Presburger Arithmetic and Its Use in Verification

7.1. A PARALLEL VERSION OF COOPER’S ALGORITHM
•Parallelexecutionisdoneuntilacertaindepth.
•Sizeofformulas(numberofliterals)canbeusedtodividetasksrunningin
parallel.
•Anoperationisdonesequentiallyifthesizeofabranchistoosmallfor
parallelism.
We have done an experiment with parallel execution on a binary tree and present
results in the next section.
7.1.3 Experimental Results
We have a version of binary tree and a parallel map function and we attempt to
measure speedups on different oWorkload functions:
type BinTree =
| Leaf of int
| Node of BinTree ∗ BinTree
let pmap depth oWorkload tree =
let rec pmapUtil t d =
match t with
| Leaf(n) −> if oWorkload(n) then Leaf(1) else Leaf(0)
| _whend=0−> map oWorkload t
| Node(t1, t2) −>
let t1’ =Task.Factory.StartNew( fun() −> pmapUtil t1 (d−1))
let t2’ =Task.Factory.StartNew( fun() −> pmapUtil t2 (d−1))
Task.WaitAll(t1’, t2’)
Node(t1’.Result, t2’.Result)
pmapUtil oWorkload tree depth
Workload functions are simple for-loops running a specified number of times:
// Workload functions
val oConstant : int −> bool
val oLog : int −> bool
val oLinear : int −> bool
Figure 7.2 shows that speedup factors are really bad when workloads are not
heavy enough and a good speedup of 6× is obtained with a rather big workload.
Consider our scenario of manipulating Presburger formulas, trees are of big size
but tasks at each node are tiny, and at most we manipulate small Terms by their
arithmetic operations. Therefore, it hardly pays off if we try to parallelize every
operation on Formula.
The experiment with Cooper elimination is performed on 10 Pigeon-Hole Presburger
formulas (see Section 6.1 for their construction), and each of them consists of
32 disjunctions which can be resolved in parallel. Their properties are summarized
in Table 7.1.
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