Branching Space-Times and Parallel Processing

There is a remarkable similarity between some mathematical objects used in the Branching Space-Times framework and those appearing in computer science in the fields of event structures for concurrent processing and Chu spaces. This paper introduces the similarities and formulates a few open questions for further research, hoping that both BST theorists and computer scientists can benefit from the project

The Glory of the Past and Geometrical Concurrency

This paper contributes to the general understanding of the geometrical model of concurrency that was named higher dimensional automata (HDAs) by Pratt. In particular we investigate modal logics for such models and their expressive power in terms of the bisimulation that can be captured. The geometric model of concurrency is interesting from two main reasons: its generality and expressiveness, and the natural way in which autoconcurrency and action refinement are captured. Logics for this model, though, are not well investigated, where a simple, yet adequate, modal logic over HDAs was only recently introduced. As this modal logic, with two existential modalities, during and after, captures only split bisimulation, which is rather low in the spectrum of van Glabbeek and Vaandrager, the immediate question was what small extension of this logic could capture the more fine-grained hereditary history preserving bisimulation (hh)? In response, the work in this paper provides several insights. One is the fact that the geometrical aspect of HDAs makes it possible to use for capturing the hh-bisimulation, a standard modal logic that does not employ event variables, opposed to the two logics (over less expressive models) that we compare with. The logic that we investigate here uses standard past modalities and extends the previously introduced logic (called HDML) that had only forward, action-labelled, modalities. Besides, we try to understand better the above issues by introducing a related model that we call ST-configuration structures, which extend the configuration structures of van Glabbeek and Plotkin. We relate this model to HDAs, and redefine and prove the earlier results in the light of this new model. These offer a different view on why the past modalities and geometrical concurrency capture the hereditary history preserving bisimulation. Additional correlating insights are also gained.Comment: 17 pages, 7 figure

Bifinite Chu Spaces

This paper studies colimits of sequences of finite Chu spaces and their ramifications. Besides generic Chu spaces, we consider extensional and biextensional variants. In the corresponding categories we first characterize the monics and then the existence (or the lack thereof) of the desired colimits. In each case, we provide a characterization of the finite objects in terms of monomorphisms/injections. Bifinite Chu spaces are then expressed with respect to the monics of generic Chu spaces, and universal, homogeneous Chu spaces are shown to exist in this category. Unanticipated results driving this development include the fact that while for generic Chu spaces monics consist of an injective first and a surjective second component, in the extensional and biextensional cases the surjectivity requirement can be dropped. Furthermore, the desired colimits are only guaranteed to exist in the extensional case. Finally, not all finite Chu spaces (considered set-theoretically) are finite objects in their categories. This study opens up opportunities for further investigations into recursively defined Chu spaces, as well as constructive models of linear logic

Distributed Computation as Hierarchy

This paper presents a new distributed computational model of distributed systems called the phase web that extends V. Pratt's orthocurrence relation from 1986. The model uses mutual-exclusion to express sequence, and a new kind of hierarchy to replace event sequences, posets, and pomsets. The model explicitly connects computation to a discrete Clifford algebra that is in turn extended into homology and co-homology, wherein the recursive nature of objects and boundaries becomes apparent and itself subject to hierarchical recursion. Topsy, a programming environment embodying the phase web, is available from www.cs.auc.dk/topsy.Comment: 16 pages, 3 figure

Sculptures in Concurrency

We give a formalization of Pratt's intuitive sculpting process for higher-dimensional automata (HDA). Based on this, we show that sculptures, Pratt's Chu spaces, and Johansen's ST-structures are in close correspondence. We also develop an algorithm to decide whether a HDA can be sculpted and use this to show that some natural acyclic HDA are not sculptures. We believe that our result shed new light on the intuitions behind sculpting as a method of modeling concurrent behavior, showing the precise reaches of its expressiveness. We also show that there are sculptures whose unfoldings cannot besculpted, and that sculptures are the same as Euclidean cubical complexes. This exposes a close connection between geometric and combinatorial models for concurrency which may be of use for both areas

Nominal Models of Linear Logic

PhD thesisMore than 30 years after the discovery of linear logic, a simple fully-complete model has still not been established. As of today, models of logics with type variables rely on di-natural transformations, with the intuition that a proof should behave uniformly at variable types. Consequently, the interpretations of the proofs are not concrete. The main goal of this thesis was to shift from a 2-categorical setting to a first-order category. We model each literal by a pool of resources of a certain type, that we encode thanks to sorted names. Based on this, we revisit a range of categorical constructions, leading to nominal relational models of linear logic. As these fail to prove fully-complete, we revisit the fully-complete game-model of linear logic established by Melliès. We give a nominal account of concurrent game semantics, with an emphasis on names as resources. Based on them, we present fully complete models of multiplicative additive tensorial, and then linear logics. This model extends the previous result by adding atomic variables, although names do not play a crucial role in this result. On the other hand, it provides a nominal structure that allows for a nominal relationship between the Böhm trees of the linear lambda-terms and the plays of the strategies. However, this full-completeness result for linear logic rests on a quotient. Therefore, in the final chapter, we revisit the concurrent operators model which was first developed by Abramsky and Melliès. In our new model, the axiomatic structure is encoded through nominal techniques and strengthened in such a way that full completeness still holds for MLL. Our model does not depend on any 2-categorical argument or quotient. Furthermore, we show that once enriched with a hypercoherent structure, we get a static fully complete model of MALL