## About

155

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Introduction

Education

September 1998 - March 2002

September 1991 - June 1993

September 1983 - September 1989

## Publications

Publications (155)

I am describing the square of opposition, in particular, and, Aristotelian Diagrams, in general. Then I describe how one can create a mathematical universe to host them. Based on this work, I introduce fuzzy Aristotelian Diagrams and describe a mathematical formulation of them. In addition, I outline the cdharacteristrics of a mathematical universe...

Computational thinking is a teaching methodology that has been successfully applied to STEM education. However, its use outside STEM education seems to be quite limited. This study aims to report a small-scale experiment in which a small group of students were taught elements of the Greek language using STEM education methodology. Since computation...

The aim of this work is to broaden the theory of computation by making use of fuzzy sets and multisets, which are generalization of ordinary sets. We study fuzzy multiset finite automata (FMFA), which are characterized by the incorporation of vagueness when it comes to the choice of the next state and multiple occurrences of the same input symbol....

Vagueness is a quality that is usually ignored or considered a deficiency for any system, in the broad sense of the word. However, vagueness seems to be a fundamental property of our world and so systems that are precisely defined are rather the exception than the norm (e.g., in most cases we ignore some details that are supposed not to affect the...

Although programming languages are expressive tools, their expressive power is quite limited. A direct consequence of this is that novice users have difficulty grasping the basic programming constructs mainly because there is a discrepancy between natural languages and programming languages (e.g., How do we express common human tasks in Python?). A...

Teaching mathematics to middle-school pupils is always quite challenging. One has to introduce pupils to formal thinking and teach them the importance of this kind of thinking. Naturally, any tool that can assist the work of a teacher is more than welcome. GeoGebra is a tool that has been created by a mathematician for math teachers. This tool has...

There are many things, ideas, and entities that we encounter every single day that can be described as vague, that is, one cannot precisely classify them as members of a specific class of objects. Although many hold the opinion that vagueness is either a deficiency of spoken languages or something that exists merely because we do not have all relev...

It is undeniable that information flow is crucial for the dissemination of information and knowledge. However, information flows in a vague environment; that is, an environment full of noise. Starting from a “pure” model of information flow, I propose a model that incorporates vagueness. This model has the capacity to describe information loss, and...

The basic set operations between fuzzy sets are defined using the min and max functions; however, later on, new operators were introduced that used other functions, which, nevertheless, had similar properties to functions min and max. The resulting fuzzy set theories are more suitable for the description and processing of specific data sets. Crisp...

The book starts with the assumption that vagueness is a fundamental property of this world. From a philosophical account of vagueness via the presentation of alternative mathematics of vagueness, the subsequent chapters explore how vagueness manifests itself in the various exact sciences: physics, chemistry, medicine, computer science, and engineer...

After a brief discussion of vagueness and its importance and short
introductions to fuzzy sets and category theory, I discuss categories of
fuzzy structures (i.e., categories whose objects are fuzzy structures) and
I present two versions of fuzzy categories: one that is based on the theory
of fuzzy graphs and one that is based on the idea that cate...

We report the results of an EU-funded project and its impact on participating teachers. Teachers participating in the project had practically no previous knowledge of game-based education whereas they got a thorough understanding of it at the end of the project. In addition, and most importantly, teachers had to guide pupils in designing and implem...

The authors show how information communication and technologies (ICTs) can be used to boost the economy of a country that emerges from a deep crisis. In particular, they discuss how the economy can change by incorporating ICTS in all areas of economic activity. In addition, they examine how Greece, which is a typical example of a country thar emerg...

Computers are widely used in physics and other natural sciences to simulate physical phenomena. Thus, people routinely use computers to model many and different physical systems. In addition, computers have been used to solve difficult problems by simulating successful practices employed by living organism. However, it seems not everything can be s...

Digital documents tend to replace printed documents in many ways. For publishers it is cheaper to produce them and readers can get them instantly. However, the spectrum of “printed” material is quite big and so there is no way to have one kind of digital document that can satisfy all those involved in the publishing business (e.g., authors, readers...

The proliferation of the use of websites and their counterparts for mobile devices has created a number of problems that did not occur in the past. Typically, one needs to create many and different accounts for all websites and apps she wants to use. However, this means one has to do a lot of booking work. Fortunately, authentication and authorizat...

Generally, an object is vague when its properties cannot precisely defined. The ontic view of vagueness is the idea that vagueness is a fundamental property of Nature. This simply means that everything is vague: animals, plants, molecules, atoms, etc. Furthermore, if atoms and molecules are vague, then the subject matter of chemistry is vague. Howe...

Geometry and topology are two different branches of mathematics that deal with the same objects. This chapter discusses fuzzy topology, and since metrics induce topologies, it begins with fuzzy metric spaces. Fuzzy metric spaces were introduced by Ivan Kramosil and Jirí Michálek. In their approach, the distance between two points is not just a real...

Logic is about the process of reaching an answer by thinking about known facts. And logic in general can be divided into formal logic, informal logic, symbolic logic, and mathematical logic. Aristotle invented logic, but it took more than two millennia to have a mathematical description of logic and its rules. In logic, we can combine statements or...

This chapter describes a fuzzy restriction that acts as an elastic constraint on the values that may be assigned to a variable, and introduces possibility measures and necessity measures. A theory that is based on possibility and necessity measures is usually called a possibility theory. The chapter discusses the possibility theory and its relation...

Many and different real or conceptual devices capable of performing computations have been proposed. Most of them operate in a crisp environment and in a crisp manner. However, there are some devices that profit from the use of vagueness in their overall operation. This chapter describes such devices and the related theory. It discusses fuzzy autom...

Fuzzy numbers are a mathematical formulation of vague statements about real numbers. They are a special kind of fuzzy sets that have been introduced by Zadeh. A triangular fuzzy number is a fuzzy set whose graph is a triangle. Trapezoidal fuzzy numbers are, of course, fuzzy sets but since their core contains more than one element, they cannot be cl...

This book aims to fill a gap in fuzzy mathematics literature by presenting recent developments and also discussing what has been done so far in the field. The book contains 12 chapters. Chapter 1 introduces the reader to the basic ideas behind fuzzy sets. The authors explain what vagueness is and how it is different from other related ideas such as...

Abstract algebra is a very important field of mathematics. Many researchers working on fuzzy mathematics tried to introduce various fuzzy versions of known algebraic structures such as groups, rings, fields, etc. This chapter begins with classical concepts and notions in abstract algebra. It starts with groupoids and semigroups with definitions. Ro...

In the context of fuzzy mathematics, one follows a different route in geometry and its relation to reality. Indeed, when one studies or measures real objects, he/she soon realizes that the very process of measurement will often render expectations and theoretically exact predictions as improbable. In fact, fuzzy geometry has been studied from diffe...

Fuzzy relations are important tools that are used in fuzzy modeling, fuzzy diagnosis, and fuzzy control, which explains why it is useful to have a good understanding of fuzzy relations and their properties. Fuzzy relations extend crisp relations just like fuzzy sets extend crisp sets. This chapter discusses the properties of relations and explains...

The importance of fuzzy calculus becomes strongly evident if one thinks of the role played by classical calculus concepts in the mathematical modeling of real‐world phenomena. There is an imprecision or vagueness often characterizing the gathered experimental information or even the theoretical background needed to quantitatively describe a particu...

Based on the assumption that all mathematics can be built up from sets, one has to first define fuzzy sets and their variations. This chapter explores truth values and their algebras and presents basic ideas and results of lattice theory. In 1965, Zadeh published a paper entitled “Fuzzy Sets” where he introduced his fuzzy sets. In classical set the...

This chapter presents a brief review of random variables, also known as stochastic variables, and their properties as well as the classical statistical notions of point estimation, interval estimation, hypothesis testing, and regression. It introduces fuzzy‐valued random variables and adopts the Kruse–Meyer approach in which a fuzzy random variable...

Brown and Gurr have introduced a model of Petri Nets that is based on de~Paiva's Dialectica categories. This model was refined in an unpublished technical report, where Petri nets with multiplicities, instead of {\em elementary} nets (i.e., nets with multiplicities zero and one only) were considered. In this note we expand this modelling to deal wi...

Bigraphs and their algebra is a model of concurrency. Fuzzy bigraphs are a generalization of birgraphs intended to be a model of concurrency that incorporates vagueness. More specifically, this model assumes that agents are similar, communication is not perfect, and, in general, everything is or happens to some degree.

Trail-And-Error machines have been proposed by Hintikka and Mutanen as an alternative formulation of the notion of (mechanical) computation. These machines extend the capabilities of the Turing machine and widen the theory of computation.

Bigraphs and their algebra is a model of concurrency. Fuzzy bigraphs are a generalization of birgraphs intended to be a model of concurrency that incorporates vagueness. More specifically, this model assumes that agents are similar, communication is not perfect, and, in general, everything is or happens to some degree.

This is a short introduction to JavaScript programming that has been designed
for people who want to learn how to create simple or not so simple animations with
the HTML5 element. The text assumes that the reader is familiar
with certain basic things (the use of a browser and a text editor), which I believe
are common knowledge nowadays.

The economy of any country can thrive, decline, or remain steady. A declining economy is a troubled one, and it is necessary to find ways to reverse its course. Naturally, there is no single recipe to put a troubled economy back on track, but innovation and technology always help an economy to boost itself. In particular, information and computer t...

