L. De Raedt and W. Van Laer. Inductive constraint logic. In Klaus P. Jantke, Takeshi Shinohara, and Thomas Zeugmann, editors, Proceedings of the 6th International Workshop on Algorithmic Learning Theory, volume 997 of Lecture Notes in Artificial Intelligence, pages 80--94. Springer-Verlag, 1995.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....complex clause is generated (with many variables and many literals sharing variables) and then refined; a large proportion of the 3 The Bongard data sets contain problems related to those used by M. Bongard [6] for research on pattern recognition and were introduced as an ILP benchmark by [9]. LA tree size original disjoint packed speedup ratio (nodes) total comp exec total comp exec net exec 592 examples 0 27 2.0 4.2 2.26 0.52 1.79 0.35 0.12 1.1 4.3 1 15 4.95 9.13 5.06 1.54 3.04 0.77 0.21 1.6 7.3 2 11 15.12 17.81 9.12 4.06 6.01 1.79 0.18 2.5 22.6 1194 examples 0 58 6.29 11.97 ....

L. De Raedt and W. Van Laer. Inductive constraint logic. In Klaus P. Jantke, Takeshi Shinohara, and Thomas Zeugmann, editors, Proceedings of the 6th International Workshop on Algorithmic Learning Theory, volume 997 of Lecture Notes in Artificial Intelligence, pages 80--94. Springer-Verlag, 1995.

....deals with examples that are interpretations. 1 Introduction Clustering is an important datamining task. Several good clustering algorithms exist in the attribute value representation (see e.g. COBWEB [8] AUTOCLASS [11] However, many recent classi cation algorithms (such as TILDE [1] ICL [5]) use a more expressive rst order representation. For the clustering task, the use of a rst order representation is more dicult because the search for a good clustering is typically guided by one of two heuristics: an evaluation measure (also called objective function) on clusters or a distance ....

L. De Raedt and W. Van Laer. Inductive constraint logic. In Klaus P. Jantke, Takeshi Shinohara, and Thomas Zeugmann, editors, Proceedings of the 6th International Workshop on Algorithmic Learning Theory, volume 997 of Lecture Notes in Articial Intelligence, pages 80-94. Springer-Verlag, 1995.

....including standard procedures) and a toy grammar learning problem. The grammar problem introduces the use of background information to qualify grammar rules that may be of independent interest for other settings. We also describe quantitative experiments with examples drawn for Bongard problems (De Raedt Van Laer, 1995). These experiments study the performance of the system by varying the size of programs and example set as well as comparing it to other systems. In all these good performance is demonstrated by LogAn H . 2. Learning from Interpretations Learning from interpretations has seen growing interest in ....

....of the system by varying the size of programs and example set as well as comparing it to other systems. In all these good performance is demonstrated by LogAn H . 2. Learning from Interpretations Learning from interpretations has seen growing interest in recent years (De Raedt Dzeroski, 1994; De Raedt Van Laer, 1995; Blockeel De Raedt, 1998) Unlike the standard ILP setting, where examples are atoms, examples in this framework are interpretations of the underlying first order language (i.e. first order structures) We introduce the setup informally through examples. Formal definitions can be found in ....

[Article contains additional citation context not shown here]

De Raedt, L., & Van Laer, W. (1995). Inductive constraint logic. Proceedings of the Conference on Algorithmic Learning Theory (pp. 80--94). LNAI 997.

....perform full ACL we must incorporate the learning of integrity constraints to ensure that negative examples are not abductively derivable in the learned theory. One way to address this issue is to integrate into the algorithm a system for characterizing induction, such as Claudien [RB93] or ICL [RL95] which will take as input together with the background theory and positive (or all) training examples, the abductive assumptions Delta (or Delta ) and generate clauses to be used as integrity constraints. 5 Experiments The main purpose of the experiments reported in this section is to ....

L. De Raedt and W. Van Lear. Inductive Constraint Logic. In Proceedings of the 5th International Workshop on Algorithmic Learning Theory, 1995.

....queries accepts specifications of the form rmode(n : A 1 ; A n ) where the A i are atoms 1 . Typically, n = 1 and the specification will be of the form rmode(n : atom) This 1 Alternatively, Warmr s language bias can be specified in Dlab format [19] as in Claudien [16] and ICL [18]. format, originally proposed for Progol [42] and later adapted to Tilde [8] indicates which atoms can be added to a query, the maximal number of times the atom can be added (n 0) and the modes and types of the variables in it. A variable V in input mode, denoted with V , has to occur ....

