A. Hutchinson. Metrics on terms and clauses. In Proceedings of the 9th European Conference on Machine Learning, Lecture Notes in Arti cial Intelligence, pages 138-145. Springer-Verlag, 1997.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... prediction setting (see e.g. 14] Other work on structure prediction includes Ramon and De Raedt s instance based function learning [21] The distance measure we propose is not the only possible one; as mentioned before many distance measures for structural values have been proposed, see e.g. [15, 9, 20]. 8 Conclusions We have discussed the task of hierarchical multi classi cation, which extends both multi classi cation and hierarchical classi cation, and which we believe to cover an interesting range of applications. We have presented an algorithm that extends the decision tree approach ....

A. Hutchinson. Metrics on terms and clauses. In Proceedings of the 9th European Conference on Machine Learning, Lecture Notes in Arti cial Intelligence, pages 138-145. Springer-Verlag, 1997.

....inequality and the positive definiteness property) as a consequence their use may lead to sub optimal inconsistent results. A first step in defining a good distance between first order objects is to define a distance between first order atoms. Some proposals exist for this, e.g. 10] and [8], but they do not handle variables in a satisfactory way. In this paper we propose a better measure for distances between non ground atoms and prove that it has all the desirable properties of a metric. As in [8] the distance between two atoms is based on their distance to the least general ....

....a distance between first order atoms. Some proposals exist for this, e.g. 10] and [8] but they do not handle variables in a satisfactory way. In this paper we propose a better measure for distances between non ground atoms and prove that it has all the desirable properties of a metric. As in [8], the distance between two atoms is based on their distance to the least general generalisation. However, our distances are pairs (F; V ) where F accounts for the differences between the functors of both atoms and V for the difference due to the variables. positive definite:d(x; y) 0 and ....

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A. Hutchinson. Metrics on terms and clauses. In Proceedings of the 9th European Conference on Machine Learning, Lecture Notes in Artificial Intelligence, pages 138--145. Springer-Verlag, 1997.

....building lattice structures and shows some experiments with this algorithm. Section 5 contains concluding remarks and directions for future work. 2 Preliminaries In this section we introduce a height based distance measure on a join semilattice following an approach similar to those described in [1] and [5] for a survey of metrics on partially ordered sets see [2] Definition 1 (Semi distance, Quasi metric) A semi distance (quasi metric) is a mapping d : O Theta O on a set of objects O with the following properties (a; b; c 2 O) 1. d(a; a) 0 and d(a; b) 0. 2. d(a; b) d(b; ....

....By placing certain restrictions on the hypothesis language the number of literals in the lgg clause can be limited by a polynomial function independent on n. Currently we use ij determinate clauses in our experiments (actually 22 determinate) memb(A, A] memb(A, 3,A] memb(2, 2] memb(1,[1]) memb(A, B,C D] memb(A, A] memb(A, C D] memb(A, 3,A] memb(1, 3,1] memb(1, 2,3,1] memb(2, 3,2] memb(A, A] memb(b, b] memb(a, a] memb(A, B,A C] memb(A, A] memb(a, a C] memb(A, A C] memb(a, b,a,b] memb(b, c,b] memb(A, B,C D] memb(A, C D] ....

[Article contains additional citation context not shown here]

A. Hutchinson. Metrics on terms and clauses. In M. van Someren and G. Widmer, editors, Machine Learning: ECML-97, volume 1224 of Lecture Notes in Artificial Intelligence, pages 138--145. Springer-Verlag, 1997.

....is no chain from A to B whose length is greater than size(B) Gamma size(A) Unfortunately the size function does not satisfy the second formal property of a height function and consequently the function d v is not a quasi metric. There is a simplified version of size, proposed by Hutchinson [6] based on the number of functional symbols in the atom. Though formally a height, this function does not account properly for the variables in the atoms and consequently it is improper for the minimality condition in the algorithm. Similarly to the propositional case a coverage based function can ....

....that the Horn clause subsumption semi lattice contains infinite chains. Therefore the definition of a formal height function (if such exists) is not a trivial task. There are also other approaches to define a proper metric on Horn clauses. An approach based on the Hausdorff metric is proposed in [6]. Practically the GSL algorithm needs an evaluation function representing the similarity (or distance) between the clauses with respect to their role in the concept learning problem. Thus similarly to the case of propositional and atomic languages a coverage based height function could be a good ....

A. Hutchinson. Metrics on terms and clauses. In M. van Someren and G. Widmer, editors, Machine Learning: ECML-97, volume 1224 of Lecture Notes in Artificial Intelligence, pages 138--145. Springer-Verlag, 1997.

.... 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 ....

A. Hutchinson. Metrics on terms and clauses. In Proceedings of the 9th European Conference on Machine Learning, Lecture Notes in Artificial Intelligence, pages 138--145. Springer-Verlag, 1997.

