Publications 2020


16.  Dynamic logic: new trends and applications

Martins, Manuel A. and Sedlár, Igor


This book constitutes the proceedings of the Third International Workshop on Dynamic Logic, DaLí 2019, held in Prague, Czech Republic in October 2020. Due to COVID-19 the workshop has been held online. The 17 full papers presented together with 6 short papers were carefully reviewed and selected from 31 submissions. The theoretical relevance and practical potential of dynamic logic is a topic of interest in a number of scientific venues, from wide-scope software engineering conferences to modal logic specific events. The DaLí 2020 workshop is exclusively dedicated to Dynamic logic and aims at filling this gap and creating a heterogeneous community of colleagues, from Academia to Industry, from Mathematics to Computer Science. | doi

Book Chapters

15.  A four-valued hybrid logic with non-dual modal operators

Costa, Diana and Martins, Manuel A.

Dynamic logic: new trends and applications

Springer International Publishing

Hybrid logics are an extension of modal logics where it is possible to refer to a specific state, thus allowing the description of what happens at specific states, equalities and transitions between them. This makes hybrid logics very desirable to work with relational structures. However, as the amount of information grows, it becomes increasingly more common to find inconsistencies. Information collected about a particular hybrid structure is not an exception. Rather than discarding all the data congregated, working with a paraconsistent type of logic allows us to keep it and still make sensible inferences. In this paper we introduce a four-valued semantics for hybrid logic, where contradictions are allowed both at the level of propositional variables and accessibility relations. A distinguishing feature of this new logic is the fact that the classical equivalence between modal operators will be broken. A sound and complete tableau system is also presented. | doi | Peer Reviewed

14.  Reversal Fuzzy Switch Graphs

Campos, Suene and Santiago, Regivan and Martins, Manuel A. and Figueiredo, Daniel

Formal Methods: Foundations and Applications. SBMF 2020.


Fuzzy Switch Graphs (FSG) generalize the notion of Fuzzy Graphs by adding high-order arrows and aggregation functions which update the fuzzy values of arrows whenever a zero-order arrow is crossed. In this paper, we propose a more general structure called Reversal Fuzzy Switch Graph (RFSG), which promotes other actions in addition to updating the fuzzy values of the arrows, namely: activation and deactivation of the arrows. RFSGs are able to model dynamical aspects of some systems which generally appear in engineering, computer science and some other fields. The paper also provides a logic to verify properties of the modelled system and closes with an application. | doi | Peer Reviewed


13.  Product preservation and stable units for reflections into idempotent subvarieties

Xarez, João J. and Xarez, Isabel A.

Categories and General Algebraic Structures with Applications

Shahid Beheshti University

We give a necessary and sufficient condition for the preservation of finite products by a reflection of a variety of universal algebras into an idempotent subvariety. It is also shown that simple and semi-left-exact reflections into subvarieties of universal algebras are the same. It then follows that a reflection of a variety of universal algebras into an idempotent subvariety has stable units if and only if it is simple and the above-mentioned condition holds. | Peer Reviewed

12.  Boolean dynamics revisited through feedback interconnections

Chaves, Madalena and Figueiredo, Daniel and Martins, Manuel A.

Natural Computing


Boolean models of physical or biological systems describe the global dynamics of the system and their attractors typically represent asymptotic behaviors. In the case of large networks composed of several modules, it may be difficult to identify all the attractors. To explore Boolean dynamics from a novel viewpoint, we will analyse the dynamics emerging from the composition of two known Boolean modules. The state transition graphs and attractors for each of the modules can be combined to construct a new asymptotic graph which will (1) provide a reliable method for attractor computation with partial information; (2) illustrate the differences in dynamical behavior induced by the updating strategy (asynchronous, synchronous, or mixed); and (3) show the inherited organization/structure of the original network’s state transition graph. | doi | Peer Reviewed

11.  A fuzzy modal logic for fuzzy transition systems

Jain, Manisha and Madeira, Alexandre and Martins, Manuel A.

Electronic Notes in Theoretical Computer Science


This paper intends to contribute with a new fuzzy modal logic to model and reason about transition systems involving uncertainty in behaviours. Our formalism supports fuzziness at transitions and on the proposition symbols assignment levels. Against of other approaches in the literature, our bisimulation and bisimilarity notions generalise the analogous standard notions of classic modal logic and of process algebras. Moreover, the outcome of our logic is also fuzzy, with the semantic interpretation of connectives supported by the Gödel algebra. | doi | Peer Reviewed

10.  Introducing synchrony in fuzzy automata

Gomes, Leandro and Madeira, Alexandre and Barbosa, Luis Soares

Electronic Notes in Theoretical Computer Science


This paper introduces a sort of automata and associated languages, often arising in modelling natural phenomena, in which both vagueness and simultaneity are taken as first class citizens. This requires a fuzzy semantics assigned to transitions and a precise notion of a synchronous product to enforce the simultaneous occurrence of actions. The expected relationships between automata and languages are revisited in this setting; in particular it is shown that any subset of a fuzzy synchronous language with the suitable signature forms a synchronous Kleene algebra. | doi | Peer Reviewed

