In this series of talks I will discuss the application of methods of combinatorial algebraic topology to theoretical distributed computing. In the first talk I will introduce the formal simplicial concepts of a task and a protocol and illustrate it on some of the central tasks of distributed computing.

I will present several theorems which connect topological properties of the associated structures with the computability issues of the related tasks.

## CiNii 図書 - Directed algebraic topology and concurrency

The second talk will be devoted to topology of protocol complexes. In the third, and the last talk, I will talk about topology of the Weak Symmetry Breaking task. Specifically I will be concentrating on the combinatorial theory of abstract simplicial path subdivisions. This will be based on the work of Castaneda, Rajsbaum and myself.

This talk will mostly consist of an assembly of some known, but scattered, results, both in computer science and in topology, and some of their simple consequences. A striking fact is the existence of relatively efficient methods for analyzing concurrent models with mutual exclusion binary semaphores only , based on safe Petri nets or prime event structures, whereas things seem to be much more intricate in classical approaches to concurrency, when it comes to dealing with more subtle synchronization primitives, such as, for instance, counting semaphores.

In fact even, in the latter case, the state space is geometrically a non-positive curvature NPC space or cubical complex, in the discrete setting. As observed by Ghrist et al. Also, as a direct consequence of work by Chepoi, Santocanale, Ardila and other authors, NPC cubical complexes are the same as prime event structures introduced to model, precisely, concurrent processes with mutual exclusion some 25 years before, and equivalent to safe Petri nets, introduced and heavily studied even before.

We explore some of the consequences of these facts in the rest of the talk, if time permits. Starting from the usual domains of directed algebraic topology like preordered spaces, d-spaces, etc. I shall present a formal setting for directed algebraic topology in a category C, based on an abstract cylinder endofunctor of C and natural transformations between its powers. The setting can be easily extended to categories of diagrams in C, or categories of algebras over C of a given type, in both cases.

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Moreover, it has strong analogies with the structure of a monad over C, a standard basis of categorical algebra. Concurrency is an ubiquitous question in computer science which actually dates back to the early ages of the discipline.

The most spread model in hardware architectures is the so-called shared memory. If no care is taken in writing concurrent programs one can observe erratic behaviour running them. The fact that such errors due to concurrency are nondeterministic make them much harder to treat.

While graphs are widely recognized as a good framework for representing sequential programs there is no consensus about the nature of their counterpart in parallel programming. Plethora of models for concurrency have been proposed. The specificity of the ones we are about to describe is that they make use of topology to formalize the fact that the actions performed along an execution trace can be locally performed in another order.

We introduce the notion of control flow graph for sequential processes then remark that parallel composition of process should corresponds to cartesian product. However the cartesian product in the category of graphs in the category if graphs do not fit, so topological models are introduced and compared.

The directed geometric realization and generalized cubical areas, from which all the relevant examples can be obtained, are described. The invariants of directed algebraic topology are then presented. The fundamental category arises as as straightforward generalization of the notion of fundamental groupoid, some examples of computations are given.

The category of components, which is used to reduce the size of the description of the fundamental categories, is built in the case of loop-free categories. Some perspectives beyond the loop-free case are given.

## Directed Algebraic Topology and Concurrency

Then we come to the question of factorization, which was raised by R. Milner in the early 90's. In this talk, I will discuss topological abstraction of HDAs, i. I will introduce two preorder relations for HDAs, called homeomorphic abstraction and topological abstraction. Clarendon Press. Oxford Roman, S. An Introduction to the Language of Category theory. Amadio Univ. Plan of the course for What kind of concurrency in this course? Restricted Dijkstra PV language Precubical sets as a generalization of graphs Control-flow graph, conservative programs, and precubical control flow Locally ordered spaces Directed realization of graphs and precubical sets Isothetic regions The fundamental category Seifert and van Kampen theorem for the fundamental category [optional] The category of components Factorization Some other topological models French and English French.

Material Slides and rather obsolete lecture notes can be found here. Bibliography Books Fajstrup, L. Brown, R. Topology and Groupoids.

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Higgins, P. Categories and Groupoids. Van Nostrand-Reinhold Articles Dijkstra, E. Basic Category Theory Awodey, S. 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. Author David Epstein. Author David Wootton.

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