Translating from LTL to automata

1. Translating from LTL to automata

2. Why translating? • 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.

3. T p T p p q p T T From formulas to Buchi automta Fp p U q GFp Gp Now try yourself: FGp, a U (b U c), X(p U (q Æ r))

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

5. Preprocessing • Convert into normal form, where negation only applies to propositional variables. • ¬G becomes F¬. • ¬F becomes G¬. • ¬( U )becomes(¬) R (¬), • ¬(  R )becomes(¬) U (¬).

6. Preprocessing • 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)

7. Example • 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 ) )

8. And now... • We need to build an automaton that accepts exactly those words that satisfy .

9. Content • 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

10. The Local Automaton • Closure of : all the subformulas ofand 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)

11. The Local Automaton / Alphabet, states • 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

12. The Local Automaton / Transition relation 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∈ sφ1∈ t • Fφ1∈ sφ1 ∈ 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.

13. The Local Automaton / Initial + final states • The initial states IL • ... is the set of states that include the formula • The accepting states FL • ... is the set of all states

14. Example: Local Automaton for Fp • Closure of Fp • cl(Fp) = {Fp, p, ¬Fp, ¬p} • SL= {{Fp, p}, {¬Fp, p}, {Fp, ¬p}, {¬Fp, ¬p}}

15. Local Automaton forFp {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).

16. Local Automaton forFp (labels) {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.

17. The Eventuality Automaton • Eventuality automaton is supposed to check that the eventualities are realized • Check formulas of the form • φ1U φ2 • Fφ // special case of U

18. The Eventuality Automaton/ Alphabet, states • 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

19. The Eventuality Automaton/ Transition relation, initial state • The transition ρF • Lett ∈ ρF(s,a) • ForFφ :Fφ ∈ t φ ∉ a • For φ1U φ2: φ1U φ2∈ t  φ2∉ a • The initial state : {}

20. The Eventuality Automaton/ accepting states • 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}

21. The Eventuality Automaton/ accepting states • The acceptance condition, formaly: • Let ei be an eventuality condition i’ Ui • 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 {}

22. ExampleEventuality automaton {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.

23. Composing the two automata • 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 {}

24. Example Composing the two automata 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

25. Example Composing the two automata 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

26. Optimizations... 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:

27. Conclusion • 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