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12
Inverse entailment and Progol
, 1995
"... This paper firstly provides a reappraisal of the development of techniques for inverting deduction, secondly introduces ModeDirected Inverse Entailment (MDIE) as a generalisation and enhancement of previous approaches and thirdly describes an implementation of MDIE in the Progol system. Progol ..."
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Cited by 631 (59 self)
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This paper firstly provides a reappraisal of the development of techniques for inverting deduction, secondly introduces ModeDirected 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 reassessment of previous techniques in terms of inverse entailment leads to new results for learning from positive data and inverting implication between pairs of clauses.
An Efficient Subsumption Algorithm for Inductive Logic Programming
 IN PROCEEDINGS OF THE ELEVENTH INTERNATIONAL CONFERENCE ON MACHINE LEARNING
, 1994
"... In this paper we investigate the efficiency of ` subsumption (` ` ), the basic provability relation in ILP. As D ` ` C is NPcomplete 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. ..."
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Cited by 31 (3 self)
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In this paper we investigate the efficiency of ` subsumption (` ` ), the basic provability relation in ILP. As D ` ` C is NPcomplete 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 klocal 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 klocal Horn clauses, an ILPproblem 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 ILPlearning algorithm, can be improved by these ideas.
Learning Recursive Theories in the Normal ILP Setting
, 2003
"... 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 separateandparallel ..."
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Cited by 13 (9 self)
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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 separateandparallel 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 laboratorysized and realworld 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.
Least Generalizations and Greatest Specializations of Sets of Clauses
 Journal of Artificial Intelligence Research
, 1996
"... 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 o ..."
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Cited by 12 (0 self)
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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 nonexistence 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 nontautologous functionfree clause (among other, not necessarily functionfree clauses). Secondly, we show that such a least generali...
The Subsumption Theorem in Inductive Logic Programming: Facts and Fallacies
 Advances in Inductive Logic Programming. IOS
, 1995
"... . 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 ..."
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Cited by 9 (2 self)
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. 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 wellknown refutationcompleteness of resolution is just a special case of this theorem. We also show that the subsumption theo...
The logic of learning: a brief introduction to Inductive Logic Programming
 University of Manchester
, 1998
"... 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 ..."
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Cited by 7 (0 self)
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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 for Several Forms of Resolution
 In Proc. CSN95
, 1996
"... 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 th ..."
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Cited by 3 (1 self)
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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 refutationcompleteness: the one can be proved from the other. The same is true for linear resolution. For input resolution, t...
The Equivalence of the Subsumption Theorem and the Refutationcompleteness for Unconstrained Resolution
 Proceedings of the Asean Computer Science Conference(ACSC95), volume 1023 of Lecture Notes in Computer Science
, 1995
"... 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]. ..."
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Cited by 3 (2 self)
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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 wellknown refutationcompleteness of resolution is an immediate consequence of this theorem. On the other hand, we also...
Learning in Clausal Logic: A Perspective on Inductive Logic Programming
 Computational Logic: Logic Programming and Beyond, volume 2407 of Lecture Notes in Computer Science
, 2002
"... 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 ..."
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Cited by 3 (0 self)
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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 topdown 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 adhoc 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 reallife applications. 1