Presburger Arithmetic and Its Use in Verification

CHAPTER 6.
EXPERIMENTS ON PRESBURGER ARITHMETIC
to test with controllably big inputs which are particularly helpful in context
of parallel execution.
Here we describe our formulation of Pigeon Hole Principle: given N pigeons and
K holes, if N ≤ K there exists a way to assign the pigeons to the holes where no
hole has more than one pigeon; otherwise, no such assignment exists. Certainly,
there are many ways to formulate Pigeon Hole Principle in logic, here we employ a
very simple interpretation.
Let x ik be the predicate where pigeon i is in hole k. ThepredicateF shows that
if pigeon i is in the hole k, thereisnootherpigeoninthathole:
F(i, k) = x ik ⇒
∧
1≤ j≤ N
j≠i
¬x jk
The predicate G shows that each pigeon is assigned one hole:
∨
G(i) =
x ik
1≤ k≤ K
The predicate H shows that each pigeon is in only one hole:
H(i, k) = x ik ⇒
∧
The principle is formulated as follows:
∧
Pigeon(N, K) = F(i, k) ∧
1≤ i≤ N
1≤ k≤ K
1≤ l≤ K
l≠k
∧
1≤ i≤ N
¬x il
G(i) ∧
∧
1≤ i≤ N
1≤ k≤ K
H(i, k)
According to the principle, satisfiability of Pigeon(i, k) iseasydeterminedby
values of N and K. Andsizeofthesepredicatescaneasilyscaledbythebigvalues
of arguments which are really helpful for testing and optimizing parallel algorithms.
Up till now we examine the Pigeon Hole Principle in a propositional setting, to
inject linear constraints and quantifiers we consider the following encodings:
x ik ←⊤ x ik ≥ 0 x ik = 0 2 | x ik
x ik ←⊥ x ik < 0 x ik ≠ 0 ¬(2 | x ik )
and define quantified formulas in the form of ∃x 11 , ..., x NK . Pigeon(N, K). We call
these formulas Pigeon-Hole Presburger formulas to attribute their origins. So these
formulas contain N ∗ K variables and N ∗ K quantifiers; we can omit some quantifiers to
adjust the number of quantifiers by our needs. Different encodings give us different
preferences of constraints which in turn demonstrate some interesting aspects in
decision procedures. This procedure of generating Presburger formulas is easy to
extend by using other encodings and there are many other ways to formulate the
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