This paper surveys recent results concerning the inference of deterministic finite automata (DFAs). The results discussed determine the extent to which DFAs can be feasibly inferred, and highlight a number of interesting approaches in computational learning theory. 1

Some extensions are considered of Gold's influential model of language learning by machine from positive data. Studied are criteria of successful learning featuring convergence in the limit to vacillation between several alternative correct grammars. The main theorem of this paper is that there are classes of languages that can be learned if convergence in the limit to up to (n+1) exactly correct grammars is allowed but which cannot be learned if convergence in the limit is to no more than n grammars, where the no more than n grammars can each make finitely many mistakes. This contrasts sharply with results of Barzdin and Podnieks and, later, Case and Smith, for learnability from both positive and negative data. A subset principle from a 1980 paper of Angluin is extended to the vacillatory and other criteria of this paper. This principle, provides a necessary condition for circumventing overgeneralization in learning from positive data. It is applied to prove another theorem to the eff...

We prove that the set of all recursive functions cannot be inferred using first-order queries in the query language containing extra symbols [+; !]. The proof of this theorem involves a new decidability result about Presburger arithmetic which is of independent interest. Using our machinery, we show that the set of all primitive recursive functions cannot be inferred with a bounded number of mind changes, again using queries in [+; !]. Additionally, we resolve an open question in [7] about passive versus active learning. 1) Introduction This paper presents new results in the area of query inductive inference (introduced in [7]); in addition, there are results of interest in mathematical logic. Inductive inference is the study of inductive machine learning in a theoretical framework. In query inductive inference, we study the ability of a Query Inference Machine 1 Supported, in part, by NSF grants CCR 88-03641 and 90-20079. 2 Also with IBM Corporation, Application Solutions...

A pattern is a finite string of constant and variable symbols. The language generated by a pattern is the set of all strings of constant symbols which can be obtained from the pattern by substituting non-empty strings for variables. We study the learnability of one-variable pattern languages in the limit with respect to the update time needed for computing a new single hypothesis and the expected total learning time taken until convergence to a correct hypothesis. Our results are as follows. First, we design a consistent and set-driven learner that, using the concept of descriptive patterns, achieves update time O(n 2 log n), where n is the size of the input sample. The best previously known algorithm for computing descriptive one-variable patterns requires time O(n 4 log n) (cf. Angluin [2]). Second, we give a parallel version of this algorithm that requires time O(log n) and O(n 3 = log n) processors on an EREW-PRAM. Third, using a modified version of the sequential algorithm a...

Abstract. It is empirically known that most incremental learning systems are order dependent, i.e. provide results that depend on the particular order of the data presentation. This paper aims at uncovering the reasons behind this, and at specifying the conditions that would guarantee order independence. It is shown that both an optimality and a storage criteria are sufficient for ensuring order independence. Given that these correspond to very strong requirements however, it is interesting to study necessary, hopefully less stringent, conditions. The results obtained prove that these necessary conditions are equally difficult to meet in practice. Besides its main outcome, this paper provides an interesting method to transform an history dependent bias into an history independent one. 1

It was previously shown by Barzdin and Podnieks that one does not increase the power of learning programs for functions by allowing learning algorithms to converge to a finite set of correct programs instead of requiring them to converge to a single correct program. In this paper we define some new, subtle, but natural concepts of mind change complexity for function learning and show that, if one bounds this complexity for learning algorithms, then, by contrast with Barzdin and Podnieks result, there are interesting and sometimes complicated tradeoffs between these complexity bounds, bounds on the number of final correct programs, and learning power. CR Classification Number: I.2.6 (Learning – Induction). 1

In the past 40 years, research on inductive inference has developed along different lines, e.g., in the formalizations used, and in the classes of target concepts considered. One common root of many of these formalizations is Gold’s model of identification in the limit. This model has been studied for learning recursive functions, recursively enumerable languages, and recursive languages, reflecting different aspects of machine learning, artificial intelligence, complexity theory, and recursion theory. One line of research focuses on indexed families of recursive languages — classes of recursive languages described in a representation scheme for which the question of membership for any string in any of the given languages is effectively decidable with a uniform procedure. Such language classes are of interest because of their naturalness. The survey at hand picks out important studies on learning indexed families (including basic as well as recent research), summarizes and illustrates the corresponding results, and points out links to related fields such as grammatical inference, machine learning, and artificial intelligence in general.

this paper initiates a study in which it is demonstrated that certain concepts (represented by functions) can be learned, but only in the event that certain relevant subconcepts (also represented by functions) have been previously learned. In other words, the Soar project presents empirical evidence that learning how to learn is viable for computers and this paper proves that doing so is the only way possible for computers to make certain inferences.

This paper deals with two problems: 1) what makes languages to be learnable in the limit by natural strategies of varying hardness; 2) what makes classes of languages to be the hardest ones to learn. To quantify hardness of learning, we use intrinsic complexity based on reductions between learning problems. Two types of reductions are considered: weak reductions mapping texts (representations of languages) to texts, and strong reductions mapping languages to languages. For both types of reductions, characterizations of complete (hardest) classes in terms of their algorithmic and topological potentials have been obtained. To characterize the strong complete degree, we discovered a new and natural complete class capable of "coding" any learning problem using density of the set of rational numbers. We have also discovered and characterized rich hierarchies of degrees of complexity based on "core" natural learning problems. The classes in these hierarchies contain "mul...

The present paper deals with the best-case, worst-case and average-case behavior of Lange and Wiehagen's (1991) pattern language learning algorithm with respect to its total learning time. Pattern languages have been introduced by Angluin (1980) and are defined as follows: Let A = f0; 1; : ::g be any non--empty finite alphabet containing at least two elements. Furthermore, let X = fx i i 2 INg be an infinite set of variables such that A " X =;. Patterns are non--empty strings over A [ X. L(), the language generated by pattern is the set of strings which can be obtained by substituting non-null strings from A 3 for the variables of the pattern. Lange and Wiehagen's (1991) algorithm learns the class of all pattern languages in the limit from text. We analyze this algorithm with respect to its total learning time behavior, i.e., the overall time taken by the algorithm until convergence. For every pattern containing k different variables it is shown that the total learning time is O(jj 2 log jAj (jAj + k)) in the best-case and unbounded in the worst-case. Furthermore, we estimate the expectation of the total learning time. In particular, it is shown that Lange and Wiehagen's algorithm possesses an expected total learning time of O(2 k