The Geometry of Timed PV Programs

AbstractWe introduce a real-time extension of the PV language: A timed PV program consists of a number of timed automata which synchronize by locking and releasing common resources. We give a geometric semantics to such programs in terms of local po-spaces, and we work towards making the established geometric techniques available for detecting deadlocks and unsafe configurations in timed PV programs

Cut-off Theorems for the PV-model

We prove cut-off results for deadlocks and serializability of a $PV$-thread $T$ run in parallel with itself: For a $PV$ thread $T$ which accesses a set $\mathcal{R}$ of resources, each with a maximal capacity $\kappa:\mathcal{R}\to\mathbb{N}$, the PV-program $T^n$, where $n$ copies of $T$ are run in parallel, is deadlock free for all $n$ if and only if $T^M$ is deadlock free where $M=\Sigma_{r\in\mathcal{R}}\kappa(r)$. This is a sharp bound: For all $\kappa:\mathcal{R}\to\mathbb{N}$ and finite $\mathcal{R}$ there is a thread $T$ using these resources such that $T^M$ has a deadlock, but $T^n$ does not for $n<M$. Moreover, we prove a more general theorem: There are no deadlocks in $p=T1|T2|\cdots |Tn$ if and only if there are no deadlocks in $T_{i_1}|T_{i_2}|\cdots |T_{i_M}$ for any subset $\{i_1,\ldots,i_M\}\subset [1:n]$. For $\kappa(r)\equiv 1$, $T^n$ is serializable for all $n$ if and only if $T^2$ is serializable. For general capacities, we define a local obstruction to serializability. There is no local obstruction to serializability in $T^n$ for all $n$ if and only if there is no local obstruction to serializability in $T^M$ for $M=\Sigma_{r\in\mathcal{R}}\kappa(r)+1$. The obstructions may be found using a deadlock algorithm in $T^{M+1}$. These serializability results also have a generalization: If there are no local obstructions to serializability in any of the $M$-dimensional sub programs, $T_{i_1}|T_{i_2}|\cdots |T_{i_M}$, then $p$ is serializable

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

Enriched categories and models for spaces of dipaths

Partially ordered sets, causets, partially ordered spaces and their local counterparts are now often used to model systems in computer science and theoretical physics. The order models `time\u27 which is often not globally given. In this setting directed paths are important objects of study as they correspond to an evolving state or particle traversing the system. Many physical problems rely on the analysis of models of the path space of space-time manifold. Many problems in concurrent systems use `spaces\u27 of paths in a system. Here we review some ideas from algebraic topology that suggest how to model the dipath space of a pospace by a simplicially enriched category

On the Expressiveness of Higher Dimensional Automata: (Extended Abstract)

In this paper I compare the expressive power of several models of concurrency based on their ability to represent causal dependence. To this end, I translate these models, in behaviour preserving ways, into the model of higher dimensional automata, which is the most expressive model under investigation. In particular, I propose four different translations of Petri nets, corresponding to the four different computational interpretations of nets found in the literature.I also extend various equivalence relations for concurrent systems to higher dimensional automata. These include the history preserving bisimulation, which is the coarsest equivalence that fully respects branching time, causality and their interplay, as well as the ST-bisimulation, a branching time respecting equivalence that takes causality into account to the extent that it is expressible by actions overlapping in time. Through their embeddings in higher dimensional automata, it is now well-defined whether members of different models of concurrency are equivalent

On the expressiveness of higher dimensional automata

In this paper I compare the expressive power of several models of concurrency based on their ability to represent causal dependence. To this end, I translate these models, in behaviour preserving ways, into the model of higher dimensional automata (HDA), which is the most expressive model under investigation. In particular, I propose four different translations of Petri nets, corresponding to the four different computational interpretations of nets found in the literature. I also extend various equivalence relations for concurrent systems to HDA. These include the history preserving bisimulation, which is the coarsest equivalence that fully respects branching time, causality and their interplay, as well as the ST-bisimulation, a branching time respecting equivalence that takes causality into account to the extent that it is expressible by actions overlapping in time. Through their embeddings in HDA, it is now well-defined whether members of different models of concurrency are equivalent. (c) 2006 Elsevier B.V. All rights reserved