....for each case in the case base the intersection (i.e. what is common) with the problem. By means of a predefined subsumption ordering these intersections are ordered. The cases that are most similar to a problem are those that yield most specific anti unification results. Other approaches (e.g. Hut97] calculate for each case the minimal number of changes that would have to be done to transform it into the problem. The weighted sum of changes then indicates the distance between the case and the problem. While these metrics may be dynamically adapted in changing environments (the case base ....

Alan Hutchinson. Metrics on Terms and Clauses. In Proceedings of ECML-97, pages 138--146. Springer, 1997.

....atoms under subsumption. A size function s(a; b) on this set can be defined as the number of different functional symbols (a constant is considered a functional symbol of arity zero) occurring in the substitution mapping a onto b (a = b) A family of similar size functions is introduced in [1], where they are called a size of substitution. Although well defined these functions do not account properly for the variables in the atoms and consequently cannot be used with non ground atoms. Theorem 1. Let (A; be a join semi lattice and s a size function. Let al.so d(a; b) s(inffa; bg; ....

....element form the well known definition of member (the recursive clause contains a redundant literal) The generated tree structure can be seen as an example of conceptual clustering of first order atoms, where the hypotheses are Horn clauses. memb(A, A] memb(A, 3,A] memb(2, 2] memb(1,[1]) memb(A, B,C D] memb(A, A] memb(A, C D] memb(A, 3,A] memb(1, 3,1] memb(1, 2,3,1] memb(2, 3,2] memb(A, A] memb(b, b] memb(a, a] memb(A, B,A C] memb(A, A] memb(a, a C] memb(A, A C] memb(a, b,a,b] memb(b, c,b] memb(A, B,C D] memb(A, C D] ....

[Article contains additional citation context not shown here]

A. Hutchinson. Metrics on terms and clauses. In M. van Someren and G. Widmer, editors, Machine Learning: ECML-97, volume 1224 of Lecture Notes in Artificial Intelligence, pages 138--145. Springer-Verlag, 1997.

....of the distances. For instance, the distance could be the Euclidean distance d 1 between the values of one or more numerical attributes, or it could be the distance d 2 as measured by a first order distance measure such as used in RIBL [ Emde and Wettschereck, 1996 ] or KBG [ Bisson, 1992 ] or [ Hutchinson, 1997 ] Given the distance at the level of the examples, the principles of instance based learning can be used to compute the prototypes. e.g. d 1 would result in a prototype function p 1 that would simply compute the mean for the cluster, whereas d 2 could result in function p 2 that would compute ....

A. Hutchinson. Metrics on terms and clauses. In Proceedings of the 9th European Conference on Machine Learning, Lecture Notes in Artificial Intelligence, pages 138--145. Springer-Verlag, 1997.

....As for algorithmic aspects, the clustering algorithm selected here is the simplest possible, and more elaborate methods, such as the one used in KBG and SPRITE, should be examined. Furthermore, three other distance measures for first order logic domains have been proposed. These are described in [13], 15] and [16] and may be well worth a try. It would also be useful to try the similarity metric and cluster selection techniques used here in the context of a top down strategy as employed in C0.5. In addition, the cluster selection technique introduced here which is only one way of generating ....

A. Hutchinson. Metrics on Terms and Clauses. In M. Someren and G. Widmer, editors, Machine Learning: ECML-97 (Proc. Ninth European Conference on Machine Learning), volume 1224 of LNAI, pages 138--

....inequality and the positive definiteness property) 1 , as a consequence their use may lead to sub optimal inconsistent results. A first step in defining a good distance between first order objects is to define a distance between first order atoms. Some proposals exist for this, e.g. 10] and [8], but they do not handle variables in a satisfactory way. In this paper we propose a better measure for distances between non ground atoms and prove that it has all the desirable properties of a metric. As in [8] the distance between two atoms is based on their distance to the least general ....

....a distance between first order atoms. Some proposals exist for this, e.g. 10] and [8] but they do not handle variables in a satisfactory way. In this paper we propose a better measure for distances between non ground atoms and prove that it has all the desirable properties of a metric. As in [8], the distance between two atoms is based on their distance to the least general generalisation. However, our distances are pairs (F; V ) where F accounts for the differences between the functors of both atoms and V for the difference due to the variables. 1 positive definite:d(x; y) 0 and ....

[Article contains additional citation context not shown here]

A. Hutchinson. Metrics on terms and clauses. In Proceedings of the 9th European Conference on Machine Learning, Lecture Notes in Artificial Intelligence, pages 138--145. Springer-Verlag, 1997.

