linear temporal logic (ltl)...overview overview5.2 introduction modelling parallel systems linear...
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Overview overview5.2
Introduction
Modelling parallel systems
Linear Time Properties
Regular Properties
Linear Temporal Logic (LTL)
syntax and semantics of LTLautomata-based LTL model checking ←−←−←−complexity of LTL model checking
Computation-Tree Logic
Equivalences and Abstraction
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LTL model checking problem ltlmc3.2-19
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LTL model checking problem ltlmc3.2-19
given: finite transition system TTT over APAPAP(without terminal states)LTL-formula ϕϕϕ over APAPAP
question: does T |= ϕT |= ϕT |= ϕ hold ?
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LTL model checking problem ltlmc3.2-19
given: finite transition system TTT over APAPAP(without terminal states)LTL-formula ϕϕϕ over APAPAP
question: does T |= ϕT |= ϕT |= ϕ hold ?
basic idea: try to refute T |= ϕT |= ϕT |= ϕ
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LTL model checking problem ltlmc3.2-19
given: finite transition system TTT over APAPAP(without terminal states)LTL-formula ϕϕϕ over APAPAP
question: does T |= ϕT |= ϕT |= ϕ hold ?
basic idea: try to refute T |= ϕT |= ϕT |= ϕ by searchingfor a path πππ in TTT s.t.
π �|= ϕπ �|= ϕπ �|= ϕ
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LTL model checking problem ltlmc3.2-19
given: finite transition system TTT over APAPAP(without terminal states)LTL-formula ϕϕϕ over APAPAP
question: does T |= ϕT |= ϕT |= ϕ hold ?
basic idea: try to refute T |= ϕT |= ϕT |= ϕ by searchingfor a path πππ in TTT s.t.
π �|= ϕπ �|= ϕπ �|= ϕ, i.e., π |= ¬ϕπ |= ¬ϕπ |= ¬ϕ
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The LTL model checking problem ltlmc3.2-19a
given: finite transition system TTT over APAPAPLTL-formula ϕϕϕ over APAPAP
question: does T |= ϕT |= ϕT |= ϕ hold ?
1. construct an NBA AAA for Words(¬ϕ)Words(¬ϕ)Words(¬ϕ)
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The LTL model checking problem ltlmc3.2-19a
given: finite transition system TTT over APAPAPLTL-formula ϕϕϕ over APAPAP
question: does T |= ϕT |= ϕT |= ϕ hold ?
1. construct an NBA AAA for Words(¬ϕ)Words(¬ϕ)Words(¬ϕ)2. search a path πππ in TTT with
trace(π) ∈Words(¬ϕ)trace(π) ∈ Words(¬ϕ)trace(π) ∈ Words(¬ϕ)
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The LTL model checking problem ltlmc3.2-19a
given: finite transition system TTT over APAPAPLTL-formula ϕϕϕ over APAPAP
question: does T |= ϕT |= ϕT |= ϕ hold ?
1. construct an NBA AAA for Words(¬ϕ)Words(¬ϕ)Words(¬ϕ)2. search a path πππ in TTT with
trace(π) ∈Words(¬ϕ)trace(π) ∈ Words(¬ϕ)trace(π) ∈ Words(¬ϕ) = Lω(A)= Lω(A)= Lω(A)
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The LTL model checking problem ltlmc3.2-19a
given: finite transition system TTT over APAPAPLTL-formula ϕϕϕ over APAPAP
question: does T |= ϕT |= ϕT |= ϕ hold ?
1. construct an NBA AAA for Words(¬ϕ)Words(¬ϕ)Words(¬ϕ)2. search a path πππ in TTT with
trace(π) ∈Words(¬ϕ)trace(π) ∈ Words(¬ϕ)trace(π) ∈ Words(¬ϕ) = Lω(A)= Lω(A)= Lω(A)↑↑↑construct the product-TS T ⊗AT ⊗AT ⊗Asearch a path in the product that meets
the acceptance condition of AAA
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Automata-based LTL model checking ltlmc3.2-18
finite transitionsystem TTT LTL formula ϕϕϕ
LTL model checking
does T |= ϕT |= ϕT |= ϕ hold ?
yes no11 / 527
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Automata-based LTL model checking ltlmc3.2-18
finite transitionsystem TTT LTL formula ϕϕϕ
NBA AAA for ¬ϕ¬ϕ¬ϕ“bad behaviors”
LTL model checking
does T |= ϕT |= ϕT |= ϕ hold ?
yes no12 / 527
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Automata-based LTL model checking ltlmc3.2-18
finite transitionsystem TTT LTL formula ϕϕϕ
NBA AAA for ¬ϕ¬ϕ¬ϕ“bad behaviors”
LTL model checking
via persistence checkingT ⊗ A |=T ⊗A |=T ⊗A |= “♦�♦�♦� no final state” ?
yes no13 / 527
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Automata-based LTL model checking ltlmc3.2-18
finite transitionsystem TTT LTL formula ϕϕϕ
NBA AAA for ¬ϕ¬ϕ¬ϕ“bad behaviors”
LTL model checking
via persistence checkingT ⊗ A |=T ⊗A |=T ⊗A |= “♦�♦�♦� no final state” ?
