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

ISBN-10: 3319153978

ISBN-13: 9783319153971

ISBN-10: 3319153986

ISBN-13: 9783319153988

This monograph offers an software of options and strategies from algebraic topology to types of *concurrent processes* in desktop technology and their analysis.

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

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

The audience for this publication comprises graduate scholars, researchers and practitioners within the box, mathematicians and laptop scientists alike.

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

**Example text**

Its code is the same as above but encapsulated in a while loop, and similarly there is one consumer which iteratively consumes values. Notice that the corresponding program is not conservative, and hence cannot be described in the formalism of the book. Such a program could still be taken in account by rewriting the producer-consumer code using monitors [92], and extending the techniques developed here to handle those, but this is outside the scope of this book. , have capacity 1. 3 Asynchronous Semantics 47 synchronization primitive, and one can show interesting properties in this particular ˇ p of a conservative case.

U ∼ s and (s/t) . u ∼ t s u ∼ t/s ∼ s/t t ∼ s t and moreover such a path u is unique up to dihomotopy. Dual properties are satisfied when the backward cube property is verified. Proof The proofs of 1, 2, and 3 are performed by induction on the derivation of dihomotopy and the length of paths, using the forward cube property. In 4, the required morphisms are obtained by starting from the homotopy between s . s and t . t , and replacing all possible half cubes using the forward cube property. Uniqueness up to homotopy of u follows from 3.

22 (Swiss flag) Consider the following program p: Pa ; Pb ; Vb ; Va || Pb ; Pa ; Va ; Vb with a, b ∈ R mutexes (κa = κb = 1). 2 State Spaces for Conservative Resources x04 Vb x03 Va x02 Pa x01 Pb x00 Pa Pa x14 Pb x24 Vb x34 x13 x23 x33 x12 x22 x32 x11 x21 x31 x10 Pb x20 Vb x30 Va Va x44 Vb x43 Va x42 Pa x41 Pb x40 33 x04 Vb x03 Va x02 Pa x01 Pb x00 Pa Pa x14 Pb x24 Vb x34 x13 x33 x11 x31 x10 Pb x20 Vb x30 Va Va x44 Vb x43 Va x42 Pa x41 Pb x40 The beginning and end vertex are respectively sp = x00 and tp = x44 .

### Directed Algebraic Topology and Concurrency by Lisbeth Fajstrup, Eric Goubault, Emmanuel Haucourt, Samuel Mimram, Martin Raussen

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