Commutativity of Reducers

In the Map-Reduce programming model for data parallel computation in a cloud environment, the reducer phase is responsible for computing an output key-value pair, given a sequence of input values associated with a particular key. Because of non-determinism in transmitting key-value pairs over the network, the inputs may not arrive at a reducer in a fixed order. This gives rise to the reducer commutativity problem, that is, is the reducer computation is independent of the order of its inputs? Commutativity of reducers is a desirable property, absence of which may lead to correctness violations and hard-to-find bugs. In this paper, we study the reducer commutativity problem formally. To model real-world reducers, we introduce the notion of an integer reducer, a syntactic subset of integer programs. We show that, in spite of syntactic restrictions, deciding commutativity of integer reducers over unbounded sequences of integer values is undecidable. It remains undecidable even with input sequences of a fixed length. The problem, however, is decidable for reducers over unbounded input sequences if the integer values are bounded. We also propose an efficient reduction of commutativity checking to conventional assertion checking and report experimental results from checking commutativity using various off-the-shelf program analyzers.