In computer science, concurrency is a property of systems in which several computations are executing simultaneously, and potentially interacting with each other. The computations may be executing on multiple cores in the same chip, preemptively timeshared threads on the same processor, or executed on physically separated processors. A number of mathematical models have been developed for general concurrent computation including Petri nets, process calculi, the synchronous model and the Actor model.
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Because computations in a concurrent system can interact with each other while they are executing, the number of possible execution paths in the system can be extremely large, and the resulting outcome can be indeterminate. Concurrent use of shared resources can be a source of indeterminacy leading to issues such as deadlock, and starvation.^{[1]}
The design of concurrent systems often entails finding reliable techniques for coordinating their execution, data exchange, memory allocation, and execution scheduling to minimize response time and maximise throughput.
Concurrency theory has been an active field of research in theoretical computer science. One of the first proposals was Carl Adam Petri's seminal work on Petri Nets in the early 1960s. In the years since, a wide variety of formalisms have been developed for modeling and reasoning about concurrency.
A number of formalisms for modeling and understanding concurrent systems have been developed, including:^{[2]}
Some of these models of concurrency are primarily intended to support reasoning and specification, while others can be used through the entire development cycle, including design, implementation, proof, testing and simulation of concurrent systems.
The proliferation of different models of concurrency has motivated some researchers to develop ways to unify these different theoretical models. For example, Lee and SangiovanniVincentelli have demonstrated that a socalled "taggedsignal" model can be used to provide a common framework for defining the denotational semantics of a variety of different models of concurrency,^{[4]} while Nielsen, Sassone, and Winskel have demonstrated that category theory can be used to provide a similar unified understanding of different models.^{[5]}
The Concurrency Representation Theorem in the Actor model provides a fairly general way to represent concurrent systems that are closed in the sense that they do not receive communications from outside. (Other concurrency systems, e.g., process calculi can be modeled in the Actor model using a twophase commit protocol.^{[6]}) The mathematical denotation denoted by a closed system S is constructed increasingly better approximations from an initial behavior called ⊥_{S} using a behavior approximating function progression_{S} to construct a denotation (meaning ) for S as follows:^{[7]}
In this way, S can be mathematically characterized in terms of all its possible behaviors.
Various types of temporal logic^{[8]} can be used to help reason about concurrent systems. Some of these logics, such as linear temporal logic and computational tree logic, allow assertions to be made about the sequences of states that a concurrent system can pass through. Others, such as action computational tree logic, HennessyMilner logic, and Lamport's temporal logic of actions, build their assertions from sequences of actions (changes in state). The principal application of these logics is in writing specifications for concurrent systems.^{[1]}
Concurrent programming encompasses the programming languages and algorithms used to implement concurrent systems. Concurrent programming is usually considered to be more general than parallel programming because it can involve arbitrary and dynamic patterns of communication and interaction, whereas parallel systems generally have a predefined and wellstructured communications pattern. The base goals of concurrent programming include correctness, performance and robustness. Concurrent systems such as operating systems are generally designed to operate indefinitely and not terminate unexpectedly. Some concurrent systems implement a form of transparent concurrency, in which concurrent computational entities may compete for and share a single resource, but the complexities of this competition and sharing are shielded from the programmer.
Because they use shared resources, concurrent systems in general require the inclusion of some kind of arbiter somewhere in their implementation (often in the underlying hardware), to control access to those resources. The use of arbiters introduces the possibility of indeterminacy in concurrent computation which has major implications for practice including correctness and performance. For example arbitration introduces unbounded nondeterminism which raises issues with model checking because it causes explosion in the state space and can even cause models to have an infinite number of states.
