Directed paths have been used by several authors to describe concurrent executions of a program. Spaces of directed paths in an appropriate state space contain executions with all possible legal schedulings. It is interesting to investigate whether one obtains different topological properties of such a space of executions if one restricts attention to schedulings with “nice” properties, e.g. involving synchronisations. This note shows that this is not the case, i.e. that one may operate with nice schedulings without inflicting any harm. Several of the results in this note had previously been obtained by Ziemiański in Ziemiański (2017. Applicable Algebra in Engineering, Communication and Computing28 497–525; 2020a. Journal of Applied and Computational Topology4 (1) 45–78). We attempt to make them accessible for a wider audience by giving an easier proof for these findings by an application of quite elementary results from algebraic topology; notably the nerve lemma.

The paper defines and studies the genus of finite state deterministic automata (FSA) and regular languages. Indeed, an FSA can be seen as a graph for which the notion of genus arises. At the same time, an FSA has a semantics via its underlying language. It is then natural to make a connection between the languages and the notion of genus. After we introduce and justify the the notion of the genus for regular languages, the following questions are addressed. First, depending on the size of the alphabet, we provide upper and lower bounds on the genus of regular languages: we show that under a relatively generic condition on the alphabet and the geometry of the automata, the genus grows at least linearly in terms of the size of the automata. Second, we show that the topological cost of the powerset determinization procedure is exponential. Third, we prove that the notion of minimization is orthogonal to the notion of genus. Fourth, we build regular languages of arbitrary large genus: the notion of genus defines a proper hierarchy of regular languages.

Lambda-calculi with vectorial structures have been studied in various ways, but their semantics remain largely uninvestigated. The main contribution of this paper is to provide a categorical framework for the semantics of such algebraic lambda-calculi. We first develop a categorical analysis of a general simply typed lambda-calculus endowed with the structure of a module. We study the problems arising from the addition of a fixed-point combinator and show how to modify the equational theory to solve them. The categorical analysis carries nicely over to the modified language. We provide various concrete models for both the case without fixpoints and for the case with them.

In this paper, we continue the study of the geometry of Brownian motions which are encoded by Kolmogorov–Chaitin random reals (complex oscillations). We unfold Kolmogorov–Chaitin complexity in the context of Brownian motion and specifically to phenomena emerging from the random geometric patterns generated by a Brownian motion.

This paper is a part of the ongoing program of analysing the complexity of various problems in computable analysis in terms of the complexity of the associated index sets. In the framework of effectively enumerable topological spaces, we investigate the following question: given an effectively enumerable topological space whether there exists a computable numbering of all its computable elements. We present a natural sufficient condition on the family of basic neighbourhoods of computable elements that guarantees the existence of a principal computable numbering. We show that weakly-effective ω–continuous domains and the natural numbers with the discrete topology satisfy this condition. We prove weak and strong analogues of Rice's theorem for computable elements. Then, we construct principal computable numberings of partial majorant-computable real-valued functions and co-effectively closed sets and calculate the complexity of index sets for important problems such as root verification and function equality. For example, we show that, for partial majorant-computable real functions, the equality problem is Π11-complete.

We formalize in Constructive Type Theory the Lambda Calculus in its classical first-order syntax, employing only one sort of names for both bound and free variables, and with α-conversion based upon name swapping. As a fundamental part of the formalization, we introduce principles of induction and recursion on terms which provide a framework for reproducing the use of the Barendregt Variable Convention as in pen-and-paper proofs within the rigorous formal setting of a proof assistant. The principles in question are all formally derivable from the simple principle of structural induction/recursion on concrete terms. We work out applications to some fundamental meta-theoretical results, such as the Church–Rosser Theorem and Weak Normalization for the Simply Typed Lambda Calculus. The whole development has been machine checked using the system Agda.

We define representations for downward-closed subsets of a rich family of well-quasi-orders, and more generally for closed subsets of an even richer family of Noetherian topological spaces. This includes the cases of finite words, of multisets, of finite trees, notably. Those representations are given as finite unions of ideals, or more generally of irreducible closed subsets. All the representations we explore are computable, in the sense that we exhibit algorithms that decide inclusion, and compute finite unions and finite intersections. The origin of this work lies in the need for computing finite representations of sets of successors of the downward closure of one state, or more generally of a downward-closed set of states, in a well-structured transition system, and this is where we start: we define adequate notions of completions of well-quasi-orders, and more generally, of Noetherian spaces. For verification purposes, we argue that the required completions must be ideal completions, or more generally sobrifications, that is, spaces of irreducible closed subsets.

