We add mobility to Place-Transition Petri nets: tokens are names for places, and an input token of a transition can be used in its postset to specify a destination. Mobile Petri nets are then further extended to dynamic nets by adding the possibility of creating new nets during the firing of a transition. In this way, starting from Petri nets, we define a simple hierarchy of nets with increasing degrees of dynamicity. For each class in this hierarchy, we provide its encoding in the former class.

Our work was largely inspired by the join-calculus of Fournet and Gonthier, which turns out to be a (well-motivated) particular case of dynamic Petri nets. The main difference is that, in the preset of a transition, we allow both non-linear patterns (name unification) and (locally) free names for input places (that is, we remove the locality constraint, and preserve reflexion).

The logic of Bunched Implications, through both its intuitionistic version (BI) and one of its classical versions, called Boolean BI (BBI), serves as a logical basis to spatial or separation logic frameworks. In BI, the logical implication is interpreted intuitionistically whereas it is generally interpreted classically in spatial or separation logics, as in BBI. In this paper, we aim to give some new insights into the semantic relations between BI and BBI. Then we propose a sound and complete syntactic constraints based framework for the Kripke semantics of both BI and BBI, a sound labelled tableau proof system for BBI, and a representation theorem relating the syntactic models of BI to those of BBI. Finally, we deduce as our main, and unexpected, result, a sound and faithful embedding of BI into BBI.

In this paper we study quasi-metric spaces using domain theory. Our main objective in this paper is to study the maximal point space problem for quasi-metric spaces. Here we prove that quasi-metric spaces that satisfy certain completeness properties, such as Yoneda and Smyth completeness, can be modelled by continuous dcpo's. To achieve this goal, we first study the partially ordered set of formal balls (BX, ⊑) of a quasi-metric space (X, d). Following Edalat and Heckmann, we prove that the order properties of (BX, ⊑) are tightly connected to topological properties of (X, d). In particular, we prove that (BX, ⊑) is a continuous dcpo if (X, d) is algebraic Yoneda complete. Furthermore, we show that this construction gives a model for Smyth-complete quasi-metric spaces. Then, for a given quasi-metric space (X, d), we introduce the partially ordered set of abstract formal balls (BX, ⊑, ≺). We prove that if the conjugate space (X, d−1) of a quasi-metric space (X, d) is right K-complete, then the ideal completion of (BX, ⊑, ≺) is a model for (X, d). This construction provides a model for any Yoneda-complete quasi-metric space (X, d), as well as the Sorgenfrey line, Kofner plane and Michael line.

Since Melliès showed that λσ (a calculus of explicit substitutions) does not preserve the strong normalization of the β-reduction, it has become a challenge to find a calculus satisfying the following properties: step-by-step simulation of the β-reduction, confluence on terms with metavariables, strong normalization of the calculus of substitutions and preservation of the strong normalization of the λ-calculus. We present here such a calculus. The main novelty of this calculus (given with de Bruijn indices) is the use of labels that represent updating functions and correspond to explicit weakening. A typed version is also presented.

We introduce the notion of strongly concatenable process as a refinement of concatenable processes (Degano et al. 1996), which can be expressed axiomatically via a functor [Qscr ](_) from the category of Petri nets to an appropriate category of symmetric strict monoidal categories, in the precise sense that, for each net N, the strongly concatenable processes of N are isomorphic to the arrows of [Qscr ](N). In addition, we identify a coreflection right adjoint to [Qscr ](_) and characterize its replete image, thus yielding an axiomatization of the category of net computations.

The investigation into the foundational aspects of linguistic mechanisms for programming long-running transactions (such as the scope operator of WS-BPEL) has recently renewed the interest in process algebraic operators that, due to the occurrence of a failure, interrupt the execution of one process, replacing it with another one called the failure handler. We investigate the decidability of termination problems for two simple fragments of CCS (one with recursion and one with replication) extended with one of two such operators, the interrupt operator of CSP and the try-catch operator for exception handling. More precisely, we consider the existential termination problem (existence of one terminated computation) and the universal termination problem (all computations terminate). We prove that, as far as the decidability of the considered problems is concerned, under replication there is no difference between interrupt and try-catch (universal termination is decidable while existential termination is not), while under recursion this is not the case (existential termination is undecidable while universal termination is decidable only for interrupt). As a consequence of our undecidability results, we show the existence of an expressiveness gap between a fragment of CCS and its extension with either the interrupt or the try-catch operator.

