Other Mathematics Commons^™

Relation Lifting, With An Application To The Many-Valued Cover Modality, Marta Bílková, Alexander Kurz, Daniela Petrişan, Jirí Velebil

Engineering Faculty Articles and Research

We introduce basic notions and results about relation liftings on categories enriched in a commutative quantale. We derive two necessary and sufficient conditions for a 2-functor T to admit a functorial relation lifting: one is the existence of a distributive law of T over the “powerset monad” on categories, one is the preservation by T of “exactness” of certain squares. Both characterisations are generalisations of the “classical” results known for set functors: the first characterisation generalises the existence of a distributive law over the genuine powerset monad, the second generalises preservation of weak pullbacks.

The results presented in this paper …

Epistemic Updates On Algebras, Alexander Kurz, Alessandra Palmigiano

Engineering Faculty Articles and Research

We develop the mathematical theory of epistemic updates with the tools of duality theory. We focus on the Logic of Epistemic Actions and Knowledge (EAK), introduced by Baltag-Moss-Solecki, without the common knowledge operator. We dually characterize the product update construction of EAK as a certain construction transforming the complex algebras associated with the given model into the complex algebra associated with the updated model. This dual characterization naturally generalizes to much wider classes of algebras, which include, but are not limited to, arbitrary BAOs and arbitrary modal expansions of Heyting algebras (HAOs). As an application of this dual characterization, we …

Nominal Coalgebraic Data Types With Applications To Lambda Calculus, Alexander Kurz, Daniela Petrişan, Paula Severi, Fer-Jan De Vries

Engineering Faculty Articles and Research

We investigate final coalgebras in nominal sets. This allows us to define types of infinite data with binding for which all constructions automatically respect alpha equivalence. We give applications to the infinitary lambda calculus.

Dynamic Sequent Calculus For The Logic Of Epistemic Actions And Knowledge, Giuseppe Greco, Alexander Kurz, Alessandra Palmigiano

Engineering Faculty Articles and Research

"Dynamic Logics (DLs) form a large family of nonclassical logics, and perhaps the one enjoying the widest range of applications. Indeed, they are designed to formalize change caused by actions of diverse nature: updates on the memory state of a computer, displacements of moving robots in an environment, measurements in models of quantum physics, belief revisions, knowledge updates, etc. In each of these areas, DL-formulas express properties of the model encoding the present state of affairs, as well as the pre- and post-conditions of a given action. Actions are semantically represented as transformations of one model into another, encoding the …

Nominal Regular Expressions For Languages Over Infinite Alphabets, Alexander Kurz, Tomoyuki Suzuki, Emilio Tuosto

Engineering Faculty Articles and Research

We propose regular expressions to abstractly model and study properties of resource-aware computations. Inspired by nominal techniques – as those popular in process calculi – we extend classical regular expressions with names (to model computational resources) and suitable operators (for allocation, deallocation, scoping of, and freshness conditions on resources). We discuss classes of such nominal regular expressions, show how such expressions have natural interpretations in terms of languages over infinite alphabets, and give Kleene theorems to characterise their formal languages in terms of nominal automata.

Nominal Computation Theory (Dagstuhl Seminar 13422), Mikołaj Bojanczyk, Bartek Klin, Alexander Kurz, Andrew M. Pitts

Engineering Faculty Articles and Research

This report documents the program and the outcomes of Dagstuhl Seminar 13422 “Nominal Computation Theory”. The underlying theme of the seminar was nominal sets (also known as sets with atoms or Fraenkel-Mostowski sets) and they role and applications in three distinct research areas: automata over infinite alphabets, program semantics using nominal sets and nominal calculi of concurrent processes.

Completeness For The Coalgebraic Cover Modality, Clemens Kupke, Alexander Kurz, Yde Venema

Engineering Faculty Articles and Research

We study the finitary version of the coalgebraic logic introduced by L. Moss. The syntax of this logic, which is introduced uniformly with respect to a coalgebraic type functor, required to preserve weak pullbacks, extends that of classical propositional logic with a so-called coalgebraic cover modality depending on the type functor. Its semantics is defined in terms of a categorically defined relation lifting operation.