We define a quantum model for multiparty communication complexity and prove a simulation theorem between the classical and quantum models. As a result, we show that if the quantum k-party communication complexity of a function f is Ω(n/2k), its classical k-party communication is Ω(n/2k/2). Finding such an f would allow us to prove strong classical lower bounds for k ≥ log n players and make progress towards solving a major open question about symmetric circuits.

We present a type system to guarantee termination of π-calculus processes that exploits input/output capabilities and subtyping, as originally introduced by Pierce and Sangiorgi, in order to analyse the usage of channels.

Our type system is based on Deng and Sangiorgi's level-based analysis of processes. We show that the addition of i/o-types makes it possible to typecheck processes where a form of level polymorphism is at work. We discuss to what extent this programming idiom can be treated by previously existing proposals. We demonstrate how our system can be extended to handle the encoding of the simply-typed λ-calculus, and discuss questions related to type inference.

Ruitenburg’s Theorem says that every endomorphism f of a finitely generated free Heyting algebra is ultimately periodic if f fixes all the generators but one. More precisely, there is N ≥ 0 such that fN+2 = fN, thus the period equals 2. We give a semantic proof of this theorem, using duality techniques and bounded bisimulation ranks. By the same techniques, we tackle investigation of arbitrary endomorphisms of free algebras. We show that they are not, in general, ultimately periodic. Yet, when they are (e.g. in the case of locally finite subvarieties), the period can be explicitly bounded as function of the cardinality of the set of generators.

We consider term rewriting under sharing in the form of compression by singleton tree grammars (STG), which is more general than the term dags. Algorithms for the subtasks of rewriting are analysed: finding a redex for rewriting by locating a position for a match, performing a rewrite step by constructing the compressed result and executing a sequence of rewrite steps. The first main result is that locating a match of a linear term s in another term t can be performed in polynomial time if s, t are both STG-compressed. This generalizes results on matching of STG-compressed terms, matching of straight-line-program-compressed strings with character-variables, where every variable occurs at most once, and on fully compressed matching of strings. Also, for the case where s is directed-acyclic-graph (DAG)-compressed, it is shown that submatching can be performed in polynomial time. The general case of compressed submatching can be computed in non-deterministic polynomial time, and an algorithm is described that may be exponential in the worst case, its complexity is nO(k), where k is the number of variables with double occurrences in s and n is the size of the input. The second main result is that in case there is an oracle for the redex position, a sequence of m parallel or single-step rewriting steps under STG-compression can be performed in polynomial time. This generalizes results on DAG-compressed rewriting sequences. Combining these results implies that for an STG-compressed term rewrite system with left-linear rules, m parallel or single-step term rewrite steps can be performed in polynomial time in the input size n and m.

We present an algebra of discrete timed input/output automata that may execute in the context of different clock granularities – which we call timed machines; this algebra includes a refinement operator through which a machine can be extended with new states and transitions in order to accommodate a finer clock granularity as required to interoperate with other machines, and an extension of the traditional product of timed input–output automata to the situation in which the granularities of the two machines are not the same. Over this algebra, we then define an algebra of networks of timed machines that includes operations through which networks can be modified at run time, thus offering a model for systems of interconnected components that can dynamically bind to other systems and, therefore, cannot be adjusted at design time to ensure that they operate in a timed homogeneous setting. We investigate important properties of timed machines such as consistency – in the sense that a machine can be ensured to generate a non-empty language, and feasibility – in the sense that a machine can be ensured to generate a non-empty language no matter what inputs it receives, and propose techniques for checking if timed machines are consistent or are feasible. We generalise those properties to networks of timed machines, and investigate how consistency and feasibility of networks can be proved through properties that can be checked at design time without having to compute, at run time, the product of the machines that operate on those networks, which would not be practical.

In this article, we indicate how the category theoretical approach to tree automata, due to Betti and Kasangian, can be fruitfully combined with Walters’ categorical approach to context-free grammars to provide a simple way of establishing the well-known correspondence between context-free languages and the behaviors of non-deterministic tree automata. The connecting link between the two notions is provided by the theory of relational presheaves.

