Lisbeth Fajstrup

Loops, ditopology and deadlocks

In Fajstrup et al. (1998a) the authors proposed a fast algorithm for deadlock detection, which was based on a geometric model of concurrency. We propose here an extension of this approach to deal with recursive processes (loops) and branchings.

Directed Algebraic Topology and Concurrency

This monograph presents an application of concepts and methods from algebraic topology to models of concurrent processes in computer science and their analysis. Taking well-known discrete models for concurrent processes in resource management as a point of departure, the book goes on to refine combinatorial and topological models. In the process, it develops tools and invariants for the new discipline directed algebraic topology, which is driven by fundamental research interests as well as by applications, primarily in the static analysis of concurrent programs. The state space of a concurrent program is described as a higher-dimensional space, the topology of which encodes the essential properties of the system. In order to analyse all possible executions in the state space, more than just the topological properties have to be considered: Execution paths need to respect a partial order given by the time flow. As a result, tools and concepts from topology have to be extended to take privileged directions into account. The target audience for this book consists of graduate students, researchers and practitioners in the field, mathematicians and computer scientists alike.

Tate cohomology of periodic k-theory with reality is trivial

We calculate the RO(Z/2)-graded spectrum for Atiyahs periodic K-theory with reality and the Tate cohomology associated to it. The latter is shown to be trivial.

Infinitely running concurrent processes with loops from a geometric viewpoint

Abstract This report gives a formal topological semantics to inductively defined concurrent systems and investigates the properties of such systems. We allow loops and infinitely running computations, which is new in the topological investigations of concurrency. In this more general setting, we prove the equivalent to the result from [2] that deadlocks and unsafe points can be found using a finite number of deloopings.

Trace spaces of directed tori with rectangular holes

In [4] the trace space of parallel non-looped, non-branching processes is given as a prod-simplicial complex derived from an index category. For looped processes, the state space is a torus and the trace space is a disjoint union of tracespaces of deloopings. The index category for the trace space of the deloopings is developed from the once delooped case. When just one process is looped, the index category is generated as words in a regular language. The automaton is constructed.

Cubical local partial orders on cubically subdivided spaces: existence and construction

The geometric models of higher dimensional automata (HDA) and Dijkstras PV-model are cubically subdivided topological spaces with a local partial order. If a cubicalization of a topological space is free of immersed cubic Mobius bands, then there are consistent choices of direction in all cubes, such that any n-cube in the cubic subdivision is dihomeomorphic to [0, 1]n with the induced partial order from Rn. After subdivision once, any cubicalized space has a cubical local partial order. In particular, all triangularized spaces have a cubical local partial order. This implies in particular that the underlying geometry of an HDA may be quite complicated.

On the Hierarchy of d-structures

In this paper we identify and study several lattice structures in the context of directed topology. The set of d-structures on a topological space is a Heyting algebra. The implication is constructed explicitely. There is a Galois connection between the lattice of subsets of the space and the lattice of d-structures which clarifies the idea of removing a subset of the space which has only constant dipaths. Hence, variation of d-structures and variation of the “forbidden area” may be considered in one structure. Moreover, the lattice of d-structures gives rise to a lattice of directed paths between a fixed pair of points. That lattice permits us to discuss a perspective on generalised persistence. Furthermore, we consider a lattice structure in the hierarchy of structures on the n-cube.

Directed Topological Models of Concurrency

We study topological models for concurrent programs with the aim of importing tools and techniques coming from algebraic topology to ease verification of concurrent programs. In those models, the state space of a program is described as a topological space, and an execution corresponds naturally to a path in this space. To rensure that models reflect order properties, we are led to enrich the concept of a topological space so that it takes causality into account. We shall focus our attention on directed paths, i.e., the ones respecting causality.

Truly Concurrent Models of Programs with Resources

The graph-based semantics introduced in the previous chapter is often not informative enough, because it does not take into account whether two actions commute or not. In this chapter, we introduce truly concurrent models which incorporate this information. We begin by extending our programming languages with resources and restrict ourselves to conservative programs, in which resource consumption only depends on the current state. We then generalize the semantics to asynchronous graphs, which explicitly describe the commutation of two actions and to precubical sets, which can more generally express the commutation of n actions. Finally, links with other classical models for concurrency are mentioned.

The Category of Components

The components of a directed space are not easy to define. We explain what the problem is and why some obvious ideas fail. Then the components of a d-space without loops is defined. It is a quotient of the fundamental category. It is defined for a general category without non-trivial isomorphisms. The component category is the quotient under a system of morphisms, the weak isomorphisms. Another construction, which gives an isomorphic fundamental category, is obtained by inverting the weak isomorphisms. An algorithm which determines a less “quotiented” category, precomponents, for simple programs is given, as well as several examples.