César Luis Alonso

A rational function decomposition algorithm by near-separated polynomials

Abstract In this paper we present an algorithm for decomposing rational functions over an arbitrary coefficient field. The algorithm requires exponential time, but is more efficient in practice than the previous ones, including the polynomial time algorithm. Moreover, our algorithm is easier to implement. We also present some applications of rational function decomposition: (1) faithful re-parameterizing unfaithfully parameterized curves, (2) computing intermediate fields in a simple purely transcendental field extension K , and (3) providing a birationality test for subfields of K (x). Several examples are computed using an implementation of our algorithm using MAPLE V.

An implicitization algorithm with fewer variables

Abstract In this paper we present a general implicitization algorithm for rational parameterizations, using Grobner Bases, which: (i) is valid for general parametric varieties (i.e. allowing both rational or polynomial parameterizations), (ii) computes the greatest ideal of polynomials vanishing over the variety and (iii) uses only as many variables as the number of parameters plus coordinates. We give examples of the performance of our algorithm in the CoCoa system, comparing the obtained results with other algorithms.

Model selection in genetic programming

In this paper we discuss the problem of model selection in Genetic Programming. We present empirical comparisons between classical statistical methods (AIC, BIC) adapted to Genetic Programming and the Structural Risk Minimization method (SRM) based on Vapnik-Chervonenkis theory (VC), for symbolic regression problems with added noise. We also introduce a new model complexity measure for the SRM method that tries to measure the non-linearity of the model. The experimentation suggests practical advantages of using VC-based model selection with the new complexity measure, when using genetic training.

Reconsidering algorithms for real parametric curves

Complex parametric curves have been subject of a symbolic algorithm approach in recent years. In this paper we analyze the theoretical applicability of some of these algorithms to the real parametric curve case. In particular, we show how several results are valid both over the real and the complex numbers, as they hold equivalently over a real curve and its complexification. Therefore, the standard algorithms for the complex case can be applied to obtain real answers in the real case. A second issue in our paper is the study of the very different behaviour of the real parametric mapping and we characterize here the properties of being (almost) injective or surjective.

Penalty functions for genetic programming algorithms

Very often symbolic regression, as addressed in Genetic Programming (GP), is equivalent to approximate interpolation. This means that, in general, GP algorithms try to fit the sample as better as possible but no notion of generalization error is considered. As a consequence, overfitting, code-bloat and noisy data are problems which are not satisfactorily solved under this approach. Motivated by this situation we review the problem of Symbolic Regression under the perspective of Machine Learning, a well founded mathematical toolbox for predictive learning. We perform empirical comparisons between classical statistical methods (AIC and BIC) and methods based on Vapnik-Chrevonenkis (VC) theory for regression problems under genetic training. Empirical comparisons of the different methods suggest practical advantages of VC-based model selection. We conclude that VC theory provides methodological framework for complexity control in Genetic Programming even when its technical results seems not be directly applicable. As main practical advantage, precise penalty functions founded on the notion of generalization error are proposed for evolving GP-trees.

A NEW LINEAR GENETIC PROGRAMMING APPROACH BASED ON STRAIGHT LINE PROGRAMS: SOME THEORETICAL AND EXPERIMENTAL ASPECTS

Tree encodings of programs are well known for their representative power and are used very often in Genetic Programming. In this paper we experiment with a new data structure, named straight line program (slp), to represent computer programs. The main features of this structure are described, new recombination operators for GP related to slps are introduced and a study of the Vapnik-Chervonenkis dimension of families of slps is done. Experiments have been performed on symbolic regression problems. Results are encouraging and suggest that the GP approach based on slps consistently outperforms conventional GP based on tree structured representations.

An evolutionary strategy for the multidimensional 0-1 knapsack problem based on genetic computation of surrogate multipliers

In this paper we present an evolutionary algorithm for the multidimensional 0–1 knapsack problem. Our algorithm incorporates a heuristic operator which computes problem-specific knowledge. The design of this operator is based on the general technique used to design greedy-like heuristics for this problem, that is, the surrogate multipliers approach of Pirkul (see [7]). The main difference with work previously done is that our heuristic operator is computed following a genetic strategy -suggested by the greedy solution of the one dimensional knapsack problem- instead of the commonly used simplex method. Experimental results show that our evolutionary algorithm is capable of obtaining high quality solutions for large size problems requiring less amount of computational effort than other evolutionary strategies supported by heuristics founded on linear programming calculation of surrogate multipliers.

Adaptation, Performance and Vapnik-Chervonenkis Dimension of Straight Line Programs

We discuss here empirical comparation between model selection methods based on Linear Genetic Programming. Two statistical methods are compared: model selection based on Empirical Risk Minimization (ERM) and model selection based on Structural Risk Minimization (SRM). For this purpose we have identified the main components which determine the capacity of some linear structures as classifiers showing an upper bound for the Vapnik-Chervonenkis (VC) dimension of classes of programs representing linear code defined by arithmetic computations and sign tests. This upper bound is used to define a fitness based on VC regularization that performs significantly better than the fitness based on empirical risk.

Heuristic Generation of the Initial Population in Solving Job Shop Problems by Evolutionary Strategies

In this work we confront the job shop scheduling problem by means of Genetic Algorithms. Our contribution is mainly the generation of a heuristic initial population from domain specific knowledge provided by a probabilitic method. Experimental results show that a Genetic Algorithm that uses a heuristic initial population outperforms not only the same algorithm when using a random initial population, but also other search strategies that exploit the same class of heuristic information.

FRAC: A Maple Package for Computing in the Rational Function Field K(X)

In this paper we present the programs package FRAC (= Funciones RACionales) which is designed for performing computations in the rational function field. The main objects in FRAC are rational functions over the field of rational numbers, but extensions to other computable fields can be done in a “natural” way. The key tool is using functional decomposition algorithms. We motivate the interest to work with rational function decomposition by presenting applications to computer science, engineering (CAD), pure mathematics or robotics. We also present some simple examples in order to illustrate the use of FRAC. Finally, we include the synopsis of the main procedures of FRAC.