Lecture Notes in Computer Science 3472

19 Model Checking 571
The alternating Büchi automaton we are going to construct from a property
φ uses the set of all sub-formulas and their negations as the set of states.
Definition 19.8. The set of sub-formulas Sub (φ) ofapropertyφ is inductively
defined on the structure of φ by:
Sub (true) ={true}
Sub (¬φ) ={¬φ}∪Sub (φ)
Sub (X (φ)) = {X (φ)}∪Sub (φ)
Sub (φ1 ∨ φ2) ={φ1 ∨ φ2}∪Sub (φ1) ∪ Sub (φ2)
Sub (U (φ1,φ2)) = {U (φ1,φ2)}∪Sub (φ1) ∪ Sub (φ2)
The transition function of an alternating Büchi automaton maps states to
positive Boolean combinations of states. Since properties in LTL may use the
negation modality ¬φ and negation is not allowed in positive Boolean functions,
negationofpropertiesisexpressedbynegationofstates.Forthisreasonthe
negatives of the sub-formulas are included into the set of states of the alternating
Büchi automaton. To connect the negation of properties to the negation of
positive Boolean combinations of states the following construction is used:
Definition 19.9. The dual φ of a positive Boolean formula is defined inductively
on the structure of a formula φ as follows:
true = false
false = true
¬φ = φ
φ1 ∨ φ2 = φ1 ˙∧φ2
φ1 ∧ φ2 = φ1 ˙∨φ2
X (φ) =¬X (φ)
U (φ1,φ2) =¬U (φ1,φ2)
Given an LTL formula φ, one can directly build an alternating Büchi automaton
Aφ =(Σ,Q,δ,q0, F ), such that L(Aφ) is exactly the set of computations
satisfying the property φ. The set of states Q is defined as the set of sub-formulas
of φ and their negations. The set of actions is defined as Σ =2AP .Thesetof
accepting states F consists of all formulas φ which have got the form ¬U (φ1,φ2).
The transition function δ is inductively defined on the structure of φ as follows:
�
true if p ∈ A
δ (p, A) =
false if p �∈ A
δ (φ1 ∨ φ2, A) =δ (φ1, A) ˙∨δ (φ2, A)
δ (φ1 ∧ φ2, A) =δ (φ1, A) ˙∧δ (φ2, A)
δ (¬φ, A) =δ (φ, A)
δ (X (φ) , A) =φ
δ (U (φ1,φ2) , A) =δ (φ2, A) ˙∨ (δ (φ1, A) ˙∧U (φ1,φ2) , A)
The idea behind this recursive definition is: whenever a composed formula is
to check it is transformed into a Boolean combination of new formulas. In this