In the theory of denotational semantics of programming languages, several authors have constructed various kinds of universal domains. We present here a categorical generalization of a well-known result in model theory, which we use to characterize large classes of reasonable categories that contain universal homogeneous objects. The existence of such objects is characterized by the condition that the finite objects in the category satisfy the amalgamation property. We derive from this the existence and uniqueness of universal homogeneous domains for several categories of bifinite domains, with embedding-projection-pairs as morphisms. We also obtain universal homogeneous objects for various categories of stable bifinite domains. In contrast, several categories of event domains and concrete domains and the category of all coherent Scott-domains do not contain universal homogeneous objects. Finally, we show that all our constructions can be performed effectively.

This paper gives a bird's-eye view of the various ingredients that make up a modern, model-checking-based approach to performability evaluation: Markov reward models, temporal logics and continuous stochastic logic, model-checking algorithms, bisimulation and the handling of non-determinism. A short historical account as well as a large case study complete this picture. In this way, we show convincingly that the smart combination of performability evaluation with stochastic model-checking techniques, developed over the last decade, provides a powerful and unified method of performability evaluation, thereby combining the advantages of earlier approaches.

We axiomatize permutation equivalence in term rewriting systems and Klop’s orthogonal Combinatory Reduction Systems (Klop 1980). The axioms for the former are provided by the general approach proposed by Meseguer (Meseguer 1992). The latter need extra axioms modelling the interplay between reductions and the operation of substitution.

As a consequence of this work, the definition of permutation equivalence is rid of residual calculi, which are heavy in general.

We review the conceptual development of (true) concurrency and branching time starting from Petri nets and proceeding via Mazurkiewicz traces, pomsets, bisimulation, and event structures up to higher dimensional automata (HDAs), whose acyclic case may be identified with triadic event structures and triadic Chu spaces. Acyclic HDAs may be understood as extending the two truth values of Boolean logic with a third value $\T$ expressing transition. We prove the necessity of such a third value under mild assumptions about the nature of observable events, and show that the expansion of any complete Boolean basis $L$ to $L_{\T}$ with a third literal $\that a$ expressing $a=\T$ forms an expressively complete basis for the representation of acyclic HDAs. The main contribution is a new value $\cancel$ of cancellation, which is a sibling of $\T$, serving to distinguish $a(b+c)$ from $ab+ac$ while simplifying the extensional definitions of termination $\tick\A$ and sequence $\A\B$. We show that every HDAX (acyclic HDA with $\cancel$) is representable in the expansion of $L_{\T}$ to $L_{\T\cancel}$ with a fourth literal $\can a$ expressing $a=\cancel$.

We present a calculus of mobile processes without prefix or summation, called the solos calculus. Using two different encodings, we show that the solos calculus can express both action prefix and guarded summation. One encoding gives a strong correspondence, but uses a match operator; the other yields a slightly weaker correspondence, but uses no additional operators. We also show that the expressive power of the solos calculus is still retained by the sub-calculus where actions carry at most two names. On the other hand, expressiveness is lost in the solos calculus without match and with actions carrying at most one name.

Fokkink and Zantema (Fokkink and Zantema 1994) have shown that bisimulation equivalence has a finite equational axiomatization over the language of Basic Process Algebra with the binary Kleene star operation (BPA*). In light of this positive result on the mathematical tractability of bisimulation equivalence over BPA*, a natural question to ask is whether any other (pre)congruence relation in van Glabbeek's linear time/branching time spectrum is finitely (in)equationally axiomatizable over it. In this paper, we prove that, unlike bisimulation equivalence, none of the preorders and equivalences in van Glabbeek's linear time/branching time spectrum, whose discriminating power lies in between that of ready simulation and that of completed traces, has a finite equational axiomatization. This we achieve by exhibiting a family of (in)equivalences that holds in ready simulation semantics (which is the finest semantics that we consider) and whose instances cannot all be proved by means of any finite set of (in)equations that is sound in completed trace semantics (which is the coarsest semantics that is appropriate for the language BPA*). To this end, for every finite collection of (in)equations that are sound in completed trace semantics, we build a model in which some of the (in)equivalences of the family under consideration fail. The construction of the model mimics the one used by Conway (Conway 1971, p. 105) in his proof of a result, originally due to Redko, to the effect that infinitely many equations are needed to axiomatize equality of regular expressions.

Our non-finite axiomatizability results apply to the language BPA* over an arbitrary non-empty set of actions. In particular, we show that completed trace equivalence is not finitely based over BPA* even when the set of actions is a singleton. Our proof of this result may be adapted to the standard language of regular expressions to yield a solution to an open problem posed by Salomaa (Salomaa 1969, p. 143).

