TSO-to-TSO linearizability is undecidable

Abstract

TSO-to-TSO linearizability is a variant of linearizability for concurrent libraries on the total store order (TSO) memory model. It is proved in this paper that TSO-to-TSO linearizability for a bounded number of processes is undecidable. We first show that the trace inclusion problem of a classic-lossy single-channel system, which is known undecidable, can be reduced to the history inclusion problem of specific libraries on the TSO memory model. Based on the equivalence between history inclusion and extended history inclusion for these libraries, we then prove that the extended history inclusion problem of libraries is undecidable on the TSO memory model. By means of extended history inclusion as an equivalent characterization of TSO-to-TSO linearizability, we finally prove that TSO-to-TSO linearizability is undecidable for a bounded number of processes. Additionally, we prove that all variants of history inclusion problems are undecidable on TSO for a bounded number of processes.

Decidability of Liveness on the TSO Memory Model

“…Bouajjani et al [4] prove that robustness is decidable on TSO. Our previous work [20] proves that TSOto-TSO linearizability [5], a correctness condition of concurrent libraries on TSO, is undecidable on TSO. Our previous work [21] proves that a bounded version of TSO-to-SC linearizability [8] is decidable on TSO.…”

“…Abdulla et al [1] reduce the cyclic post correspondence problem (CPCP) [19], a known undecidable problem, into checking whether a specific lossy channel machine has an infinite execution that visits a specific state infinitely often. Our undecidability proof of lock-freedom and wait-freedom is obtained by reducing the checking of the lossy channel machine problem into checking lock-freedom and wait-freedom for a specific library, based on a close connection of concurrent programs on TSO and lossy channel machines [2,20].…”

“…Our previous work [20] considers safety properties of libraries. Both [20] and this paper use the collaboration of two methods to simulate one lossy channel machine transition.…”

“…Our previous work [20] considers safety properties of libraries. Both [20] and this paper use the collaboration of two methods to simulate one lossy channel machine transition. Our idea for simulating lossy channel machine transitions with libraries extends that of [20], since each library constructed using the latter contains executions violating liveness, which makes such libraries not suitable for their reduction to liveness.…”

“…Both [20] and this paper use the collaboration of two methods to simulate one lossy channel machine transition. Our idea for simulating lossy channel machine transitions with libraries extends that of [20], since each library constructed using the latter contains executions violating liveness, which makes such libraries not suitable for their reduction to liveness. The library of [20] contains a method that never returns and thus, do not need to consider "fixing methods to processes".…”