This paper axiomatises the structure of bigraphs, and proves that the resulting theory is complete. Bigraphs are graphs with double structure, representing locality and connectivity. They have been shown to represent dynamic theories for the $\pi$-calculus, mobile ambients and Petri nets in a way that is faithful to each of those models of discrete behaviour. While the main purpose of bigraphs is to understand mobile systems, a prerequisite for this understanding is a well-behaved theory of the structure of states in such systems. The algebra of bigraph structure is surprisingly simple, as this paper demonstrates; this is because bigraphs treat locality and connectivity orthogonally.

In order to model the behaviour of open concurrent systems by means of Petri nets, we introduce open Petri nets, a generalisation of the ordinary model where some places, designated as open, represent an interface between the system and the environment. Besides generalising the token game to reflect this extension, we define a truly concurrent semantics for open nets by extending the Goltz–Reisig process semantics of Petri nets. We introduce a composition operation over open nets, characterised as a pushout in the corresponding category, suitable for modelling both interaction through open places and synchronisation of transitions. The deterministic process semantics is shown to be compositional with respect to such a composition operation. If a net $Z_3$ results as the composition of two nets $Z_1$ and $Z_2$, having a common subnet $Z_0$, then any two deterministic processes of $Z_1$ and $Z_2$ that ‘agree’ on the common part, can be ‘amalgamated’ to produce a deterministic process of $Z_3$. Conversely, any deterministic process of $Z_3$ can be decomposed into processes of the component nets. The amalgamation and decomposition operations are shown to be inverse to each other, leading to a bijective correspondence between the deterministic processes of $Z_3$ and the pair of deterministic processes of $Z_1$ and $Z_2$ that agree on the common subnet $Z_0$. Technically, our result is similar to the amalgamation theorem for data-types in the framework of algebraic specification. A possible application field of the proposed constructions and results is the modelling of interorganisational workflows, recently studied in the literature. This is illustrated by a running example.

It is interesting to note that the very first papers related to isomorphism of types were written before the notion itself appeared. Their subjects were the study of equality of terms defined on numbers, the isomorphism of objects in certain categories and the invertibility of $\lambda$-terms, but not the isomorphism of types ‘as such’. One may cite the so-called ‘Tarsky High School Algebra Problem’: whether all identities between terms built from $+, x, \uparrow$, variables and constants are derivable from basic ‘high school equalities’, like $(xy)^z=x^zy^z$. The earliest publications related to this problem date from the 1940s, cf. Birkhoff (1940) – an extensive bibliography may be found in Burris and Yeats (2002).

A number of program transformations currently of interest can be derived from Wadler's ‘free theorems’ for calculi approximating modern functional languages. Although delicate but fundamental issues arise in proving the correctness of free theorems-based program transformations, these issues have usually been left unaddressed in the correctness proofs appearing in the literature. As a result, most such proofs are incomplete, and most free theorems-based transformations are applied to programs in calculi for which they are not actually known to be correct.

The purpose of this paper is three-fold. First, we raise and clarify some of the issues that must be addressed when constructing correctness proofs for free theorems-based program transformations. Second, we offer a principled approach to developing such proofs. Third, we use Pitts' recent work on parametricity and observational equivalence to show how our approach can be used to give the first proof that transformations based on the Acid Rain theorems preserve observational equivalence of programs in a polymorphic lambda calculus supporting FPC-style fixpoints and algebraic data types. Correctness of the foldr-build rule, the destroy-unfoldr rule, and the hylofusion program transformation for this calculus follows immediately. The same approach is expected to yield complete correctness proofs for free theorems-based transformations in calculi that even more closely resemble languages with which programmers are concerned in practice.

