synchronous parallel (BSP) model Petri nets Process calculi Calculus of communicating systems (CCS) Communicating sequential processes (CSP) model π-calculus Tuple spaces, e.g., Linda Simple Concurrent Object-Oriented Programming (SCOOP) Reo Coordination Language Trace monoids 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. Some of these are based on message passing, while others have different mechanisms for concurrency. The proliferation of different models of concurrency has motivated some researchers to develop ways to unify these different theoretical models. For example, Lee and Sangiovanni-Vincentelli have demonstrated that a so-called "tagged-signal" model can be used to provide a common framework for defining the denotational semantics of a variety of different models of concurrency, while Nielsen, Sassone, and Winskel have demonstrated that category theory can be used to provide a similar unified understanding of different models. 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 two-phase commit protocol.) The mathematical denotation denoted by a closed system is constructed increasingly better approximations from an initial behavior called using a behavior approximating function to construct a denotation (meaning ) for as follows: In this way, can be mathematically characterized in terms of all its possible behaviors. Logics Various types of temporal logic can be used to help reason about concurrent systems. Some of these logics, such as linear temporal logic and computation 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, Hennessy–Milner 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. Practice Concurrent programming encompasses 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 well-structured communications pattern. The base goals of concurrent programming include correctness, performance and robustness. Concurrent systems such as Operating systems