Publications (27)3.71 Total impact
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ABSTRACT: We present an application of constraint logic programming to create multiplechoice questions for math quizzes. Constraints are used for the configuration of the generator, giving the user some flexibility to customize the forms of the expressions arising in the exercises. Constraints are also used to control the application of the buggy rules in the derivation of plausible wrong solutions to the quiz questions. We developed a prototype based on the core system of AGILMAT [18]. For delivering math quizzes to students, we used an automatic evaluation feature of Mooshak [8] that was improved to handle math expressions. The communication between the two systems  AgilmatQuiz and Mooshak  relies on a specially designed LaTeX based quiz format. This tool is being used at our institution to create quizzes to support assessment in a PreCalculus course for first year undergraduate students. 
Conference Paper: Guarding Thin Orthogonal Polygons Is Hard
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ABSTRACT: An orthogonal polygon of P is called "thin" if the dual graph of the partition obtained by extending all edges of P towards its interior until they hit the boundary is a tree. We show that the problem of computing a minimum guard set for either a thin orthogonal polygon or only its vertices is NPhard, indeed APXhard, either for guards lying on the boundary or on vertices of the polygon. For guards lying anywhere in the polygon, we show that computing an optimal guard set for the vertices of such a polygon is NPhard.  [Show abstract] [Hide abstract]
ABSTRACT: The paper addresses a variant of the stable marriage problem that models a job recruitment problem in which applicants are strictly ordered by priority but their preference lists may have ties. Some applicants may hold a post initially. These posts may be assigned to other applicants if their holders get another post. By reducing the problem to a sequence of maximum cardinality bipartite matching problems, combined with an effective propagation of the stability constraints, we show that applicantoptimal stable matchings may be found efficiently. 
Conference Paper: A Web Application for Mathematics Education.
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ABSTRACT: AGILMAT is a web application designed to help students learn Mathematics, with focus on highschool algebra and calculus drills. A modular and extensible architecture and a wizardbased configuration interface decoupled from the system core are major design features of AGILMAT. The drill expressions are specified by grammars and constraints imposed by default profiles and user options, so that AGILMAT may support distinct learning levels and stages. The core system uses symbolic manipulation and automated reasoning to provide correct answers for the drills. The paper shows how AGILMAT may be used to create and customize drills automatically.  [Show abstract] [Hide abstract]
ABSTRACT: We present a conditional rewrite system for arithmetic and membership univariate constraints over real numbers, designed for computer assisted learning (CAL) in elementary math. Two fundamental principles guided the design of the proposed rewrite rules: cognitive fidelity (emulating steps students should take) and correctness, aiming that stepbystep solutions to problems look like ones carried out by students. In order to gain more flexibility to modify rules, add new ones and customize solvers, the rules are written in a specification language and then compiled to Prolog. The rewrite system is complete for a relevant subset of problems found in highschool math textbooks.  [Show abstract] [Hide abstract]
ABSTRACT: 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. 
Conference Paper: On Visibility Problems in the Plane  Solving Minimum Vertex Guard Problems by Successive Approximations.
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ABSTRACT: We address the problem of stationing guards in vertices of a simple polygon in such a way that the whole polygon is guarded and the number of guards is minimum. It is known that this is an NPhard Art Gallery Problem with relevant practical applications. In this paper we present an approximation method that solves the problem by successive approximations, which we introduced in [21]. We report on some results of its experimental evaluation and describe two algorithms for characterizing visibility from a point, that we designed for its implementation.  [Show abstract] [Hide abstract]
ABSTRACT: For a long time, term orderings defined by polynomial interpretations have been considered far too restrictive to be used for computeraided 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 under lying 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 mono tonic orderings. From these criteria, we obtain new techniques both for mechanically checking termination using a given polynomial interpretation, and for finding such in terpretations with full automation. With regards to automated search, we propose an original method for solving Diophantine constraints. We implemented these techniques into the CiME rewrite tool, and we provide ex periments that show how useful polynomial orderings actually are in practice. 
Conference Paper: AGILMAT  a Web Application for Math Education
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ABSTRACT: To understand what people do when they do mathematics and write programs emulating that process is a continuous research topic in artificial intelligence, automated reasoning and symbolic computation. We present the first release of an web application for athematics education that is being developed within AGILMAT (Automatic Generation of Interactive Drills for Mathematics Learning, POSI/CHS/48565/2002) project. AGILMAT aims at developing a system to automatically create and solve mathematics exercises that is flexible enough to be easily customizable to different curricula and users. Its major guiding principles are: the abstraction and formal representation of the problems that may be actually solved by algebraic algorithms covered by the curricula; the customization of these models by adding further constraints; and designing flexible solvers that emulate the steps students usually take to solve the generated drills. 
Conference Paper: QuadraticTime LinearSpace Algorithms for Generating Orthogonal Polygons with a Given Number of Vertices
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ABSTRACT: We propose InflatePaste { 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 InflateCut { a technique we introduced in (12) that works by cutting rectangles. 


