Kernel methods in general and support vector machines in particular have been successful in various learning tasks on data represented in a single table. Much 'real-world ' data, however, is structured- it has no natural representation in a single table. Usually, to apply kernel methods to 'realworld' data, extensive pre-processing is performed toembed the data into areal vector space and thus in a single table. This survey describes several approaches ofdefining positive definite kernels on structured instances directly.

This paper provides a foundation for inductive learning based on the use of higherorder logic for knowledge representation. In particular, the paper (i) provides a systematic individuals-as-terms approach to knowledge representation for inductive learning, and demonstrates the utility of types and higher-order constructs for this purpose; (ii) gives a systematic way of constructing predicates for use in induced definitions; (iii) widens the applicability of decision-tree algorithms beyond the usual attribute-value setting to the classification of individuals with complex structure; and (iv) shows how to induce definitions which are comprehensible and have predictive power. The paper contains ten illustrative applications involving a variety of types to which a decision-tree learning system is applied. The e#ectiveness of the approach is further demonstrated by applying the learning system to two larger benchmark applications. 1 Introduction Inductive learning focuses on tec...

This paper provides a study of probabilistic modelling, inference and learning in a logic-based setting. We show how probability densities, being functions, can be represented and reasoned with naturally and directly in higher-order logic, an expressive formalism not unlike the (informal) everyday language of mathematics. We give efficient inference algorithms and illustrate the general approach with a diverse collection of applications. Some learning issues are also considered.

. This paper presents the case for the use of higher-order logic as a foundation for computational logic. A suitable polymorphicallytyped, higher-order logic is introduced and its syntax and proof theory briefly described. In addition, a metric space of closed terms suitable for knowledge representation purposes is presented. The approach to representing individuals is illustrated with some examples, as is the technique of programming with abstractions. The paper concludes by placing the results in the wider context of previous and current research in the use of higher-order logic in computational logic. 1 Introduction In 1974, Robert Kowalski published the seminal idea that predicate logic could be used as a programming language [Kow74]. The setting for Kowalski's idea was first-order logic, in fact, the Horn clause fragment of that logic. Since 1974, there has been an explosion of research activity that has pushed the fundamental idea in many di#erent directions. Once the idea that ...

Predicate construction is a guided search process: a space of predicates must be enumerated in some systematic way in order to find, according to some criterion, a suitable predicate for some purpose. Often the search space is large and heuristics are used to guide the search. Typically, predicate construction is studied and implemented in the context of first-order logic. In this paper, I outline the foundations of predicate construction in higherorder logic. The main contribution is a method of incremental construction of predicates on higher-order terms that represent individuals. Applications of the approach to machine learning are indicated. 1 1 Introduction Predicate construction is a guided search process: a space of predicates must be enumerated in some systematic way in order to find, according to some criterion, a suitable predicate for some purpose. Often the search space is large and heuristics are used to guide the search. Typically, predicate construction is studied and...

this paper is contained in [Llo00]. Accounts of the application of the logical setting to inductive 1

Introduction This extended abstract outlines a submission to The Predictive Toxicology Challenge for 20002001 [PTC]. The Challenge is to obtain models that predict the outcome of biological tests for the toxicity of chemicals using information related to chemical structure only. The models reported here are based on the approach outlined in [BGCL00]. (Much more detail is given in [BGCL01].) In essence, the learning system is a decision-tree system. However, it is rather more general than conventional decision-tree learners in that it allows the individuals that are to be classified to be represented as certain terms in a higher-order logic rather than as simple feature vectors. The logic allows the use of sets, multisets, lists, graphs and so on, to represent individuals, thereby capturing complex structural information about the individuals [Llo01]. This information is highly relevant to the main aim of the Challenge, which is to predict toxicity from the chemical structure

the definition of learning systems with