The articles in this part of this issue of the journal are a selection of the papers originally presented to the Fifth Workshop on Coalgebraic Methods in Computer Science. CMCS was held in April 2002, in Grenoble, France as a satellite conference of ETAPS. The conference proceedings were published in Electronic Notes in Theoretical Computer Science65 (1). There were fifteen contributed papers and two invited talks. Six papers were selected by the Program Committee after a meeting for consideration of this issue, and revised versions were then sent to referees. In consultation with them, I selected the four published here. Most of them were later revised, so, in general, the papers appearing here are improved versions of the conference versions.

We investigate a new denotational model of linear logic based on the purely relational model. In this semantics, webs are equipped with a notion of ‘finitary’ subsets satisfying a closure condition and proofs are interpreted as finitary sets. In spite of a formal similarity, this model is quite different from the usual models of linear logic (coherence semantics, hypercoherence semantics, the various existing game semantics…). In particular, the standard fix-point operators used for defining the general recursive functions are not finitary, although the primitive recursion operators are. This model can be considered as a discrete analogue of the Köthe space semantics introduced in a previous paper: we show how, given a field, each finiteness space gives rise to a vector space endowed with a linear topology, a notion introduced by Lefschetz in 1942, and we study the corresponding model where morphisms are linear continuous maps (a version of Girard's quantitative semantics with coefficients in the field). In this way we obtain a new model of the recently introduced differential lambda-calculus.

This paper presents general syntactic conditions ensuring the strong normalisation and the logical consistency of the Calculus of Algebraic Constructions, an extension of the Calculus of Constructions with functions and predicates defined by higher-order rewrite rules. On the one hand, the Calculus of Constructions is a powerful type system in which one can formalise the propositions and natural deduction proofs of higher-order logic. On the other hand, rewriting is a simple and powerful computation paradigm. The combination of the two allows, among other things, the development of formal proofs with a reduced size and more automation compared with more traditional proof assistants. The main novelty is to consider a general form of rewriting at the predicate-level that generalises the strong elimination of the Calculus of Inductive Constructions.

By the Final Coalgebra Theorem of Aczel and Mendler, every endofunctor of the category of sets has a final coalgebra, which, however, may be a proper class. We generalise this to all ‘well-behaved’ categories ${\frak K}$. The role of the category of classes is played by a free cocompletion ${\frak K}^\infty$ of ${\frak K}$ under transfinite colimits, that is, colimits of ordinal-indexed chains. Every endofunctor $F$ of ${\frak K}$ has a canonical extension to an endofunctor $F^\infty$ of ${\frak K}^\infty$, which is proved to have a final coalgebra (and an initial algebra). Based on this, we prove a general solution theorem: for every endofunctor of a locally presentable category ${\frak K}$ all guarded equation-morphisms have unique solutions. The last result does not need the extension ${\frak K}^\infty$: the solutions are always found within the category ${\frak K}$.

The logic of bunched implications, BI, provides a logical analysis of a basic notion of resource that is rich enough, for example, to form the logical basis for ‘pointer logic’ and ‘separation logic’ semantics for programs that manipulate mutable data structures. We develop a theory of semantic tableaux for BI, so providing an elegant basis for efficient theorem proving tools for BI. It is based on the use of an algebra of labels for BI's tableaux to solve the resource-distribution problem, the labels being the elements of resource models. For BI with inconsistency, $\bot$, the challenge consists in dealing with BI's Grothendieck topological models within such a proof-search method, based on labels. We prove soundness and completeness theorems for a resource tableaux method TBI with respect to this semantics and provide a way to build countermodels from so-called dependency graphs. Then, from these results, we can define a new resource semantics of BI, based on partially defined monoids, and prove that this semantics is complete. Such a semantics, based on partiality, is closely related to the semantics of BI's (intuitionistic) pointer and separation logics. Returning to the tableaux calculus, we propose a new version with liberalised rules for which the countermodels are closely related to the topological Kripke semantics of BI. As consequences of the relationships between semantics of BI and resource tableaux, we prove two new strong results for propositional BI: its decidability and the finite model property with respect to topological semantics.

