Mathematical Structures In Computer Science Research Articles

Two-level type theory and applications

AbstractWe define and develop two-level type theory (2LTT), a version of Martin-Löf type theory which combines two different type theories. We refer to them as the ‘inner’ and the ‘outer’ type theory. In our case of interest, the inner theory is homotopy type theory (HoTT) which may include univalent universes and higher inductive types. The outer theory is a traditional form of type theory validating uniqueness of identity proofs (UIP). One point of view on it is as internalised meta-theory of the inner type theory. There are two motivations for 2LTT. Firstly, there are certain results about HoTT which are of meta-theoretic nature, such as the statement that semisimplicial types up to level n can be constructed in HoTT for any externally fixed natural number n. Such results cannot be expressed in HoTT itself, but they can be formalised and proved in 2LTT, where n will be a variable in the outer theory. This point of view is inspired by observations about conservativity of presheaf models. Secondly, 2LTT is a framework which is suitable for formulating additional axioms that one might want to add to HoTT. This idea is heavily inspired by Voevodsky’s Homotopy Type System (HTS), which constitutes one specific instance of a 2LTT. HTS has an axiom ensuring that the type of natural numbers behaves like the external natural numbers, which allows the construction of a universe of semisimplicial types. In 2LTT, this axiom can be assumed by postulating that the inner and outer natural numbers types are isomorphic. After defining 2LTT, we set up a collection of tools with the goal of making 2LTT a convenient language for future developments. As a first such application, we develop the theory of Reedy fibrant diagrams in the style of Shulman. Continuing this line of thought, we suggest a definition of $(\infty,1)$ -category and give some examples.

Preface to the MSCS Issue 31.1 (2021) Homotopy Type Theory and Univalent Foundations

This issue of Mathematical Structures in Computer Science is Part I of a Special Issue dedicated to the emerging field of Homotopy Type Theory and Univalent Foundations.

Preface to special issue: ICTAC 2015

This issue of Mathematical Structures in Computer Science (MSCS) contains a selection of papers presented at the 12th International Colloquium on Theoretical Aspects of Computing (ICTAC 2015), which took place in Cali, Colombia, on October 29–31, 2015.

Preface

This special issue of Mathematical Structures in Computer Science is dedicated to the memory of Barry Cooper (October 9, 1943–October 26, 2015). Barry's life in research had a huge impact on the development of the world of computability. Aside from his enormous achievements in classical computability theory, he contributed immensely to the promotion of interdisciplinary developments related to computability, in particular, by founding the Association for Computability in Europe (CiE) and its annual, highly successful conference series.

Preface to the special issue: Continuity, computability, constructivity: from logic to algorithms 2013

This issue of Mathematical Structures in Computer Science is composed mainly of papers submitted by participants of the Workshop ‘Continuity, Computability, Constructivity: From Logic to Algorithms,’ held in Gregynog, a conference centre of the University of Wales located in the beautiful nature of Mid Wales, in the last week of June 2013. In addition, several colleagues accepted our invitation to contribute to this volume.

Foreword: special issue on coalgebraic logic

The second Dagstuhl seminar on coalgebraic logics took place from October 7–12, 2012, in the Leibniz Forschungszentrum Schloss Dagstuhl, following a successful earlier one in December 2009. From the 44 researchers who attended and the 30 talks presented, this collection highlights some of the progress that has been made in the field. We are grateful to Giuseppe Longo and his interest in a special issue in Mathematical Structures in Computer Science.

Preface

This special issue of Mathematical Structures in Computer Science is devoted to the fourteenth Italian Conference on Theoretical Computer Science (ICTCS) held at University of Palermo, Italy, from 9th to 11th September 2013. ICTCS is the conference of the Italian Chapter of the European Association for Theoretical Computer Science and covers a wide spectrum of topics in Theoretical Computer Science, ranging from computational complexity to logic, from algorithms and data structure to programming languages, from combinatorics on words to distributed computing. For this reason, the contributions here included come from very different areas of Theoretical Computer Science. In fact this special issue is motivated by the desire to give people who have presented their ideas at the 14th ICTCS the opportunity to publish papers on their work. Submitted papers have been subject to a careful and severe reviewing process and 11 of them were selected for this special issue.

Preface to special issue: EXPRESS 2011

This issue of Mathematical Structures in Computer Science contains a selection of papers presented at the 18th International Workshop on Expressiveness in Concurrency (EXPRESS'11), a satellite event of CONCUR'11, held on September 5th, 2011 in Aachen, Germany.

Preface to the special issue: Computing with infinite data: topological and logical foundations

This special issue of Mathematical Structures in Computer Science is composed mainly of papers submitted by participants of the Dagstuhl Seminar on Computing with Infinite Data: Topological and Logical Foundations. The workshop took place in the Schloss Dagstuhl - Leibniz Center for Informatics in the first half of October 2011.

