1. Translating Linear Temporal Logic into Büchi Automata
Presented by Choi, Chang-Beom

2. Content Overview Translating LTL formula into Büchi Automata
Linear Temporal Logic
Büchi Automata
Translating LTL formula into Büchi Automata
Local Automaton
Eventuality Automaton
Model Automata
Further Study
Reference
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST

3. Overview Model checking or
Specify requirement properties and build system model
Generate possible states from the model and then check whether given requirement properties are satisfied within the state space OK
Target
Program or
Model Check
Requirement
Properties
(F W)
Error Trace Found
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST

4. Overview A process of Model Checking Modeling Specification
Build a model of program or system
Specification
Describe requirement properties
Verification
Checking that a model of the program or system satisfies a given specification
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST

5. Overview How can we model check of a program or system? Modeling
Build a Büchi automaton for a given program or system
Specification
Describe requirement properties using Temporal Logic
Verification
Automatically (semi-automatic)
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST

6. Overview Process of Model Checking
Model Checker
Requirement Properties
Target Program
(F W)
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST

7. Overview Linear Temporal Logic
LTL is an extension of propositional logic geared to reasoning about infinite sequences of states
Time is viewed as linear
Each time instant has a unique successor
The sequences considered are isomorphic to the natural numbers and each state is a propositional interpretation
The living being always, eventually breathe. 5 10 …
Time
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST

8. Overview Syntax of Linear Temporal Logic
The formulas of linear temporal logic built from a set of atomic propositions P are following
true, false, p, and ¬p, ∀ p ∈P;
φ1 ∧ φ2, and φ1∨ φ2 are LTL formulas;
○ φ1, φ1 U φ2, and φ1 Ũ φ2 are LTL formulas
Sequence σ = σ[0…] = σ0σ[1…]
σ[i] = si
σ[…i] = s0s1…si
σ[i…] = sisi+1…
The definition of sequence is from “Recognizing Safety and Livness, B. Alpern, F. Schneider”
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST

9. Overview Temporal Operators
Reads : “next”
Means : at next state
Operator U
Reads : “strong until”
Means : second argument holds at the current or a future position, and first argument has to hold until that position
Operator Ũ
Reads : “weak until”
Means : first argument be true until its second argument is true (does not require that the second argument ever become true)
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST

10. Overview Semantic of Linear Temporal Logic
Definition A transition system M = (S,→,L) is a set of states S endowed with a transition relation → (a binary relation on S), such that every s ∈ S has some s’ ∈ S with s → s’, and a labeling function L :S → P(Atoms)
Atoms : Atomic Propositions (Atomic description) e.g. : Atoms = {p, q}, P(Atoms)={{}, {p}, {q}, {p, q}}
L(s) : contains all atoms which are true in state s
e.g. : L(s0) = {p, q}, L(s1) = {q, r}, L(s2) = {r} s0
p, q s2 s1
q, r r
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST

11. Overview Semantic of Linear Temporal Logic
Definition
A path in a model M = (S,→,L) is an infinite sequence of sate s1, s2, s3, … in S such that, for each i > 1, si → si+1. We write the path π as s1 → s2 → …
π ≡ σ = σ[0…] = σ[0]σ[1…]
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST

12. Overview Semantic of Linear Temporal Logic
Operator U
Reads : “strong until”
Means : second argument holds at the current or a future position, and first argument has to hold until that position
σ[1…] ⊨ p, σ[1…] ⊭ q, σ[1…] ⊨ p U q
σ[2…] ⊨ p, σ[2...] ⊭ q, σ[2…] ⊨ p U q
σ[3…] ⊭ p, σ[3…] ⊨ q, σ[3…] ⊨ p U q
σ[4…] ⊭ p, σ[4…] ⊨ q, σ[4...] ⊨ p U q
σ[5...] ⊭ p, σ[5...] ⊭ q, σ[5...] ⊭ p U q …
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST

13. Overview Semantic of Linear Temporal Logic
Operator Ũ
Reads : “weak until”
Means : first argument be true until its second argument is true (does not require that the second argument ever become true) p
σ[1…] ⊨ p, σ[1…] ⊭ q, σ[1…] ⊨ p U q
σ[2…] ⊨ p, σ[2…] ⊭ q, σ[2…] ⊨ p U q
σ[3…] ⊨ p, σ[3…] ⊭ q, σ[3…] ⊨ p U q
σ[4…] ⊨ p, σ[4…] ⊭ q, σ[4…] ⊨ p U q
σ[5…] ⊨ p, σ[5…] ⊭ q, σ[5…] ⊨ p U q q
p Ũ q … 1 2 3 4 5 6 7
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST

