Presburger Arithmetic and Its Use in Verification

6.2. SIMPLIFICATION OF PRESBURGER FORMULAS
Figure 6.1. AN-sequenceKripkestructure[15].
principle in logic, resulting in various subsets of Presburger formulas. We are open
for extending the procedure for new subsets of Presburger fragments in the future.
Detailed source code of the formulation can be found in Appendix B.4.
For Presburger fragments arise from the model checker, we consider Kripke
structures by concatenating N identical automata which is illustrated in Figure 6.1.
The DC formula is □(l < 5 ⇒ ∫ p < 3)where□φ is defined by ¬(true ⌢ (¬φ) ⌢ true).
Resulting Presburger fragments are quite big and they are used as the input for the
simplification process which is discussed in the next section.
6.2 Simplification of Presburger formulas
In Section 5.2, we have discussed an algorithm to quickly simplify Presburger fragments.
This section presents experimental results of the algorithm and compares
them with previous approaches. Our experiment is conducted as follows:
•Fourside-conditionPresburgerformulasaregeneratedfrom2-sequence,3-
sequence, 4-sequence and 5-sequence automata with the associated DC formula(see
6.1 for details).
•SimplifiedformulasarerecordedbyrunningasimplificationprocessbyHansen
et al. (HB’s) [15] and one by our algorithm (Ours).
•SimplifiedformulasarefedintotheSMT-solverZ3forquantifierelimination
and evaluation.
Our experimental results are summarized in Table 6.1. These results show that
our simplification process at least reduces 4.5% ofnumberofquantifiersmorethan
the other method, and the deepest nesting of quantifiers is also 10.5% smaller.
And it is also clear that less complex formulas are easier for Z3 to solve. This
brings the hope that recognizing patterns of some Presburger fragments and quickly
simplifying them is important to assist decision procedures. By performing this
experiment, we demonstrate that further simplification of Presburger fragments is
possible and meaningful to reduce stress on decision procedures. We are going to
incorporate more elements to simplify formulas as much as possible before solving
them.
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