yes no +++ error indication14 / 527
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Safety and LTL model checking ltlmc3.2-20
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Safety and LTL model checking ltlmc3.2-20
safety property EEE LTL-formula ϕϕϕ
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Safety and LTL model checking ltlmc3.2-20
safety property EEE LTL-formula ϕϕϕ
NFA for thebad prefixes for EEEL(A) ⊆ (2AP)+L(A) ⊆ (2AP)+L(A) ⊆ (2AP)+
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Safety and LTL model checking ltlmc3.2-20
safety property EEE LTL-formula ϕϕϕ
NFA for thebad prefixes for EEEL(A) ⊆ (2AP)+L(A) ⊆ (2AP)+L(A) ⊆ (2AP)+
NBA for the“bad behaviors”
Lω(A) = Words(¬ϕ)Lω(A) = Words(¬ϕ)Lω(A) = Words(¬ϕ)
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Safety and LTL model checking ltlmc3.2-20
safety property EEE LTL-formula ϕϕϕ
NFA for thebad prefixes for EEEL(A) ⊆ (2AP)+L(A) ⊆ (2AP)+L(A) ⊆ (2AP)+
NBA for the“bad behaviors”
Lω(A) = Words(¬ϕ)Lω(A) = Words(¬ϕ)Lω(A) = Words(¬ϕ)
Tracesfin(T ) ∩ L(A) = ∅Tracesfin(T ) ∩ L(A) = ∅Tracesfin(T ) ∩ L(A) = ∅
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Safety and LTL model checking ltlmc3.2-20
safety property EEE LTL-formula ϕϕϕ
NFA for thebad prefixes for EEEL(A) ⊆ (2AP)+L(A) ⊆ (2AP)+L(A) ⊆ (2AP)+
NBA for the“bad behaviors”
Lω(A) = Words(¬ϕ)Lω(A) = Words(¬ϕ)Lω(A) = Words(¬ϕ)
Tracesfin(T ) ∩ L(A) = ∅Tracesfin(T ) ∩ L(A) = ∅Tracesfin(T ) ∩ L(A) = ∅ Traces(T ) ∩ Lω(A) = ∅Traces(T ) ∩ Lω(A) = ∅Traces(T ) ∩ Lω(A) = ∅
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Safety and LTL model checking ltlmc3.2-20
safety property EEE LTL-formula ϕϕϕ
NFA for thebad prefixes for EEEL(A) ⊆ (2AP)+L(A) ⊆ (2AP)+L(A) ⊆ (2AP)+
NBA for the“bad behaviors”
Lω(A) = Words(¬ϕ)Lω(A) = Words(¬ϕ)Lω(A) = Words(¬ϕ)
Tracesfin(T ) ∩ L(A) = ∅Tracesfin(T ) ∩ L(A) = ∅Tracesfin(T ) ∩ L(A) = ∅ Traces(T ) ∩ Lω(A) = ∅Traces(T ) ∩ Lω(A) = ∅Traces(T ) ∩ Lω(A) = ∅
invariant checkingin the productT ⊗A |= �¬FT ⊗A |= �¬FT ⊗A |= �¬F ?
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Safety and LTL model checking ltlmc3.2-20
safety property EEE LTL-formula ϕϕϕ
NFA for thebad prefixes for EEEL(A) ⊆ (2AP)+L(A) ⊆ (2AP)+L(A) ⊆ (2AP)+
NBA for the“bad behaviors”
Lω(A) = Words(¬ϕ)Lω(A) = Words(¬ϕ)Lω(A) = Words(¬ϕ)
Tracesfin(T ) ∩ L(A) = ∅Tracesfin(T ) ∩ L(A) = ∅Tracesfin(T ) ∩ L(A) = ∅ Traces(T ) ∩ Lω(A) = ∅Traces(T ) ∩ Lω(A) = ∅Traces(T ) ∩ Lω(A) = ∅
invariant checkingin the productT ⊗A |= �¬FT ⊗A |= �¬FT ⊗A |= �¬F ?
persistence checkingin the productT ⊗A |= ♦�¬FT ⊗A |= ♦�¬FT ⊗A |= ♦�¬F ?
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Safety and LTL model checking ltlmc3.2-20
safety property EEE LTL-formula ϕϕϕ
NFA for thebad prefixes for EEEL(A) ⊆ (2AP)+L(A) ⊆ (2AP)+L(A) ⊆ (2AP)+
NBA for the“bad behaviors”
Lω(A) = Words(¬ϕ)Lω(A) = Words(¬ϕ)Lω(A) = Words(¬ϕ)
Tracesfin(T ) ∩ L(A) = ∅Tracesfin(T ) ∩ L(A) = ∅Tracesfin(T ) ∩ L(A) = ∅ Traces(T ) ∩ Lω(A) = ∅Traces(T ) ∩ Lω(A) = ∅Traces(T ) ∩ Lω(A) = ∅
invariant checkingin the productT ⊗A |= �¬FT ⊗A |= �¬FT ⊗A |= �¬F ?
