Linearizability is the gold standard among algorithm designers for deducing the correctness of a distributed algorithm using implemented shared objects from the correctness of the corresponding algorithm using atomic versions of the same objects. We show that linearizability does not suffice for this purpose when processes can exploit randomization, and we discuss the existence of alternative correctness conditions. This paper makes the following contributions: 1. Various examples demonstrate that using well-known linearizable implementations of objects (e.g., snapshots) in place of atomic objects can change the probability distribution of the outcomes that the adversary is able to generate. In some cases, an oblivious adversary can create a probability distribution of outcomes for an algorithm with implemented, linearizable objects, that not even a strong adversary can generate for the same algorithm with atomic objects. 2. A new correctness condition for shared object implementations, called strong linearizability, is defined. We prove that a strong adversary (i.e., one that sees the outcome of each coin flip immediately) gains no additional power when atomic objects are replaced by strongly linearizable implementations. In general, no strictly weaker correctness condition suffices to ensure this. We also show that strong linearizability is a local and composable property. 3. In contrast to the situation for the strong adversary, for a natural weaker adversary (one that cannot see a process' coin flip until its next operation on a shared object) we prove that there is no correspondingly general correctness condition. Specifically, any linearizable implementation of counters from atomic registers and load-linked/store-conditional objects, that satisfies a natural locality property, necessarily gives the weak adversary more power than it has with atomic counters.

Abstract. We present a deterministic obstruction-free implementation of leader election from O( √ n) atomic O(log n)-bit registers in the standard asynchronous shared memory system with n processes. We provide also a technique to transform any deterministic obstruction-free algorithm, in which any process can finish if it runs for b steps without interference, into a randomized wait-free algorithm for the oblivious adversary, in which the expected step complexity is polynomial in n and b. This transformation allows us to combine our obstruction-free algorithm with the leader election algorithm by Giakkoupis and Woelfel [21], to obtain a fast randomized leader election (and thus test-and-set) implementation from O( √ n) O(log n)-bit registers, that has expected step complexity O(log * n) against the oblivious adversary. Our algorithm provides the first sub-linear space upper bound for obstruction-free leader election. A lower bound of Ω(log n) has been known since 1989 [29]. Our research is also motivated by the long-standing open problem whether there is an obstruction-free consensus algorithm which uses fewer than n registers.

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