....interpretations paradigm has proven to be particularly suitable for the design of upgrades to popular attribute value learning techniques. In that respect, Apriori Warmr is only one of the more recent additions to a sequence of similar upgrades [15] Explora [32] Claudien [16] CN2 [10] ICL [18], C4.5 [49] Tilde [8] and [33] C0.5 [14] 6.2 Clausal discovery Association rules A 1 : A k ) A k 1 : A n , as introduced in Section 3.1.2, can easily be confused with a clauses A 1 : A k A k 1 : A n : both are interpreted as if then rules with atoms A i . We first ....

L. De Raedt and W. Van Laer. Inductive constraint logic. In Proceedings of the 5th Workshop on Algorithmic Learning Theory, volume 997 of Lecture Notes in Artificial Intelligence. Springer-Verlag, 1995.

....a playing strategy from the database. For this purpose, we generated 100 games of a player using an optimal strategy (rook side) vs. a player that plays randomly. From all positions and their associated optimal moves and bad mistakes (dropping the rook or stale mating) we had the ILP system ICL [8] learn predicates that check whether a given move is optimal or a bad mistake. These predicates were then used in an artificial player that generates all legal moves, rules out all moves that were deemed bad mistakes, and randomly plays one of the remaining moves that was judged to be an optimal ....

Luc De Raedt and WimVan Laer. Inductive constraint logic. In Proceedings of the 5th Workshop on Algorithmic Learning Theory (ALT-95). SpringerVerlag, 1995.

....a playing strategy from the database. For this purpose, we generated 100 games of a player using an optimal strategy (rook side) vs. a player that plays randomly. From all positions and their associated optimal moves and bad mistakes (dropping the rook or stale mating) we had the ILP system ICL [De Raedt and Van Laer, 1995] learn predicates that check whether a given move is optimal or a bad mistake. These predicates were then used in an artificial player that generates all legal moves, rules out all moves that were deemed bad mistakes, and randomly plays one of the remaining moves that was judged to be an optimal ....

....that might help to focus the learner on interesting concepts. Most available chess databases, like the KRKN and the KRKPa7 datasets from the UCI repository of machine learning databases, conform to this format. However, research in the field of Inductive Logic Programming (ILP) Muggleton, 1992; De Raedt, 1995] has lead to the development of algorithms that are able to make use of background knowledge in full first order horn clause logic. Roughly speaking, these algorithms are concerned with the induction of PROLOG programs. The ability to use background knowledge in the form of PROLOG clauses allows ....

Luc De Raedt and Wim Van Laer. Inductive constraint logic. In Proceedings of the 5th Workshop on Algorithmic Learning Theory (ALT95) . Springer-Verlag, 1995.

....a complementary notion of classification via the satisfaction of the constraints and (iv) allows us to learn from incomplete data, e.g. from positive data only where the constraints provide implicitly negative data. Although there exist several learning systems [ De Raedt and Bruynooghe, 1993; De Raedt and Van Laer, 1995; Muggleton, 1995 ] that learn (or can learn) integrity constraints none of these does so in a strongly integrated fashion as we are proposing here. In fact, most of the constraints learned in practical domains (see e.g. De Raedt and Dehaspe, 1996 ] have the form of definite clauses (the same ....

....also learn integrity constraints in the form of denials. CLAUDIEN ( De Raedt and Bruynooghe, 1993 ] De Raedt and Dehaspe, 1996 ] learns from interpretations: it takes as input a set of interpretations P and learns a set of constraints C that hold in each of the given interpretations. ICL [ De Raedt and Van Laer, 1995 ] takes an additional set of interpretations N requiring that at least one constraint in IC is violated by each interpretation in N . To sum up, some explanatory ILP systems can take into account ICs given by the user. These are used in the learning phase, but seldom (if ever) together with the ....

L. De Raedt and V. Van Laer. Inductive constraint logic. In Proc. Sixth International Workshop on Algorithmic Learning Theory, pages 80--94, 1995.

