Presburger Arithmetic and Its Use in Verification

CHAPTER 6.
EXPERIMENTS ON PRESBURGER ARITHMETIC
No. of quantifiers Deepest nesting of quantifiers Solving time
(Original/HB’s/Ours) (Original/HB’s/Ours) (HB’s/Ours)
2-seq 3190/729/583 54/25/19 0.31/0.28 s
3-seq 10955/2823/2352 83/39/30 0.9/0.78 s
4-seq 26705/7638/5165 113/54/42 2.27/1.47 s
5-seq 53284/16830/14501 144/70/55 56.47/6.32 s
Table 6.1. Experimental results of two simplification algorithms.
6.3 Optimization of Cooper’s algorithm
Before going to implement a parallel variant of Cooper’s algorithm, we consider
some design choices for the algorithm itself. These design choices affect both sequential
and parallel versions, but to keep things simple we justify the selection by
abenchmarkofthesequentialalgorithmonly. Therearesomeimprovementswhich
have been done to contribute to a better implementation of Cooper’s procedure:
• Reduce all literals as much as possible. By reducing, we mean all constraints
are evaluated to truth values whenever possible, and their coefficients
are reduced to the smaller values. This process will lead to early evaluation of
some parts of a formula, and huge coefficients are avoided as much as possible.
• Keep a formula in a uniform way. Thisimprovementreliesonourexperience
of GNF, and uniform formulas not only have small depths of recursion
but also collect many formulas in their collection and introduce more chances
of concurrency, which is important for the purpose of parallel execution later
on.
The test set consists of ten hand-annotated Presburger formulas, each one has
3-4 nested universal quantifiers and a disjunction of 3-4 equalities. The benchmark
is performed in the following manner: each test is run 100 times and execution time
is calculated by average execution time of 100 sessions. Moreover, some sessions
were executed before the benchmark, so every function is ensured in the warm state
in the time of benchmarking.
Here we consider two important optimizations which have been mentioned in
Section 4.2: eliminating blocks of quantifiers and using heterogeneous coefficients of
literals. Theideaofeliminatingblocksofquantifiersisclear,insteadofmanipulating
ahugeformula,wesplittheworkintomanysmallpiecesandmanipulatemany
small formulas. As illustrated in Figure 6.2, pushing blocks of quantifiers into small
formulas results in 10−20× performance gain compared to the baseline version. The
result is significant because smaller formulas easily fit into caches, so manipulating
them is much faster in practice. Using heterogeneous coefficients adds up a slight
performance gain as demonstrated in the figure. Their advantage is two-fold; we save
one traversal through the formula to collect all coefficients so resulting coefficients
are smaller and resulting quantifier-free formulas are easier to evaluate in the later
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