Presburger Arithmetic and Its Use in Verification

Chapter 6
Experiments on Presburger Arithmetic
This chapter is organized as a series of experiments on different aspects of Presburger
Arithmetic. The chapter starts with discussion of generating various Presburger
fragments. These formulas are used as inputs for testing different algorithms later
on. The next part deals with simplification of Presburger formulas; it pops up
from the fact that side-condition Presburger fragments are complex and able to be
reduced by some cheap quantifier elimination. After that, we discuss design and
implementation of Cooper’s algorithm in a sequential manner. Some design choices
are made which have influence on both sequential and parallel versions, and we
attempt to do a benchmark to decide which option is good for the procedure.
6.1 Generation of Presburger fragments
Our main source of Presburger formulas is from the model checker of Duration
Calculus. However, as discussed in Chapter 5, generated Presburger fragments
are in a huge size even for a very small model-checking problem, and testing and
optimizing decision procedures on those formulas are pretty difficult. Also there
is no common benchmark suite for Presburger formulas; therefore, we attempt to
generate test formulas which are controllable in terms of size and complexity and
postpone working on the real-world formulas until a later phase. Our test formulas
are from two sources:
• Hand-annotated Presburger formulas: in most of the cases, these formulas
are able to be quickly solved by hand. They serve the purpose of ensuring
the correctness of decision procedures and testing some facets of Presburger
formulas which are easier to be constructed by hand.
• Automatically-generated Presburger formulas: we generate these formulas by
formulation of Pigeon Hole Principle, and the detailed procedure is presented
later in this section. We use this formulation to generate formulas whose sizes
are controllable and satisfiability is predetermined. These formulas allow us
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