Translating Linear Temporal Logic into Büchi
AutomataPresented by Choi, Chang-
Beom
Content• Overview
– Linear Temporal Logic– Büchi Automata
• Translating LTL formula into Büchi Automata– Local Automaton– Eventuality Automaton– Model Automata
• Further Study
• ReferenceTranslating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab,
KAIST 2/42
Overview
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST 3/42
• Model checking– 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
Error TraceFound
or
TargetProgram
RequirementProperties
Model Check
Overview• A process of Model Checking–Modeling• 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 4/42
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 5/42
Model Checker
OverviewProcess of Model Checking
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST 6/42
Target Program Requirement
Properties
OverviewLinear 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
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST 7/42
The living being always, eventually breathe.
Time0 105 …
OverviewSyntax 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 8/42
OverviewTemporal Operators
• Operator ○ – 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 9/42
OverviewSemantic of Linear Temporal Logic
• DefinitionA 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 se.g. : L(s0) = {p, q}, L(s1) = {q, r}, L(s2) = {r}
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST 10/42
p, qq, r r
s0
s1 s2
• 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 11/42
OverviewSemantic 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
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST 12/42
σ[1…] ⊨ p, σ[1…] ⊭ q, σ[1…] ⊨ p U qσ[2…] ⊨ p, σ[2...] ⊭ q, σ[2…] ⊨ p U qσ[3…] ⊭ p, σ[3…] ⊨ q, σ[3…] ⊨ p U qσ[4…] ⊭ p, σ[4…] ⊨ q, σ[4...] ⊨ p U qσ[5...] ⊭ p, σ[5...] ⊭ q, σ[5...] ⊭ p U q
…
OverviewSemantic 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)
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST 13/42
p
q
p Ũ q0 1 2 3 4 5 6 7 …
σ[1…] ⊨ p, σ[1…] ⊭ q, σ[1…] ⊨ p U qσ[2…] ⊨ p, σ[2…] ⊭ q, σ[2…] ⊨ p U qσ[3…] ⊨ p, σ[3…] ⊭ q, σ[3…] ⊨ p U qσ[4…] ⊨ p, σ[4…] ⊭ q, σ[4…] ⊨ p U qσ[5…] ⊨ p, σ[5…] ⊭ q, σ[5…] ⊨ p U q
OverviewSemantic of Linear Temporal Logic
• ㅁ φ – 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 14/42
OverviewSemantic of Linear Temporal Logic
OverviewSemantic 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 15/42
OverviewSemantic 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
• ㅁ φ = ¬ ⋄¬ φ
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST 16/42
Content• Overview
– Linear Temporal Logic– Büchi Automata
• Translating LTL formula into Büchi Automata– Local Automaton– Eventuality Automaton– Model Automata
• Further Study
• ReferenceTranslating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab,
KAIST 17/42
OverviewBüchi Automata
• 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 18/42
OverviewBüchi Automata
• Definition– A = (Σ, S, S0, ρ, F)
Σ: alphabet (set of program states)S : set of automaton statesS0 : set of initial stateρ : a transition function (S xΣx S) F : a set of accepting states
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST 19/42
– 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 20/42
OverviewBüchi Automata
OverviewBüchi Automata
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST 21/42
• S = {q0, q1}• S0 = {q0}• ρ = {(q0,true, q0), (q0, P, q1), (q1,true, q1)• F = {q1}
run : q0, q1, q1, q1, …
Content• Overview
– Linear Temporal Logic– Büchi Automata
• Translating LTL formula into Büchi Automata– Local Automaton– Eventuality Automaton– Model Automata
• Conclusion and Further Study
• ReferenceTranslating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab,
KAIST 22/42
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 23/42
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 24/42
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 ∈ sTranslating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab,
KAIST 25/42
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• ∀ ㅁ φ1 ∈ cl(φ) we have ㅁ φ1 ∈ s iff φ1 ∈ s and ㅁ φ1 ∈ 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 26/42
• 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 27/42
Translating LTL formula into Büchi Automata
