Ana Paula Tomás

Mechanically Proving Termination Using Polynomial Interpretations

For a long time, term orderings defined by polynomial interpretations were scarcely used in computer-aided termination proof of TRSs. But recently, the introduction of the dependency pairs approach achieved considerable progress w.r.t. automated termination proof, in particular by requiring from the underlying ordering much weaker properties than the classical approach. As a consequence, the noticeable power of a combination dependency pairs/polynomial orderings yielded a regain of interest for these interpretations. We describe criteria on polynomial interpretations for them to define weakly monotonic orderings. From these criteria, we obtain new techniques both for mechanically checking termination using a given polynomial interpretation and for finding such interpretations with full automation. With regard to automated search, we propose an original method for solving Diophantine constraints. We implemented these techniques into the CiME rewrite tool, and we provide some experimental results that show how useful polynomial orderings actually are in practice.

Quadratic-Time Linear-Space Algorithms for Generating Orthogonal Polygons with a Given Number of Vertices

We propose Inflate-Paste – a new technique for generating orthogonal polygons with a given number of vertices from a unit square based on gluing rectangles. It is dual to Inflate-Cut – a technique we introduced in [12] that works by cutting rectangles.

Approximation algorithms to minimum vertex cover problems on polygons and terrains

We propose an anytime algorithm to compute successively better approximations of the optimum of Minimum Vertex Guard. Though the presentation is focused on polygons, the work may be directly extended to terrains along the lines of [4]. A major idea in our approach is to explore dominance of visibility regions to first detect pieces that are more difficult to guard.

Generating Random Orthogonal Polygons

We propose two different methods for generating random orthogonal polygons with a given number of vertices. One is a polynomial time algorithm and it is supported by a technique we developed to obtain polygons with an increasing number of vertices starting from a unit square. The other follows a constraint programming approach and gives great control on the generated polygons. In particular, it may be used to find all n-vertex orthogonal polygons with no collinear edges that can be drawn in an \(\frac{n}{2} \times \frac{n}{2}\) grid, for small n, with symmetries broken.

A CLP-Based Tool for Computer Aided Generation and Solving of Maths Exercises

We propose an interesting application of Constraint Logic Programming to automatic generation and explanation of mathematics exercises. A particular topic in mathematics is considered to investigate and illustrate the advantages of using the CLP paradigm. The goal is to develop software components that make the formulation and explanation of exercises easier. We describe exercises by grammars which enables us to get specialized forms almost for free, by imposing further conditions through constraints. To define the grammars we concentrate on the solving procedures that are taught instead of trying to abstract an exercise template from a sample of similar exercises. Prototype programs indicate that Constraint Logic Programming frameworks may be adequate to implement such a tool. These languages have the right expressiveness to encode control on the system in an elegant and declarative way.

Partitioning Orthogonal Polygons by Extension of All Edges Incident to Reflex Vertices: Lower and Upper Bounds on the Number of Pieces

Given an orthogonal polygon P, let |Π(P)| be the number of rectangles that result when we partition P by extending the edges incident to reflex vertices towards INT(P). In [4] we have shown that |Π(P)| ≤ 1+r+r 2, where r is the number of reflex vertices of P. We shall now give sharper bounds both for max p |Π(P)| and min p |Π(P)|. Moreover, we characterize the structure of orthogonal polygons in general position for which these new bounds are exact. We also present bounds on the area of grid n-ogons and characterize those having the largest and the smallest area.

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.

From Elliott-MacMahon to an Algorithm for General Linear Constraints on Naturals

We describe a new algorithm for solving a conjunction of linear diophantine equations, inequations and disequations in natural numbers. We derive our algorithm from one proposed by Elliott in 1903 for solving a single homogeneous equation. This algorithm was then extended to solve homogeneous systems of equations by MacMahon. We show how it further extends to an algorithm which solves general linear constraints in nonnegative integers and allows a parallel implementation. This algorithm provides a parametric representation of the solutions from which minimal solutions may be extracted immediately. Moreover, it may be easily implemented in parallel. It has however one drawback: it is redundant which means that the same minimal solution is usually generated many times. We show how this redundancy may be eliminated at the cost of an increase in the space complexity.