persistence checkingin the productT ⊗A |= ♦�¬FT ⊗A |= ♦�¬FT ⊗A |= ♦�¬F ?
error indication:π ∈ Pathsfin(T )π ∈ Pathsfin(T )π ∈ Pathsfin(T )
s.t. trace(π) ∈ L(A)trace(π) ∈ L(A)trace(π) ∈ L(A)23 / 527
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Safety and LTL model checking ltlmc3.2-20
safety property EEE LTL-formula ϕϕϕ
NFA for thebad prefixes for EEEL(A) ⊆ (2AP)+L(A) ⊆ (2AP)+L(A) ⊆ (2AP)+
NBA for the“bad behaviors”
Lω(A) = Words(¬ϕ)Lω(A) = Words(¬ϕ)Lω(A) = Words(¬ϕ)
Tracesfin(T ) ∩ L(A) = ∅Tracesfin(T ) ∩ L(A) = ∅Tracesfin(T ) ∩ L(A) = ∅ Traces(T ) ∩ Lω(A) = ∅Traces(T ) ∩ Lω(A) = ∅Traces(T ) ∩ Lω(A) = ∅
invariant checkingin the productT ⊗A |= �¬FT ⊗A |= �¬FT ⊗A |= �¬F ?
persistence checkingin the productT ⊗A |= ♦�¬FT ⊗A |= ♦�¬FT ⊗A |= ♦�¬F ?
error indication:π ∈ Pathsfin(T )π ∈ Pathsfin(T )π ∈ Pathsfin(T )
s.t. trace(π) ∈ L(A)trace(π) ∈ L(A)trace(π) ∈ L(A)
error indication:prefix of a path πππ
s.t. trace(π) ∈ Lω(A)trace(π) ∈ Lω(A)trace(π) ∈ Lω(A)24 / 527
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Safety vs LTL model checking ltlmc3.2-10
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Safety vs LTL model checking ltlmc3.2-10
T |=T |=T |= safety property EEE
iff Tracesfin(T ) ∩ L(A) = ∅Tracesfin(T ) ∩ L(A) = ∅Tracesfin(T ) ∩ L(A) = ∅
where AAA is an NFA for the bad prefixes
T |=T |=T |= LTL-formula ϕϕϕ
iff Traces(T ) ∩ Lω(A) = ∅Traces(T ) ∩ Lω(A) = ∅Traces(T ) ∩ Lω(A) = ∅
where AAA is an NBA for ¬ϕ¬ϕ¬ϕ
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Safety vs LTL model checking ltlmc3.2-10
T |=T |=T |= safety property EEE
iff Tracesfin(T ) ∩ L(A) = ∅Tracesfin(T ) ∩ L(A) = ∅Tracesfin(T ) ∩ L(A) = ∅
iff there is no path fragment 〈s0, q0〉 〈s1, q1〉 . . . 〈sn, qn〉〈s0, q0〉 〈s1, q1〉 . . . 〈sn, qn〉〈s0, q0〉 〈s1, q1〉 . . . 〈sn, qn〉in T ⊗ AT ⊗AT ⊗A s. t. qn ∈ Fqn ∈ Fqn ∈ F
T |=T |=T |= LTL-formula ϕϕϕ
iff Traces(T ) ∩ Lω(A) = ∅Traces(T ) ∩ Lω(A) = ∅Traces(T ) ∩ Lω(A) = ∅
iff there is no path 〈s0, q0〉 〈s1, q1〉 〈s2, q2〉 . . .〈s0, q0〉 〈s1, q1〉 〈s2, q2〉 . . .〈s0, q0〉 〈s1, q1〉 〈s2, q2〉 . . .in T ⊗ AT ⊗AT ⊗A s.t. qi ∈ Fqi ∈ Fqi ∈ F for infinitely many i ∈ Ni ∈ Ni ∈ N
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Safety vs LTL model checking ltlmc3.2-10
T |=T |=T |= safety property EEE
iff Tracesfin(T ) ∩ L(A) = ∅Tracesfin(T ) ∩ L(A) = ∅Tracesfin(T ) ∩ L(A) = ∅
iff there is no path fragment 〈s0, q0〉 〈s1, q1〉 . . . 〈sn, qn〉〈s0, q0〉 〈s1, q1〉 . . . 〈sn, qn〉〈s0, q0〉 〈s1, q1〉 . . . 〈sn, qn〉in T ⊗ AT ⊗AT ⊗A s. t. qn ∈ Fqn ∈ Fqn ∈ F
iff T ⊗A |= �¬FT ⊗ A |= �¬FT ⊗ A |= �¬F
T |=T |=T |= LTL-formula ϕϕϕ
iff Traces(T ) ∩ Lω(A) = ∅Traces(T ) ∩ Lω(A) = ∅Traces(T ) ∩ Lω(A) = ∅
iff there is no path 〈s0, q0〉 〈s1, q1〉 〈s2, q2〉 . . .〈s0, q0〉 〈s1, q1〉 〈s2, q2〉 . . .〈s0, q0〉 〈s1, q1〉 〈s2, q2〉 . . .in T ⊗ AT ⊗AT ⊗A s.t. qi ∈ Fqi ∈ Fqi ∈ F for infinitely many i ∈ Ni ∈ Ni ∈ N
