Presburger Arithmetic and Its Use in Verification

Chapter 7
Parallel execution of decision procedures
In this chapter, parallel versions of Cooper’s algorithm and the Omega Test are
discussed with many details regarding their concurrency, efficiency and scalability.
While Cooper’s algorithm is presented in a complete procedure, the Omega Test is
used as a partial process assisting simplification of complex Presburger formulas.
In each section, some test sets are selected for experimentation and experimental
results are interpreted. These test sets are carefully chosen from sources of Presburger
fragments in Section 6.1 and experiments if not explicitly mentioned are
performed on the 8-core 2.40GHz Intel Xeon workstation with 8GB shared physical
memory. Each test is chosen so that its execution time is smaller than 12 seconds.
Measurement of execution time is done in the following manner:
•Benchmarkfunctionsareexecutedinalow-loadenvironment.
•Somesessionsareperformedseveraltimessothateveryfunctionisawarm
state before benchmarking.
•Eachtestisrun10 times and average execution time is recorded.
7.1 A parallel version of Cooper’s algorithm
In this section, we discuss parallelism in Cooper’s algorithm and sketch some ideas
about how to exploit it. We come up with design and implementation for the
procedure and present experimental results.
7.1.1 Finding concurrency
As discussed in Section 6.3, eliminating blocks of quantifiers is good because many
small formulas are faster to manipulate. Other than that there is a chance of concurrency
in distributing blocks of quantifiers, where small formulas are independent
of each other and it is possible to eliminate quantifiers in parallel. Our idea of
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