We propose a refinement and a simplification of the behavioural semantics of session types, based on the concepts of compliance and sub-behaviour from the theory of web contracts. We introduce three relations on a suitable class of behaviours with higher-order input/output, called ‘session behaviours’. Such relations, depending on each other, represent the idea of sub-behaviour from the point of view of a client, a server or a peer, respectively. A restriction of the intersection of the first two relations characterizes the ‘usual’ sub-behaviour relation from the literature. We then device an algorithmic formal system for three subtyping relations (dubbed CSP-subtyping) for session types that takes into account the role played by a user of a channel during an interaction, so extending Gay and Hole subtyping theory. We show that our session behaviours and sub-behaviour relations provide sound and complete semantics for CSP-subtyping, and for Gay and Hole subtyping as a by-product.

In object-oriented programming, there are many notions of ‘collection with members in X’. This paper offers an axiomatic theory of collections based on monads in the category of sets and total functions. Heuristically, the axioms defining a collection monad state that each collection has a finite set of members of X, that pure 1-element collections exist and that a collection of collections flattens to a single collection whose members are the union of the members of the constituent collections. The relationship between monads and universal algebra leads to a formal definition of collection implementation in terms of tree-processing. Ideas from elementary category theory underly the classification of collections. For example, collections can be zipped if and only if the monad's endofunctor preserves pullbacks. Or, a collection can be uniquely specified by its shape and list of data if the morphisms of the Kleisli category of the monad are all deterministic, and the converse holds for commutative monads. Again, a collection monad is ordered if the monad's functor preserves equalizers of monomorphisms (so, in particular, if collections can be zipped the monad is ordered). Every implementable monad is ordered. It is shown using the well-ordering principle that a collection monad is ordered if and only if its functor admits an appropriated list-valued natural transformation that lists the members of each collection.

Tile systems offer a general paradigm for modular descriptions of concurrent systems, based on a set of rewriting rules with side-effects. Monoidal double categories are a natural semantic framework for tile systems, because the mathematical structures describing system states and synchronizing actions (called configurations and observations, respectively, in our terminology) are monoidal categories having the same objects (the interfaces of the system). In particular, configurations and observations based on net-process-like and term structures are usually described in terms of symmetric monoidal and cartesian categories, where the auxiliary structures for the rearrangement of interfaces correspond to suitable natural transformations. In this paper we discuss the lifting of these auxiliary structures to double categories. We notice that the internal construction of double categories produces a pathological asymmetric notion of natural transformation, which is fully exploited in one dimension only (for example, for configurations or for observations, but not for both). Following Ehresmann (1963), we overcome this biased definition, introducing the notion of generalized natural transformation between four double functors (rather than two). As a consequence, the concepts of symmetric monoidal and cartesian (with consistently chosen products) double categories arise in a natural way from the corresponding ordinary versions, giving a very good relationship between the auxiliary structures of configurations and observations. Moreover, the Kelly–Mac Lane coherence axioms can be lifted to our setting without effort, thanks to the characterization of two suitable diagonal categories that are always present in a double category. Then, symmetric monoidal and cartesian double categories are shown to offer an adequate semantic setting for process and term tile systems.

The notion of a natural model of type theory is defined in terms of that of a representable natural transfomation of presheaves. It is shown that such models agree exactly with the concept of a category with families in the sense of Dybjer, which can be regarded as an algebraic formulation of type theory. We determine conditions for such models to satisfy the inference rules for dependent sums Σ, dependent products Π and intensional identity types Id, as used in homotopy type theory. It is then shown that a category admits such a model if it has a class of maps that behave like the abstract fibrations in axiomatic homotopy theory: They should be stable under pullback, closed under composition and relative products, and there should be weakly orthogonal factorizations into the class. It follows that many familiar settings for homotopy theory also admit natural models of the basic system of homotopy type theory.

In this paper we will discuss various aspects of computable/constructive analysis, namely semantics, proofs and computations. We will present some of the problems and solutions of exact real arithmetic varying from concrete implementations, representation and algorithms to various models for real computation. We then put these models in a uniform framework using realisability, which opens the door to the use of type theoretic and coalgebraic constructions both in computing and reasoning about these computations. We will indicate that it is often natural to use constructive logic to reason about these computations.

Working in homotopy type theory, we provide a systematic study of homotopy limits of diagrams over graphs, formalized in the Coq proof assistant. We discuss some of the challenges posed by this approach to the formalizing homotopy-theoretic material. We also compare our constructions with the more classical approach to homotopy limits via fibration categories.

We formulate a typed version of call-by-value λ-calculus containing variants of Felleisen's control operators A and C that provide explicit access to continuations and logically extend the propositions-as-types correspondence to classical propositional logic. We give an equational theory for this calculus, which is shown to be sound and complete with respect to a class of categorical models based on continuation-passing-style semantics.

