Learning Recursive Functions Refutably

Learning of recursive functions refutably means that for every recursive function, the learning machine has either to learn this function or to refute it, i.e., to signal that it is not able to learn it. Three modi of making precise the notion of refuting are considered. We show that the corresponding types of learning refutably are of strictly increasing power, where already the most stringent of them turns out to be of remarkable topological and algorithmical richness. All these types are closed under union, though in different strengths. Also, these types are shown to be different with respect to their intrinsic complexity; two of them do not contain function classes that are “most difficult” to learn, while the third one does. Moreover, we present characterizations for these types of learning refutably. Some of these characterizations make clear where the refuting ability of the corresponding learning machines comes from and how it can be realized, in general. For learning with anomalies refutably, we show that several results from standard learning without refutation stand refutably. Then we derive hierarchies for refutable learning. Finally, we show that stricter refutability constraints cannot be traded for more liberal learning criteria

On Learning of Functions Refutably

Learning of recursive functions refutably informally means that for every recursive function, the learning machine has either to learn this function or to refute it, that is to signal that it is not able to learn it. Three modi of making precise the notion of refuting are considered. We show that the corresponding types of learning refutably are of strictly increasing power, where already the most stringent of them turns out to be of remarkable topological and algorithmical richness. Furthermore, all these types are closed under union, though in different strengths. Also, these types are shown to be different with respect to their intrinsic complexity; two of them do not contain function classes that are “most difficult” to learn, while the third one does. Moreover, we present several characterizations for these types of learning refutably. Some of these characterizations make clear where the refuting ability of the corresponding learning machines comes from and how it can be realized, in general.For learning with anomalies refutably, we show that several results from standard learning without refutation stand refutably. From this we derive some hierarchies for refutable learning. Finally, we prove that in general one cannot trade stricter refutability constraints for more liberal learning criteria

Can Finite Samples Detect Singularities of Real-Valued Functions?

Consider the following type of problem: There is an unknown function, f : R n ! R m , there is also a black-box that on query x (2 R n ) returns f(x). Is there an algorithm that, using probes to the black-box, can figure out analytic information about f? (For an example: "Is f a polynomial? ", "Is f a second order differentiable at x = (0; 0; : : : ; 0)?" etc.). Clearly, for examples as these, if we bound the number of probes an algorithm has to settle for, no algorithm can carry the task. On the other hand, if one allows an infinite iteration of a `probe compute and guess' process, then, (quite surprisingly) for many such questions, there are algorithms that are guaranteed to be correct in all but finitely many of their guesses. We call such questions Decidable In the Limit, (DIL). We analyze the class of DIL problems and provide a necessary and sufficient condition for the membership of a decision problem in this class. We offer an algorithm for any DIL problem, and apply it to several types of learning tasks. We introduce a an extension of the usual Inductive Inference learning model - Inductive Inference with a Cheating Teacher. In this model the teacher may choose to present to the learner, not only a language belonging to the agreed - upon family of languages, but also an arbitrary language outside this family. In such a case we require that the learner will be able to eventually detect the faulty choice made by the teacher. We show that such strong type of learning is possible, and there exist learning algorithms that will fail only on arbitrarily small sets of faulty languages. Furthermore, if an a-priori probability distribution P , according to which f is being chosen, is available to the algorithm, then it can be strengthened into a finite A prelimi..