9.  Systematic maximum sum rank codes

Almeida, Paulo and Martínez-Peñas, Umberto and Napp, Diego

Finite Fields and Their Applications


In the last decade there has been a great interest in extending results for codes equipped with the Hamming metric to analogous results for codes endowed with the rank metric. This work follows this thread of research and studies the characterization of systematic generator matrices (encoders) of codes with maximum rank distance. In the context of Hamming distance these codes are the so-called Maximum Distance Separable (MDS) codes and systematic encoders have been fully investigated. In this paper we investigate the algebraic properties and representation of encoders in systematic form of Maximum Rank Distance (MRD) codes and Maximum Sum Rank Distance (MSRD) codes. We address both block codes and convolutional codes separately and present necessary and sufficient conditions for an encoder in systematic form to generate a code with maximum (sum) rank distance. These characterizations are given in terms of certain matrices that must be superregular in a extension field and that preserve superregularity after some transformations performed over the base field. We conclude the work presenting some examples of Maximum Sum Rank convolutional codes over small fields. For the given parameters the examples obtained are over smaller fields than the examples obtained by other authors. | doi | Peer Reviewed

8.  Cartesian closed exact completions in topology

Clementino, Maria Manuel and Hofmann, Dirk and Ribeiro, Willian

Journal of Pure and Applied Algebra


Using generalized enriched categories, in this paper we show that Rosický’s proof of cartesian closedness of the exact completion of the category of topological spaces can be extended to a wide range of topological categories over Set, like metric spaces, approach spaces, ultrametric spaces, probabilistic metric spaces, and bitopological spaces. In order to do so we prove a sufficient criterion for exponentiability of (T, V)-categories and show that, under suitable conditions, every injective (T, V)-category is exponentiable in (T, V)-Cat. | doi | Peer Reviewed

7.  Automatic adjoint differentiation for gradient descent and model calibration

Goloubentsev, Dmitri and Lakshtanov, Evgeny

International Journal of Wavelets, Multiresolution and Information Processing

World Scientific Publishing

In this work, we discuss the Automatic Adjoint Differentiation (AAD) for functions of the form G=12∑m1(Eyi−Ci)2, which often appear in the calibration of stochastic models. We demonstrate that it allows a perfect SIMDa parallelization and provides its relative computational cost. In addition, we demonstrate that this theoretical result is in concordance with numerical experiments. a Single Input Multiple Data. | doi | Peer Reviewed

6.  Hausdorff coalgebras

Hofmann, Dirk and Nora, Pedro

Applied Categorical Structures


As composites of constant, (co)product, identity, and powerset functors, Kripke polynomial functors form a relevant class of $mathsf{Set}$-functors in the theory of coalgebras. The main goal of this paper is to expand the theory of limits in categories of coalgebras of Kripke polynomial functors to the context of quantale-enriched categories. To assume the role of the powerset functor we consider "powerset-like" functors based on the Hausdorff $mathsf{V}$-category structure. As a starting point, we show that for a lifting of a $mathsf{SET}$-functor to a topological category $mathsf{X}$ over $mathsf{Set}$ that commutes with the forgetful functor, the corresponding category of coalgebras over $mathsf{X}$ is topological over the category of coalgebras over $mathsf{Set}$ and, therefore, it is "as complete" but cannot be "more complete". Secondly, based on a Cantor-like argument, we observe that Hausdorff functors on categories of quantale-enriched categories do not admit a terminal coalgebra. Finally, in order to overcome these "negative" results, we combine quantale-enriched categories and topology emph{`a la} Nachbin. Besides studying some basic properties of these categories, we investigate "powerset-like" functors which simultaneously encode the classical Hausdorff metric and Vietoris topology and show that the corresponding categories of coalgebras of "Kripke polynomial" functors are (co)complete. | doi | Peer Reviewed

5.  Promover o raciocínio geométrico em alunos com Perturbação do Espectro do Autismo através de um ambiente digital

Santos, Maria Isabel Gomes dos and Breda, Ana Maria Reis d’Azevedo and Almeida, Ana Margarida Pisco

