Commutativity Condition Refinement

We present a technique for automatically generatingcommutativity conditions from (abstract-level) data-structurespecifications. We observe that one can pose the commutativityquestion in a way that does not introduce additional quantifiers,via a mechanized lifting of a (potentially partial) specificationto an equivalent total specification. We then describe an al-gorithm that iteratively refines an under-approximation of thecommutativity (and non-commutativity) condition for two data-structure methods m and n into an increasingly precise version.Our algorithm terminates if/when the entire state space hasbeen considered, and can be aborted at any time to obtain apartial, sound commutativity condition. We prove soundness,relative completeness, describe how to obtain input predicates,and discuss heuristics to improve qualitative aspects of ouralgorithm’s output.We have implemented our technique in a tool called SERVOIS,which uses an SMT solver. We show that we can automaticallygenerate symbolic commutativity conditions for a range of datastructures including Set, HashTable, Accumulator, Counter, andStack. This work has the potential to impact a variety of contexts,in particular, multicore software where it has been realized thatcommutativity is at the heart of increased concurrency.