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

This monograph provides an software of thoughts and strategies from algebraic topology to types of concurrent processes in laptop technological know-how and their analysis.

Taking recognized discrete versions for concurrent strategies in source administration as some degree of departure, the booklet is going directly to refine combinatorial and topological types. within the procedure, it develops instruments and invariants for the recent self-discipline directed algebraic topology, that's pushed through primary learn pursuits in addition to by way of functions, essentially within the static research of concurrent programs.

The kingdom house of a concurrent application is defined as a higher-dimensional area, the topology of which encodes the fundamental homes of the procedure. that allows you to examine all attainable executions within the nation area, greater than “just” the topological houses need to be thought of: Execution paths have to recognize a partial order given by the point move. hence, instruments and ideas from topology need to be prolonged to take privileged directions into account.

The target market for this e-book involves graduate scholars, researchers and practitioners within the box, mathematicians and machine scientists alike.

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However, because of the way threads and accesses to memory are implemented in practice, it happens that the variable might also contain 1, or even a completely unrelated value: in most programming languages concurrent access to shared memory is unspecified. In order to achieve reasonably predictable behavior when using shared memory, most operating systems provide a construction, called a mutex (short for mutual exclusion), which is a resource that can be held by at most one thread. Given such a mutex a, a thread can perform two operations on it [32]: • lock the resource, which is modeled by the instruction Pa , • release the resource, which is modeled by the instruction Va .

Above, the positions only “potentially” have these properties because they might not be reachable, we will generally omit mentioning it in the following. , the way synchronization primitives are used, but does not consider the values manipulated by the program, which explains why it cannot find all such positions. 25. 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 .

3. 38 The asynchronous semantics G y:=2); (Pa ; z:=x; Va p = (x:=1 Pa ; z:=y; Va ) where a is a mutex, is Va Pa z:=y z:=x Pa ∼ sp Va y:=2 x:=1 tp Pa y:=2 Va x:=1 z:=y z:=x Va Pa which, as expected, encodes the fact that the actions x:=1 and y:=2 commute, but the actions z:=x and z:=y do not. t. the semantics of programs. However, we will often be interested in shorter examples, not involving manipulation of data, but still exhibiting a behavior which is interesting from the point of view of concurrency.

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