Formal Relationships Between Geometrical and Classical Models for Concurrency

A wide variety of models for concurrent programs has been proposed during the past decades, each one focusing on various aspects of computations: trace equivalence, causality between events, conflicts and schedules due to resource accesses, etc. More recently, models with a geometrical flavor have been introduced, based on the notion of cubical set. These models are very rich and expressive since they can represent commutation between any bunch of events, thus generalizing the principle of true concurrency. While they seem to be very promising - because they make possible the use of techniques from algebraic topology in order to study concurrent computations - they have not yet been precisely related to the previous models, and the purpose of this paper is to fill this gap. In particular, we describe an adjunction between Petri nets and cubical sets which extends the previously known adjunction between Petri nets and asynchronous transition systems by Nielsen and Winskel

Generalized Asynchronous Systems

The paper is devoted to a mathematical model of concurrency the special case of which is asynchronous system. Distributed asynchronous automata are introduced here. It is proved that the Petri nets and transition systems with independence can be considered like the distributed asynchronous automata. Time distributed asynchronous automata are defined in standard way by the map which assigns time intervals to events. It is proved that the time distributed asynchronous automata are generalized the time Petri nets and asynchronous systems.Comment: 8 page

Trace Spaces: an Efficient New Technique for State-Space Reduction

State-space reduction techniques, used primarily in model-checkers, all rely on the idea that some actions are independent, hence could be taken in any (respective) order while put in parallel, without changing the semantics. It is thus not necessary to consider all execution paths in the interleaving semantics of a concurrent program, but rather some equivalence classes. The purpose of this paper is to describe a new algorithm to compute such equivalence classes, and a representative per class, which is based on ideas originating in algebraic topology. We introduce a geometric semantics of concurrent languages, where programs are interpreted as directed topological spaces, and study its properties in order to devise an algorithm for computing dihomotopy classes of execution paths. In particular, our algorithm is able to compute a control-flow graph for concurrent programs, possibly containing loops, which is "as reduced as possible" in the sense that it generates traces modulo equivalence. A preliminary implementation was achieved, showing promising results towards efficient methods to analyze concurrent programs, with very promising results compared to partial-order reduction techniques

History-Preserving Bisimilarity for Higher-Dimensional Automata via Open Maps

We show that history-preserving bisimilarity for higher-dimensional automata has a simple characterization directly in terms of higher-dimensional transitions. This implies that it is decidable for finite higher-dimensional automata. To arrive at our characterization, we apply the open-maps framework of Joyal, Nielsen and Winskel in the category of unfoldings of precubical sets.Comment: Minor updates in accordance with reviewer comments. Submitted to MFPS 201

Обобщенные асинхронные системы

The paper consider a mathematical model of a concurrent system, the special case of which is an asynchronous system. Distributed asynchronous automata are introduced here. It is proved that Petri nets and transition systems with independence can be considered as distributed asynchronous automata. Time distributed asynchronous automata are defined in a standard way by correspondence which relates events with time intervals. It is proved that the time distributed asynchronous automata generalize time Petri nets and asynchronous systems.Работа посвящена математической модели параллельной системы, частным случаем которой является асинхронная система. В ней введены дистрибутивные асинхронные автоматы. Доказано, что сети Петри и системы переходов с отношением независимости можно рассматривать как дистрибутивные асинхронные автоматы. Стандартным образом, посредством отображения, сопоставляющего событиям временные интервалы, определяются временные дистрибутивные асинхронные автоматы. Доказано, что временные дистрибутивные асинхронные автоматы обобщают временные сети Петри и асинхронные системы

Cubical Sets and Trace Monoid Actions

This paper is devoted to connections between trace monoids and cubical sets. We prove that the category of trace monoids is isomorphic to the category of generalized tori and it is a reflective subcategory of the category of cubical sets. Adjoint functors between the categories of cubical sets and trace monoid actions are constructed. These functors carry independence preserving morphisms in the independence preserving morphisms. This allows us to build adjoint functors between the category of weak asynchronous systems and the category of higher dimensional automata

A Categorical Approach to Verifying Consistency in Concurrent Systems

A concurrent system involves several executing components. Such a system usually allows to carry out multiple tasks at the same time, which can speed up the computational work of software substantially. The recent research findings demonstrate that process-oriented programming languages provide a suitable means for developing concurrent systems. However, design and implementation are at different levels of abstraction in software development process. It is challenging to incorporate knowledge and experience to control the consistency between these phases in developing concurrent systems. The potential inconsistencies arising would introduce errors to the production of concurrent systems, which would prove fatal to the systems in areas with zero tolerance for failure. To tackle such a challenge, the goal of this research is to propose an innovative categorical framework for designing, implementing and verifying the consistency of communications. This framework is inspired by Hoare's vision of category theory and obtained research results towards validating the vision. In this framework, Communicating Sequential Processes(CSP) and Erasmus are used for design and implementation. In addition, abstract interpretation is employed to extract process communications from implementation. Furthermore, several novel rules to analyze semantics of abstraction of implementation are proposed for Erasmus. Finally, category theory is utilized as an innovative means to model and verify consistency of process communications. The framework is illustrated by using several running examples