Weak until, release and positive normal form Convert LTL formula into PNF , Weak unit, release

1. Weak until, release and
positive normal form
Nadeem Qasmi
Sadain Iqrar
Abdur Rehman Abbasi

2. Syntax of LTL (Linear temporal logic)
formula
• LTL formulae over the set AP of atomic proposition are formed according to the
following grammar*
where a ∈ AP.
* Backus Naur form (BNF)

3. Positive Normal Form of LTL formula
• Also called as Canonical Form
• BNF to positive normal form transformation
• Rules:
• Negation can only occur adjacent to Atomic prepositions
• For every operator, it’s dual operator is needed to be incorporated

4. Negation Rule
• This is done by successively “pushing” negations “inside” the formula at hand
￢true  false
￢false  true
￢￢ϕ  ϕ
￢(ϕ ∧ ψ)  ￢ϕ∨ ￢ψ
￢ Oϕ  O￢ϕ
These rewrite rules are lifted to the derived operators as follows:
￢(ϕ ∨ ψ)  ￢ϕ ∧ ￢ψ and ￢ ♦ ϕ  ￢ϕ and ￢ ϕ  ♦￢ϕ.

5. Negation Rule
• Example:
• Consider the LTL formula ￢ (( a Ub ) ∨ Oc). This formula is not in
PNF, but can be transformed into an equivalent LTL formula in weak-
until PNF as follows:
￢ (( aUb) ∨ Oc)
≡ ♦￢(( aUb ) ∨ Oc)
≡ ♦(￢ (aUb) ∧ ￢ Oc)
≡ ♦((a∧￢b) W (￢a∧￢b) ∧ O￢c)

6. Dual Operator Rule
• For every operator it’s dual must be incorporated
• For BNF operators Dual operators are
ϕ ::= true | a | ϕ1 ^ ϕ2 | ￢ϕ | Oϕ | ϕ1 Uϕ2
• Duals:
false = ￢ true,
true = ￢ false,
￢ (ϕ ∨ ψ) ≡ ￢ ϕ∧￢ ψ (Disjunction)
￢ (ϕ∧ ψ) ≡ ￢ ϕ ∨ ￢ ψ (Conjunction)
￢ O ϕ ≡ O￢ ϕ, (Dual of itself)

7. Weak Until as Dual of Until
• Consider the “Until” Operator
￢ (ϕUψ) ≡ ((ϕ∧￢ψ) U (￢ ϕ ∧ ￢ ψ)) ∨ ( ϕ ∧￢ ψ)
• This observation provides the motivation to introduce the operator W (called weak until or
unless) as the dual of U. It is defined by:
Φ W ψ ≡ (ϕU ψ) ∨ ϕ.
• Until and Weak-Until/Unless Duality
￢ ( ϕU ψ) ≡ (ϕ∧￢ ψ) W (￢ ϕ∧￢ ψ)
￢ ( ϕWψ) ≡ (ϕ∧￢ ψ) U (￢ ϕ∧￢ ψ)

8. Weak Until
• It is interesting to observe that W and U satisfy the same
expansion law:

9. Lemma 5.19
• Weak-Until is the Greatest Solution of the Expansion Law
• The formulation “greatest LT property with the indicated condition (*) is to be understood
in the following sense:
(1) P ⊇ Words ( ϕ W ψ) satisfies (*).
(2) Words ( ϕ W ψ) ⊇ P for all LT properties P satisfying condition (*).

10. Lemma 5.19

11. Positive Normal Form for LTL (Weak-Until
PNF)
• For a ∈ AP, the set of LTL formulae in weak-until positive normal form
(weak-until PNF, for short, or simply PNF) is given by:

12. Theorem 5.22.
• Existence of Equivalent Weak-Until PNF Formulae
• For each LTL formula there exists an equivalent LTL formula in weak-until
PNF.
￢(ϕU ψ) ≡ (￢ψ)W(￢ϕ ∧ ￢ψ)

13. Release Operator
• To avoid the exponential blowup in transforming an LTL formula in PNF, another temporal
modality is used that is dual to the until operator: the so-called binary release-operator,
denoted R . It is defined by
Φ R ψ ≡ ￢ (￢ ϕ U￢ ψ).
• Formula ϕ R ψ holds for a word if ψ always holds, a requirement that is released as soon
as ϕ becomes valid.
• The weak-until and the until operator are obtained by:
Φ W ψ ≡ (￢ ϕ ∨ ψ ) R (ϕ ∨ ψ) , ϕ U ψ ≡ ￢ (￢ ϕR ￢ ψ).
Φ R ψ ≡ (￢ ϕ ∧ ψ ) W ( ϕ ∧ ψ ).

14. Release Operator
• The expansion law for release reads as follows:
Φ R ψ ≡ ψ ∧ (ϕ ∨ O( ϕ R ψ))
• For a ∈ AP, LTL formulae in release positive normal form (release PNF, or
simply PNF) are given by

15. Question ?