Presburger Arithmetic and Its Use in Verification

5.2. SIMPLIFICATION OF PRESBURGER FORMULAS
and a conjunction guard are proposition logic fragments in the following forms respectively:
∧
L i ⇒ ∨ P j
i j
∧
L i ∧ ∧ P j
i j
We consider GNF in the Presburger setting where literals are comparison and
divisibility constraints and guards consist of propositional connectives of quantified
and quantifier-free formulas. First, GNF supports an easy way to express NNF [15]:
P ∧ Q ≡ ∧ i∈φ
∧P ∧ Q
P ∨ Q ≡ ∧ i∈φ
⇒ P ∨ Q
¬( ∧ i
L i ⇒ ∨ j
P j ) ≡ ∧ i
L i ∧ ∧ j
¬P j
¬( ∧ i
L i ∧ ∧ j
P j ) ≡ ∧ i
L i ⇒ ∨ j
¬P j
As we can see from above equations, the implication guard and the conjunction
guard are dual of each other. Transformation into NNF is done by simply pushing
negation inside inner guarded formulas only. A formula is said to be in GNF if
conjunction guards only occur in implication guards and vice versa. A guarded
formula is put into GNF by applying equivalences:
∧
L i ⇒ ( ∧ L ′ k ⇒ ∨ P l ′ ) ∨ ∨ P j ≡ ∧ L i ∧ ∧ L ′ k ⇒ ∨ P j ∨ ∨ P l ′ (5.1)
i
k l j i k j l
∧
L i ∧ ( ∧ L ′ k ∧ ∧
i k l
P ′
l ) ∧ ∧ j
P j ≡ ∧ i
L i ∧ ∧ k
L ′ k ∧ ∧ j
P j ∨ ∧ l
P ′
l (5.2)
Application of above rules helps to reduce the nesting of guarded formulas; inthe
next section, we study how to use GNF in connection with PA and simplify formulas
as much as possible.
5.2.2 Equation-based simplification
Given a Presburger formula in the GNF, we sketch a simple algorithm to do cheap
simplification on the formula. The first set of rules is for simplifying quantifier
guards with equations inside:
∃x.(nx = t ∧ ∧ i
L i ⇒ ∨ j
P j ) ≡⊤ (5.3)
∀x.(nx = t ∧ ∧ i
L i ∧ ∧ j
P j ) ≡⊥ (5.4)
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