Some theorems on passing from local to global presence of properties of functions

DOI10.2168/LMCS-12(4:3)2016zbMATH Open1445.03052arXiv1609.04254MaRDI QIDQ2974787

Publication date: 11 April 2017

Published in: Logical Methods in Computer Science (Search for Journal in Brave)

Abstract: When given a class of functions and a finite collection of sets, one might be interested whether the class in question contains any function whose domain is a subset of the union of the sets of the given collection and whose restrictions to all of them belong to this class. The collections with the formulated property are said to be strongly join permitting for the given class (the notion of join permitting collection is defined in the same way, but without the words "a subset of"). Three theorems concerning certain instances of the problem are proved. A necessary and sufficient condition for being strongly join permitting is given for the case when, for some

n

, the class consists of the potentially partial recursive functions of

n

variables, and the collection consists of sets of

n

-tuples of natural numbers. The second theorem gives a sufficient condition for the case when the class consists of the continuous partial functions between two given topological spaces, and the collection consists of subsets of the first of them (the condition is also necessary under a weak assumption on the second one). The third theorem is of a similar character but, instead of continuity, it concerns computability in the spirit of the one in effective topological spaces.

Full work available at URL: https://arxiv.org/abs/1609.04254

zbMATH Keywords

approximation separability enumeration continuous function partial recursive function restriction recursively enumerable set topological space recursive function primitive recursive function enumeration operator computable function recursive operator

Mathematics Subject Classification ID

Continuous maps (54C05) Computation over the reals, computable analysis (03D78)