Timestamped StackThe TS stack is a linearizable highperformance concurrent stack which works by attaching timestamps to elements, rather than building a linked datastructure in memory. The top element of the stack is then the element with the latest timestamp, not the element at the head of a linked list. Insights from the stack theorem (see below) allow the TS Stack to have multiple top elements, implemented with unordered timestamps, which can be removed in parallel. Thereby the contention on the head of the stack is reduced. Additionally timestamps allow highly efficient elimination, which is key for highperformance concurrent stacks. The graph to the right shows the performance of the TS Stack in comparison with a Treiber Stack and an elimination backoff stack (EB Stack) in a highcontention producerconsumer benchmark on a 64core AMD Opteron ccNUMA machine. More information about the TS stack is available in our POPL15 paper. The TS Stack is implemented as part of the Scal project. 

Stack TheoremProving linearizability of the TS stack is difficult because existing proof techniques turn out to be insufficient (e.g. the TS stack does not provide linearization points in the code). We present a generic stack theorem which uses partial order information provided by the code to prove linearizability. The stack theorem is based on the following insight: Linearizability requires that the order in which elements are pushed is preserved. Concurrently pushed elements, however, are not ordered and thus no order has to be preserved. We show in the stack theorem that a partial order on elements, as illustrated to the left, is sufficient for a linearizable stack implementation. If pop operations always remove one of the top elements in the partial order, then the implementation is linearizable with respect to stack semantics. The stack theorem is proven in Isabelle HOL (sources). More details are available in our POPL 2015 paper and artifact. In the TS Stack implementation we use the timestamps to encode the partial order on elements. By searching and removing the element with the latest timestamp we ensure that the TS Stack satisfies all conditions of the stack theorem. Therefore the TS Stack is linearizable with respect to stack semantics. 