By Lisbeth Fajstrup, Eric Goubault, Emmanuel Haucourt, Samuel Mimram, Martin Raussen

This monograph offers an program of strategies and techniques from algebraic topology to types of *concurrent processes* in laptop technology and their analysis.

Taking famous discrete versions for concurrent procedures in source administration as some degree of departure, the booklet is going directly to refine combinatorial and topological types. within the technique, it develops instruments and invariants for the recent self-discipline *directed algebraic topology*, that is pushed by way of basic examine pursuits in addition to by means of functions, basically within the static research of concurrent programs.

The country house of a concurrent software is defined as a higher-dimensional house, the topology of which encodes the basic houses of the procedure. as a way to examine all attainable executions within the kingdom area, greater than “just” the topological houses must be thought of: Execution paths have to admire a partial order given by the point movement. for that reason, instruments and ideas from topology need to be prolonged to take privileged *directions* into account.

The audience for this ebook involves graduate scholars, researchers and practitioners within the box, mathematicians and desktop scientists alike.

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**Extra resources for Directed Algebraic Topology and Concurrency**

**Example text**

28 because it is unreachable for similar reasons as above (however, because of the specific programming language we chose, it can be shown that all deadlocks are discovered by the algorithm). Finally, consider the following program p whose pruned transition graph is shown on the right: 36 3 Truly Concurrent Models of Programs with Resources true p = sp while true do skip skip ¬true tp The end position tp is unreachable because the condition ¬true is never true. However, it is not discovered by the algorithm.

We will provide examples of programs with resources in the next sections, and many other can be found in [33]. 1 Conservative Programs In order to study the resource consumption of a program, we introduce the following notion which expresses the overall effect of a program on the resources. , the difference between the number of Va instructions and the number of Pa instructions encountered in an execution of p. It is defined by induction on p by Δ (A) Δ (Pa ) Δ (p;q) = = 0 −δ a = Δ (p) + Δ (q) Δ (if b then p else q) Δ (while b do p) = = Δ (skip) Δ (Va ) Δ (p||q) Δ (p) 0 = = = 0 δa Δ (p) + Δ (q) whenever Δ (p) = Δ (q) whenever Δ (p) = 0 where A is an arbitrary action.

2). 28 because it is unreachable for similar reasons as above (however, because of the specific programming language we chose, it can be shown that all deadlocks are discovered by the algorithm). Finally, consider the following program p whose pruned transition graph is shown on the right: 36 3 Truly Concurrent Models of Programs with Resources true p = sp while true do skip skip ¬true tp The end position tp is unreachable because the condition ¬true is never true. However, it is not discovered by the algorithm.