Computable obstructions to wait-free computability

Summary.

We present an algorithmic test for deterministic wait-free solvability of decision tasks in asynchronous distributed systems whose processes communicate via read-write shared memory. Input to the test is a formal representation of the decision task as a triple $({\cal I},{\cal O},\Delta)$, where ${\cal I}$ and ${\cal O}$ are simplicial complexes specifying the inputs and outputs of the task and $\Delta$ is the input-output relation of the task. The form of ${\cal I}$, ${\cal O}$, and $\Delta$ fixes the system size (i.e., number of processes). The result of the test is either (1) that there is no wait-free solution to the decision task for the given system size or (2) inconclusive. Incompleteness of the test is unavoidable since wait-free solvability of decision tasks is undecidable for a system of size at least three. The test is shown to detect the impossibility of wait-free consensus for all systems, and experimental results show that the test detects the impossibility of wait-free set consensus for systems of size at most five. A more complete description of the efficacy of the test remains open.

The key new ingredient underlying the test is a simplicial complex ${\cal T}$, the task complex\/, associated to $\Delta$. There is a simplicial projection map $\alpha$ from ${\cal T}$ to ${\cal I}$, and $\alpha$ induces a homomorphism $\alpha_*$ from $H_*({\cal T})$ to $H_*({\cal I})$, where $H_*$ denotes simplicial homology. Failure of $\alpha_*$ to surject on $H_*({\cal I})$ implies that no wait-free protocol can solve the task. Put another way, the elements of $H_*({\cal I})$ that are not in the image of $\alpha_*$ are obstructions\/ to solvability of the task. These obstructions are computable when using suitable homology coefficients.

By passing to quotients of ${\cal T}$ and ${\cal I}$ by well-behaved group actions, the test can be adapted to check the impossibility of solution of a decision task by any wait-free protocol that is symmetric or anonymous relative to the group.