Presburger Arithmetic and Its Use in Verification

CHAPTER 7.
PARALLEL EXECUTION OF DECISION PROCEDURES
Figure 7.4. Speedup factors in Cooper evaluation.
7.2.1 Finding concurrency
As mentioned in Section 4.2.2, converting Presburger formulas into DNF is inevitable
for the Omega Test. DNF is a significant bottleneck of the algorithm because
formulas may grow exponentially. Take a formula ∧ m
i=1 L i ∧ ∧ n
j=1 (L1 j ∨ L2 j )
as an example, its DNF version is a disjunction of 2 n conjunctions of m + n literals.
DNF is a challenge for employing the Omega Test, but there is a chance of
massive concurrency in this procedure. In the above example, 2 n conjunctions are
totally independent and ready for being resolved in parallel. To do so we rely on
the following rule (which is similar to Equation 7.2):
∃ x 1 ...x n . ∨ ∧
L ij ≡ ∨ ∃ x 1 ...x n . ∧ L ij (7.3)
i j i
j
Using DNF, we can easily divide a formula into many smaller ones and apply the
Omega Test independently. The parallel pattern used here is Data Parallelism,
when we have the same operation working on a big collection of data. As discussed
in Section 2.2, F# provide very good constructs for data parallelism such as
Parallel.For(Each), Array.Parallel and PSeq module.
Looking into details of the Omega Test procedure, it is easy to recognize that
their three basic blocks could be executed in parallel because there is no requirement
for order of executions among Real Shadow, Dark Shadow and Gray Shadow. We
can easily spawn three independent Tasks, butthereisalittlecoordinationwhich
should be done between these tasks. First of all, if Real Shadow’s task completes
first and returns false, the other tasks should be terminated immediately. Similarly,
if Dark Shadow’s task returns true, the other running tasks should be stopped
right away. In the worst-case scenario, the result is returned from Gray Shadow’s
task. It is likely that this task is the longest running one so total execution time is
comparable to execution time of this task. This idea of three parallel coordinated
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