Authors: Frank Stephan^{*} and Thomas Zeugmann^{**}
Source:
Theoretical Computer Science Vol. 288, Issue 2, 2002, 309-341. (Special Issue for ALT '99). Abstract. Blum and Blum (1975) showed that a class B of suitable recursive approximations to the halting problem K is reliably EX-learnable but left it open whether or not B is in NUM. By showing B to be not in NUM we resolve this old problem. Moreover, variants of this problem obtained by approximating any given recursively enumerable set A instead of the halting problem K are studied. All corresponding function classes U(A) are still EX-inferable but may fail to be reliably EX-learnable, for example if A is non-high and hypersimple. Blum and Blum (1975) considered only approximations to K defined by monotone complexity functions. We prove this condition to be necessary for making learnability independent of the underlying complexity measure. The class B' of all recursive approximations to K generated by all total complexity functions is shown to be not even behaviorally correct learnable for a class of natural complexity measures. On the other hand, there are complexity measures such that B' is EX-learnable. A similar result is obtained for all classes U(A)'. For natural complexity measures, B is shown to be not robustly learnable, but again there are complexity measures such that B and, more generally, every class U(A) is robustly EX-learnable. This result extends the criticism of Jain, Smith and Wiehagen (1998), since the classes defined by artificial complexity measures turn out to be robustly learnable while those defined by natural complexity measures are not robustly learnable.
^{*}Supported by the Deutsche Forschungsgemeinschaft (DFG) under Heisenberg grant no.Ste 967/1-1.
^{**}
Supported by the
Grant-in-Aid for Scientific Research in Fundamental Areas from the
Japanese Ministry of Education,
Science, Sports, and Culture under grant no. 10558047.
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