iff T ⊗A |= ♦�¬FT ⊗ A |= ♦�¬FT ⊗ A |= ♦�¬F28 / 527
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Safety vs LTL model checking ltlmc3.2-10
T |=T |=T |= safety property EEE
iff Tracesfin(T ) ∩ L(A) = ∅Tracesfin(T ) ∩ L(A) = ∅Tracesfin(T ) ∩ L(A) = ∅
iff there is no path fragment 〈s0, q0〉 〈s1, q1〉 . . . 〈sn, qn〉〈s0, q0〉 〈s1, q1〉 . . . 〈sn, qn〉〈s0, q0〉 〈s1, q1〉 . . . 〈sn, qn〉in T ⊗ AT ⊗AT ⊗A s. t. qn ∈ Fqn ∈ Fqn ∈ F
iff T ⊗A |= �¬FT ⊗ A |= �¬FT ⊗ A |= �¬F ←−←−←− invariant checking
T |=T |=T |= LTL-formula ϕϕϕ
iff Traces(T ) ∩ Lω(A) = ∅Traces(T ) ∩ Lω(A) = ∅Traces(T ) ∩ Lω(A) = ∅
iff there is no path 〈s0, q0〉 〈s1, q1〉 〈s2, q2〉 . . .〈s0, q0〉 〈s1, q1〉 〈s2, q2〉 . . .〈s0, q0〉 〈s1, q1〉 〈s2, q2〉 . . .in T ⊗ AT ⊗AT ⊗A s.t. qi ∈ Fqi ∈ Fqi ∈ F for infinitely many i ∈ Ni ∈ Ni ∈ N
iff T ⊗A |= ♦�¬FT ⊗ A |= ♦�¬FT ⊗ A |= ♦�¬F ←−←−←− persistence checking29 / 527
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From LTL to NBA ltlmc3.2-thm-LTL-2-NBA
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From LTL to NBA ltlmc3.2-thm-LTL-2-NBA
For each LTL formula ϕϕϕ over APAPAP there is anNBA AAA over the alphabet 2AP2AP2AP such that
• Words(ϕ) = Lω(A)Words(ϕ) = Lω(A)Words(ϕ) = Lω(A)
• size(A) = O(exp(|ϕ|)
)size(A) = O
(exp(|ϕ|)
)size(A) = O
(exp(|ϕ|)
)
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From LTL to NBA ltlmc3.2-thm-LTL-2-NBA
For each LTL formula ϕϕϕ over APAPAP there is anNBA AAA over the alphabet 2AP2AP2AP such that
• Words(ϕ) = Lω(A)Words(ϕ) = Lω(A)Words(ϕ) = Lω(A)
• size(A) = O(exp(|ϕ|)
)size(A) = O
(exp(|ϕ|)
)size(A) = O
(exp(|ϕ|)
)proof: ... later ...
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NBA for LTL formulas ltlmc3.2-3
q0q0q0 q1q1q1 qFqFqF
truetrue ¬a¬a¬a Lω(A) =Lω(A) =Lω(A) = ?
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NBA for LTL formulas ltlmc3.2-3
q0q0q0 q1q1q1 qFqFqF
truetrue ¬a¬a¬a Lω(A) =Lω(A) =Lω(A) = Words(©¬a)Words(©¬a)Words(©¬a)
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NBA for LTL formulas ltlmc3.2-3
q0q0q0 q1q1q1 qFqFqF
truetrue ¬a¬a¬a Lω(A) =Lω(A) =Lω(A) = Words(©¬a)Words(©¬a)Words(©¬a)
q0q0q0 qFqFqF trueaaa
p0p0p0 pFpFpF truebbb Lω(A) =Lω(A) =Lω(A) = ?
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NBA for LTL formulas ltlmc3.2-3
q0q0q0 q1q1q1 qFqFqF
truetrue ¬a¬a¬a Lω(A) =Lω(A) =Lω(A) = Words(©¬a)Words(©¬a)Words(©¬a)
q0q0q0 qFqFqF trueaaa
p0p0p0 pFpFpF truebbbLω(A) =Lω(A) =Lω(A) = Words(a ∨ b)Words(a ∨ b)Words(a ∨ b)
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NBA for LTL formulas ltlmc3.2-3
q0q0q0 q1q1q1 qFqFqF
truetrue ¬a¬a¬a Lω(A) =Lω(A) =Lω(A) = Words(©¬a)Words(©¬a)Words(©¬a)
q0q0q0 qFqFqF trueaaa
p0p0p0 pFpFpF truebbbLω(A) =Lω(A) =Lω(A) = Words(a ∨ b)Words(a ∨ b)Words(a ∨ b)
qFqFqF q1q1q1
aaa
bbbbbb
Lω(A) =Lω(A) =Lω(A) = ?
