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 lim-computablefunction 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 "not-so-nearly" minimal size, e.g., to be within a lim-computable 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 lim-computable 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 lim-computability, the power of the resultant learning criteria form finely graded, infinitely ramifying, infinite hierarchies intermediate between the computable and the lim-computable cases. Some of these hierarchies, for the natural notations determining them, are shown to be optimally tight. (shrink)

Identification of programs for computable functions from their graphs and identification of grammars for recursively enumerable languages from positive data are two extensively studied problems in the recursion theoretic framework of inductive inference.In the context of function identification, Freivalds et al. have shown that only those collections of functions, , are identifiable in the limit for which there exists a 1-1 computable numbering ψ and a discrimination function d such that1. for each , the number of indices i such (...) that ψi = ƒ is exactly one and2. for each , there are only finitely many indices i such that ƒ and ψi agree on the first d arguments.A similar characterization for language identification in the limit has turned out to be difficult. A partial answer is provided in this paper. Several new techniques are introduced which have found use in other investigations on language identification. (shrink)

Limiting identification of r.e. indexes for r.e. languages (from a presentation of elements of the language) and limiting identification of programs for computable functions (from a graph of the function) have served as models for investigating the boundaries of learnability. Recently, a new approach to the study of "intrinsic" complexity of identification in the limit has been proposed. This approach, instead of dealing with the resource requirements of the learning algorithm, uses the notion of reducibility from recursion theory to compare (...) and to capture the intuitive difficulty of learning various classes of concepts. Freivalds, Kinber, and Smith have studied this approach for function identification and Jain and Sharma have studied it for language identification. The present paper explores the structure of these reducibilities in the context of language identification. It is shown that there is an infinite hierarchy of language classes that represent learning problems of increasing difficulty. It is also shown that the language classes in this hierarchy are incomparable, under the reductions introduced, to the collection of pattern languages. Richness of the structure of intrinsic complexity is demonstrated by proving that any finite, acyclic, directed graph can be embedded in the reducibility structure. However, it is also established that this structure is not dense. The question of embedding any infinite, acyclic, directed graph is open. (shrink)

Limiting identification of r.e. indexes for r.e. languages and limiting identification of programs for computable functions have served as models for investigating the boundaries of learnability. Recently, a new approach to the study of "intrinsic" complexity of identification in the limit has been proposed. This approach, instead of dealing with the resource requirements of the learning algorithm, uses the notion of reducibility from recursion theory to compare and to capture the intuitive difficulty of learning various classes of concepts. Freivalds, (...) Kinber, and Smith have studied this approach for function identification and Jain and Sharma have studied it for language identification. The present paper explores the structure of these reducibilities in the context of language identification. It is shown that there is an infinite hierarchy of language classes that represent learning problems of increasing difficulty. It is also shown that the language classes in this hierarchy are incomparable, under the reductions introduced, to the collection of pattern languages. Richness of the structure of intrinsic complexity is demonstrated by proving that any finite, acyclic, directed graph can be embedded in the reducibility structure. However, it is also established that this structure is not dense. The question of embedding any infinite, acyclic, directed graph is open. (shrink)