Divisibility monoids are a natural algebraic generalisation of Mazurkiewicz trace monoids, namely monoids in which the distributivity of the underlying lattices is kept as a hypothesis, but the relations between the generators are not necessarily supposed to be commutations. Garside monoids are a natural algebraic generalisation of spherical Artin monoids, namely monoids in which the existence of common multiples is kept as a hypothesis, but the relations between the generators are not necessarily supposed to be of Coxeter type. Here, we show that the quasi-centre of these monoids can be studied and described in a similar way, and then we exhibit the intersection between the two classes of monoids.

The existence of semi-pullbacks for stochastic relations over analytic spaces is addressed. It is shown by means of a measure extension that the theory of measurable selectors for measurable relations may be employed for its solution. The category of stochastic relations is shown not to have (weak) pullbacks.

We were all taught in high school that two objects $A$ and $B$ are isomorphic iff there exist two functions $f$ and $g$ such that

We develop a coinductive calculus of streams based on the presence of a final coalgebra structure on the set of streams (infinite sequences of real numbers). The main ingredient is the notion of stream derivative, which can be used to formulate both coinductive proofs and definitions. In close analogy to classical analysis, the latter are presented as behavioural differential equations. A number of applications of the calculus are presented, including difference equations, analytical differential equations, continued fractions, and some problems from discrete mathematics and combinatorics.

Type isomorphisms are pairs of functions $f:A\rightarrow B$ and $g:B\rightarrow A$ that are mutually inverse with respect to convertibility. They can be used in the context of type checking to perform type checking modulo type isomorphisms, by adopting the rule $$\frac{a:A}{a:B}$$ whenever $f:A\rightarrow B$ and $g:B\rightarrow A$ have been declared as type isomorphisms. Type isomorphisms may be viewed as a special instance of coercions that provide embeddings of one type into another. Indeed, type systems for coercive subtyping feature a rule $$\frac{a:A}{a:B}$$ whenever there exists a coercion $f:A\rightarrow B$. By declaring $f$ and $g$ as coercions for every type isomorphism $f:A\rightarrow B$ and $g:B\rightarrow A$, one can simulate type-checking modulo type isomorphisms. However, the proposed encoding relies on the possibility of declaring coercions $f:A\rightarrow B$ and $g:B\rightarrow A$ simultaneously. Such coercions, which we call back-and-forth coercions, are only allowed provided $f$ and $g$ are mutually inverse. In principle, type isomorphisms, viewed as back-and-forth coercions, could be used in the context of proof assistants based on dependent type theory in order to relate equivalent representations of mathematical notions, for example, the polar and cartesian representations of complex numbers. However, the coercions that map one representation into another are not mutually inverse because of the intensional nature of dependent type theories. Consequently, the standard concept of type isomorphisms has limited applicability in the context of proof assistants.

In order to circumvent this problem, we develop a computational interpretation of implicit coercions that allows for the definition of back-and-forth coercions without requiring them to be mutually inverse. We illustrate the usefulness of our approach in a number of formal developments that require us to navigate between different representations of mathematical objects or structures. We also discuss important meta-theoretical properties of our interpretation.

A class $K$ of coalgebras for an endofunctor $T\,{:}\,\Set\to\Set$ is a behavioural covariety if it is closed under disjoint unions and images of bisimulation relations (hence closed under images and domains of coalgebraic morphisms, including subcoalgebras). $K$ may be thought of as the class of all coalgebras that satisfy some computationally significant property. In any logical system suitable for specifying properties of state-transition systems in the Hennessy–Milner style, each formula will define a class of models that is a behavioural covariety.