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NBA for LTL formulas ltlmc3.2-3
q0q0q0 q1q1q1 qFqFqF
truetrue ¬a¬a¬a Lω(A) =Lω(A) =Lω(A) = Words(©¬a)Words(©¬a)Words(©¬a)
q0q0q0 qFqFqF trueaaa
p0p0p0 pFpFpF truebbbLω(A) =Lω(A) =Lω(A) = Words(a ∨ b)Words(a ∨ b)Words(a ∨ b)
qFqFqF q1q1q1
aaa
bbbbbb
Lω(A) =Lω(A) =Lω(A) = Words(�a)Words(�a)Words(�a)
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NBA for LTL formulas ltlmc3.2-4
q0q0q0 q1q1q1
¬a¬a¬a aaa
aaa
¬a¬a¬a Lω(A) =Lω(A) =Lω(A) = ?
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NBA for LTL formulas ltlmc3.2-4
q0q0q0 q1q1q1
¬a¬a¬a aaa
aaa
¬a¬a¬a Lω(A) =Lω(A) =Lω(A) = Words(�♦a)Words(�♦a)Words(�♦a)
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NBA for LTL formulas ltlmc3.2-4
q0q0q0 q1q1q1
¬a¬a¬a aaa
aaa
¬a¬a¬a Lω(A) =Lω(A) =Lω(A) = Words(�♦a)Words(�♦a)Words(�♦a)
q0q0q0 q1q1q1
¬a ∨ b¬a ∨ b¬a ∨ b ¬b¬b¬b
a ∧ ¬ba ∧ ¬ba ∧ ¬b
bbb Lω(A) =Lω(A) =Lω(A) = ?
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NBA for LTL formulas ltlmc3.2-4
q0q0q0 q1q1q1
¬a¬a¬a aaa
aaa
¬a¬a¬a Lω(A) =Lω(A) =Lω(A) = Words(�♦a)Words(�♦a)Words(�♦a)
q0q0q0 q1q1q1
¬a ∨ b¬a ∨ b¬a ∨ b ¬b¬b¬b
a ∧ ¬ba ∧ ¬ba ∧ ¬b
bbb Lω(A) =Lω(A) =Lω(A) = ?
e.g., ∅ ∅ ∅ ∅ . . . = ∅ω∅ ∅ ∅ ∅ . . . = ∅ω
∅ ∅ ∅ ∅ . . . = ∅ω
({a} {b})ω({a} {b})ω({a} {b})ω}
are accepted by AAA
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NBA for LTL formulas ltlmc3.2-4
q0q0q0 q1q1q1
¬a¬a¬a aaa
aaa
¬a¬a¬a Lω(A) =Lω(A) =Lω(A) = Words(�♦a)Words(�♦a)Words(�♦a)
q0q0q0 q1q1q1
¬a ∨ b¬a ∨ b¬a ∨ b ¬b¬b¬b
a ∧ ¬ba ∧ ¬ba ∧ ¬b
bbb Lω(A) =Lω(A) =Lω(A) = Words(�(a→ ♦b))Words(�(a→ ♦b))Words(�(a→ ♦b))
e.g., ∅ ∅ ∅ ∅ . . . = ∅ω∅ ∅ ∅ ∅ . . . = ∅ω
∅ ∅ ∅ ∅ . . . = ∅ω
({a} {b})ω({a} {b})ω({a} {b})ω}
are accepted by AAA
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NBA for LTL formula ltlmc3.2-5
q0q0q0 q1q1q1 q1q1q1
true aaa true
aaa ¬a¬a¬a
Lω(A) =Lω(A) =Lω(A) = ?
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NBA for LTL formula ltlmc3.2-5
q0q0q0 q1q1q1 q1q1q1
true aaa true
aaa ¬a¬a¬a
Lω(A) =Lω(A) =Lω(A) = Words(♦�a)Words(♦�a)Words(♦�a)
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NBA for LTL formula ltlmc3.2-5
q0q0q0 q1q1q1 q1q1q1
true aaa true
aaa ¬a¬a¬a
Lω(A) =Lω(A) =Lω(A) = Words(♦�a)Words(♦�a)Words(♦�a)
possible runs for {a}ω{a}ω{a}ω
q0 q0 q0 q0 q0 q0 ...q0 q0 q0 q0 q0 q0 ...q0 q0 q0 q0 q0 q0 ... not acceptingq0 q1 q1 q1 q1 q1 ...q0 q1 q1 q1 q1 q1 ...q0 q1 q1 q1 q1 q1 ... acceptingq0 q0 q1 q1 q1 q1 ...q0 q0 q1 q1 q1 q1 ...q0 q0 q1 q1 q1 q1 ... acceptingq0 q0 q0 q1 q1 q1 ...q0 q0 q0 q1 q1 q1 ...q0 q0 q0 q1 q1 q1 ... accepting
...
...
...
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NFA and NBA for safety properties ltlmc3.2-6
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NFA and NBA for safety properties ltlmc3.2-6
Let AAA be an NFA for the language of all bad prefixesfor a safety property EEE .
