Miguel Filgueiras

A fast method for finding the basis of nonnegative solutions to a linear Diophantine equation

Abstract We present a complete characterization of the set of minimal solutions of a single linear Diophantine equation in three unknowns over the natural numbers. This characterization, for which we give a geometric interpretation, is based on well-known properties of congruences and we use it as the foundation of direct algorithms for solving this particular kind of equation. These direct algorithms and an enumeration procedure are then put together to build an algorithm for solving the general case of a Diophantine equation over the naturals. We also put forth a statistical method for comparing algorithms for solving Diophantine equations which is more sound than comparisons based on times observed for small sets of equations. From an extensive comparison with algorithms described by other authors it becomes clear that our algorithm is the fastest known to date for a class of equations. Typically the equations in this class have a small number of unknowns in one side, the maximum value for their coefficients being greater than 3.

Fast Methods for Solving Linear Diophantine Equations

We present some recent results from our research on methods for finding the minimal solutions to linear Diophantine equations over the naturals. We give an overview of a family of methods we developed and describe two of them, called Slopes algorithm and Rectangles algorithm. From empirical evidence obtained by directly comparing our methods with others, and which is partly presented here, we are convinced that ours are the fastest known to date when the equation coefficients are not too small (ie., greater than 2 or 3).

Solving Linear Diophantine Equations Using the Geometric Structure of the Solution Space

In the development of algorithms for finding the minimal solutions of systems of linear Diophantine equations, little use has been made (to our knowledge) of the results by Stanley using the geometric properties of the solution space. Building upon these results, we present a new algorithm, and we suggest the use of geometric properties of the solution space in finding bounds for searching solutions and in having a qualitative evaluation of the difficulty in solving a given system.

An Algorithm for Solving Systems of Linear Diophantine Equations in Naturals

A new algorithm for fording the minimal solutions of systems of linear Diophantine equations has recently been published. In its description the emphasis was put on the mathematical aspects of the algorithm. In complement to that, in this paper another presentation of the algorithm is given which may be of use for anyone wanting to implement it.

A New Method for Solving Linear Constraints on the Natural Numbers

In the recent past much attention has been given to the handling of constraints. The focus of this paper is solving linear constraints on the natural numbers, a problem that is also of great importance in AC-unification. We describe a new algorithm that is faster than the methods we compared it with, and that may well contradict the view that this kind of algorithms is too expensive to be of practical use, for instance in the implementation of Constraint Logic Programming languages.

Some Comments on a Logic Programming Approach to Natural Language Semantics

The aim of this paper is to present a criticism of, and to introduce some new ideas about the approach to natural language semantics that has been put forward in [Porto, Filgueiras]. The main points we will discuss are the distribution of properties among entities, the distinction between modifiers and predications, the concept of part of an object, and the way the Principle of Compositionality is applied.

Implementing a symbolic music processing system

A description is given of the ongoing work on the implementation of a music processing system, ScEX. Although originally thought of as a system for typesetting music (and hence the name, for ‘score (T)eX’), in fact it can be taken as a more general system that processes textual descriptions of music pieces and produces, for instance, printable scores, or input for music rendering systems. The architecture of the system, based on the existence of different representations for isolated voices and mixtures of voices, is discussed, which may be of use for developers of similar systems. Hopefully, it will be clear how beneficial for the implementation work was the use of well established compiler design notions and techniques, the exploitation of program reversibility, and the selection of a very high‐level language (Prolog) as the implementation language. This description may then be taken as a case study in the use of such techniques, and make interesting reading for software developers in general.

A Successful Case of Computer Aided Translation

This paper reports on the techniques used to successfully translate with the help of a computer a Mathematics textbook from Portuguese to English. Even if these techniques are simple and this case has some peculiarities, it seems that the same method will be useful in other cases as well, and, more importantly, that this experience is a good motivation for some research to be done on the possible refinements of the method.

Solving Linear Constraints on Finite Domains Through Parsing

In this paper we present results from ongoing research which allows the use of parsing methods to solve a particular kind of constraints, namely linear constraints on finite domains. Solving this kind of constraints is equivalent to solving systems of linear Diophantine equations on a finite subset of the naturals. We associate, to such a system, a definite-clause grammar that can be used to enumerate its solutions, and define a class of grammars, the connected grammars, for which the set of successful derivations covers the set of non-negative solutions of the associated system. This definition is based on a study of cycles in context-free grammars using compiler construction concepts and techniques.

Cooperating Rewrite Processes Revisited

As a first step in the study of the properties of the parsing techniques based on cooperating rewrite processes used in the Spiral Natural Language system, this paper presents a description of these techniques from a formal point of view.