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Parsimony Hierarchies for Inductive Inference
 Journal of Symbolic Logic
"... Freivalds defined an acceptable programming system independent criterion for learning programs for functions in which the final programs were required to be both correct and "nearly" minimal size, i.e, within a computable function of being purely minimal size. Kinber showed that this parsimony requi ..."
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Freivalds defined an acceptable programming system independent criterion for learning programs for functions in which the final programs were required to be both correct and "nearly" minimal size, i.e, within a computable function of being purely minimal size. Kinber showed that this parsimony requirement on final programs limits learning power. However, in scientific inference, parsimony is considered highly desirable. A limcomputable function is (by definition) one calculable by a total procedure allowed to change its mind finitely many times about its output. Investigated is the possibility of assuaging somewhat the limitation on learning power resulting from requiring parsimonious final programs by use of criteria which require the final, correct programs to be "notsonearly" minimal size, e.g., to be within a limcomputable function of actual minimal size. It is shown that some parsimony in the final program is thereby retained, yet learning power strictly increases. Considered, then, are limcomputable functions as above but for which notations for constructive ordinals are used to bound the number of mind changes allowed regarding the output. This is a variant of an idea introduced by Freivalds and Smith. For this ordinal notation complexity bounded version of limcomputability, the power of the resultant learning criteria form finely graded, infinitely ramifying, infinite hierarchies intermediate between the computable and the limcomputable cases. Some of these hierarchies, for the natural notations determining them, are shown to be optimally tight.
Counting Extensional Differences in BCLearning
 Proceedings of the 5th International Colloquium on Grammatical Inference (ICGI 2000), Springer Lecture Notes in A. I. 1891
, 2000
"... . Let BC be the model of behaviourally correct function learning as introduced by Barzdins [4] and Case and Smith [8]. We introduce a mind change hierarchy for BC, counting the number of extensional differences in the hypotheses of a learner. We compare the resulting models BCn to models from th ..."
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. Let BC be the model of behaviourally correct function learning as introduced by Barzdins [4] and Case and Smith [8]. We introduce a mind change hierarchy for BC, counting the number of extensional differences in the hypotheses of a learner. We compare the resulting models BCn to models from the literature and discuss confidence, team learning, and finitely defective hypotheses. Among other things, we prove that there is a tradeoff between the number of semantic mind changes and the number of anomalies in the hypotheses. We also discuss consequences for language learning. In particular we show that, in contrast to the case of function learning, the family of classes that are confidently BClearnable from text is not closed under finite unions. Keywords. Models of grammar induction, inductive inference, behaviourally correct learning. 1 Introduction Gold [10] introduced an abstract model of learning computable functions, where a learner receives increasing amounts of data ...
Counting Extensional Differences in BCLearning
"... Let BC be the model of behaviourally correct function learning as introduced by Barzdins [4] and Case and Smith [8]. We introduce a mind change hierarchy for BC, counting the number of extensional differences in the hypotheses of a learner. We compare the resulting models BC n to models from the lit ..."
Abstract
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Let BC be the model of behaviourally correct function learning as introduced by Barzdins [4] and Case and Smith [8]. We introduce a mind change hierarchy for BC, counting the number of extensional differences in the hypotheses of a learner. We compare the resulting models BC n to models from the literature and discuss confidence, team learning, and finitely defective hypotheses. Among other things, we prove that there is a tradeoff between the number of semantic mind changes and the number of anomalies in the hypotheses. We also discuss consequences for language learning. In particular we show that, in contrast to the case of function learning, the family of classes that are confidently BClearnable from text is not closed under finite unions.