J. Ramon and M. Bruynooghe, A framework for defining distances between first order logic objects, in Proceedings of 8th International Workshop, ILP98, pp.271-280, 1998  Home/Search   Document Details and Download   Summary   Related Articles   Check

....in advance, but they strongly depend on the problem considered. In other words, examples are mapped to distance space only after a certain viewpoint is given. Consequently, problem solution can be done using the similarity based methods. In previous works, many similarity measures are proposed [1 7]. There is a similarity measure [1] which is derived from some assumptions. It assumes that a similarity between objects is the unique function of commonali # Partially supported by Toyota Physical Chemical Research Institute. ties and differences between objects. The commonalities (resp. ....

J. Ramon and M. Bruynooghe, A framework for defining distances between first order logic objects, in Proceedings of 8th International Workshop, ILP98, pp.271-280, 1998

....attention has been paid to studying distance measures in first order languages. The basic idea is to apply the highly successful instance based algorithms to relational data using first order logic descriptions. Various approaches have been proposed in this area. Some of the most recent ones are [1, 4, 6, 7]. These approaches as well as most of the others define a simple metric on atoms and then extend it to sets of atoms (clauses or models) using the Hausdorff metric or other similarity functions. Because of the complexity of the functions involved and the problems with the computability of the ....

J. Ramon, M. Bruynooghe, and W. V. Laer. A framework for defining a distance between first-order logic objects. Technical Report CW 263, Katholieke Universiteit Leuven, 1998.

....the nearest example that is taken into account, but some number of the nearest neighbours. The most common classification amongst those instances is returned as the classification of the query. There are several approaches to measuring distances between first order objects, recent ones include [3] [6] and [8] We will want to learn some rules from the data to make query answering faster and to aid in explaining our actions to the user if they ask. But although rules are quick to return answers, they take time to learn so we only want to learn those rules which are of the highest quality . ....

J. Ramon and M. Bruynooghe. A framework for defining distances between firstorder logic objects. In D. Page, editor, Proceedings of the 8th International Conference on Inductive Logic Programming, volume 1446 of Lecture Notes in Artificial Intelligence, pages 271--280. Springer-Verlag, 1998.

....distance measures in first order languages. The basic idea is to apply the highly successful instance based algorithms to relational data described in the much more expressive language of first order logic. Various approaches have been proposed in this area. Some of the most recent ones are [1, 5 7]. These approaches as well as most of the others define a simple metric on atoms and then extend it to sets of atoms (clauses or models) using the Hausdorff metric or other similarity functions. Because of the complexity of the functions involved and problems with the computability of the models ....

J. Ramon, M. Bruynooghe, and W. V. Laer. A framework for defining a distance between first-order logic objects. Technical Report CW 263, Katholieke Universiteit Leuven, 1998.

....while dividing the other 0.5 over the different argument positions. In future work we will experimentally compare this new distance with those of [8] and [10] in distances for sets of atoms which are parameterised with a distance between atoms. We have described such distances in [11]. Acknowledgements We thank Shan Hwei Nienhuys Cheng for reading the draft and the useful comments. We also thank Luc De Raedt for the interesting discussions. Wim 6 Van Laer and Maurice Bruynooghe are supported by the Fund of Scientific Research, Flanders. This work is supported by the ....

J. Ramon and M. Bruynooghe. A framework for defining distances between firstorder logic objects. Technical Report CW 263, Department of Computer Science, Katholieke Universiteit Leuven, 1998.

.... If t is a term, then t=ffl = t. if t = f(t 1 ; t n ) then t= i Delta u) t i =u. In instance based learning a distance is needed between the examples. As we work with structured terms as examples, we must use a distance between terms such as the distances defined in [11] 7] [13], 14] In this paper we use the simple distance defined in [11] Definition 1 (distance d nc between terms) If t 1 and t 2 are terms, then if t 1 = t 2 , d nc (t 1 ; t 2 ) 0. if t 1 = p(x 1 ; xn ) and t 2 = q(y 1 ; ym ) with p 6= q or n 6= m, then d nc (t 1 ; t 2 ) ....

J. Ramon and M. Bruynooghe. A framework for defining distances between firstorder logic objects. In Proceedings of the 8th International Conference on Inductive Logic Programming, Lecture Notes in Artificial Intelligence, pages 271--280. Springer-Verlag, 1998.