Assuming that the forgetful functor on $T$-coalgebras has a right adjoint, providing for the construction of cofree coalgebras, and letting $\G^T$ be the comonad arising from this adjunction, we show that behavioural covarieties $K$ are (isomorphic to) the Eilenberg–Moore categories of coalgebras for certain comonads $\G^K$ naturally associated with $\G^T$. These are called pure subcomonads of $\G^T$, and a categorical characterisation of them is given that involves a pullback condition on the naturality squares of a transformation from $\G^K$ to $\G^T$. We show that there is a bijective correspondence between behavioural covarieties of $T$-coalgebras and isomorphism classes of pure subcomonads of $\G^T$.

This paper introduces guarded and strongly guarded monads as a unified model of a variety of different term algebras covering fundamental examples such as initial algebras, final coalgebras, rational terms and term graphs. We develop a general method for obtaining finitary guarded monads that allows us to define and prove properties of the rational and term graph monads. Furthermore, our treatment of rational equations extends the traditional approach to allow right-hand sides of equations to be infinite terms, term graphs or other such coalgebraic structures. As an application, we use these generalised rational equations to sketch part of the correctness of the term graph implementation of functional programming languages.

We present a modular correspondence between various categorical structures and their internal languages in terms of extensional dependent type theories à la Martin-Löf. Starting from lex categories, through regular ones, we provide internal languages of pretopoi and topoi and some variations of them, such as, for example, Heyting pretopoi.

With respect to the internal languages already known for some of these categories, such as topoi, the novelty of these calculi is that formulas corresponding to subobjects can be regained as particular types that are equipped with proof-terms according to the isomorphism ‘propositions as mono types’, which was invisible in previously described internal languages.

Constructive type theory is an expressive programming language in which both algorithms and proofs can be represented. A limitation of constructive type theory as a programming language is that only terminating programs can be defined in it. Hence, general recursive algorithms have no direct formalisation in type theory since they contain recursive calls that satisfy no syntactic condition guaranteeing termination. In this work, we present a method to formalise general recursive algorithms in type theory. Given a general recursive algorithm, our method is to define an inductive special-purpose accessibility predicate that characterises the inputs on which the algorithm terminates. The type-theoretic version of the algorithm is then defined by structural recursion on the proof that the input values satisfy this predicate. The method separates the computational and logical parts of the definitions and thus the resulting type-theoretic algorithms are clear, compact and easy to understand. They are as simple as their equivalents in a functional programming language, where there is no restriction on recursive calls. Here, we give a formal definition of the method and discuss its power and its limitations.

We develop a multidimensional syntax for cut-free proofs of Multiplicative Linear Logic. This syntax is essentially equivalent to the traditional formalism of proof-nets; the interest of the multi-dimensional formalism consists in its explicit relationship with the formalism of bordisms. Bordisms are compact manifolds with boundary, which are treated as morphisms between the ‘incoming’ and ‘outgoing’ boundary components (composition is given by glueing bordisms along matching boundaries). The category of bordisms has recently become important in contemporary mathematics, in particular, because of developments in topological quantum theory and quantum gravity. A semantics of MLL underlying the multi-dimensional syntax is based on a certain category of bordisms, which we call ‘coherent space-times’. The resulting model has an extremely intuitive geometric description. The dual multiplicative connectives $\otimes$ and $\smash{\raise5.99pt\hbox{\rotatebox{-180}{\&}}}$ correspond simply to disjoint unions and connected sums of bordisms. Following ideas from topological quantum field theory, we also discover deep relationships between this new model and the author's coherent phase spaces model (Slavnov 2003), which is based on the context of symplectic geometry.

We show how coalgebras can be presented by operations and equations. This is a special case of Linton's approach to algebras over a general base category ${\cal X}$, namely where ${\cal X}$ is taken as the dual of sets. Since the resulting equations generalise coalgebraic coequations to situations without cofree coalgebras, we call them coequations. We prove a general co-Birkhoff theorem describing covarieties of coalgebras by means of coequations. We argue that the resulting coequational logic generalises modal logic. This relies on the fact that coalgebraic operations respect an appropriate notion of bisimulation and can be considered as modal operators.