We add extensional equalities for the functional and product types to the typed λ-calculus with, in addition to products and terminal object, sums and bounded recursion (a version of recursion that does not allow recursive calls of infinite length). We provide a confluent and strongly normalizing (thus decidable) rewriting system for the calculus that stays confluent when allowing unbounded recursion. To do this, we turn the extensional equalities into expansion rules, and not into contractions as is done traditionally. We first prove the calculus to be weakly confluent, which is a more complex and interesting task than for the usual λ-calculus. Then we provide an effective mechanism to simulate expansions without expansion rules, so that the strong normalization of the calculus can be derived from that of the underlying, traditional, non-extensional system. These results give us the confluence of the full calculus, but we also show how to deduce confluence directly form our simulation technique without using the weak confluence property.

The analysis of the dynamics of Biological Regulatory Networks (BRNs) requires innovative methods to cope with the state-space explosion. This paper settles an original approach for deciding reachability properties based on Process Hitting, which is a framework suitable for modelling dynamical complex systems. In particular, Process Hitting has been shown to be of interest in providing compact models of the dynamics of BRNs with discrete values. Process Hitting splits a finite number of processes into so-called sorts and describes the way each process is able to act upon (that is, to ‘hit’) another one (or itself) in order to ‘bounce’ it as another process of the same sort with further actions.

By using complementary abstract interpretations of the succession of actions in Process Hitting, we build a very efficient static analysis to over- and under-approximate reachability properties, which avoids the need to build the underlying states graph. The analysis is proved to have a low theoretical complexity, in particular when the number of processes per sorts is limited, while a very large number of sorts can be managed.

This makes such an approach very promising for the scalable analysis of abstract complex systems. We illustrate this through the analysis of a large BRN of 94 components. Our method replies quasi-instantaneously to reachability questions, while standard model-checking techniques regularly fail because of the combinatoric explosion of behaviours.

The delay datatype was introduced by Capretta (Logical Methods in Computer Science, 1(2), article 1, 2005) as a means to deal with partial functions (as in computability theory) in Martin-Löf type theory. The delay datatype is a monad. It is often desirable to consider two delayed computations equal, if they terminate with equal values, whenever one of them terminates. The equivalence relation underlying this identification is called weak bisimilarity. In type theory, one commonly replaces quotients with setoids. In this approach, the delay datatype quotiented by weak bisimilarity is still a monad–a constructive alternative to the maybe monad. In this paper, we consider the alternative approach of Hofmann (Extensional Constructs in Intensional Type Theory, Springer, London, 1997) of extending type theory with inductive-like quotient types. In this setting, it is difficult to define the intended monad multiplication for the quotiented datatype. We give a solution where we postulate some principles, crucially proposition extensionality and the (semi-classical) axiom of countable choice. With the aid of these principles, we also prove that the quotiented delay datatype delivers free ω-complete pointed partial orders (ωcppos).

Altenkirch et al. (Lecture Notes in Computer Science, vol. 10203, Springer, Heidelberg, 534–549, 2017) demonstrated that, in homotopy type theory, a certain higher inductive–inductive type is the free ωcppo on a type X essentially by definition; this allowed them to obtain a monad of free ωcppos without recourse to a choice principle. We notice that, by a similar construction, a simpler ordinary higher inductive type gives the free countably complete join semilattice on the unit type 1. This type suffices for constructing a monad, which is isomorphic to the one of Altenkirch et al. We have fully formalized our results in the Agda dependently typed programming language.

In a controversial paper (De Millo et al. 1979) at the end of the 1970's, R. A. De Millo, R. J. Lipton and A. J. Perlis argued against formal verifications of programs, mostly motivating their position by an analogy with proofs in mathematics, and, in particular, with the impracticality of a strictly formalist approach to this discipline. The recent, impressive achievements in the field of interactive theorem proving provide an interesting ground for a critical revisiting of their theses. We believe that the social nature of proof and program development is uncontroversial and ineluctable, but formal verification is not antithetical to it. Formal verification should strive not only to cope with, but to ease and enhance the collaborative, organic nature of this process, eventually helping us to master the growing complexity of scientific knowledge.

We refine the simulation technique introduced in Di Cosmo and Kesner (1997) to show strong normalisation of $\l$-calculi with explicit substitutions via termination of cut elimination in proof nets (Girard 1987). We first propose a notion of equivalence relation for proof nets that extends the one in Di Cosmo and Guerrini (1999), and show that cut elimination modulo this equivalence relation is terminating. We then show strong normalisation of the typed version of the $\ll$-calculus with de Bruijn indices (a calculus with full composition defined in David and Guillaume (1999)) using a translation from typed $\ll$ to proof nets. Finally, we propose a version of typed $\ll$ with named variables, which helps to give a better understanding of the complex mechanism of the explicit weakening notation introduced in the $\ll$-calculus with de Bruijn indices (David and Guillaume 1999).

