Memory consistency models define the order in which accesses to shared memory in a concurrent system may be observed to occur. Such models are a necessity since program order is not a reliable indicator of execution order, due to microarchitectural features or compiler transformations. Concurrent programming, already a challenging task, is thus made even harder when weak memory effects must be addressed. A rigorous specification of weak memory models is therefore essential to make this problem tractable for developers of safety- and security-critical, low-level software. In this paper we survey the field of formalisations of weak memory models, including their specification, their effects on execution, and tools and inference systems for reasoning about code. To assist the discussion we also provide an introduction to two styles of formal representation found commonly in the literature (using a much simplified version of Intel's x86 as the example): a step-by-step construction of traces of the system (operational semantics); and with respect to relations between memory events (axiomatic semantics). The survey covers some long-standing hardware features that lead to observable weak behaviours, a description of historical developments in practice and in theory, an overview of computability and complexity results, and outlines current and future directions in the field.

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