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Translating from LTL to automata

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- Want to write the specification in some logic.
- Want to check that an automaton (or a Kripke structure) satisfies this property.
- The check (“model-checking”) will be based on automata operations – hence we need to translate the property to automata.

T

p

T

p

p

q

p

T

T

Fp

p U q

GFp

Gp

Now try yourself:

FGp, a U (b U c), X(p U (q Æ r))

- So now we need to show an algorithmic translation from LTL to Buchi
- It will work in two stages:
- Translate to Generalized Buchi
- Degeneralization.

- Convert into normal form, where negation only applies to propositional variables.
- ¬G becomes F¬.
- ¬F becomes G¬.
- ¬( U )becomes(¬) R (¬),
- ¬( R )becomes(¬) U (¬).

- Convert to Negation Normal Form
- Push negations over propositional conenctives, and eliminate operators other thanÆ,Ç

- Eliminate G
- Replace G by (False R).
- (in general we can stay withU, R, X)

- Translate (GFP ) ! ( GFQ )
- Eliminate implication ¬(GF P ) Ç (GF Q )
- Eliminate G, F:¬( False R ( True U P ) ) Ç ( False R ( True U Q ) )
- Push negation inwards:(True U (False R¬P ) ) Ç ( False R ( True U Q ) )

- We need to build an automaton that accepts exactly those words that satisfy .

- The construction continues as follows:
- Build the Local Automaton
- This automaton guarantees that the word satisfies all conditions imposed by the formula

- Build the Eventuality Automaton
- Eventualities : formulas of the form Fφand φ1U φ2
- The problem is that nothing prevents us from postponing forever the time at which (eventuality) formula will be true

- Compose them

- Build the Local Automaton

- Closure of : all the subformulas ofand their negations.
- Formally: cl() is the smallest set of formulas satisfying the following conditions
- φ ∈ cl(φ)
- φ1∈ cl(φ) ⇒¬φ1 ∈ cl(φ)
- φ1∧ φ2∈ cl(φ) ⇒φ1 , φ2∈ cl(φ)
- φ1∨ φ2∈ cl(φ) ⇒ φ1 , φ2∈ cl(φ)
- Xφ1∈ cl(φ) ⇒φ1 ∈ cl(φ)
- Fφ1∈ cl(φ) ⇒ φ1 ∈ cl(φ)
- φ1U φ2∈ cl(φ) ⇒φ1 , φ2∈ cl(φ)
- φ1R φ2∈ cl(φ) ⇒φ1 , φ2∈ cl(φ)

- The local automaton is L = (∑, SL, ρL, IL, FL)
- The alphabet ∑
- ∑ µ 2cl(φ)
- ∑ elements are consistent: for s2∑ and f ∈ cl(φ): f ∈ s ¬f ∉ s

- The states SL
- All propositionally consistent subsets s of cl(φ):
- φ1 ∈ s¬φ1 ∉ s

- All propositionally consistent subsets s of cl(φ):

The edges:

- ρL(s, a) must check the next state is compatible with the semantics of the temporal operators.
- Let t ∈ ρL(s, a). Then:
- Xφ1∈ sφ1∈ t
- Fφ1∈ sφ1 ∈ sor Fφ1∈ t
- φ1U φ2∈ s (φ2∈ s) or (φ1∈ sand φ1U φ2∈ t)
- φ1R φ2∈ s (φ1 ⋀φ2∈ s) or (φ2∈ sand φ1R φ2∈ t)
The labeling on the edges:

- For a state s;, s is the label on all the outgoing edges from s.

- The initial states IL
- ... is the set of states that include the formula

- The accepting states FL
- ... is the set of all states

- Closure of Fp
- cl(Fp) = {Fp, p, ¬Fp, ¬p}

- SL= {{Fp, p}, {¬Fp, p}, {Fp, ¬p}, {¬Fp, ¬p}}

{Fp, p}

{Fp, ¬p}

{¬Fp, p}

{¬Fp, ¬p}

- Recall the defnition: (Fp∈ s) (p∈ s or Fp∈ t)(t is the target state)

Top-right: Since p s then t can only be such thatFp 2 t.