P systems are computing conceptual computing devices that are at least as powerful as Turing machines. However, until recently it was not known how one can encode any recursive function as a P~system. Here we propose a new encoding of recursive as P~systems with graph-like structure, which is the main difference with previous documented attempts. T...

The motion planning problem is a fundamental problem in robotics, so that every autonomous robot should be able to deal with it. A number of solutions have been proposed and a probabilistic one seems to be quite reasonable. However, here we propose a more adoptive solution that uses fuzzy set theory and we expose this solution next to a sort survey...

Podcasting and vodcasting are audio and video files, respectively. These files can be accessed by subscribers at any time of day. Initially, the technology was used for information and entertainment. Later on, it became clear that this technology could be useful in education. There are many advantages in the use of these technologies. Yet, there ar...

The motion planning problem is a fundamental problem in robotics, so that every autonomous robot should be able to deal with it. A number of solutions have been proposed and a probabilistic one seems to be quite reasonable. However, here we propose a more adoptive solution that uses fuzzy set theory and we expose this solution next to a sort survey...

Vagueness is something everyone is familiar with. In fact, most people think that vagueness is closely related to language and exists only there. However, vagueness is a property of the physical world. Quantum computers harness superposition and entanglement to perform their computational tasks. Both superposition and entanglement are vague process...

Podcasting and vodcasting are audio and video files, respectively. These files can be accessed by subscribers at any time of day. Initially, the technology was used for information and entertainment. Later on, it became clear that this technology could be useful in education. There are many advantages in the use of these technologies yet there are...

The economy of any country can thrive, decline or remain steady. A declining economy is a troubled one and it is necessary to find ways to reverse its course. Naturally, there is no single recipe to put a troubled economy back on track but innovation and technology always help an economy to boost itself. In particular, information and computer tech...

Ideograms (symbols that represent a word or idea) have a great communicative value. They refer to concepts in a simple manner, thus making it easier the understanding of complex ideas where such concepts appear. In addition, ideograms simplify the notation, which is often cumbersome for equations in physics. We propose ideograms for essential conce...

Vagueness refers to situations where it is not clear whether an object has a property or not (e.g., properties that manifest vagueness include height, spiciness, and beauty). Fuzzy set theory is one possible mathematical model of vagueness and fuzzy logic is the logic associated with fuzzy sets. Typically, elements belong to fuzzy sets to a degree,...

Fractal geometry has found many applications in science and technology. Some time ago, it was
used to study urban development. However, something that has not been addressed so far, to the best of
our knowledge, is whether a drastic extension of some urban area also changes drastically the box-counting
dimension of the area. In addition, it is not...

Ideograms (symbols that represent a word or idea) have great communicative value. They refer to concepts in a simple manner, easing the understanding of related ideas. Moreover, ideograms can simplify the often cumbersome notation used in the fields of Physics and physical Chemistry. Nonetheless only a few specific ideograms for these fields have b...

A presentation of the Asana Math font at the ATypI 2016 conference

Although Internet Addiction is not listed in the Diagnostic and Statistical Manual of Mental Disorders, it is an increasingly prevalent problem that affects a lot of people, including pupils. Erasmus+ Strategic partnerships, which are co-funded by the Erasmus+ Program of the European Union, allow schools to form partnerships that work on projects t...

Vagueness is a linguistic phenomenon as well as a property of physical objects. Fuzzy set theory is a mathematical model of vagueness that has been used to define vague models of computation. The prominent model of vague computation is the fuzzy Turing machine. This conceptual computing device gives an idea of what computing under vagueness means,...

Multisets and Multirelations are extensions of ordinary sets and relations while fuzzy multisets and fuzzy multirelations are fuzzy extensions of these concept. Although many generalizations tend to be meaningless, these are quite useful as one can model things we see everyday. In addition, these structures have found uses in the theory of computat...

A description of a new font (and its accompanying LaTeX package) that introduces new symbols for setting Physics texts.

Since categories are graphs with additional "structure", one should start
from fuzzy graphs in order to define a theory of fuzzy categories. Thus is
makes sense to introduce categories whose morphisms are associated with a
plausibility degree that determines to what extend it is possible to "go" from
one object to another one. These categories are...

Turing machines form the core of computability theory, or recursion theory as it is also known. This chapter introduces basic notions and results and readers already familiar with them can safely skip it. The exposition is based on standard references [18, 39, 67, 83, 109]. In the discussion that follows, the symbol ℕ will stand for the set of posi...

Fuzzy Turing machines are not the only way to perform computation in a vague environment. Other models of fuzzy computation are inspired by biological phenomena or, more generally, by natural phenomena. Fuzzy P systems and the fuzzy chemical abstract machine are such models of computation. Some of these models have been studied in some detail while...