....only a fraction of all train cases, may offer a small number of prototypes as explanations. Alternatively, one might apply mFOIL [6, 10] and use the m estimate to guide the search towards more general rules (larger m in the estimate of the accuracy prefers rules that cover more examples) ICL [5] is also an interesting candidate ILP system to apply to this problem. Acknowledgements Saso Dzeroski is an ERCIM (European Research Consortium for Informatics and Mathematics) fellow at ICS FORTH. This work started during his visit to GMD (German National Research Center for Information ....

De Raedt, L., and Van Laer, V. Inductive constraint logic. In Proc. Sixth International Workshop on Algorithmic Learning Theory, pages 80--94. Springer, Berlin, 1995.

....7 , we tried both a 5 fold leave one structure out (mesh xv5) and a random 10 fold crossvalidation (meshxv10) We calculated standard error of the estimate, and accuracy. For accuracy we rounded Maccent s predictions to the nearest class. The table below summarizes results for Maccent and ICL [ De Raedt and Van Laer, 1995 ] Accuracy Standard error Algorithm mesh xv5 mesh xv10 mesh xv5 mesh xv10 ICL 0.50 0:70( Sigma0:05) 2:82( Sigma1:19) 1:84( Sigma1:03) Maccent 0.25 0:44( Sigma0:11) 2:83( Sigma0:42) 1:54( Sigma0:33) The above table shows roughly the same picture for mesh xv5 and mesh xv10: Maccent scores a lot ....

L. De Raedt and W. Van Laer. Inductive constraint logic. In Proceedings of the 5th Workshop on Algorithmic Learning Theory, volume 997 of Lecture Notes in Artificial Intelligence. Springer-Verlag, 1995.

....the rules induced by FOIL from red and prop are still overly specific. An interesting direction for further work would be to apply mFOIL [8] and use the m estimate to guide the search towards more general rules (larger m in the estimate of the accuracy prefers rules that cover more examples) ICL [4] is also an interesting candidate to apply to this problem. Neural networks can be applied to the prop propositional version and C4.5 to the representation of Section 4.2. Some combinations of the representations used here may deserve further investigation using propositional or ILP methods. The ....

De Raedt, L., and Van Laer, V. Inductive constraint logic. In Proc. Sixth International Workshop on Algorithmic Learning Theory, pages 80--94. Springer, Berlin, 1995.

....by the theory for the target concept. However, recently the model based view of the ILP learning problem, which has originally been advocated for what has been called descriptional ILP [De Raedt and Dzeroski, 1994; Wrobel and Dzeroski, 1995] has also been adapted for classification learning [De Raedt and Van Laer, 1995; Blockeel and De Raedt, 1997] In this framework, examples are interpretations, for which the learned theory has to be true [De Raedt, 1996] Many ILP learning problems can be formulated in both settings, which would yield different estimates, when the size of the example space is measured by ....

....these static approaches have enjoyed less popularity than the dynamic approaches for identifying relevant feature subsets. In ILP research, the opposite is the case: Research on what has been termed declarative bias has flourished [N edellec et al. 1996; Cohen, 1994; Dehaspe and De Raedt, 1996; Ade et al. 1995] , while there are almost no approaches for dynamically reducing the size of the hypothesis space. A notable exception is [Lavra c et al. 1995] where an approach for propositional literal selection [Gamberger, 1995] is used in a first order framework by transforming the first order problem into ....

Luc De Raedt and Wim Van Laer. Inductive constraint logic. In Proceedings of the 5th Workshop on Algorithmic Learning Theory (ALT95) . Springer-Verlag, 1995.

....specifically in ICL. Though we do not report on any experiments in this short note, it is our intention to report on such experiments at the workshop. Currently, we are carrying out experiments on the mutagenesis data. 2 Inductive Constraint Logic: overview An overview of ICL can be found in [ 8 ] and [ 1 ] Here, we shortly describe the framework of ICL and expand on some practical aspects. 2.1 Framework We will use some notions of first order logic and model theory (for an introduction, see [ 12; 11 ] Definitions of the concepts used here can be found in [ 8 ] ICL is a ....

....of ICL can be found in [ 8 ] and [ 1 ] Here, we shortly describe the framework of ICL and expand on some practical aspects. 2. 1 Framework We will use some notions of first order logic and model theory (for an introduction, see [ 12; 11 ] Definitions of the concepts used here can be found in [ 8 ] ICL is a classification system that learns a clausal theory which discriminates as good as possible between two classes of examples (let s say positives and negatives) Examples are seen as interpretations I of the target theory T . A positive example (interpretation) P is a model of the ....