It has recently become evident that categories of representations of Hopf algebras provide fundamental examples of monoidal categories. In this expository paper, we examine such categories as models of (multiplicative) linear logic. By varying the Hopf algebra, it is possible to model several variants of linear logic. We present models of the original commutative logic, the noncommutative logic of Lambek and Abrusci, the braided variant due to the author, and the cyclic logic of Yetter. Hopf algebras provide a unifying framework for the analysis of these variants. While these categories are monoidal closed, they lack sufficient structure to model the involutive negation of classical linear logic. We recall work of Lefschetz and Barr in which vector spaces are endowed with an additional topological structure, called linear topology. The resulting category has a large class of reflexive objects, which form a *-autonomous category, and so model the involutive negation. We show that the monoidal closed structure of the category of representations of a Hopf algebra can be extended to this topological category in a natural and simple manner. The models we obtain have the advantage of being nondegenerate in the sense that the two multiplicative connectives, tensor and par, are not equated. It has been recently shown by Barr that this category of topological vector spaces can be viewed as a subcategory of a certain Chu category. In an Appendix, Barr uses this equivalence to analyze the structure of its tensor product.

We present strategic port graph rewriting as a basis for the implementation of visual modelling tools. The goal is to facilitate the specification and programming tasks associated with the modelling of complex systems. A system is represented by an initial graph and a collection of graph rewrite rules, together with a user-defined strategy to control the application of rules. The traditional operators found in strategy languages for term rewriting have been adapted to deal with the more general setting of graph rewriting, and some new constructs have been included in the strategy language to deal with graph traversal and management of rewriting positions in the graph. We give a formal semantics for the language, and describe its implementation: the graph transformation and visualisation tool Porgy.

Tries, a form of string-indexed look-up structure, are generalized, in a manner first discovered by Wadsworth, to permit indexing by terms built according to an arbitrary signature. The construction is parametric with respect to the type of data to be stored as values; this is essential, because the recursion that defines tries appeals from one value type to others. ‘Trie’ (for any fixed signature) is then a functor, and the corresponding look-up function is a natural isomorphism.

The trie functor is in principle definable by the ‘initial fixed point’ semantics of Smyth and Plotkin. We simplify the construction, however, by introducing the ‘category-cpo’, a class of category within which calculations can retain some domain-theoretic flavor. Our construction of tries extends easily to many-sorted signatures.

Induction is a well-established proof principle that is taught in most undergraduate programs in mathematics and computer science. In computer science, it is used primarily to reason about inductively defined datatypes such as finite lists, finite trees and the natural numbers. Coinduction is the dual principle that can be used to reason about coinductive datatypes such as infinite streams or trees, but it is not as widespread or as well understood. In this paper, we illustrate through several examples the use of coinduction in informal mathematical arguments. Our aim is to promote the principle as a useful tool for the working mathematician and to bring it to a level of familiarity on par with induction. We show that coinduction is not only about bisimilarity and equality of behaviors, but also applicable to a variety of functions and relations defined on coinductive datatypes.

Relational structures offer a common framework for handling graphs and hypergraphs of various kinds. Operations like disjoint union, the creation of new relations by means of quantifier-free formulas, and relabellings of relations make it possible to denote them using algebraic expressions. It is known that every monadic second-order property of a structure is verifiable in time proportional to the size of such an algebraic expression defining it. We prove here that this result remains true if we also use in these algebraic expressions a fusion operation that fuses all elements of the domain satisfying some unary predicate. The value mapping from these algebraic expressions to the structures they denote is a monadic second-order definable transduction, which means that the structure is definable inside the tree representing the algebraic expression by monadic second-order formulas. It follows (by using results of other articles) that, with this fusion operation, we cannot generate more graph families, but we can generate them with less unary auxiliary predicates. We also obtain clear-cut characterizations of Vertex Replacement and Hyperedge Replacement context-free graph grammars in terms of four types of operations, amongst which is the fusion of vertices satisfying a specified predicate.

The purpose of this paper is to develop the basic theory of stably compact spaces (viz. compact, locally compact, coherent sober spaces) and introduce in an accessible manner and with a minimum of prerequisites some significant new lines of investigation and application arising from recent research, which has arisen primarily in the theoretical computer science community. Three primary themes have developed:

1. (i) the property of stable compactness is preserved under a large variety of constructions involving powerdomains, hyperspaces and function spaces;

2. (ii) the underlying de Groot duality of stably compact spaces, which finds varied expression, is reflected by duality theorems involving the just mentioned constructions; and

3. (iii) the notion of inner and outer pavings is a useful and natural tool for such studies of stably compact spaces.

Since Pnueli’s seminal paper in 1977, Temporal Logic has been used as a formalism for specifying and verifying the correctness of reactive systems. In this paper, we show that, besides its expressive power, Temporal Logic enjoys a very strong structural property: it is categorical on processes. That is, we show how temporal specifications (as theories) can be embedded in categories of process behaviour, and out of this adjunction we build an institution that is categorical in the sense of Meseguer. This characterisation means that temporal logic is, in a sense, ‘sound and complete’ with respect to process specification and interconnection techniques.

In another paper (Abramsky et al. 1999), we have developed Abramsky's analysis of Girard's Geometry of Interaction programme in detail. In this paper, our goal is to study the data ow based computational aspects of that analysis. We introduce unique decomposition categories that provide a suitable categorical framework for such computational analysis. The current study also serves to establish connections with the work on proof nets and paths by Girard and Danos and Regnier in this categorical setting. The latter goal is partially achieved here by the presentation of categorical models for dynamic algebras.