Limit computation of some discontinuous functions (数学解析の計算機上での理論的展開とその遂行可能性 研究集会報告集)

Co nputability on the continuum has been investigated over the years, but we are interested in the mathematical approach such as the one by Pour-El and Richards [2]. In their approach, classical mathematics is accepted. The point is to see how computable objects and operators look like in ordinary mathematics. Co nputabilities of real numbers and continuous (real) functions were defined by Grzegorczyk. Areal number is computable if it can be effectively represented by arational sequence. Acontinuous function is computable if it preserves sequential computability and it is effectively uniformly continuous. We do. however, compute and draw graphs of discontinuous function. We can let Mathematica draw graphs of such functions. The problem arising ill so doing is the computation of the value at ajump point. This is because it is not in general decidable if areal number is ajump point, that is., $a=0$ is not decidable even for acomputable $a$ . One approach to this problem was proposed by Pour-El and Richards. It was a $\mathrm{f}$ unctional approach, that is, afunction is regarded as computable if it can be effectively approximated by rational coefficient polynomials with respect to the norm of afunction space, such as aBanach space or aFrechet space. hi such acase., afunction is regarded as computable as apoint in aspace. This is sufficient in order to draw arough graph of the function, but does not supply us with information 011 computation of individual values. One way of computing the value at a ju mp point is to do it in terms of limiting recursive functional. Another is to change the topology of the $\mathrm{d}$ omain. Here we will report the former method. Of course, there are many other ways of dealing with the computability of discontinuous functions. but we will concentrate on our treatment. For aquick reference of Pour-El and Richards approach, [7] is available. In Section 2., acounterexample of acomputable sequence of real numbers whose values of the integer part function do not form acomputable sequence. In Section 3., the limiting recursive functional is defined according to $\mathrm{G}\mathrm{o}\mathrm{l}\mathrm{d}’.\mathrm{s}$ idea. Ill Section 4. examples of discontinuous functions which call be computed in terms of limiting recursive functional. This report is concluded by listing some problems yet to be worked out (Section 5) 数理解析研究所講究録 1286巻 2002年 79-84