Weak until, release and positive normal form

Nadeem Qasmi

Sadain Iqrar

Abdur Rehman Abbasi

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)

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

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.

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)

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)

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)

Weak Until

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

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 (*).

Lemma 5.19

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:

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)

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 ).

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

Question ?