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When series of computable functions with varying domains are computable
Author(s) -
Kalantari Iraj,
Welch Larry
Publication year - 2013
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.201200053
Subject(s) - computable function , lebesgue measure , mathematics , intersection (aeronautics) , computable analysis , series (stratigraphy) , domain (mathematical analysis) , measure (data warehouse) , function (biology) , computable number , discrete mathematics , open set , lebesgue integration , space (punctuation) , pure mathematics , computer science , mathematical analysis , data mining , paleontology , evolutionary biology , engineering , biology , aerospace engineering , operating system
In this paper we study the behavior of computable series of computable partial functions with varying domains (but each domain containing all computable points), and prove that the sum of the series exists and is computable exactly on the intersection of the domains when a certain computable Cauchyness criterion is met. In our point‐free approach, we name points via nested sequences of basic open sets, and thus our functions from a topological space into the reals are generated by functions from basic open sets to basic open sets. The construction of a function that produces the sum of a series requires working with an infinite array of pairs of basic open sets, and reconciling the varying domains. We introduce a general technique for using such an array to produce a set function that generates a well‐defined point function and apply the technique to a series to establish our main result. Finally, we use the main finding to construct a computable, and thus continuous, function whose domain is of Lebesgue measure zero and which is nonextendible to a continuous function whose domain properly includes the original domain. (We had established existence of such functions with domains of measure less than ε for any ɛ > 0 , in an earlier paper.)