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accesses to shared AfP-hard algorithm analysis approach artifact race atomic operations branch conditions branch-free program combinations of branches conflicting statements constructs is monotonic contain a race contain synchronization control flow data race Definition e2 occur e2 overlap event ordering event pairs events are ordered events forming races exact solution execute concurrently exist executions fixed input fork/join MC91 free program infeasible races lock covers message passing Non-Monotonically Synchronized number of events Offset-Span labeling on-the-fly ordered critical sections ordered or semi-ordered ordering relationships original program pairs of events partial order particular input polynomial possible input Post and Wait post Ct post mortem post(y post/wait no clear Post/Wait/Clear problem race conditions race detection race no yes race statement races between events schedule shared memory parallel shared variable single semaphore statement instances synchronization operations taxonomy Theorem trary shared memory type of synchronization unordered races updates XisF XisT y/n general race yes y/n y/n