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NFA and NBA for safety properties ltlmc3.2-6
Let AAA be an NFA for the language of all bad prefixesfor a safety property EEE . Then:
Lω(A) = E =(2AP
)ω \ ELω(A) = E =(2AP
)ω \ ELω(A) = E =(2AP
)ω \ E
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NFA and NBA for safety properties ltlmc3.2-6
Let AAA be an NFA for the language of all bad prefixesfor a safety property EEE . Then:
Lω(A) = E =(2AP
)ω \ ELω(A) = E =(2AP
)ω \ ELω(A) = E =(2AP
)ω \ E
Example: EEE === “never aaa twice in a row”
q0q0q0 q1q1q1 q2q2q2aaa
true
aaa
true55 / 527
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NFA and NBA for safety properties ltlmc3.2-6
Let AAA be an NFA for the language of all bad prefixesfor a safety property EEE . Then:
Lω(A) = E =(2AP
)ω \ ELω(A) = E =(2AP
)ω \ ELω(A) = E =(2AP
)ω \ E = Words(¬ϕ)= Words(¬ϕ)= Words(¬ϕ)
Example: EEE === “never aaa twice in a row”
q0q0q0 q1q1q1 q2q2q2aaa
true
aaa
true
ϕ = �(a→©¬a)ϕ = �(a→©¬a)ϕ = �(a→©¬a)
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NFA and NBA for safety properties ltlmc3.2-6
Let AAA be an NFA for the language of all bad prefixesfor a safety property EEE . Then:
Lω(A) = E =(2AP
)ω \ ELω(A) = E =(2AP
)ω \ ELω(A) = E =(2AP
)ω \ E = Words(¬ϕ)= Words(¬ϕ)= Words(¬ϕ)
wrong, if L(A) =L(A) =L(A) = language of minimal bad prefixes
Example: EEE === “never aaa twice in a row”
q0q0q0 q1q1q1 q2q2q2aaa
true
aaa
true
ϕ = �(a→©¬a)ϕ = �(a→©¬a)ϕ = �(a→©¬a)
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NFA and NBA for safety properties ltlmc3.2-6
Let AAA be an NFA for the language of all bad prefixesfor a safety property EEE . Then:
Lω(A) = E =(2AP
)ω \ ELω(A) = E =(2AP
)ω \ ELω(A) = E =(2AP
)ω \ E = Words(¬ϕ)= Words(¬ϕ)= Words(¬ϕ)
wrong, if L(A) =L(A) =L(A) = language of minimal bad prefixes
Example: EEE === “never aaa twice in a row”
q0q0q0 q1q1q1 q2q2q2aaa
¬a¬a¬a
aaa q3q3q3
true
true Lω(A) = ∅Lω(A) = ∅Lω(A) = ∅
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NFA and NBA for safety properties ltlmc3.2-6
Let AAA be an NFA for the language of all bad prefixesfor a safety property EEE . Then:
Lω(A) = E =(2AP
)ω \ ELω(A) = E =(2AP
)ω \ ELω(A) = E =(2AP
)ω \ E = Words(¬ϕ)= Words(¬ϕ)= Words(¬ϕ)
wrong, if L(A) =L(A) =L(A) = language of minimal bad prefixeseven if AAA is a non-blocking DFA
Example: EEE === “never aaa twice in a row”
q0q0q0 q1q1q1 q2q2q2aaa
¬a¬a¬a
aaa q3q3q3
true
true Lω(A) = ∅Lω(A) = ∅Lω(A) = ∅
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NFA and NBA for safety properties ltlmc3.2-6
Let AAA be an NFA for the language of all bad prefixesfor a safety property EEE . Then:
Lω(A) = E =(2AP
)ω \ ELω(A) = E =(2AP
)ω \ ELω(A) = E =(2AP
)ω \ E = Words(¬ϕ)= Words(¬ϕ)= Words(¬ϕ)
wrong, if L(A) =L(A) =L(A) = language of minimal bad prefixeseven if AAA is a non-blocking DFA
Example: EEE === “never aaa twice in a row”
q0q0q0 q1q1q1 q2q2q2aaa
¬a¬a¬a
aaa
¬a¬a¬aq3q3q3
true
true Lω(A) = ∅Lω(A) = ∅Lω(A) = ∅
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LTL model checking ltlmc3.2-2a
finite transitionsystem TTT
LTL model checking
persistence checkingT ⊗A |= ♦�¬FT ⊗ A |= ♦�¬FT ⊗ A |= ♦�¬F ?
LTL formula ϕϕϕ
NBA AAA for ¬ϕ¬ϕ¬ϕ
yes no +++ counterexample61 / 527
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LTL model checking ltlmc3.2-2a
finite transitionsystem TTT
LTL model checking
persistence checkingT ⊗A |= ♦�¬FT ⊗ A |= ♦�¬FT ⊗ A |= ♦�¬F ?
LTL formula ϕϕϕ
NBA AAA for ¬ϕ¬ϕ¬ϕ
yes no +++ counterexample
later
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LTL model checking ltlmc3.2-38
given: finite TS TTT , LTL-formula ϕϕϕquestion: does T |= ϕT |= ϕT |= ϕ hold ?
