This paper firstly provides a re-appraisal of the development of techniques for inverting deduction, secondly introduces Mode-Directed Inverse Entailment (MDIE) as a generalisation and enhancement of previous approaches and thirdly describes an implementation of MDIE in the Progol system. Progol is implemented in C and available by anonymous ftp. The re-assessment of previous techniques in terms of inverse entailment leads to new results for learning from positive data and inverting implication between pairs of clauses.

In this paper we investigate the efficiency of `-- subsumption (` ` ), the basic provability relation in ILP. As D ` ` C is NP--complete even if we restrict ourselves to linked Horn clauses and fix C to contain only a small constant number of literals, we investigate in several restrictions of D. We first adapt the notion of determinate clauses used in ILP and show that `--subsumption is decidable in polynomial time if D is determinate with respect to C. Secondly, we adapt the notion of k--local Horn clauses and show that `-- subsumption is efficiently computable for some reasonably small k. We then show how these results can be combined, to give an efficient reasoning procedure for determinate k--local Horn clauses, an ILP--problem recently suggested to be polynomial predictable by Cohen (1993) by a simple counting argument. We finally outline how the `--reduction algorithm, an essential part of every lgg ILP--learning algorithm, can be improved by these ideas.

The main operations in Inductive Logic Programming (ILP) are generalization and specialization, which only make sense in a generality order. In ILP, the three most important generality orders are subsumption, implication and implication relative to background knowledge. The two languages used most often are languages of clauses and languages of only Horn clauses. This gives a total of six different ordered languages. In this paper, we give a systematic treatment of the existence or non-existence of least generalizations and greatest specializations of finite sets of clauses in each of these six ordered sets. We survey results already obtained by others and also contribute some answers of our own. Our main new results are, firstly, the existence of a computable least generalization under implication of every finite set of clauses containing at least one non-tautologous function-free clause (among other, not necessarily function-free clauses). Secondly, we show that such a least generali...

Induction of recursive theories in the normal ILP setting is a difficult learning task whose complexity is equivalent to multiple predicate learning. In this paper we propose computational solutions to some relevant issues raised by the multiple predicate learning problem. A separate-andparallel -conquer search strategy is adopted to interleave the learning of clauses supplying predicates with mutually recursive definitions. A novel generality order to be imposed on the search space of clauses is investigated, in order to cope with recursion in a more suitable way. The consistency recovery is performed by reformulating the current theory and by applying a layering technique, based on the collapsed dependency graph. The proposed approach has been implemented in the ILP system ATRE and tested on some laboratory-sized and real-world data sets. Experimental results demonstrate that ATRE is able to learn correct theories autonomously and to discover concept dependencies. Finally, related works and their main differences with our approach are discussed.

. The subsumption theorem is an important theorem concerning resolution. Essentially, it says that if a set of clauses \Sigma logically implies a clause C, then either C is a tautology, or a clause D which subsumes C can be derived from \Sigma with resolution. It was originally proved in 1967 by Lee. In Inductive Logic Programming, interest in this theorem is increasing since its independent rediscovery by Bain and Muggleton. It provides a quite natural "bridge" between subsumption and logical implication. Unfortunately, a correct formulation and proof of the subsumption theorem are not available. It is not clear which forms of resolution are allowed. In fact, at least one of the current forms of this theorem is false. This causes a lot of confusion. In this paper we give a careful proof of the subsumption theorem for unconstrained resolution, and show that the well-known refutationcompleteness of resolution is just a special case of this theorem. We also show that the subsumption theo...

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This paper is intended to provide an introduction to ILP. We will both review some of the established approaches to Horn clause induction (Section 2), and recent work on induction of integrity constraints (Section 3). 2 Horn clause induction

The Subsumption Theorem is the following completeness result for resolution: if # is a set of clauses and C is a clause, then # logically implies C i# C is a tautology, or there exists a clause D which subsumes C, and which can be derived from # by some form of resolution. Di#erentversions of this theorem exist, depending on the kind of resolution we use. It provides a more #direct" form of completeness than the better known refutationcompleteness, which often makes the Subsumption Theorem better suited for theoretical research. In this paper weinvestigate for which forms of resolution the theorem holds, and for which it does not. We collect results earlier obtained by others, and contribute some results of our own. The main results of the paper are as follows. For #unconstrained" resolution, the Subsumption Theorem holds, and is equivalent to the refutation-completeness: the one can be proved from the other. The same is true for linear resolution. For input resolution, t...

The subsumption theorem is an important theorem concerning resolution. Essentially, it says that a set of clauses \Sigma logically implies a clause C, iff C is a tautology, or a clause D which subsumes C can be derived from \Sigma with resolution. It was originally proved in 1967 by Lee in [Lee67]. In Inductive Logic Programming, interest in this theorem is increasing since its independent rediscovery by Bain and Muggleton [BM92]. It provides a quite natural "bridge" between subsumption and logical implication. Unfortunately, a correct formulation and proof of the subsumption theorem are not available. It is not clear which forms of resolution are allowed. In fact, at least one of the current forms of this theorem is false. This causes a lot of confusion. In this paper, we give a careful proof of the subsumption theorem for unconstrained resolution, and show that the well-known refutationcompleteness of resolution is an immediate consequence of this theorem. On the other hand, we also...

Abstract. Inductive logic programming is a form of machine learning from examples which employs the representation formalism of clausal logic. One of the earliest inductive logic programming systems was Ehud Shapiro’s Model Inference System [90], which could synthesise simple recursive programs like append/3. Many of the techniques devised by Shapiro, such as top-down search of program clauses by refinement operators, the use of intensional background knowledge, and the capability of inducing recursive clauses, are still in use today. On the other hand, significant advances have been made regarding dealing with noisy data, efficient heuristic and stochastic search methods, the use of logical representations going beyond definite clauses, and restricting the search space by means of declarative bias. The latter is a general term denoting any form of restrictions on the syntactic form of possible hypotheses. These include the use of types, input/output mode declarations, and clause schemata. Recently, some researchers have started using alternatives to Prolog featuring strong typing and real functions, which alleviate the need for some of the above ad-hoc mechanisms. Others have gone beyond Prolog by investigating learning tasks in which the hypotheses are not definite clause programs, but for instance sets of indefinite clauses or denials, constraint logic programs, or clauses representing association rules. The chapter gives an accessible introduction to the above topics. In addition, it outlines the main current research directions which have been strongly influenced by recent developments in data mining and challenging real-life applications. 1