Existing parallelizing compiler have been shown to detect and exploit only small protions of parallelism in real application programs. A core problem is difficulty to collect sufficient information from the program to utilize the underlying multiprocessor architecture, Symbolic analysis is the key component to support compilers in parameterizing codes by the number os processorys as well as array sizes and to enable performance analyzers examining programs with program unknowns. We present a variety of novel symbolic analysis techniques that are crucial for a new generation of parallelizing compilers and performance analysis tools to handle non-linear array index functiosn, complex loop boudns, and deal with unknown problem ,array and machine sizes which is critical for programming languages such as High Perfornace Fortran. This includes counting solutions to a system of constraints, computing lower and upper bonds of symbolic expressions, and comparing symbolic expressions. We have implemented all of these techniques as part of a state-of-the-art parallelizing compiler and a performance estimator. Examples will be shown that demonstrate the effectiveness of our symbolic analysis.
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