Bolema: Boletim de Educação Matemática

UNESP - Universidade Estadual Paulista

O raciocínio e a construção do sentido espacial são capacidades essenciais em todos os processos de aprendizagem e compreensão matemática de crianças com desenvolvimento típico. Em crianças com Perturbação do Espectro do Autismo (PEA), estas capacidades se tornam ainda muito mais significativas, considerando o papel relevante que desempenham para uma vida independente bem-sucedida. O uso da tecnologia é referido como uma forma eficaz de trabalhar o conteúdo acadêmico com crianças com PEA, possibilitando a criação de ambientes criativos e construtivos onde se podem desenvolver atividades diferenciadas, significativas e de qualidade. No entanto, o desenvolvimento de aplicações tecnológicas para crianças e jovens com PEA continua a merecer pouca atenção, nomeadamente as que dizem respeito ao desenvolvimento do pensamento geométrico. O objetivo deste artigo é relatar os principais resultados obtidos com crianças com PEA utilizando o ambiente digital Learning Environment on Mathematics for Autistic Children desenvolvido, particularmente, no que se refere à promoção das suas capacidades matemáticas fundamentais em geometria. O “PISA 2015 Mathematics Framework” foi a base teórica utilizada para a recolha dos dados, cuja análise, para além de situar o pensamento geométrico dos alunos participantes entre parcialmente estruturado e estruturado, apontou, também, para o redesenho de algumas das atividades implementadas no ambiente digital, tendo em vista a promoção do pensamento geométrico. | doi | Peer Reviewed

4.  Complete j-MDP convolutional codes

Almeida, Paulo J. and Lieb, Julia

IEEE Transactions on Information Theory


Maximum distance profile (MDP) convolutional codes have been proven to be very suitable for transmission over an erasure channel. In addition, the subclass of complete MDP convolutional codes has the ability to restart decoding after a burst of erasures. However, there is a lack of constructions of these codes over fields of small size. In this paper, we introduce the notion of complete j-MDP convolutional codes, which are a generalization of complete MDP convolutional codes, and describe their decoding properties. In particular, we present a decoding algorithm for decoding erasures within a given time delay T and show that complete T-MDP convolutional codes are optimal for this algorithm. Moreover, using a computer search with the MAPLE software, we determine the minimal binary and non-binary field size for the existence of (2, 1, 2) complete j-MDP convolutional codes and provide corresponding constructions. We give a description of all (2, 1, 2) complete MDP convolutional codes over the smallest possible fields, namely F13 and F16 and we also give constructions for (2, 1, 3) complete 4-MDP convolutional codes over F128 obtained by a randomized computer search. | doi | Peer Reviewed

3.  Représentation de la courbe intersection d’une quadrique et d’une quartique

Breda, Ana and Trocado, Alexandre and Neves, António and Santos, José dos


Cet article présente l’implémentation dans le langage Geogebra de deux algorithmes pour calculer la courbe intersection d’une quadrique et d’une quartique. Nous présentons deux méthodes de calcul et de visualisation de cette courbe avec GeoGebra. La première utilise un logiciel de Calcul formel pour obtenir une paramétrisation de la courbe et sa représentation. La seconde est basée sur le calcul d’une projection plane de la courbe intersection, sur la détermination de son allure et de ses singularités, et sur son relèvement dans l’espace à trois dimensions. Ce problème met en évidence les difficultés innérentes à l’implémentaion dans Geogebra d’un algorithme géométrique basé sur les équations algébriques définissant les surfaces. Nous nous intéressons également à l’utilisation de ces techniques de calcul comme outil pédagogique dans l’enseignement supérieur.

2.  Mapification of n-dimensional abstract polytopes and hypertopes

Breda d'Azevedo, António

Ars Mathematica Contemporanea

The n-dimensional abstract polytopes and hypertopes, particularly the regular ones, have gained great popularity over recent years. The main focus of research has been their symmetries and regularity. The planification of a polyhedron helps its spatial construction, yet it destroys symmetries. No “planification” of n-dimensional polytopes do exist, however it is possible to make a “mapification” of an n-dimensional polytope; in other words it is possible to construct a restrictedly-marked map representation of an abstract polytope on some surface that describes its combinatorial structures as well as all of its symmetries. There are infinitely many ways to do this, yet there is one that is more natural that describes reflections on the sides of (n-1)-simplices (flags or n-flags) with reflections on the sides of n-gons. The restrictedly-marked map representation of an abstract polytope is a cellular embedding of the flag graph of a polytope. We illustrate this construction with the 4-cube, a regular 4-polytope with automorphism group of size 384. This paper pays a tribute to Lynne James’ last work on map representations. | doi | Peer Reviewed

1.  Strong map-symmetry of SL(3, K) and PSL(3, K) for every finite field K

Breda d'Azevedo, António and Catalano, Domenico A.

Journal of Algebra and Its Applications

World Scientific Publishing

In this paper, we show that for any finite field K, any pair of map-generators (that is when one of the generators is an involution) of SL(3,K) and PSL(3,K) has a group automorphism that inverts both generators. In the theory of maps, this corresponds to say that any regular oriented map with automorphism group SL(3,K) or PSL(3,K) is reflexible, or equivalently, there are no chiral regular maps with automorphism group SL(3,K) or PSL(3,K). As remarked by Leemans and Liebeck, also SU(3,K) and PSU(3,K) are not automorphism groups of chiral regular maps. These two results complete the work of the above authors on simples groups supporting chiral regular maps. | doi | Peer Reviewed
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