C o - l e a r n a b i l i t y and F I N - i d e n t i f i a b i l i t y of e n u m e r a b l e classes of t o t a l r e c u r s i v e functions * Rfisirj~ Freivalds t
Dace Gobleja ~
MarekKarpinski w
Carl H . S m i t h ~
Abstract Co-learnability is an inference process where instead of producing the final result,'the strategy produces all the natural numbers but one, and the omitted number is an encoding of the correct result. It has been proved in [1] that co-learnability of Goedel numbers is equivalent to EX-identifiability. We consider co-learnability of indices in recursively enumerable (r.e.) numberings. The power of co-learnability depends on the numberings used. Every r.e. class of total recursive functions is co-learnable in some r.e. numbering. FIN-identifiable classes are co-learnable in all r.e. numberings, and classes containing a function being accumulation point are not colearnable in some r.e. numberings. Hence it was conjectured in [1] that only FIN-identifiable classes are co-learnable in all r.e. numberings. The conjecture is disproved in this paper using a sophisticated construction by V.L.Selivanov.
*The research of the two first authors was supported by the grant No. 93.599 from Latvian Science Council. The fourth author is supported, in part, by National Science Foundation Grant 9301339. ~Institute of Mathematics and Computer Science, University of Latvia, Ralna bulv. 29, Riga, Latvia, e-mail: {rusins}@mii.lu.lv tInstitute of Mathematics and Computer Science, University of Latvia, Raina bulv. 29, Riga, Latvia, e-maih {dgobleja}~mii.lu. lv w of Computer Science, University of Bonn, 53117 Bonn, and the International Computer Science Institute, Berkeley, California. Research supported in part by the DFG Grant KA 673/4-1, by the ESPRIT BR Grants 7097 and ECUS030, e-mail: karpinaki@cs .bonn. edu 82 of Computer Science, University of Maryland, CoUege Park, MD, U.S.A., e-mail:
[email protected]
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In practical problems of machine learning rather often the learning algorithm starts from a large finite set of possible formulas, refute all of them but one, and produces the remaining one as the final result of the learning process. A natural recursion-theoretical counterpart of this approach was considered in [1] and named co-learning. We say that a strategy F co-learns r-indices for U (where r is a numbering for U or for a superclass of U) if for arbitrary function f in U, the strategy F outputs all natural numbers but one, and the missing one is a correct r-index for f. This definition reminds the well-known definition of the finite identification. We say that a partial recursive strategy G FIN-identifies r-indices for U if for arbitrary function f in U, the strategy G outputs a natural number being a correct r-index for f. Since the partial recursive strategy G outputs the result after having seen only a finite initial fragment of the graph of the function f , it is obvious that the FIN-identifiability can be defined in the following way as well, and the two definitions for recursively enumerable classes U are equivalent. A strategy H FIN-identifies U if for arbitrary function f in U, the strategy G outputs a natural number being a Goedel number of f. We consider only recursively enumerable classes of total recursive functions in this paper. A class U of total recursive functions is called recursively enumerabte if there is a total recursive function of two arguments r(i, x) such that
u = {r0, r l , r 2 , . . . } for r (x) =
r(i,x).
The abovementioned 2-argument function r ( i , x ) provides a recursively enumerable numbering r for the class U. For every non-trivial r.e. class U there are very many r.e. numberings, and their properties can essentially differ. We say that a numbering c~ is reducible to the numbering r of the same class U ((7 < r) if there is a total recursive function h such that, for arbitrary i, the functions ai and rh(i) are the same. Reducibility of the numberings are traditionally interpreted as the existence of a compiler which transforms arbitrary programs in the programming language cr into equivalent programs in the programming language r.
We consider equivalence of numberings (~ ~ r) defined as
< r) & (r