[Article contains additional citation context not shown here]

L. De Raedt and W. Van Laer. Inductive constraint logic. In Proceedings of the 5th Workshop on Algorithmic Learning Theory, Lecture Notes in Artificial Intelligence. Springer-Verlag, 1995.

....were the decision tree inducer C4.5 [16] and the rule induction program RIPPER [6] For regression, the regression tree induction program M5 [21] a re implementation of M5 [17] was used. It can construct linear models in the leaves of the tree. Relational learning systems applied include ICL [8], which induces classification rules, SRT [15] and TILDE [1] The latter are capable of inducing both classification and regression trees. ICL is an upgrade of CN2 [5] to first order logic, TILDE is an upgrade of C4.5, and SRT is an upgrade of CART [3] TILDE cannot construct linear models in the ....

De Raedt, L., and Van Laer, W. 1995. Inductive constraint logic. In Proc. 6th Intl. Workshop on Algorithmic Learning Theory, pages 80--94. Springer, Berlin.

....a result which has yet seldomly been achieved within artificial intelligence. 5. 2 Recent and Ongoing Leuven Work The, from a practical point of view, most relevant recent results by the Leuven ILP group are concerned with the development of two efficient induction systems: CLAUDIEN [13] and ICL [15]. This progress was made possible by theoretical advances in computational learning theory [14] in bias specification [1] and in the knowledge representation for examples and hypotheses used, i.e. by using the so called non monotonic semantics for inductive logic programming [60] The CLAUDIEN ....

L. De Raedt and W. Van Laer. Inductive constraint logic. In Proceedings of the 5th Workshop on Algorithmic Learning Theory, Lecture Notes in Artificial Intelligence. Springer-Verlag, 1995.

....thus generalises over regression and classification. 4.2 STOPPING CRITERIA Stopping criteria are often based on significance tests. In the classification context a 2 test is often used to check whether the class distributions in the subtrees differ significantly [ Clark and Niblett, 1989; De Raedt and Van Laer, 1995 ] Since regression and clustering use variance as a heuristic for choosing the best split, a reasonable heuristic for the stopping criterion seems to be the F test. If a set of examples is split into two subsets, the variance should decrease significantly, i.e. F = SS= n Gamma 1) SSL SSR ....

L. De Raedt and W. Van Laer. Inductive constraint logic. In Proceedings of the 5th Workshop on Algorithmic Learning Theory, volume 997 of Lecture Notes in Artificial Intelligence. Springer-Verlag, 1995.

....for distances between objects in an attribute value representation (see e.g. 9, ch. 4] Recently there is a growing interest in using more expressive first order representations of objects and in upgrading propositional learning systems into first order learning systems (e.g. TILDE [2] ICL [5] and CLAUDIEN [4] Some ad hoc similarity measures exist for distances between first order objects [6] but they do not have all the desirable mathematical properties (e.g. the triangle inequality and the positive definiteness property) 1 , as a consequence their use may lead to ....

L. De Raedt and W. Van Laer. Inductive constraint logic. In Proceedings of the 5th Workshop on Algorithmic Learning Theory, volume 997 of Lecture Notes in Artificial Intelligence. Springer-Verlag, 1995.

....then c is valid iff c is true in M(B [ E) where M(B [ E) denotes the minimal Herbrand model of B [ E) Every clause describes the data in some way, and the set of all valid clauses forms a maximally informative description of the data. ILP systems using this setting exist (e.g. Claudien[1] icl[2]) but are far less numerous than systems using the classical setting. There is a straightforward relationship between finding an intensional definition for a relation, and classification of tuples. If R is considered to represent some class of tuples (v 1 ; v k ) then the expression of R ....

....system. This is the easiest way to use ILP for knowledge discovery in a relational database: nothing has to be changed to the ILP engine itself. 3. 2 A link to databases at the Prolog level As they work with a logical representation, many ILP systems are implemented in Prolog (e.g. Claudien[1] icl[2]) or, when they are implemented in another 1 It should be mentioned here that most ILP systems work with positive and negative examples, while in a database only positive examples are given. The closed world assumption has to be made in this case. Recently, however, more attention has been ....

L. De Raedt and W. Van Laer. Inductive constraint logic. In Proceedings of the 5th Workshop on Algorithmic Learning Theory, Lecture Notes in Artificial Intelligence. Springer-Verlag, 1995.