The fusion of computability theory (CP) with fuzzy set theory (FST) demands a good knowledge of both theories. Thus, this introductory chapter tries to familiarize readers with a number of concepts and ideas that are necessary for the rest of this book. In particular, I start with a review of the events that lead to what is now known as computer sc...

Nowadays, scripting programming languages like Python, Perl and Ruby are
widely used in system programming, scientific computing, etc. Although solving
a particular problem in these languages requires less time, less programming
effort, and less concepts to be taught to achieve the desired goal, still they
are not used as teaching tools. Therefore,...

The notion of fuzziness lies at the core of fuzzy computability theory. Thus, one should have a basic understanding of the ideas involved. This chapter serves both as a crash course in fuzzy set theory, for those readers that have no previous knowledge of the concepts involved, and as a précis of fuzzy set theory, for those readers familiar with th...

The Turing machine is the archetypal conceptual computing device and as such it is almost always used to define new models of computations by augmenting the functionality of the machine. For example, as was explained in Sect. 2.2.2, a probabilistic model of computation is defined by introducing a probabilistic version of the Turing machine, that is...

This book provides the first full length exploration of fuzzy computability. It describes the notion of fuzziness and features the basic ideas of computability theory. It then presents the various approaches to fuzzy computability. This text provides a glimpse into the different approaches in this area, which is important for researchers in order t...

In this review I give a brief overview of the use of vagueness in computing. In particular, I give a description of what vagueness is and how, in the form of fuzzy set theory, it has been applied to define (conceptual) models of computation. In particular, I give an overview of fuzzy Turing machine, fuzzy P systems, and the fuzzy chemical abstract...

Hypercomputation is about the feasibility of machines and systems that are either more expressive or computationally more powerful than the Turing machine. A number of researchers and thinkers have put forth a number of supposedly knock-out arguments against hypercomputation. Nevertheless, these arguments are not unwavering as they seem to be and h...

Typically, when a computer performs a task, it can be seen as a calculation or a reckoning. For example, consider a simple arcade video game where the machine continuously gets input from the user and computes the new position of some "characters" that move on a board, etc. A particularly interesting aspect of computation is that the majority of pe...

An orthogonal approach to the fuzzification of both multisets and hybrid sets
is presented. In particular, we introduce L-multi-fuzzy and L-fuzzy hybrid
sets, which are general enough and in spirit with the basic concepts of fuzzy
set theory. In addition, we study the properties of these structures. Also, the
usefulness of these structures is exami...

Dialectica categories are a very versatile categorical model of linear logic.
These have been used to model many seemingly different things (e.g., Petri nets
and Lambek's calculus). In this note, we expand our previous work on fuzzy
petri nets to deal with fuzzy topological systems. One basic idea is to use as
the dualizing object in the Dialectica...

By using the representational power of Chu spaces we define the notion of a
generalized topological space (or GTS, for short), i.e., a mathematical
structure that generalizes the notion of a topological space. We demonstrate
that these topological spaces have as special cases known topological spaces.
Furthermore, we develop the various topological...

Scala is a highly expressive, concise and scalable language. It is also the most prominent method of the new and exciting methodology known as object-functional programming. In this book, the authors show how Scala grows to the needs of the programmer, whether professional or hobbyist. They teach Scala with a step-by-step approach and explain how t...

Roughly, the Church-Turing thesis is a hypothesis that describes exactly what can be computed by any real or feasible conceptual computing device. Generally speaking, the computational metaphor is the idea that everything, including the universe itself, has a computational nature. However, if the Church-Turing thesis is not valid, then does it make...

Hypercomputation is a relatively new branch of computer science that emerged from the idea that the Church--Turing Thesis, which is supposed to describe what is computable and what is noncomputable, cannot possible be true. Because of its apparent validity, the Church--Turing Thesis has been used to investigate the possible limits of intelligence o...

Fuzzy set theory opens new vistas in computability theory and here I show this by defining a new computational metaphor--the fuzzy chemical metaphor. This metaphor is an extension of the chemical metaphor. In particular, I introduce the idea of a state of a system as a solution of fuzzy molecules, that is molecules that are not just different but r...

This book provides a thorough description of hypercomputation. It covers all attempts at devising conceptual hypermachines and all new promising computational paradigms that may eventually lead to the construction of a hypermachine. Readers will gain a deeper understanding of what computability is, and why the Church-Turing thesis poses an arbitrar...

## Questions

Question (1)

In 2006 I published a paper entitled

**(see http://comjnl.oxfordjournals.org/content/early/2006/07/20/comjnl.bxl029.extract ) where I introduced this term to describe a version of fuzzy multisets. I guess the authors of the paper you mention should have chosen a new name or at least explain that they are talking about something new or entirely different,***Fuzzifying P Systems*