Conference Paper: Partitioning Orthogonal Polygons by Extension of All Edges Incident to Reflex Vertices: Lower and Upper Bounds on the Number of Pieces
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ABSTRACT: Given an orthogonal polygon P , let j (P )j be the number of rectangles that result when we partition P by extending the edges incident to reex vertices towards INT(P ). In (4) we have shown that j (P )j 1 + r + r 2 , where r is the number of reex vertices of P. We shall now give sharper bounds both for maxP j (P )j and minP j (P )j. 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 nogons and characterize those having the largest and the smallest area. 
Conference Paper: Approximation Algorithms to Minimum Vertex Cover Problems on Polygons and Terrains
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ABSTRACT: 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 rst detect pieces that are more dicult to guard.  [Show abstract] [Hide abstract]
ABSTRACT: 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 de.ne 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. 
Conference Paper: Lecture Notes in Computer Science
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ABSTRACT: We propose two dieren t methods for generating random or thogonal polygons with a given number of vertices. One is a polynomial time algorithm and it is supported by a technique we developed to ob tain 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 nd all nvertex orthogonal polygons with no collinear edges that can be drawn in an n 2 n 2 grid, for small n, with symmetries broken. 
Conference Paper: Solving Optimal Location of Traffic Counting Points at Urban Intersections in CLP(FD).
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ABSTRACT: We present an application of Constraint Logic Programming (CLP) for finding the minimum number and location of countposts at urban roundabouts so as to obtain origindestination data at minimum cost. By finding nice mathematical properties, we were able to model this problem as a constraint satisfaction problem in finite domains, and use CLP(FD) systems to solve it, with almost no implementation effort and very quickly. 
Conference Paper: Solving Linear Diophantine Equations Using the Geometric Structure of the Solution Space.
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ABSTRACT: In the development of algorithms for nding 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 nding bounds for searching solutions and in having a qualitative evaluation of the diculty in solving a given system. word problems, or combinatorics. In terms of the development of algorithms for solving this problem, little use has been made (to our knowledge) of the results by Stanley using the geometric properties of the solution space (14, 15), in particular, his characterization of the generating function of the solutions monoid. Building upon these results, we present a new algorithm, which is a refor mulation of the Slopes Algorithm we described previously for solving a single equation (6), and we suggest the use of geometric properties of the solution space in nding bounds for searching solutions and in having a qualitative evaluation of the diculty in solving a given system. We also note that, as a direct conse quence of Stanley's results, the algorithm by Domenjoud (3) can be improved. 
Article: A Fast Method for Finding the Basis of Nonnegative Solutions to a Linear Diophantine Equation
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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 wellknown 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. 
Publication Stats
178  Citations  
3.71  Total Impact Points  
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Institutions

19972013

University of Porto
 Faculty of Sciences
Oporto, Porto, Portugal