Top left: Since p 2 s then all states can be t.

Bottom left: contradictory, hence no point in this state (can be removed)

Bottom right: since the condition above is iff relation, then we need that

(:p 2 s) and (:Fp 2 t).

{Fp, ¬p}

{Fp, p}

{Fp, p}

{Fp, p}

{Fp, ¬p}

{Fp, ¬p}

{Fp, p}

{¬Fp, ¬p}

{¬Fp, p}

{¬Fp, ¬p}

Recall: the edge labels are equivalent to the source state names.

- Eventuality automaton is supposed to check that the eventualities are realized
- Check formulas of the form
- φ1U φ2
- Fφ // special case of U

- Check formulas of the form

- Ev = (∑, 2ev(φ), ρF, {{}}, F)
- The alphabet
- ∑ µ2cl(φ)
- ∑ elements are consistent: for s2∑ and f ∈ cl(φ): f ∈ s ¬f ∉ s

- The states 2ev(φ)
- The set of subsets of the eventualities of the formula φ
- A state {e1, …, ek} means that the eventualities e1, …, ekstill have to be realized

- The transition ρF
- Lett ∈ ρF(s,a)
- ForFφ :Fφ ∈ t φ ∉ a
- For φ1U φ2: φ1U φ2∈ t φ2∉ a

- Lett ∈ ρF(s,a)
- The initial state : {}

- The acceptance condition F is complicated...
- When can we accept a state s?
- if s has an eventuality, it satisfies it.
- Examples:
- s is accepting: s ={pUq,:p, q} s = {:pUq,:p, :q}
- s is not accepting: s ={pUq, p, :q} s = {pUq, :p, :q}

- The acceptance condition, formaly:
- Let ei be an eventuality condition i’ Ui
- Suppose we have the eventuality conditions e1,...,em.
- Then F is a generalized Buchi condition:F = {Á1,...,Ám} whereÁi = {s 2 S | ei2 s !i2 s}

- In our example:
- We have two states: {} and {Fp}
- Thus,Fcontains the single state {}

{Fp, p}

{¬Fp, ¬p}

{¬Fp, p}

{Fp, ¬p}

{Fp, ¬p}

{}

{Fp}

{¬Fp, ¬p}

{¬Fp, p}

{Fp, p}

We can begin with all edges and all labels and then remove those that are incompatible with the condition we saw in the previous slide:

The condition is: Fp∈ t p∉ a

- Q: When is this automaton satisfied?
- A: When all eventualities are satisfied.

- M = (∑, SM, ρM ,NM0, FM)
- ∑ µ 2cl()
- SM = SL x 2ev(φ)(Cartesian Product)
- (p, q) ∈ ρM((s, t), a) p ∈ ρL(s, a) and q ∈ ρF(t, a)
- NM0 = Nφ x {}
- FM = NL x {}

The propositions are the ‘real’ labels.

({¬Fp, ¬p} ,Fp)

({Fp, ¬p} ,Fp)

p

:p

({Fp, p},Fp)

p

:p

({Fp, ¬p} ,{})

({Fp, p},{})

({¬Fp, ¬p} ,{})

:p

Equivalently: labels move to outgoing edges.

p

({Fp, ¬p} ,Fp)

:p

({Fp, p},Fp)

p

p

:p

p

:p

p

({Fp, ¬p} ,{})

({Fp, p},{})

:p

p

({¬Fp, ¬p} ,{})

:p

p

:p

:p

p

p

:p

p Ç:p

true

:p

p

There are optimizations that make the automaton much smaller:

If we define the alphabet ∑ as formulas over AP we can do better:

- The number of States
- Local Automaton : 2cl(φ) = O(22|φ|)
- Eventuality Automaton : 2ev(φ) = O(2|φ|)
- Composed Automata : 2cl(φ) X 2ev(φ) = O(23|φ|)
- |φ| is length of formula φ