Klaus P. Jantke

Monotonic and Nonmonotonic Inductive Inference of Functions and Patterns

Monotonic and non-monotonic reasoning is introduced into inductive inference. In inductive inference, which is a mathematical theory of algorithmic learning from possibly incomplete information, monotonicity means to construct hypotheses somehow incrementally. whereas the necessity of non-monotonic reasoning indicates that during hypothesis formation considerable belief revisions may be required. Therefore, it is of a particular interest to find areas of inductive inference where monotonic construction of hypotheses is always possible. It turned out that in the area of inductive inference of total recursive functions monotonicity can rarely be guaranteed. These results are compared to the problem of inductively inferring text patterns from finite samples. For this area, there is a universal weakly monotonic inductive inference algorithm. The computability of a stronger algorithm which is developed depends on the decidability of the inclusion problem for pattern languages. This problems remains open. Unfortunately, the latter algorithm turns out to be inconsistent, i.e. it sometimes generates hypotheses not able to reflect the information they are build upon. Consistency and monotonicity can hardly be achieved simultaneously. It arises the question under which circumstances an inductive inference algorithm for learning text patterns can be both consistent and monotonic. This problem class is characterized by closedness under intersection.

Case-Based Representation and Learning of Pattern Languages

Abstract Pattern languages seem to suit case-based reasoning particularly well. Therefore, the problem of inductively learning pattern languages is paraphrased in a case-based manner. A careful investigation requires a formal semantics for case bases together with similarity measures in terms of formal languages. Two basic semantics are introduced and investigated. It turns out that representability problems are major obstacles for case-based learnability. Restricting the attention to the so-called proper patterns avoids these representability problems. A couple of learnability results for proper pattern languages are derived both for case-based learning from only positive data and for case-based learning from positive and negative data. Under the so-called competing semantics, we show that the learnability result for positive and negative data can be lifted to the general case of arbitrary patterns. Learning under the standard semantics from positive data is closely related to monotonic language learning.

Case-based learning in inductive inference

There is proposed a formalization of case-based learning in terms of recursion-theoretic inductive inference. This approach is directly derived from some recently published case-based learning algorithms. The intention of the present paper is to exhibit the relationship between case-based learning and inductive inference and to specify this relation with mathematical precision. In particular, it is the authors intention to invoke inductive inference results for pointing to the crucial questions in case-based learning which allow to improve the power of case-based learning algorithms considerably. There are formalized several approaches to case-based learning. First, they vary in the way of presenting cases to a learning algorithm. Second, they are different with respect to the underlying semantics of case bases together with similarity measures. Third, they are distinguished by the flexibility in using similarity functions. The investigations presented relate the introduced learning types to identification types in recursion-theoretic inductive inference.

Learning Languages by Collecting Cases and Tuning Parameters

We investigate the problem of case-based learning of formal languages. Case-based reasoning and learning is a currently booming area of artificial intelligence. The formal framework for case-based learning of languages has recently been developed by [JL93] in an inductive inference manner.

Inductive Completion for Transformation of Equational Specifications

The Knuth-Bendix completion procedure is a tool for algorithmically completing term rewriting systems which are operationally incomplete in the sense that the uniqueness of normal forms is not guaranteed. As the problem of operational completeness is undecidable, one may only expect a technique applicable to an enumerable number of cases. The Knuth-Bendix completion procedure may fail either by generating a critical pair which can not be oriented to form a new rewrite rule or by generating an infinite sequence of critical pairs to be introduced as new rewrite rules. The latter case is investigated. The basic idea is to invoke inductive inference techniques for abbreviating infinitely long sequences of rules by finitely many other rules. If simple syntactic generalization does not do, there will be automatically generated auxiliary operators. This is the key idea of the present paper. It contains a calculus of five learning rules for extending Knuth-Bendix completion procedures by inductive inference techniques. These rules are shown to be correct. The problem of completeness remains open.

Inductive Synthesis of Rewrite Rules as Program Synthesis

It is rather unrealistic to expect newly designed software specifications to be always complete. Naturally, there are several forms of incompleteness. We are not concerned with explicitly known incompleteness during an unfinished design process. Instead, the present approach is focussed on specifications which are either a little incomplete by mistake or intentionally incomplete as descriptions by examples are. Completing given specifications to an operationally complete form may be understood as program synthesis.