Abstract
Inductive Logic Programming (ILP) is concerned with construction of logic programs from examples. It shares many concerns of Machine Learning (ML), but is committed to logic. As logic can help to provide a basis for elaborating such a methodology for learning, the area of ILP has attracted a wide attention of many researchers1. This paper reviews some of the methods and techniques in ML that exploit logic.

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We prove that the speed of convergence of two Markov families determines the smoothness of the conjugacy between them. One of the applications of this result that we give is that the attractors of any two quadratic foldings at the Feigenbaum accumulation point of period doubling are C2+0.11 conjugate. Our main result provides the basis for a complete unification of renormalization and smooth conjugacy results which includes both the classical theorems and more recent results about critical systems.

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In this paper, we show how the Yule-Walker type difference equations for higher order moments and cumulants, recently derived for certain types of bilinear time series models, the BL(p,0,p,1) models, by Sesay and Subba Rao (1988, 1991), could be used for tentative identification of the order of the model. The technique we use for identification is canonical correlation analysis, carried out between the linear combination of the observations and linear combination of higher powers of the observations. The methods are illustrated with real and simulated examples.

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