14. Overview Semantic of Linear Temporal Logicㅁφ
always
φ Ũ false
Requires that its argument be true always
At all future points ⋄φ
eventually
true U φ
Requires that its argument be true eventually
At some point in the future
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST

15. Overview Semantic of Linear Temporal Logic
For all, we have σ ⊨ true and σ ⊭ false
For σ ⊨ p for p ∈ P iff p ∈ L(σ[0]) = L(s0)
For σ ⊨ ¬p for p ∈ P iff p ∉ L(σ[0])
σ ⊨ φ1 ∧ φ2, iff σ ⊨ φ1 and σ ⊨ φ2
σ ⊨ φ1∨ φ2 , iff σ ⊨ φ1 or σ ⊨ φ2
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST

16. Overview Semantic of Linear Temporal Logic: Temporal logic
σ ⊨ ○ φ1, iff σ[1] ⊨ φ1
σ[0…] ⊨ ○ φ1, iff σ[1] ⊨ φ1
σ[i…]⊨ φ1 Ũ φ2 iff σ[i]⊨ φ2 ∨ (σ[i]⊨ φ1 ∧ σ[i+1…] ⊨ φ1 Ũ φ2)
σ[i…]⊨ φ1 U φ2 iff σ[i…]⊨ φ1 Ũ φ2 ∧ ∃j, j≥ i, σ[j]⊨ φ2
ㅁφ = ¬⋄¬ φ
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST

17. Content Overview Translating LTL formula into Büchi Automata
Linear Temporal Logic
Büchi Automata
Translating LTL formula into Büchi Automata
Local Automaton
Eventuality Automaton
Model Automata
Further Study
Reference
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST

18. Overview Büchi Automata
Automata which accepts infinite word
Büchi Automata m accepts the sequences of program states that are in L(m)
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST

19. Overview Büchi Automata
Definition
A = (Σ, S, S0, ρ, F) Σ: alphabet (set of program states) S : set of automaton states
S0 : set of initial state ρ : a transition function (S xΣx S) F : a set of accepting states
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST

20. Overview Büchi Automata
A = (Σ, S, S0, ρ, F)
The input of A is infinite w : a0, a1, … (∈ Σω)
A run is a sequence of states r: s0,s1, … (∈ Sω)
Initiation: s0 ∈ S0
Consecution : si+1∈ρ(si, ai)
Accepting run (r = s0,s1, … )
There is some state s ∈ F
An infinite number of integers i ∈ N such that si = s
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST

21. Overview Büchi Automata
run : q0, q1, q1, q1, …
S = {q0, q1}
S0 = {q0}
ρ = {(q0,true, q0), (q0, P, q1), (q1,true, q1)
F = {q1}
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST

22. Content Overview Translating LTL formula into Büchi Automata
Linear Temporal Logic
Büchi Automata
Translating LTL formula into Büchi Automata
Local Automaton
Eventuality Automaton
Model Automata
Conclusion and Further Study
Reference
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST

23. Translating LTL formula into Büchi Automata Process of translating LTL into Büchi Automata
Create Local Automaton
Checks that the sequence satisfies all conditions imposed by the formula
It checks conditions a step by step check on the sequence
Create Eventuality Automaton
Checks that the eventualities are realized
The problem is that nothing prevents us from postponing forever the time at which (eventuality) formula will be true
Eventualities : formulas of the form ⋄φ and φ1 U φ2
ㅁ φ ≡ (φ ∧ ○ㅁ φ)
⋄φ ≡ (φ ∨ ○ ⋄ φ)
¬(φ1 Ũ φ2 )≡ (¬φ1 ∧¬φ2 ∨ (¬φ1 ∧○¬(φ1 Ũ φ2)))
Determine which eventualities have to be realized
Compose two automaton
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST

24. Translating LTL formula into Büchi Automata Local Automaton
Closure cl()
Smallest set of formulas satisfying the following conditions
φ ∈ cl(φ)
φ1 ∧ φ2∈ cl(φ) ⇒ φ1 , φ2 ∈ cl(φ)
φ1 ∨ φ2 ∈ cl(φ) ⇒ φ1 , φ2 ∈ cl(φ)
φ1 → φ2 ∈ cl(φ) ⇒ φ1 , φ2 ∈ cl(φ)
¬ φ1 ∈ cl(φ) ⇒ φ1 ∈ cl(φ)
φ1 ∈ cl(φ) ⇒ ¬ φ1 ∈ cl(φ)
○ φ1 ∈ cl(φ) ⇒ φ1 ∈ cl(φ)
ㅁ φ1 ∈ cl(φ) ⇒ φ1 ∈ cl(φ)
⋄ φ1 ∈ cl(φ) ⇒ φ1 ∈ cl(φ)
φ1 Ũ φ2 ∈ cl(φ) ⇒ φ1 , φ2 ∈ cl(φ)
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST

25. Translating LTL formula into Büchi Automata Local Automaton
L = (∑, NL, ρL, Nφ, NL)
∑ : ∑ ⊂ 2cl(φ)
s ∈ ∑, for every f ∈ cl(φ), f ∈ s iff ¬f ∉ s
NL includes all subsets s of cl(φ) that are propositionally consistent.
For every φ1 ∈ cl(φ), we have φ1 ∈ s iff ¬φ1 ∉ s
For every φ1 ∧ φ2 ∈ cl(φ), we have φ1 ∧ φ2 ∈ s iff φ1 ∈ s and φ2 ∈ s
For every φ1 ∨ φ2 ∈ cl(φ), we have φ1 ∧ φ2 ∈ s iff φ1 ∈ s or φ2 ∈ s
For every φ1 → φ2 ∈ cl(φ), we have φ1 ∧ φ2 ∈ s iff ¬ φ1 ∈ s or φ2 ∈ s
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST

26. Translating LTL formula into Büchi Automata Local Automaton
L = (∑, NL, ρL, Nφ, NL)
If ρL(s, a) is non-empty then a = s
Symbol being read is compatible with the state of the automaton
ρL(s, a) must check the next state is compatible with the semantics of the temporal operators ( t ∈ ρL(s, a))
∀○φ1 ∈ cl(φ), we have ○φ1 ∈ s iff φ1 ∈ t
∀ㅁφ1 ∈ cl(φ) we have ㅁφ1 ∈ s iff φ1 ∈ s and ㅁφ1 ∈ t
∀⋄φ1 ∈ cl(φ) we have ⋄φ1 ∈ s iff either φ1 ∈ s, or ⋄φ1 ∈ t
∀φ1 Ũ φ2 ∈ cl(φ) we have φ1 U φ2 ∈ s iff either φ2 ∈ s, or φ1 ∈ s and φ1 Ũ φ2 ∈ t
∀φ1 U φ2 ∈ cl(φ) we have φ1 U φ2 ∈ s iff either φ1 ⋀ φ2 ∈ s, or φ2 ∈ s or φ1 U φ2 ∈ t
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST

27. Translating LTL formula into Büchi Automata Local Automaton
L = (∑, NL, ρL, Nφ, NL)
The set Nφ of initial states is the set of states that include the formula
The set NL of accpeting states is , the set of all states
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST

28. Example Local Automaton : ⋄p
Closure of ⋄p
cl(⋄p) = {⋄p, p, true,¬⋄p, ¬p, false}
NL= {{⋄p, p, true}, {⋄p, p, false}, {¬⋄p, p, true}, {¬⋄p, p, false}, {⋄p, ¬p, true}, {⋄p, ¬p, false}, {¬⋄p, ¬p, true}, {¬⋄p, ¬p, false}}
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST

29. Example Local Automaton for ⋄p
{⋄p, p, true}
{⋄p, ¬p, true}
{¬⋄p, p, true}
{¬⋄p, ¬p, true}
{¬⋄p, ¬p, false}
{¬⋄p, p, false}
σ ⊭ false
{⋄p, p, false}
{⋄p, ¬p, false}
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST

30. Example Local Automaton for ⋄p (some optimization)
{⋄p, p}
{⋄p, ¬p}
{¬⋄p, p}
{¬⋄p, ¬p}
cl(⋄p) = {⋄p, p, ¬⋄p, ¬p}
NL= {{⋄p, p}, {¬⋄p, p}, {⋄p, ¬p}, {¬⋄p, ¬p}}
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST

31. Translating LTL formula into Büchi Automata Eventuality Automaton
Eventuality automaton is supposed to check that the eventualities are realized
Check each time a formula of the form (or φ1 U φ2)
⋄φ ≡ (φ ∨ ○ ⋄ φ)
¬(φ1 Ũ φ2) ≡ (¬φ1∧¬φ2)∨ (¬φ2 ∧ ○¬(φ1 Ũ φ2))
Eventuality automaton starts by finding out which eventualities have to be realized at the initial time instant, then it checks that these are realized
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST

32. Translating LTL formula into Büchi Automata Eventuality Automaton
F = (∑, 2ev(φ), ρF, {{}}, {{}})
∑ : ∑ ⊂ 2cl(φ)
s ∈ ∑, for every f ∈ cl(φ), f ∈ s iff ¬f ∉ s
The set 2ev(φ) of states is the set of subsets of the eventualities of the formula φ (a state {e1, …, ek} means that the eventualities e1, …, ek still have to be realized)
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST

33. Translating LTL formula into Büchi Automata Eventuality Automaton
F = (∑, 2ev(φ), ρF, {{}}, {{}})
ρF(s,a), t ∈ ρF(s,a)
s = {}
∀ ⋄φ ∈ a, one has ⋄φ ∈ t iff φ ∉ a
∀ ¬(φ1 Ũ φ2 ) ∈ a, one has ¬(φ1 Ũ φ2 ) ∈ t iff ¬φ1∧¬φ2∉ a
s ≠ {}
∀ ⋄φ ∈ s, one has ⋄φ ∈ t iff φ ∉ a
∀ ¬(φ1 Ũ φ2 )∈ s, one has ¬(φ1 Ũ φ2 ) ∈ t iff ¬φ1∧¬φ2 ∉ a
Initial state : {}
Finial state : {}
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST

34. Example Eventuality automaton
{⋄p, p}
{¬⋄p, ¬p}
{¬⋄p, p}
{⋄p, ¬p}
{⋄p, ¬p} {} ⋄p
{¬⋄p, ¬p}
{¬⋄p, p}
{⋄p, p}
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST

35. Translating LTL formula into Büchi Automata Composing the two automata
M = (∑, NM, ρM ,NM0, FM)
NM = NL X 2ev(φ) (Cartesian Product)
(p, q) ∈ ρM((s, t), a) iff p ∈ ρL(s, a) and q ∈ ρF(t, a)
NM0 = Nφ X {}
FM = NL X {}
Given two Büchi automata A1 = (∑, S1, ρ1, S01, F1) and A2 = (∑, S2, ρ2, S02, F2), it is possible to build a Büchi automaton accepting the language L(A1) ∩ L(A2)
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST

36. Example Composing the two automata
({¬⋄p, ¬p} , ⋄p)
({¬⋄p, p}, ⋄p)
({⋄p, ¬p} , ⋄p)
({⋄p, p}, ⋄p)
({¬⋄p, ¬p} , {})
({¬⋄p, p}, {})
({⋄p, ¬p} , {})
({⋄p, p}, {})
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST

37. Translating LTL formula into Büchi Automata The automaton on the 2P
M = (∑, NM, ρM ,NM0, FM) ⇓
M’ = (2P, NM, ρM’ ,NM0, FM)
t ∈ ρM’(s, a) iff some b ∈ ∑ such that a = b∩P and t ∈ ρM(s, b)
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST

38. Final automaton for ⋄p p p ¬p ¬p ¬p true
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST

39. Content Overview Translating LTL formula into Büchi Automata
Linear Temporal Logic
Büchi Automata
Translating LTL formula into Büchi Automata
Local Automaton
Eventuality Automaton
Model Automata
Conclusion and Further Study
Reference
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST

40. Conclusion # 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
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST

41. Further Study Translate LTL to Büchi Automata Next Topic?
Using Alternating Büchi Automata
Tableau Method
On the fly method
Next Topic?
On-line Algorithm
More specific research on Interactive Systems
Symbolic graph representation: Ordered Binary Decision Diagram
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST

42. Reference
On the Relation of Programs and Computations to Models of Temporal Logic by Pierre Wolper, In Proc. Temporal Logic in Specification, vol. 398 of LNCS, pages Springer-Verlag, 1989.
Constructing Automata from Temporal Logic Formulas: A Tutorial by Pierre Wolper In Lectures on Formal Methods in Performance Analysis, vol of LNCS, pages Springer-Verlag, July 2001.
From Modal Logic to Deductive Databases by A. Thayse et al., Wiley, 1989
Logic in Computer Science, second edition by M. Huth and M. Ryan, Cambridge press
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST

43. Complicated Example Closure of φ = ㅁ⋄p
cl(φ) = {ㅁ⋄p, ¬ㅁ⋄p, ⋄p, ¬⋄p, p, ¬p}
NL= {{ㅁ⋄p, ⋄p, p}, {¬ㅁ⋄p, ⋄p, p}, {ㅁ⋄p, ¬⋄p, p}, {ㅁ⋄p, ⋄p, ¬ p }, {¬ㅁ⋄p, ¬⋄p, p}, {¬ㅁ⋄p, ⋄p, ¬p}, {ㅁ⋄p, ¬⋄p, ¬p}, {¬ㅁ⋄p, ¬⋄p, ¬p}}
= {{ㅁ⋄p, ⋄p, p}, {⋄¬⋄p, ⋄p, p}, {ㅁ⋄p, ¬⋄p, p}, {ㅁ⋄p, ⋄p, ¬ p }, {⋄¬⋄p, ¬⋄p, p}, {⋄¬⋄p, ⋄p, ¬p}, {ㅁ⋄p, ¬⋄p, ¬p}, {⋄¬⋄p, ¬⋄p, ¬p}}
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST

44. Local Automaton {ㅁ⋄p, ⋄p} {⋄¬⋄p, ⋄p} {ㅁ⋄p, ¬⋄p} {ㅁ⋄p, ⋄p} {⋄¬⋄p, ¬⋄p}
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST

45. Local Automaton {⋄p, p} {⋄p, p} {¬⋄p, p} {⋄p, ¬p} {¬⋄p, p} {⋄p, ¬p}
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST

46. Local Automaton {ㅁ⋄p, ⋄p, p} {⋄¬⋄p, ⋄p, p} {ㅁ⋄p, ¬⋄p, p}
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST

47. Local Automaton {ㅁ⋄p, ⋄p} {⋄¬⋄p, ⋄p} {ㅁ⋄p, ¬⋄p} {ㅁ⋄p, ⋄p} {⋄¬⋄p, ¬⋄p}
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST

48. Local Automaton {⋄p, p} {⋄p, p} {¬⋄p, p} {⋄p, ¬p} {¬⋄p, p} {⋄p, ¬p}
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST

49. Local Automaton {ㅁ⋄p, ⋄p, p} {⋄¬⋄p, ⋄p, p} {ㅁ⋄p, ¬⋄p, p}
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST

50. Eventuality Automaton
ev(φ) = {⋄¬⋄p, ⋄p}
2ev(φ) = {{}, {⋄¬⋄p}, {⋄p}, {⋄¬⋄p,⋄p} = {{¬⋄¬⋄p, ¬⋄p}, {⋄¬⋄p, ¬⋄p}, {¬⋄¬⋄p, ⋄p}, {⋄¬⋄p, ⋄p}}
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST

51. Eventuality Automaton⋄p
{¬⋄¬⋄p, ⋄p, p}
{¬⋄¬⋄p, ¬⋄p, ¬p}
{¬⋄¬⋄p, ¬⋄p, p}
{⋄¬⋄p, ¬⋄p, p}
{¬⋄¬⋄p, ⋄p, ¬ p } {}
⋄p,
⋄¬⋄p
{⋄¬⋄p, ⋄p, p}
{⋄¬⋄p, ¬⋄p, ¬p}
⋄¬⋄p
{⋄¬⋄p, ⋄p, ¬p}
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST

52. Eventuality Automaton
{¬⋄¬⋄p, ¬⋄p, ¬p}
{¬⋄¬⋄p, ⋄p, ¬p} ⋄p
{⋄¬⋄p, ¬⋄p, ¬p}
{¬⋄¬⋄p, ¬⋄p, p}
{¬⋄¬⋄p, ⋄p, p}
{⋄¬⋄p, ¬⋄p, p}
{⋄¬⋄p, ⋄p, p}
{⋄¬⋄p, ⋄p, ¬p} {}
⋄p,
⋄¬⋄p
⋄¬⋄p
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST

53. Eventuality Automaton⋄p
{¬⋄¬⋄p, ⋄p, ¬p} {}
⋄p,
⋄¬⋄p
{⋄¬⋄p, ⋄p, ¬p}
{⋄¬⋄p, ¬⋄p, p}
{¬⋄¬⋄p, ¬⋄p, p}
{⋄¬⋄p, ¬⋄p, ¬p}
⋄¬⋄p
{¬⋄¬⋄p, ¬⋄p, ¬p}
{⋄¬⋄p, ⋄p, p}
{¬⋄¬⋄p, ⋄p, p}
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST

54. Eventuality Automaton⋄p
{⋄¬⋄p, ¬⋄p, ¬p}
{⋄¬⋄p, ⋄p, ¬p}
{¬⋄¬⋄p, ⋄p, ¬p}
{¬⋄¬⋄p, ¬⋄p, ¬p} {}
⋄p,
⋄¬⋄p
{¬⋄¬⋄p, ¬⋄p, p}
{⋄¬⋄p, ¬⋄p, p}
{⋄¬⋄p, ⋄p, p}
⋄¬⋄p
{¬⋄¬⋄p, ⋄p, p}
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST

55. Combining Automata {} ⋄¬⋄p ⋄¬⋄p, ⋄p ⋄p
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST