Showing papers by "John Case published in 2011"

Optimal language learning from positive data

Abstract: [email protected]?s original paper on inductive inference introduced a notion of an optimal learner. Intuitively, a learner identifies a class of objects optimally iff there is no other learner that: requires as little of each presentation of each object in the class in order to identify that object, and, for some presentation of some object in the class, requires less of that presentation in order to identify that object. Beick considered this notion in the context of function learning, and gave an intuitive characterization of an optimal function learner. Jantke and Beick subsequently characterized the classes of functions that are algorithmically, optimally identifiable. Herein, [email protected]?s notion is considered in the context of language learning. It is shown that a characterization of optimal language learners analogous to [email protected]?s does not hold. It is also shown that the classes of languages that are algorithmically, optimally identifiable cannot be characterized in a manner analogous to that of Jantke and Beick. Other interesting results concerning optimal language learning include the following. It is shown that strong non-U-shapedness, a property involved in [email protected]?s characterization of optimal function learners, does not restrict algorithmic language learning power. It is also shown that, for an arbitrary optimal learner F of a class of languages L, F optimally identifies a subclass K of L iff F is class-preserving with respect to K.

Automatic learners with feedback queries

Abstract: Automatic classes are classes of languages for which a finite automaton can decide whether a given element is in a set given by its index. The present work studies the learnability of automatic families by automatic learners which, in each round, output a hypothesis and update a long term memory, depending on the input datum, via an automatic function, that is, via a function whose graph is recognised by a finite automaton. Many variants of automatic learners are investigated: where the long term memory is restricted to be the just prior hypothesis whenever this exists, cannot be of size larger than the size of the longest example or has to consist of a constant number of examples seen so far. Furthermore, learnability is also studied with respect to queries which reveal information about past data or past computation history; the number of queries per round is bounded by a constant. These models are generalisations of the model of feedback queries, given by Lange, Wiehagen and Zeugmann.

Measuring Learning Complexity with Criteria Epitomizers

Abstract: In prior papers, beginning with the seminal work by Freivalds et al. 1995, the notion of intrinsic complexity is used to analyze the learning complexity of sets of functions in a Gold-style learning setting. Herein are pointed out some weaknesses of this notion. Offered is an alternative based on epitomizing sets of functions -- sets, which are learnable under a given learning criterion, but not under other criteria which are not at least as powerful. To capture the idea of epitomizing sets, new reducibility notions are given based on robust learning (closure of learning under certain classes of operators). Various degrees of epitomizing sets are characterized as the sets complete with respect to corresponding reducibility notions! These characterizations also provide an easy method for showing sets to be epitomizers, and they are, then, employed to prove several sets to be epitomizing. Furthermore, a scheme is provided to generate easily very strong epitomizers for a multitude of learning criteria. These strong epitomizers are so-called self-learning sets, previously applied by Case & Koetzing, 2010. These strong epitomizers can be generated and employed in a myriad of settings to witness the strict separation in learning power between the criteria so epitomized and other not as powerful criteria!

Automatic learning of subclasses of pattern languages

Abstract: Automatic classes are classes of languages for which a finite automaton can decide membership question for the languages in the class, in a uniform way, given an index for the language. For alphabet size of at least 4, every automatic class of erasing pattern languages is contained, for some constant n, in the class of all languages generated by patterns which contain (1) every variable only once and (2) at most n symbols after the first occurrence of a variable. It is shown that such a class is automatically learnable using a learner with long-term memory bounded by the length of the first example seen. The study is extended to show the learnability of related classes such as the class of unions of two pattern languages of the above type.

Rice and Rice-Shapiro Theorems for transfinite correction grammars

Abstract: Hay and, then, Johnson extended the classic Rice and Rice-Shapiro Theorems for computably enumerable sets, to analogs for all the higher levels in the finite Ershov Hierarchy. The present paper extends their work (with some motivations presented) to analogs in the transfinite Ershov Hierarchy. Some of the transfinite cases are done for all transfinite notations in Kleene's important system of notations, . Other cases are done for all transfinite notations in a very natural, proper subsystem of , where has at least one notation for each constructive ordinal. In these latter cases it is open as to what happens for the entire set of transfinite notations in .

Properties Complementary to Program Self-Reference

Abstract: In computability theory, program self-reference is formalized by the not-necessarily-constructive form of Kleene's Recursion Theorem (krt). In a programming system in which krt holds, for any preassigned, algorithmic task, there exists a program that, in a sense, creates a copy of itself, and then performs that task using the self-copy. Interpreted in this way, such self-copying programs have usable self-knowledge. Herein, properties complementary to krt are considered. Of particular interest are those properties involving the implementation of control structures. One main result is that no property involving the implementation of denotational control structures is complementary to krt. This is in contrast to a result of Royer, which showed that implementation of if-then-else — a denotational control structure — is complementary to the constructive form of Kleene's Recursion Theorem. Examples of non-denotational control structures whose implementation is complementary to krt are then given. Some such control structures so nearly resemble denotational control structures that they might be called quasi-denotational.