.... for this comes from the fact that in attribute value learning examples are true and false interpretations (i.e. models and non models, or positive and negative examples) of a target theory, whereas in inductive logic programming, examples are true and false facts (or clauses) Recently, De Raedt and Van Laer, 1995 ] have defined a notion of conceptlearning in first order logic, in which examples are true and false interpretations of a target theory, and the target theory is a set of clauses. Each clause in the target theory can be seen as a constraint. These ideas are incorporated in a system named ICL, ....

....In the following, each example will be completed implicitly with the background knowledge. So whenever we say an example e is true (false) for a clause (or theory) we mean that M (B [ e) is true (false) for that clause (theory) This setting is illustrated in Examples 1 and 2 (taken from [ De Raedt and Van Laer, 1995 ] Example 1 The well known autolander problem (from the Irvine database) is described by a table (see figure 1) in attribute value representation (only a part is shown) This attribute value learning problem can directly be specified in terms of the framework. e.g. the interpretation I 1 = ....

[Article contains additional citation context not shown here]

L. De Raedt and W. Van Laer. Inductive constraint logic. In Proceedings of the 5th Workshop on Algorithmic Learning Theory. Springer-Verlag, 1995.

.... for this comes from the fact that in attribute value learning examples are true and false interpretations (i.e. models and non models, or positive and negative examples) of a target theory, whereas in inductive logic programming, examples are true and false facts (or clauses) Recently, [3] have defined a notion of concept learning in first order logic, in which examples are true and false interpretations of a target theory, and the target theory is a set of clauses. Each clause in the target theory can be seen as a constraint. These ideas are incorporated in a system named ICL, ....

....Examples In the following, each example will be completed implicitly with the background knowledge. So whenever we say an example e is true (false) for a clause (or theory) we mean that M(B [ e) is true (false) for that clause (theory) Our setting is illustrated in Examples 1 and 2 (taken from [3]) Example 1 The well known autolander problem (from the Irvine database) is described by a table (see figure 1) in attribute value representation (only a part is shown) This attribute value learning problem can directly be specified in terms of the framework. e.g. the interpretation I 1 = ....

[Article contains additional citation context not shown here]

L. De Raedt and W. Van Laer. Inductive constraint logic. In Proceedings of the 5th Workshop on Algorithmic Learning Theory, Lecture Notes in Artificial Intelligence. Springer-Verlag, 1995.

....1994] From a practical point of view, learning from satisfiability allows to represent incomplete examples in a very expressive representation language. Two algorithms for learning from satisfiability are presented. They are adapted from the earlier Claudien [De Raedt and Dehaspe, 1997] and ICL [De Raedt and Van Laer, 1995] systems that learn from interpretations. The first algorithm, called Claudien Sat performs characteristic concept learning, where the aim is to find a most specific hypothesis (within the concept description language) that covers a given set of positive examples. The second algorithm, called ....

....section two algorithms are proposed for learning from satisfiability. The first algorithm performs characteristic induction and is integrated in the clausal discovery engine CLAUDIEN [De Raedt and Dehaspe, 1997] the second one performs discriminant induction and is integrated in the ICL system [De Raedt and Van Laer, 1995]. 5.1 Assumptions We will assume that the set of clauses allowed in hypotheses L is finite. This is to avoid problems with non termination or infinite solutions, see [De Raedt and Dehaspe, 1997] for more details. We will also assume that a refinement operator ae (under subsumption) exists on ....

[Article contains additional citation context not shown here]

L. De Raedt and W. Van Laer. Inductive constraint logic. In Proceedings of the 5th Workshop on Algorithmic Learning Theory, volume 997 of Lecture Notes in Artificial Intelligence. Springer-Verlag, 1995.

....most specific hypothesis that covers (explains) all the examples. e.g. in Figure 1(b) the hypothesis H characterizes all the given examples. The most popular descriptive data mining technique is that of discovering association rules [1, 2, 22] The three companion systems, Claudien [12, 11] ICL [13] and Tilde [4, 5] can be considered first order upgrades of existing attribute value data mining approaches. Claudien upgrades the descriptive association rule approach, ICL upgrades the predictive production rule approach as incorporated in e.g. CN2 [9, 8] and AQ [17] and Tilde is a recent ....