Another semantics that is usually considered in process theory is trace semantics. Trace semantics is, in general, not preserved by sequential composition, and is therefore inappropriate for the language BPA*. We show that, if the set of actions is a singleton, trace equivalence and preorder are preserved by all the operators in the signature of BPA*, and coincide with simulation equivalence and preorder, respectively. In that case, unlike all the other semantics considered in this paper, trace semantics have finite, complete equational axiomatizations over closed terms.

Higher Order Matching consists of solving finite sets of equations of the form u =βηv, where u, v are simply typed terms of λ-calculus, and v is a closed term. Whether this problem is decidable is an open question. In this paper its decidability is proved when the free variables of all equations are of order at most 4 – we actually extend this result to a more general situation in which in addition to equations, disequations are also considered.

The type-theoretic explanation of modules proposed to date (for programming languages like ML) is unsatisfactory, because it does not capture that the evaluation of type-expressions is independent from the evaluation of program expressions. We propose a new explanation based on ‘programming languages as indexed categories’ and illustrate how ML can be extended to support higher order modules, by developing a category-theoretic semantics for a calculus of modules with dependent types. The paper also outlines a methodology, which may lead to a modular approach in the study of programming languages.

We pose the following question: If a physical experiment were to be completely controlled by an algorithm, what effect would the algorithm have on the physical measurements made possible by the experiment?

In a programme to study the nature of computation possible by physical systems, and by algorithms coupled with physical systems, we have begun to analyse:

1. (i) the algorithmic nature of experimental procedures; and

2. (ii) the idea of using a physical experiment as an oracle to Turing Machines.

This paper studies fundamental connections between profunctors (that is, distributors, or bimodules), open maps and bisimulation. In particular, it proves that a colimit preserving functor between presheaf categories (corresponding to a profunctor) preserves open maps and open map bisimulation. Consequently, the composition of profunctors preserves open maps as 2-cells. A guiding idea is the view that profunctors, and colimit preserving functors, are linear maps in a model of classical linear logic. But profunctors, and colimit preserving functors, as linear maps, are too restrictive for many applications. This leads to a study of a range of pseudo-comonads and of how non-linear maps in their co-Kleisli bicategories preserve open maps and bisimulation. The pseudo-comonads considered are based on finite colimit completion, ‘lifting’, and indexed families. The paper includes an appendix summarising the key results on coends, left Kan extensions and the preservation of colimits. One motivation for this work is that it provides a mathematical framework for extending domain theory and denotational semantics of programming languages to the more intricate models, languages and equivalences found in concurrent computation, but the results are likely to have more general applicability because of the ubiquitous nature of profunctors.

A reactive system can be specified by a labelled transition system, which indicates static structure, along with temporal-logic formulas, which assert dynamic behaviour. But refining the former while preserving the latter can be difficult, because:

(i) Labelled transition systems are ‘total’ – characterised up to bisimulation – meaning that no new transition structure can appear in a refinement.

(ii) Alternatively, a refinement criterion not based on bisimulation might generate a refined transition system that violates the temporal properties.

In response, Larsen and Thomson proposed modal transition systems, which are ‘partial’, and defined a refinement criterion that preserved formulas in Hennessy–Milner logic. We show that modal transition systems are, up to a saturation condition, exactly the mixed transition systems of Dams that meet a mix condition, and we extend such systems to non-flat state sets. We then solve a domain equation over the mixed powerdomain whose solution is a bifinite domain that is universal for all saturated modal transition systems and is itself fully abstract when considered as a modal transition system. We demonstrate that many frameworks of partial systems can be translated into the domain: partial Kripke structures, partial bisimulation structures, Kripke modal transition systems, and pointer-shape-analysis graphs.

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.

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.

A framework is defined within which reactive systems can be studied formally. The framework is based on s-categories, which are a new variety of categories within which reactive systems can be set up in such a way that labelled transition systems can be uniformly extracted. These lead in turn to behavioural preorders and equivalences, such as the failures preorder (treated elsewhere) and bisimilarity, which are guaranteed to be congruential. The theory rests on the notion of relative pushout, which was previously introduced by the authors.

The framework is applied to a particular graphical model, known as link graphs, which encompasses a variety of calculi for mobile distributed processes. The specific theory of link graphs is developed. It is then applied to an established calculus, namely condition-event Petri nets.

In particular, a labelled transition system is derived for condition-event nets, corresponding to a natural notion of observable actions in Petri-net theory. The transition system yields a congruential bisimilarity coinciding with one derived directly from the observable actions. This yields a calibration of the general theory of reactive systems and link graphs against known specific theories.