Relational Equivalence Proofs Between Imperative and MapReduce Algorithms

MapReduce frameworks are widely used for the implementation of distributed algorithms. However, translating imperative algorithms into these frameworks requires significant structural changes to the algorithm. As the costs of running faulty algorithms at scale can be severe, it is highly desirable to verify the correctness of the translation, i.e., to prove that the MapReduce version is equivalent to the imperative original. We present a novel approach for proving equivalence between imperative and MapReduce algorithms based on partitioning the equivalence proof into a sequence of equivalence proofs between intermediate programs with smaller differences. Our approach is based on the insight that two kinds of sub-proofs are required: (1) uniform transformations changing the controlflow structure that are mostly independent of the particular context in which they are applied; and (2) context-dependent transformations that are not uniform but that preserve the overall structure and can be proved correct using coupling invariants. We demonstrate the feasibility of our approach by evaluating it on two prototypical algorithms commonly used as examples in MapReduce frameworks: k-means and PageRank. To carry out the proofs, we use the interactive theorem prover Coq with partial proof automation. The results show that our approach and its prototypical implementation based on Coq enables equivalence proofs of non-trivial algorithms and could be automated to a large degree

Bacterial aerosol emission rates from municipal wastewater aeration tanks

In this report we describe the results of a study conducted to determine the rates of bacterial aerosol emission from the surfaces of the aeration tanks of the Metropolitan Water Reclamation District of Greater Chicago John E. Egan Water Reclamation Plant. This study was accomplished by conducting test runs in which Andersen six-stage viable samplers were used to collect bacterial aerosol samples inside a walled tower positioned above an aeration tank liquid surface at the John E. Egan Water Reclamation Plant. The samples were analyzed for standard plate counts (SPC), total coliforms (TC), fecal coliforms, and fecal streptococci. Two methods of calculation were used to estimate the bacterial emission rate. The first method was a conventional stack emission rate calculation method in which the measured air concentration of bacteria was multiplied by the air flow rate emanating from the aeration tanks. The second method was a more empirical method in which an attempt was made to measure all of the bacteria emanating from an isolated area (0.37 m2) of the aeration tank surface over time. The data from six test runs were used to determine bacterial emission rates by both calculation methods. As determined by the conventional calculation method, the average SPC emission rate was 1.61 SPC/m2/s (range, 0.66 to 2.65 SPC/m2/s). As determined by the empirical calculation method, the average SPC emission rate was 2.18 SPC/m2/s (range, 1.25 to 2.66 SPC/m2/s). For TC, the average emission rate was 0.20 TC/m2/s (range, 0.02 to 0.40 TC/m2/s) when the conventional calculation method was used and 0.27 TC/m2/s (range, 0.04 to 0.53 TC/m2/s) when the empirical calculation method was used.(ABSTRACT TRUNCATED AT 250 WORDS)</jats:p