Local Automaton
ExampleLocal 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 28/42
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST 29/42
ExampleLocal Automaton for ⋄p
{ ⋄p, p, true}
{ ⋄p, p, false}
{¬ ⋄p, p, true}
{¬ ⋄p, p, false}
{ ⋄p, ¬p, true}
{ ⋄p, ¬p, false}
{¬ ⋄p, ¬p, true}
{¬ ⋄p, ¬p, false}
σ ⊭ false
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST 30/42
ExampleLocal 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}}
• 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 31/42
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 32/42
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 33/42
Translating LTL formula into Büchi Automata
Eventuality Automaton
ExampleEventuality automaton
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST 34/42
⋄p{}
{ ⋄p, p}
{¬ ⋄p, p}{¬ ⋄p, ¬p}
{ ⋄p, ¬p}
{ ⋄p, p}{¬ ⋄p, p}
{¬ ⋄p, ¬p}{ ⋄p, ¬p}
• 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 35/42
Translating LTL formula into Büchi Automata
Composing the two automata
Example Composing the two automata
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST 36/42
({⋄p, p}, ⋄p)
({⋄p, ¬p} , ⋄p)
({¬ ⋄p, ¬p} , ⋄p) ({¬ ⋄p, p}, ⋄p)
({⋄p, p}, {})
({⋄p, ¬p} , {})
({¬ ⋄p, ¬p} , {})
({¬ ⋄p, p}, {})
• 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 37/42
Translating LTL formula into Büchi Automata
The automaton on the 2P
Final automaton for ⋄p
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST 38/42
¬pp
¬p
¬pp
true
Content• Overview
– Linear Temporal Logic– Büchi Automata
• Translating LTL formula into Büchi Automata– Local Automaton– Eventuality Automaton– Model Automata
• Conclusion and Further Study
• ReferenceTranslating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab,
KAIST 39/42
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 40/42
Further Study• Translate LTL to Büchi Automata
– 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 41/42
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 75-123. Springer-Verlag, 1989.
• Constructing Automata from Temporal Logic Formulas: A Tutorial by Pierre Wolper In Lectures on Formal Methods in Performance Analysis, vol. 2090 of LNCS, pages 261-277. Springer-Verlag, July 2001.
• From Modal Logic to Deductive Databasesby A. Thayse et al., Wiley, 1989
• Logic in Computer Science, second editionby M. Huth and M. Ryan, Cambridge press
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST 42/42
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}}
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Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST 44/42
Local Automaton
{ ㅁ⋄ p, ⋄p} { ⋄¬ ⋄p, ⋄p}
{ ㅁ⋄ p, ¬ ⋄p} { ㅁ⋄ p, ⋄p}
{ ⋄¬ ⋄p, ¬ ⋄p} { ⋄¬ ⋄p, ⋄p}
{ ㅁ⋄ p, ¬ ⋄p} { ⋄¬ ⋄p, ¬ ⋄p}
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST 45/42
Local Automaton
{ ⋄p, p} { ⋄p, p}
{¬ ⋄p, p} { ⋄p, ¬p}
{¬ ⋄p, p} { ⋄p, ¬p}
{¬ ⋄p, ¬p} {¬ ⋄p, ¬p}
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{ 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}
Local Automaton
Local Automaton
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST 47/42
{ ㅁ⋄ p, ⋄p} { ⋄¬ ⋄p, ⋄p}
{ ㅁ⋄ p, ¬ ⋄p} { ㅁ⋄ p, ⋄p}
{ ⋄¬ ⋄p, ¬ ⋄p} { ⋄¬ ⋄p, ⋄p}
{ ㅁ⋄ p, ¬ ⋄p} { ⋄¬ ⋄p, ¬ ⋄p}
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST 48/42
{ ⋄p, p} { ⋄p, p}
{¬ ⋄p, p} { ⋄p, ¬p}
{¬ ⋄p, p} { ⋄p, ¬p}
{¬ ⋄p, ¬p} {¬ ⋄p, ¬p}
Local Automaton
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST 49/42
Local Automaton
{ 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}
Eventuality Automaton• ev(φ) = {⋄¬ ⋄p, ⋄p}• 2ev(φ) = {{}, {⋄¬ ⋄p}, {⋄p},
{ ⋄¬ ⋄p,⋄p} = {{¬ ⋄¬ ⋄p, ¬ ⋄p}, {⋄¬ ⋄p,
¬ ⋄p},{¬ ⋄¬ ⋄p, ⋄p}, {⋄¬ ⋄p, ⋄p}}
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Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST 51/42
{}
⋄p
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}
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST 52/42
{}
⋄p
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}
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST 53/42
{}
⋄p
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}
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST 54/42
{}
⋄p
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}
Combining Automata
Translating LTL into Büchi Automata, Chang-Beom Choi, Provable Software Lab, KAIST 55/42
⋄¬ ⋄p
⋄¬ ⋄p, ⋄p
⋄p
{}