1. In a system that processes instructions, including checkpoint instructions that establish checkpoint synchronization points where failures that occur before checkpoint synchronization points may not be reported after the checkpoint synchronization points, a method to process checkpoint instructions that allows certain instructions subsequent to a checkpoint instruction to be executed concurrently with the execution of said checkpoint instruction, comprising the steps of: decoding said checkpoint instruction that includes an operand store step; decoding an instruction other than said checkpoint instruction and executing it concurrently with said checkpoint instruction to a point prior to completion; executing a first unit of operation of said checkpoint instruction up to a point prior to said operand store step; completing said first unit of operation only after completion of operand store operations for instructions prior to said checkpoint instruction; executing a second unit of operation of said checkpoint instruction that does not change execution results from said execution of said first unit of operation; and signaling completion of said checkpoint instruction only after completion of said operand store step for said checkpoint instruction.