....the set and which is better suited to measure the distance between sets of very different sizes is developed in section 6. In section 7 some applications from the machine learning area are discussed. We end with a brief summary in section 8. This paper is an extension of some of the material in [11]. 2 Preliminaries Let #S denote the cardinality of a set S; jnj denotes the absolute value of a number n; for a relation f A Theta B, f(x) denotes the set fyj(x; y) 2 fg, f(S) denotes the set fyjx 2 S(x; y) 2 fg, #f(A) is abbreviated as #f and f Gamma1 denotes the relation f(y; x)j(x; y) 2 ....

.... (it is determined by the distance of the most distant element of both sets to the nearest neighbour in the other set) This makes this metric unsuited for applications where one set has likely a point which is very different from all points of the other set as e.g. in Inductive Logic Programming [11]. Sum of minimal distance measure Eiter and Mannila [5] discuss the sum of minimal distances similarity measure. It is defined as: d(X; Y ) 1 2 0 X x2X Gamma min y2Y d(x; y) Delta X y2Y Gamma min x2X d(x; y) Delta 1 A However, this is in general not a metric. ....

J. Ramon and M. Bruynooghe. A framework for defining distances between first-order logic objects. In Proceedings of the Eighth International Conference on Inductive Logic Programming, Lecture Notes in Artificial Intelligence, pages 271--280. Springer-Verlag, 1998.

....predicate functor while dividing the other 0.5 over the different argument positions. In future work we will experimentally compare this new distance with those of [8] and [10] in distances for sets of atoms which are parameterised with a distance between atoms. We have described such distances in [11]. Acknowledgements We thank Shan Hwei Nienhuys Cheng for reading the draft and the useful comments. We also thank Luc De Raedt for the interesting discussions. Wim Van Laer and Maurice Bruynooghe are supported by the Fund of Scientific Research, Flanders. This work is supported by the European ....

J. Ramon and M. Bruynooghe. A framework for defining distances between firstorder logic objects. Technical Report CW 263, Department of Computer Science, Katholieke Universiteit Leuven, 1998. http://www.cs.kuleuven.ac.be/- publicaties/rapporten/CW1998.html.

....while dividing the other 0.5 over the different argument positions. In future work we will experimentally compare this new distance with those of [8] and [10] in distances for sets of atoms which are parameterised with a distance between atoms. We have described such distances in [11] and [12]. Acknowledgements We thank Luc De Raedt for the many interesting discussions. We thank also Shan Hwei Nienhuys Cheng for reading the draft and the useful comments. Wim Van Laer and Maurice Bruynooghe are supported by the Fund of Scientific Research, Flanders. This work is supported by the ....

J. Ramon and M. Bruynooghe. A framework for defining distances between first-order logic objects. Technical Report CW 263, Department of Computer Science, Katholieke Universiteit Leuven, 1998. http://www.cs.kuleuven.ac.be/publicaties/rapporten/- CW1998.html.

....predicate functor while dividing the other 0.5 over the different argument positions. In future work we will experimentally compare this new distance with those of [8] and [10] in distances for sets of atoms which are parameterised with a distance between atoms. We have described such distances in [11] and [12] Acknowledgements We thank Luc De Raedt for the many interesting discussions. We thank also Shan Hwei Nienhuys Cheng for reading the draft and the useful comments. Wim Van Laer and Maurice Bruynooghe are supported by the Fund of Scientific Research, Flanders. This work is supported by ....

J. Ramon and M. Bruynooghe. A framework for defining distances between first-order logic objects. In Proceedings of the 8th International Conference on Inductive Logic Programming, Lecture Notes in Artificial Intelligence, pages 271--280. Springer-Verlag, 1998.

.... d l ) They show that these measures can be evaluated in polynomial time, but are not distance functions (the triangle inequality is violated) Using matchings (m fi = m m with m m (A; B) the set of all matchings between A and B) one obtains another instantiation for which we prove in [13] the following theorems: A B C A 0 B B A 0 B AC B C C 0 f m AB m BC Figure 3: triangle inequality for matchings distance Theorem 5 d m (A; B) is a distance function. Proof: d m (X; Y ) 0 , X = Y follows trivially from the definition and the fact that d is a good distance. d m (X; Y ) ....

J. Ramon and M. Bruynooghe. A framework for defining distances between first-order logic objects. Technical Report CW 263, Department of Computer Science, Katholieke Universiteit Leuven, 1998. http://www.cs.kuleuven.ac.be/publicaties/rapporten/- CW1998.html.

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