As the main contributions of our paper we introduce a derivation system, and prove that it provides a sound and complete axiomatization for the collection of coalgebraically valid inequalities. Our soundness and completeness proof is algebraic, …

Strongly Complete Logics For Coalgebras, Alexander Kurz, Jiří Rosický

Engineering Faculty Articles and Research

Coalgebras for a functor model different types of transition systems in a uniform way. This paper focuses on a uniform account of finitary logics for set-based coalgebras. In particular, a general construction of a logic from an arbitrary set-functor is given and proven to be strongly complete under additional assumptions. We proceed in three parts.

Part I argues that sifted colimit preserving functors are those functors that preserve universal algebraic structure. Our main theorem here states that a functor preserves sifted colimits if and only if it has a finitary presentation by operations and equations. Moreover, the presentation of the …

Coalgebraic Logics (Dagstuhl Seminar 12411), Ernst-Erich Doberkat, Alexander Kurz

Engineering Faculty Articles and Research

This report documents the program and the outcomes of Dagstuhl Seminar 12411 “Coalgebraic Logics”. The seminar deals with recent developments in the area of coalgebraic logic, a branch of logics which combines modal logics with coalgebraic semantics. Modal logic finds its uses when reasoning about behavioural and temporal properties of computation and communication, coalgebras have evolved into a general theory of systems. Consequently, it is natural to combine both areas for a mathematical description of system specification. Coalgebraic logics are closely related to the broader categories semantics/formal methods and verification/logic.

Applications Of Extenics To 2d-Space And 3d-Space, Florentin Smarandache, Victor Vladareanu

Branch Mathematics and Statistics Faculty and Staff Publications

In this article one proposes several numerical examples for applying the extension set to 2D- and 3D-spaces. While rectangular and prism geometrical figures can easily be decomposed from 2D and 3D into 1D linear problems, similarly for the circle and the sphere, it is not possible in general to do the same for other geometrical figures.

Characterizing Conflict In Wikipedia, Nathaniel Miller

Mathematics, Statistics, and Computer Science Honors Projects

Wikipedia serves as the Internet's most widely viewed reference. In order to ensure its success, editors who create and maintain articles must resolve conflicts over appropriate article content. Previous research has measured Wikipedia conflict at two levels: single articles and categories of pages. I observe conflicts within small groups of articles, identifying their frequency, size, and intensity. Additionally, I identify individual conflicts spanning multiple articles and effects of conflict upon users' editing habits. I analyze cross-article conflict in three stages. First, I cluster a group of 1.4 million Wikipedia articles. Next, I find individual user conflicts within each article cluster …

Relation Liftings On Preorders And Posets, Marta Bílková, Alexander Kurz, Daniela Petrişan, Jiří Velebil

Engineering Faculty Articles and Research

The category Rel(Set) of sets and relations can be described as a category of spans and as the Kleisli category for the powerset monad. A set-functor can be lifted to a functor on Rel(Set) iff it preserves weak pullbacks. We show that these results extend to the enriched setting, if we replace sets by posets or preorders. Preservation of weak pullbacks becomes preservation of exact lax squares. As an application we present Moss’s coalgebraic over posets.

Generic Trace Logics, Christian Kissig, Alexander Kurz

Engineering Faculty Articles and Research

We combine previous work on coalgebraic logic with the coalgebraic traces semantics of Hasuo, Jacobs, and Sokolova.

Towards Nominal Formal Languages, Alexander Kurz, Tomoyuki Suzuki, Emilio Tuosto

Engineering Faculty Articles and Research

We introduce formal languages over infinite alphabets where words may contain binders.We define the notions of nominal language, nominal monoid, and nominal regular expressions. Moreover, we extend history-dependent automata (HD-automata) by adding stack, and study the recognisability of nominal languages.

A New Approach To Algebraic Coding Theory Through The Applications Of Soft Sets, Florentin Smarandache, Mumtaz Ali

Branch Mathematics and Statistics Faculty and Staff Publications

Algebraic codes play a signifcant role in the minimisation of data corruption which caused by defects such as inference, noise channel, crosstalk, and packet loss. In this paper, we introduce soft codes (soft linear codes) through the application of soft sets which is an approximated collection of codes. We also discuss several types of soft codes such as type-1 soft codes, complete soft codes etc. Further, we construct the soft generator matrix and soft parity check matrix for the soft linear codes. Moreover, we develop two techniques for the decoding of soft codes.