We consider the problem of determining the number of fixed points of a combinator (a closed λ-term). This appears to be a still unsolved problem. We give a partial answer by showing that if there is a fixed point in normal form, then this fixed point is unique or there are infinitely many fixed points.

We give a realizability interpretation of an intuitionistic version of Church's Simple Theory of Types (CST) which can be viewed as a formalization of intuitionistic higher-order logic. Although definable in CST we include operators for monotone induction and coinduction and provide simple realizers for them. Realizers are formally represented in an untyped lambda–calculus with pairing and case-construct. The purpose of this interpretation is to provide a foundation for the extraction of verified programs from formal proofs as an alternative to type-theoretic systems. The advantages of our approach are that (a) induction and coinduction are not restricted to the strictly positive case, (b) abstract mathematical structures and results may be imported, (c) the formalization is technically simpler than in other systems, for example, regarding the definition of realizability, which is a simple syntactical substitution, and the treatment of nested and simultaneous (co)inductive definitions.

We consider computation with real numbers that arise through a process of physical measurement. We have developed a theory in which physical experiments that measure quantities can be used as oracles to algorithms and we have begun to classify the computational power of various forms of experiment using non-uniform complexity classes. Earlier, in Beggs et al. (2014 Reviews of Symbolic Logic7(4) 618–646), we observed that measurement can be viewed as a process of comparing a rational number z – a test quantity – with a real number y – an unknown quantity; each oracle call performs such a comparison. Experiments can then be classified into three categories, that correspond with being able to return test results

$$\begin{eqnarray*} z < y\text{ or }z > y\text{ or }\textit{timeout},\\ z < y\text{ or }\textit{timeout},\\ z \neq y\text{ or }\textit{timeout}. \end{eqnarray*} $$

This article introduces an analogue of permutation classes in the context of polyominoes. For both permutation classes and polyomino classes, we present an original way of characterizing them by avoidance constraints (namely, with excluded submatrices) and we discuss how canonical such a description by submatrix-avoidance can be. We provide numerous examples of permutation and polyomino classes which may be defined and studied from the submatrix-avoidance point of view, and conclude with various directions for future research on this topic.

In this paper we show how the expressive power of a language for the description of timed processes strongly affects the discriminating power of urgent and patient actions. In a sense, it studies the interplay between syntax and semantics of time-critical systems.

We consider a notion of approximation for terms of de Groote–Saurin Λμ-calculus. Then, we introduce an intersection type assignment system for that calculus which is invariant under subject conversion. The type assignment system also induces a filter model, which is an extensional Λμ-model in the sense of Nakazawa and Katsumata. We then establish the approximation theorem, stating that a type can be assigned to a term in the system if and only if it can be assigned to same of its approximations.

‘Geometry and Concurrency’ is not (yet) a well-established domain of research, but consists rather of a collection of seemingly related techniques, algorithms and formalizations, coming from different application areas, and accumulated over a long period of time. There is currently a certain amount of effort being made to unify these, following the workshop ‘New Connections between Computer Science and Mathematics’ held at the Newton Institute in Cambridge, England in November 1995 (and sponsored by HP/BRIMS). More recently, the first workshop on the same subject was held in Aalborg, Denmark (see http://www.math.auc.dk/˜raussen/admin/workshop/workshop.html) where the articles of this issue, among others, were first sketched.

As a starter, and to put in context the papers by Lisbeth Fajstrup, Philippe Gaucher, Maurice Herlihy, Vaughan Pratt, Sergio Rajsbaum and Martin Raussen that appear later in this issue, I have written a brief historical note, or survey, which follows and naturally completes this foreword.

Last and not least, I must express my gratitude to Giuseppe Longo, who gave me the opportunity to act as a guest editor for this special issue, and who very kindly helped me through the many different steps that editing necessitates. My very warm thanks also go to the authors who did a very nice job, in a relatively short time. Finally, I wish to thank the referees of this issue, who found ‘bugs’ that I would not have found, and who suggested very useful improvements to the authors. The referees were Nir Shavit, Stefan Sokolowski, Christophe Tabacznyj, Jean Goubault-Larrecq, Arnaud Venet, Ian Mackie (who helped me with the survey article), Ronnie Brown and Tim Porter, as well as some of the authors who cross-checked other authors' articles, Philippe Gaucher, Martin Raussen and Lisbeth Fajstrup.