....As for algorithmic aspects, the clustering algorithm selected here is the simplest possible, and more elaborate methods, such as the one used in KBG and SPRITE, should be examined. Furthermore, three other distance measures for first order logic domains have been proposed. These are described in [13], 15] and [16] and may be well worth a try. It would also be useful to try the similarity metric and cluster selection techniques used here in the context of a top down strategy as employed in C0.5. In addition, the cluster selection technique introduced here which is only one way of generating ....

A. Hutchinson. Metrics on Terms and Clauses. In M. Someren and G. Widmer, editors, Machine Learning: ECML-97 (Proc. Ninth European Conference on Machine Learning), volume 1224 of LNAI, pages 138--145. Springer Verlag, 1997.

....inequality and the positive definiteness property) 1 , as a consequence their use may lead to sub optimal inconsistent results. A first step in defining a good distance between first order objects is to define a distance between first order atoms. Some proposals exist for this, e.g. 10] and [8], but they do not handle variables in a satisfactory way. In this paper we propose a better measure for distances between non ground atoms and prove that it has all the desirable properties of a metric. As in [8] the distance between two atoms is based on their distance to the least general ....

....a distance between first order atoms. Some proposals exist for this, e.g. 10] and [8] but they do not handle variables in a satisfactory way. In this paper we propose a better measure for distances between non ground atoms and prove that it has all the desirable properties of a metric. As in [8], the distance between two atoms is based on their distance to the least general generalisation. However, our distances are pairs (F; V ) where F accounts for the differences between the functors of both atoms and V for the difference due to the variables. For example, the distance between the ....

[Article contains additional citation context not shown here]

A. Hutchinson. Metrics on terms and clauses. In Proceedings of the 9th European Conference on Machine Learning, Lecture Notes in Artificial Intelligence, pages 138--145. Springer-Verlag, 1997.

....hand IBL has a number of disadvantages with respect to ILP. Notably, IBL predictions lack explanation, and there is a need to de ne a metric to describe similarity between instances. Generally, similarity metrics are hard to justify, even when they can be shown to have desirable properties (e.g. [5, 11, 12]) In comparison AP predictions are directly associated with an explicit hypothesis, which provides explanation. Also AP does not require a similarity measure since predictions are made on the basis of the hypothesis. This paper has the following structure. A formal framework for AP is provided in ....

A. Hutchinson. Metrics on terms and clauses. In M. Someren and G. Widmer, editors, Proceedings of the Ninth European Conference on Machine Learning, pages 138-145, Berlin, 1997. Springer.

....[4] The upgrading of clustering and instance based learning systems requires to develop a measure for the distance between first order objects, either described as clauses or as models of first order theories. Some proposals for distance measures between atoms and clauses exists (e.g. 12] and [9]) They use Hausdorff metrics to extend distances between atoms into distances between sets of atoms (clauses or models) This has two drawbacks. Firstly, the value of the Hausdorff metric depends very much on the most extreme value in both sets. Secondly, the similarity due to occurrences of the ....

....1 2n P a j=1 D i;j ffl Let I be such that D I = min i=1; k D i . ffl return (t I ; D I D) The execution time of this function is at most linear in the sum of the description lengths of the objects. Setting P dnc = t I with P dnc ;h = t I ; D I ) gives a good prototype function. Hutchinson [9] starts with defining a size for substitutions: Definition 18 A real valued function S on substitutions is called a size iff ffl for all substitutions , S( 0. ffl S(ffl) 0 with ffl the identity substitution. ffl for any terms u, v and w: if ab , bc and ac are substitutions such ....

[Article contains additional citation context not shown here]

A. Hutchinson. Metrics on terms and clauses. In Proceedings of the 9th European Conference on Machine Learning, Lecture Notes in Artificial Intelligence, pages 138--145. Springer-Verlag, 1997.

.... distance d 2 between the points in the three dimensional space corresponding to the lumo; logp and activity values of the two compounds, or it could be the distance d 3 as measure by a first order distance measure such as used in RIBL [ Emde and Wettschereck, 1996 ] or KBG [ Bisson, 1992a ] or [ Hutchinson, 1997 ] Given the distance at the level of the examples, the principles of instance based learning can be used as to compute the prototypes. e.g. on the mutagenesis problem, d 1 would result in a prototype function p 1 that would simply compute the average activity of the compounds in the cluster, ....

A. Hutchinson. Metrics on terms and clauses. In Proceedings of the 9th European Conference on Machine Learning, 1997.

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A. Hutchinson. Metrics on Terms and Clauses. In: M. Someren, G. Widmer, editors. Proceedings of the 9th European Conference on Machine Learning #ECML-97#, pages 138-145, 1997. Springer-Verlag.

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A. Hutchinson. Metrics on Terms and Clauses. In M. Someren and G. Widmer, editors, Machine Learning: ECML-97 #Proc. Ninth European Conference on Machine Learning#,volume 1224 of LNAI, pages 138#145. Springer Verlag, 1997.

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