Automated Theorem Prover Axiom Management, Ashley T. Holeman, Ewen Denney

STAR Program Research Presentations

Automated Theorem Provers (ATPs), are computer programs that use collections of axioms,which are logical statements assumed to be true, in order to prove conjectures. NASA uses these programs to verify safety and functional requirements in domains like Guidance, Navigation, and Control. There are about 30 axioms on each major topic including the theory of coordinate systems, elementary arithmetic and linear algebra. These axioms have been created over the duration of many projects and combined into a single file. One task is to manage the axioms by arranging them into logical sections, deleting unnecessary ones and rewriting some into a more …

Software Internationalization: A Framework Validated Against Industry Requirements For Computer Science And Software Engineering Programs, John Huân Vũ

Master's Theses

View John Huân Vũ's thesis presentation at http://youtu.be/y3bzNmkTr-c.

In 2001, the ACM and IEEE Computing Curriculum stated that it was necessary to address "the need to develop implementation models that are international in scope and could be practiced in universities around the world." With increasing connectivity through the internet, the move towards a global economy and growing use of technology places software internationalization as a more important concern for developers. However, there has been a "clear shortage in terms of numbers of trained persons applying for entry-level positions" in this area. Eric Brechner, Director of Microsoft Development Training, suggested …

On Coalgebras Over Algebras, Adriana Balan, Alexander Kurz

Engineering Faculty Articles and Research

We extend Barr’s well-known characterization of the final coalgebra of a Set-endofunctor as the completion of its initial algebra to the Eilenberg-Moore category of algebras for a Set-monad M for functors arising as liftings. As an application we introduce the notion of commuting pair of endofunctors with respect to the monad M and show that under reasonable assumptions, the final coalgebra of one of the endofunctors involved can be obtained as the free algebra generated by the initial algebra of the other endofunctor.

Families Of Symmetries As Efficient Models Of Resource Binding, Vincenzo Ciancia, Alexander Kurz, Ugo Montanari

Engineering Faculty Articles and Research

Calculi that feature resource-allocating constructs (e.g. the pi-calculus or the fusion calculus) require special kinds of models. The best-known ones are presheaves and nominal sets. But named sets have the advantage of being finite in a wide range of cases where the other two are infinite. The three models are equivalent. Finiteness of named sets is strictly related to the notion of finite support in nominal sets and the corresponding presheaves. We show that named sets are generalisd by the categorical model of families, that is, free coproduct completions, indexed by symmetries, and explain how locality of interfaces gives good …

On Universal Algebra Over Nominal Sets, Alexander Kurz, Daniela Petrişan

Engineering Faculty Articles and Research

We investigate universal algebra over the category Nom of nominal sets. Using the fact that Nom is a full re ective subcategory of a monadic category, we obtain an HSP-like theorem for algebras over nominal sets. We isolate a `uniform' fragment of our equational logic, which corresponds to the nominal logics present in the literature. We give semantically invariant translations of theories for nominal algebra and NEL into `uniform' theories and systematically prove HSP theorems for models of these theories.

Bitopological Duality For Distributive Lattices And Heyting Algebras, Guram Bezhanishvili, Nick Bezhanishvili, David Gabelaia, Alexander Kurz

Engineering Faculty Articles and Research

We introduce pairwise Stone spaces as a natural bitopological generalization of Stone spaces—the duals of Boolean algebras—and show that they are exactly the bitopological duals of bounded distributive lattices. The category PStone of pairwise Stone spaces is isomorphic to the category Spec of spectral spaces and to the category Pries of Priestley spaces. In fact, the isomorphism of Spec and Pries is most naturally seen through PStone by first establishing that Pries is isomorphic to PStone, and then showing that PStone is isomorphic to Spec. We provide the bitopological and spectral descriptions of many algebraic concepts important for the study …

Algebraic Theories Over Nominal Sets, Alexander Kurz, Daniela Petrişan, Jiří Velebil

Engineering Faculty Articles and Research

We investigate the foundations of a theory of algebraic data types with variable binding inside classical universal algebra. In the first part, a category-theoretic study of monads over the nominal sets of Gabbay and Pitts leads us to introduce new notions of finitary based monads and uniform monads. In a second part we spell out these notions in the language of universal algebra, show how to recover the logics of Gabbay-Mathijssen and Clouston-Pitts, and apply classical results from universal algebra.

