Presburger Arithmetic and Its Use in Verification

Chapter 4
Theory of Presburger Arithmetic
In this chapter, we discuss Presburger Arithmetic and its properties. A lot of research
has been conducted to decide Presburger fragments. We present two decision
procedures including Cooper’s algorithm and the Omega Test, and they play important
roles in processing Presburger fragments of our interest later on.
4.1 Overview
Presburger Arithmetic introduced by Mojzaesz Presburger in 1929 is a first-order
theory of integers which accepts + as its only operation. An expression is considered
to be a Presburger formula if it contains elements in the following forms:
• s = t, s ≠ t, s < t, s > t, s ≤ t, s ≥ t
Comparison constraints
• d | t
• ⊤ (true), ⊥ (false)
• F ∨ G, F ∧ G, ¬F
• ∃ x. F , ∀ x. F
Divisibility constraints
Propositional constants
Propositional connectives
First-order fragments
where s and t are terms, d is an integer and x is a variable. Terms consist of
integer constants and variables, they accept addition (+), subtraction (-) and multiplication
by constants. One adopted convention is abbreviation of Qx 1 .Qx 2 ...Q x n . F
as Qx 1 x 2 ...x n . F.
For example, a classic example of representing some amount of money by 3-cent
coins and 5-cent coins appears in PA as follows:
∀z.∃x ∃y. 3x + 5y = z
∀z. z ≥ 8 ⇒∃x ∃y. 3x + 5y = z
Or the clause shows existence of even numbers could be formulated:
∃x. 2 | x
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