Preface to special issue: EXPRESS 2010

This issue of Mathematical Structures in Computer Science contains a selection of papers presented at the 17th International Workshop on Expressiveness in Concurrency (EXPRESS'10), a satellite event of CONCUR'10, held on August 30th in Paris, France.

Introduction: special issue ICICTA 2012

Mathematical Structures in Computer Science bridges the gap between theoretical computer science and software design. By publishing original perspectives from all areas of computing, the journal stresses applications from logic, algebra, geometry, category theory and other areas of logic and mathematics. Through issues such as this special issue, the journal also plans to play an occasional, but important role in the fields of intelligent computation and automation.

Foreword: special issue on structure transformation

This special issue of Mathematical Structures in Computer Science is dedicated to structure transformation and contains contributions to two main research areas: extending the constructions that have been developed over nearly forty years for graph transformations to more general structures; and employing these transformations to model the operational and abstract semantics of distributed and mobile systems.

Debate on the concept of probability, and conclusions to this special issue on developments of the concepts of randomness, statistics and probability

The main promotor of this special issue, Mioara Mugur-Schächter, organised a final debate for the one day conference dedicated to the theme of this special issue of Mathematical Structures in Computer Science.

Preface to special issue: lightweight and practical formal methods in the design and analysis of safety-critical systems

The papers included in this special issue of Mathematical Structures in Computer Science were selected from a larger set we solicited from leading research groups on both sides of the Atlantic. They cover a wide spectrum of tutorials, recent results and surveys in the area of lightweight and practical formal methods in the design and analysis of safety-critical systems. All the papers we received were submitted to a rigorous process of review and revision, based on which we made our final selection.

Introduction: computability of the physical

Albert Einstein encapsulated a commonly held view within the scientific community when he wrote in his book Out of My Later Years (Einstein 1950, page 54) ‘When we say that we understand a group of natural phenomena, we mean that we have found a constructive theory which embraces them.’ This represents a dual challenge to the scientist: on the one hand, to explain the real world in a very basic, and if possible, mathematical, way; but on the other, to characterise the extent to which this is even possible. Recent years have seen the mathematics of computability play an increasingly vital role in pushing forward basic science and in illuminating its limitations within a creative coming together of researchers from different disciplines. This special issue of Mathematical Structures in Computer Science is based on the special session ‘Computability of the Physical’ at the International Conference Computability in Europe 2010, held at Ponta Delgada, Portugal, in June 2010, and it, together with the individual papers it contains, forms what we believe to be a special contribution to this exciting and developing process.

Preface to special issue: EXPRESS, ICE and SOS 2009

This special issue of Mathematical Structures in Computer Science contains a selection of papers presented at three satellite events of CONCUR'09, which was held between 31 August and 5 September 2009 in Bologna (Italy). Specifically, it contains three papers from the 16th International Workshop on Expressiveness in Concurrency (EXPRESS'09), one paper from the 2nd Interaction and Concurrency Experience (ICE'09) and two papers from the 6th Workshop on Structural Operational Semantics (SOS'09).

Preface

This special issue of Mathematical Structures in Computer Science is devoted to the theme of ‘Interactive theorem proving and the formalisation of mathematics’.The formalisation of mathematics started at the turn of the 20th century when mathematical logic emerged from the work of Frege and his contemporaries with the invention of the formal notation for mathematical statements called predicate calculus. This notation allowed the formulation of abstract general statements over possibly infinite domains in a uniform way, and thus went well beyond propositional calculus, which goes back to Aristotle and only allowed tautologies over unquantified statements.

Foreword: special issue on coalgebraic logic

The Coalgebraic Logic seminar took place between the 6th and 9th of December 2009 at the Leibniz Forschungszentrum Schloss Dagstuhl. The event was very well received, with more than 35 scientists from Europe, the United States, Canada and China attending, and with more than thirty presentations during the three days. Given the unforeseen enthusiasm of the meeting's reception, we thought that it would be a good idea for us to put together a special journal issue dedicated to this topic. We were also encouraged by the positive response we received from Guiseppe Longo, who indicated his interest in a special issue for Mathematical Structures in Computer Science, and here we are.

Preface for the special issue on domains

This special issue of Mathematical Structures in Computer Science contains six papers from the Workshop on Domains IX held at the University of Sussex (Brighton), on 22–24 September 2008. This was the ninth event in the long tradition of Domains workshops, which started in Darmstadt in 1994. Since then, workshops have been organised in Braunschweig (1996), Munich (1997), Siegen (1998), Darmstadt again (1999, 2004), Birmingham (2002) and Novosibirsk (2007).

Preface to special issue: Expressiveness in Concurrency 2008

This issue of Mathematical Structures in Computer Science contains three papers selected from the 15th International Workshop on Expressiveness in Concurrency (EXPRESS'08) held on 23 August 2008 in Toronto (Canada) as a satellite event of CONCUR'08.