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LTL model checking ltlmc3.2-38
given: finite TS TTT , LTL-formula ϕϕϕquestion: does T |= ϕT |= ϕT |= ϕ hold ?
construct an NBA AAA for ¬ϕ¬ϕ¬ϕ and the product T ⊗AT ⊗AT ⊗Acheck whether T ⊗ A |= ♦�¬FT ⊗ A |= ♦�¬FT ⊗A |= ♦�¬F ←−←−←− persistence
checkingnested DFS
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LTL model checking ltlmc3.2-38
given: finite TS TTT , LTL-formula ϕϕϕquestion: does T |= ϕT |= ϕT |= ϕ hold ?
construct an NBA AAA for ¬ϕ¬ϕ¬ϕ and the product T ⊗AT ⊗AT ⊗Acheck whether T ⊗ A |= ♦�¬FT ⊗ A |= ♦�¬FT ⊗A |= ♦�¬F ←−←−←− persistence
checkingnested DFS
IF T ⊗A |= ♦�¬FT ⊗ A |= ♦�¬FT ⊗A |= ♦�¬FTHEN return “yes”ELSE compute a counterexample
〈s0, p0〉 . . . 〈sn, pn〉 . . . 〈sn, pn〉〈s0, p0〉 . . . 〈sn, pn〉 . . . 〈sn, pn〉〈s0, p0〉 . . . 〈sn, pn〉 . . . 〈sn, pn〉for T ⊗AT ⊗AT ⊗A and ♦�¬F♦�¬F♦�¬F
return “no” and s0 . . . sn . . . sns0 . . . sn . . . sns0 . . . sn . . . sn
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Complexity of LTL model checking ltlmc3.2-38
given: finite TS TTT , LTL-formula ϕϕϕquestion: does T |= ϕT |= ϕT |= ϕ hold ?
����������������������������������construct an NBA AAA for ¬ϕ¬ϕ¬ϕ and the product T ⊗ AT ⊗AT ⊗Acheck whether T ⊗ A |= ♦�¬FT ⊗ A |= ♦�¬FT ⊗A |= ♦�¬F ←−←−←− persistence
checkingnested DFS
IF T ⊗A |= ♦�¬FT ⊗ A |= ♦�¬FT ⊗A |= ♦�¬FTHEN return “yes”ELSE compute a counterexample
〈s0, p0〉 . . . 〈sn, pn〉 . . . 〈sn, pn〉〈s0, p0〉 . . . 〈sn, pn〉 . . . 〈sn, pn〉〈s0, p0〉 . . . 〈sn, pn〉 . . . 〈sn, pn〉for T ⊗AT ⊗AT ⊗A and ♦�¬F♦�¬F♦�¬F
return “no” and s0 . . . sn . . . sns0 . . . sn . . . sns0 . . . sn . . . sn
time complexity: O(size(T ) · size(A))O(size(T ) · size(A))O(size(T ) · size(A))86 / 527
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LTL model checking ltlmc3.2-2
finite transitionsystem TTT
LTL model checking
persistence checkingT ⊗A |= ♦�¬FT ⊗ A |= ♦�¬FT ⊗ A |= ♦�¬F ?
LTL formula ϕϕϕ
NBA AAA for ¬ϕ¬ϕ¬ϕ
yes no +++ counterexample87 / 527
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LTL model checking ltlmc3.2-2
finite transitionsystem TTT
LTL model checking
persistence checkingT ⊗A |= ♦�¬FT ⊗ A |= ♦�¬FT ⊗ A |= ♦�¬F ?
LTL formula ϕϕϕ
NBA AAA for ¬ϕ¬ϕ¬ϕ
yes no +++ counterexample88 / 527
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From LTL to NBA ltlmc3.2-46
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From LTL to NBA ltlmc3.2-46
For each LTL formula ϕϕϕ there is an NBA AAA s.t.Lω(A) = Words(ϕ)Lω(A) = Words(ϕ)Lω(A) = Words(ϕ)
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From LTL to NBA ltlmc3.2-46
For each LTL formula ϕϕϕ there is an NBA AAA s.t.Lω(A) = Words(ϕ)Lω(A) = Words(ϕ)Lω(A) = Words(ϕ)
LTL formula ϕϕϕ
NBA AAA s.t.Lω(A) = Words(ϕ)Lω(A) = Words(ϕ)Lω(A) = Words(ϕ)
nondeterministicBuchi automaton
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From LTL to NBA ltlmc3.2-46
For each LTL formula ϕϕϕ there is an NBA AAA s.t.Lω(A) = Words(ϕ)Lω(A) = Words(ϕ)Lω(A) = Words(ϕ)
LTL formula ϕϕϕ
GNBA GGG s.t.Lω(G) = Words(ϕ)Lω(G) = Words(ϕ)Lω(G) = Words(ϕ)
NBA AAA s.t.Lω(A) = Lω(G)Lω(A) = Lω(G)Lω(A) = Lω(G)
generalized NBAseveral acceptance sets
nondeterministicBuchi automaton111 acceptance set
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From LTL to NBA ltlmc3.2-46
For each LTL formula ϕϕϕ there is an NBA AAA s.t.Lω(A) = Words(ϕ)Lω(A) = Words(ϕ)Lω(A) = Words(ϕ)
LTL formula ϕϕϕ
GNBA GGG s.t.Lω(G) = Words(ϕ)Lω(G) = Words(ϕ)Lω(G) = Words(ϕ)
NBA AAA s.t.Lω(A) = Lω(G)Lω(A) = Lω(G)Lω(A) = Lω(G)
generalized NBAkkk acceptance sets
