The trend towards processors with more and more parallel cores is increasing the need for software that can take advantage of parallelism. The most widespread method for writing parallel software is to use explicit threads. Writing correct multithreaded programs, however, has proven to be quite challenging in practice. The key difficulty is non-determinism. The threads of a parallel application may be interleaved non-deterministically during execution. In a buggy program, non-deterministic scheduling will lead to nondeterministic results--some interleavings will produce the correct result while others will not. We propose an assertion framework for specifying that regions of a parallel program behave deterministically despite nondeterministic thread interleaving. Our framework allows programmers to write assertions involving pairs of program states arising from different parallel schedules. We describe an implementation of our deterministic assertions as a library for Java, and evalua...