Decidability of linearizabilities for relaxed data structures

Many recent implementations of concurrent data structures relaxed their linearizability requirements for better performance and scalability. Quasi-linearizability, k-linearizability and regular-relaxed linearizability are three quantitative relaxation variants of linearizability that have been proposed as correctness conditions of relaxed data structures, yet preserving the intuition of linearizability. Quasi-linearizability has been proved undecidable. In this paper, we first show that k-linearizability is undecidable for a bounded number of processes, by reducing quasi-linearizability into it. We then show that regular-relaxed linearizability is decidable for a bounded number of processes. We also find that the number of the states of a relaxed specification is exponential to the number of the states of the underlying specification automaton (representing its relaxation strategy), and polynomial to the number of the states of the underlying quantitative sequential specification and the number of operations.