Equational Coalgebraic Logic, Alexander Kurz, Raul Leal

Engineering Faculty Articles and Research

Coalgebra develops a general theory of transition systems, parametric in a functor T; the functor T specifies the possible one-step behaviours of the system. A fundamental question in this area is how to obtain, for an arbitrary functor T, a logic for T-coalgebras. We compare two existing proposals, Moss’s coalgebraic logic and the logic of all predicate liftings, by providing one-step translations between them, extending the results in [21] by making systematic use of Stone duality. Our main contribution then is a novel coalgebraic logic, which can be seen as an equational axiomatization of Moss’s logic. The three logics are …

Dynamic Web Tools For Trigonometry, Steven J. Wilson

Innovations in Math Technology

In the last 20 years, computer technology having mathematical capability has been developed, improved, and become widely available, but textbook presentations are still largely free of any discussion that might require technology. Technology could be used in mathematical instruction for student drill and practice, for instructor demonstrations that promote conceptual understanding, or for the exploration of mathematical ideas, but software is often designed to be pedagogically generic, leaving its use to the creativity of the instructor. Technological solutions for local machines can be quite extensive, but cost and time constraints then limit availability for student use. The internet has the …

Functorial Coalgebraic Logic: The Case Of Many-Sorted Varieties, Alexander Kurz, Daniela Petrişan

Engineering Faculty Articles and Research

Following earlier work, a modal logic for T-coalgebras is a functor L on a suitable variety. Syntax and proof system of the logic are given by presentations of the functor. This paper makes two contributions. First, a previous result characterizing those functors that have presentations is generalized from endofunctors on one-sorted varieties to functors between many-sorted varieties. This yields an equational logic for the presheaf semantics of higher-order abstract syntax. As another application, we show how the move to functors between many-sorted varieties allows to modularly combine syntax and proof systems of different logics. Second, we show how to associate …

How Geometer’S Sketchpad Improves Student Learning, William George Awe

Mathematics Graduate Theses

This study investigated whether dynamic software improves learning and reviews different types of dynamic software. The study then narrows its focus to one type of dynamic software, Geometer’s Sketchpad and studies how Geometer’s Sketchpad, in itself, improves student learning. The researcher identified what dynamic software is, the importance of dynamic software and different types of dynamic software. The researcher also identifies how dynamic software is utilized in schools and researches how to improve the use of dynamic software in schools. Dynamic software, and Geometer’s Sketchpad, In particular, was found to improve learning across grade levels and with different levels of …

Pi-Calculus In Logical Form, Marcello M. Bonsangue, Alexander Kurz

Engineering Faculty Articles and Research

Abramsky’s logical formulation of domain theory is extended to encompass the domain theoretic model for picalculus processes of Stark and of Fiore, Moggi and Sangiorgi. This is done by defining a logical counterpart of categorical constructions including dynamic name allocation and name exponentiation, and showing that they are dual to standard constructs in functor categories. We show that initial algebras of functors defined in terms of these constructs give rise to a logic that is sound, complete, and characterises bisimilarity. The approach is modular, and we apply it to derive a logical formulation of pi-calculus. The resulting logic is a …

Interdependency Of Pharmacokinetic Parameters: A Chicken-And-Egg Problem? Not!, Reza Mehvar

Pharmacy Faculty Articles and Research

Pharmacokinetic (PK) software packages are widely used by scientists in different disciplines to estimate PK parameters. However, their use without a clear understanding of physiological parameters affecting the PK parameters and how different PK parameters are related to each other may result in erroneous interpretation of data. Often, mathematical relationships used for the estimation of PK parameters obscure the true physiological relationships among these parameters, prompting a discussion of which parameter came first and giving the appearance of the-chicken-and-the-egg dilemma. In this article, the author attempts to show how different PK parameters are related to physiological parameters and each other …

Coalgebras And Their Logics, Alexander Kurz

Engineering Faculty Articles and Research

"Transition systems pervade much of computer science. This article outlines the beginnings of a general theory of specification languages for transition systems. More specifically, transition systems are generalised to coalgebras. Specification languages together with their proof systems, in the following called (logical or modal) calculi, are presented by the associated classes of algebras (e.g., classical propositional logic by Boolean algebras). Stone duality will be used to relate the logics and their coalgebraic semantics."

Weak Factorizations, Fractions And Homotopies, Alexander Kurz, Jiří Rosický

Engineering Faculty Articles and Research

We show that the homotopy category can be assigned to any category equipped with a weak factorization system. A classical example of this construction is the stable category of modules. We discuss a connection with the open map approach to bisimulations proposed by Joyal, Nielsen and Winskel.