Presburger Arithmetic and Its Use in Verification

CHAPTER 7.
PARALLEL EXECUTION OF DECISION PROCEDURES
We analyze the algorithm in a case of our specific input. We have a quantified
formula with k quantifiers and a conjunctions of m literals and n disjunctions of
two literals. As discussed above, the corresponding formula in DNF consists of 2 n
conjunctions of (m + n) literals. Examiningeachconjunctionseparately,eachquantifier
elimination step causes a product between two lists of size (m+n). Therefore,
we have a worst-case complexity of Θ((m + n) 2k ). Certainly, this does not happen
often in practice, if one of these lists is empty we have size of new conjunctions
remaining linear of m + n. The worst-case complexity of the whole procedure is
Work = Θ(2 n (m + n) 2k ). Analyzing the algorithm by the DAG model of multithreading,
Span = Θ((m + n) 2k ), and we have Parallelism Factor of Θ(2 n ). We can say this
one is an instance of embarrassingly-parallel algorithms, and its parallelism is only
bounded by the number of used processors.
Regarding specific implementation details, we have written two different versions
of the algorithm: one is List-based and one is Array-based. Wewanttoexamine
performance of List and Array in the context of parallel execution although it
requires a little effort to translate from List-based version to Array-based one.
Certainly, with each version we implement a parallel variant and a sequential one.
They are running on the same codebase, there are only some small differences
regarding parallelism constructs. In the next section, we present our experiment on
asmalltestsetandsomeconclusionsaboutthealgorithmanditsparallelexecution.
7.2.3 Experimental Results
We create a testset from consistency predicates of N-sequence automata in Figure
6.1. For each automaton in 3-sequence, 4-sequence and 5-sequence automata, we
select five arbitrary consistency predicates. Therefore, we have a testset of 15 Presburger
formulas ordered by their number of quantifiers and their size. We notice
that each consistency predicates is a conjunction between literals and disjunctions
which in turn contain two literals, characteristics of the testset are summarized in
Table 7.3.
Automaton Quantifiers Literals Disjunctions
3-seq 3 8 5
4-seq 6 13 7
5-seq 10 19 9
Table 7.3. Test set for the Omega Test.
From the table we can see that at most the procedure resolves 2 5 ,2 7 or 2 9 Omega
Tests in parallel. Because these numbers are usually bigger than the number of
hardware threads, using some work balancing mechanisms would benefit the whole
procedure. In the List-based version, work balancing is done by PLINQ engine
without users’ intervention. In our Array-based variant, we have more control
over distribution of work on different cores. Here we use a partitioning technique
which divides work into balanced chunks for the executing operation.
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