates that much can be said on the subject which is almost completely independent of the methods used to compute functions, the present paper points out the need for a theory giving specific measures of the intrinsic computational difficulty of specific functions such as addition and multiplication. The author does not provide such a theory, but does state some results which form a beginning, and focuses attention on problems which deserve study. One such result generalizes Ritchie's work (Classes of predictably computable functions, XXVIII 252) and characterizes each class &", k ^ 3, of the Grzegorczyk hierarchy in terms of the time and storage required to compute its members by machines. Since there is no parallel to this theorem for k < 3 (at least in terms of computation time) the author introduces a class £f consisting of those functions which are computed by Turing machines in a number of steps bounded by a polynomial in the length (say, in decimal notation) of the input. He argues that i f is a natural class to consider, since the Turing machines could be replaced by any reasonable computing model and the class would remain unchanged. He states without proof the following interesting characterization of -S?. THEOREM. & is the least class of functions including the eleven functions s{(x) = 10* + /, / = 0, 1, • • •, 9, and x? (where ((y) is the decimal length of y), and closed under the operations of explicit transformation, composition, and limited recursion on notation (digit-by-digit recursion). Among the problems posed, one of the most interesting is to decide whether «? (of the Grzegorczyk hierarchy) is contained in £P. The author conjectures the function f(n) = the nth prime is in