....It should be mentioned that the above description of ICL and CN2 is largely simplified. First, it may be the case that the data are noisy in which case a perfect solution may not exist. Under such conditions the algorithms will employ heuristics to find an approximation of the target concept (cf. [13]) Second, the find rule algorithm as outlined above performs hill climbing : it keeps track of a single current best candidate. ICL and CN2 however perform a beam search, where one always keeps track of the m best rules. Finally, the above algorithm does not specify which literals can be added to ....

L. De Raedt and W. Van Laer. Inductive constraint logic. In Proceedings of the 5th Workshop on Algorithmic Learning Theory, volume 997 of Lecture Notes in Artificial Intelligence. Springer-Verlag, 1995.

....for distances between objects in an attribute value representation (see e.g. 9, ch. 4] Recently there is a growing interest in using more expressive first order representations of objects and in upgrading propositional learning systems into first order learning systems (e.g. TILDE [2] ICL [5] and CLAUDIEN [4] Some ad hoc similarity measures exist for distances between first order objects [6] but they do not have all the desirable mathematical properties (e.g. the triangle inequality and the positive definiteness property) 1 , as a consequence their use may lead to ....

L. De Raedt and W. Van Laer. Inductive constraint logic. In Klaus P. Jantke, Takeshi Shinohara, and Thomas Zeugmann, editors, Proceedings of the 6th International Workshop on Algorithmic Learning Theory, volume 997 of Lecture Notes in Artificial Intelligence, pages 80--94. SpringerVerlag, 1995.

....tipo(A,carretera) seccionesposteriores(A,C) ocupaciond(B,C,alta) seccionesposteriores(C,D) ocupaciond(B,D,alta) 0,0,26] Fig. 1. An incomplete listing of the rules generated by ICL. 4 Experiments and results Three ILP systems were applied to the problem formulated above. ICL [8] and TILDE [2] operate within the learning from interpretations [6] setting, while PROGOL [11] operates in the learning from entailment setting. This is the reason for the slightly different form of rules induced by the three systems, i.e. the appearance of the literals section(A) and ....

.... : velocidadd(B,A,alta) saturaciond(B,A,media) 0,0,32] noncs(A,B) saturaciond(B,A,alta) tipo(A,carretera) 0,0,30] noncs(A,B) seccionesposteriores(C,A) ocupaciond(B,C,baja) 0,0,25] noncs(A,B) ocupaciond(B,A,baja) saturaciond(B,A,baja) tipo(A,rampaincorporacion) [0,0,8] noncs(A,B) ocupaciond(B,A,baja) seccionesposteriores(A,C) saturaciond(B,C,media) 0,0,5] noncs(A,B) ocupaciond(B,A,baja) seccionesposteriores(C,A) saturaciond(B,C,alta) 0,0,11] noncs(A,B) ocupaciond(B,A,baja) saturaciond(B,A,baja) seccionesposteriores(C,A) ....

De Raedt, L., and Van Laer, V. (1995). Inductive constraint logic. Proc. Sixth International Workshop on Algorithmic Learning Theory, pp. 80-94. Berlin: Springer.

....for top down induction of logical decision trees by adapting C4.5 s heuristics. This results in the Tilde system, which is the main topic of this paper. Tilde works within the learning from interpretations paradigm introduced by [ De Raedt and Dzeroski, 1994 ] and used in the ICL system of [ De Raedt and Van Laer, 1995 ] Within Tilde, we also incorporated two other novelties w.r.t. inductive logic programming: discretization of numeric attributes (based on [ Van Laer et al. 1996; Fayyad and Irani, 1993 ] and dynamic lookahead facilities. We also report on a number of encouraging experiments, in the ....

....and touch upon related work. 2 The Learning Problem We assume familiarity with the Prolog programming language (see e.g. Bratko, 1990 ] We essentially use the learning from interpretations paradigm for inductive logic programming, introduced by [ De Raedt and Dzeroski, 1994 ] used in ICL [ De Raedt and Van Laer, 1995 ] and related to other inductive logic programming settings in [ De Raedt, 1996 ] In this paradigm, each example is a Prolog knowledge base (i.e. a set of definite clauses) encoding the specific properties of the example. Furthermore, each example is classified into one of a finite set of ....

L. De Raedt and W. Van Laer. Inductive constraint logic. In Proceedings of the 5th Workshop on Algorithmic Learning Theory, volume 997 of Lecture Notes in Artificial Intelligence. Springer-Verlag, 1995.

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