Presburger Arithmetic and Its Use in Verification

CHAPTER 7.
PARALLEL EXECUTION OF DECISION PROCEDURES
Figure 7.2. Speedup factors with different workloads.
Test No. Pigeons Holes Variables Quantifiers Literals
1 21 1 21 3 483
2 22 1 22 3 528
3 23 1 23 3 575
4 24 1 24 3 624
5 25 1 25 3 675
6 26 1 26 3 728
7 27 1 27 3 783
8 28 1 28 3 840
9 29 1 29 3 899
10 30 1 30 3 960
Table 7.1. Test set for Cooper elimination.
Each formula contains 32 smaller formulas which can be resolved in parallel.
Experimental results are presented in Figure 7.3. As can be seen from the figure,
speedup factors are around 4 − 5×; thesesuboptimalspeedupscanbeexplainedby
afast-growingnumberofcachemisseswhentheproblemsizeincreases. Someother
reasons of suboptimal speedups could be found in Section 3.2.
Cooper evaluation is done with some Pigeon-Hole Presburger formulas. They
are chosen in a way that their truth values are false, so it ensures that the algorithm
has to iterate through the whole search space to search for an answer. Some
characteristics of tests formulas are presented in Table 7.2.
Experimental results are shown in Figure 7.4. Speedup factors are good, ranging
from 5x to 8x, because search spaces are huge and the algorithm has been traversed
54