nondeterministicBuchi automaton111 acceptance set
kkk copies of GGG
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Encoding of LTL semantics in a GNBA ltlmc3.2-39
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Encoding of LTL semantics in a GNBA ltlmc3.2-39
idea: encode the semantics of the operators appearingin ϕϕϕ by appropriate components of the GNBA GGG
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Encoding of LTL semantics in a GNBA ltlmc3.2-39
idea: encode the semantics of the operators appearingin ϕϕϕ by appropriate components of the GNBA GGG
semantics of ... encoding
propositional logictruetruetrue, ¬¬¬, ∧∧∧next©©©until UUU
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Encoding of LTL semantics in a GNBA ltlmc3.2-39
idea: encode the semantics of the operators appearingin ϕϕϕ by appropriate components of the GNBA GGG
semantics of ... encoding
propositional logictruetruetrue, ¬¬¬, ∧∧∧ in the states
next©©© in the transition relation
until UUU via expansion law
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Encoding of LTL semantics in a GNBA ltlmc3.2-39
idea: encode the semantics of the operators appearingin ϕϕϕ by appropriate components of the GNBA GGG
semantics of ... encoding
propositional logictruetruetrue, ¬¬¬, ∧∧∧ in the states
next©©© in the transition relation
until UUU via expansion law
ψ1 Uψ2ψ1 Uψ2ψ1 Uψ2 ≡≡≡ ψ2 ∨ (ψ1 ∧©(ψ1 Uψ2))ψ2 ∨ (ψ1 ∧©(ψ1 Uψ2))ψ2 ∨ (ψ1 ∧©(ψ1 Uψ2))↖↖↖↗↗↗ ↑↑↑
encoded inthe states
encoded in thetransition relation
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Encoding of LTL semantics in a GNBA ltlmc3.2-39
idea: encode the semantics of the operators appearingin ϕϕϕ by appropriate components of the GNBA GGG
semantics of ... encoding
propositional logictruetruetrue, ¬¬¬, ∧∧∧ in the states
next©©© in the transition relation
until UUU expansion law, least fixed point
ψ1 Uψ2ψ1 Uψ2ψ1 Uψ2 ≡≡≡ ψ2 ∨ (ψ1 ∧©(ψ1 Uψ2))ψ2 ∨ (ψ1 ∧©(ψ1 Uψ2))ψ2 ∨ (ψ1 ∧©(ψ1 Uψ2)) ↑↑↑↖↖↖↗↗↗ ↑↑↑
encoded inthe states
encoded in thetransition relation
acceptancecondition
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LTL ��� GNBA ltlmc3.2-46a
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LTL ��� GNBA ltlmc3.2-46a
LTL formula ϕϕϕ��� GNBA GGG for Words(ϕ)Words(ϕ)Words(ϕ)
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LTL ��� GNBA ltlmc3.2-46a
LTL formula ϕϕϕ��� GNBA GGG for Words(ϕ)Words(ϕ)Words(ϕ)
states of GGG === (certain) sets of subformulas of ϕϕϕ
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LTL ��� GNBA ltlmc3.2-46a
LTL formula ϕϕϕ��� GNBA GGG for Words(ϕ)Words(ϕ)Words(ϕ)
states of GGG === (certain) sets of subformulas of ϕϕϕs.t. each word σ = A0 A1 A2... ∈ Words(ϕ)σ = A0 A1 A2... ∈Words(ϕ)σ = A0 A1 A2... ∈ Words(ϕ) can beextended to an accepting run B0 B1 B2 . . .B0 B1 B2 . . .B0 B1 B2 . . . in GGG
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LTL ��� GNBA ltlmc3.2-46a
LTL formula ϕϕϕ��� GNBA GGG for Words(ϕ)Words(ϕ)Words(ϕ)
states of GGG === (certain) sets of subformulas of ϕϕϕs.t. each word σ = A0 A1 A2... ∈ Words(ϕ)σ = A0 A1 A2... ∈Words(ϕ)σ = A0 A1 A2... ∈ Words(ϕ) can beextended to an accepting run B0 B1 B2 . . .B0 B1 B2 . . .B0 B1 B2 . . . in GGG
A0A0A0 A1A1A1 A2A2A2 A3A3A3 ......... ∈ Words(ϕ)∈ Words(ϕ)∈Words(ϕ)
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LTL ��� GNBA ltlmc3.2-46a
LTL formula ϕϕϕ��� GNBA GGG for Words(ϕ)Words(ϕ)Words(ϕ)
states of GGG === (certain) sets of subformulas of ϕϕϕs.t. each word σ = A0 A1 A2... ∈ Words(ϕ)σ = A0 A1 A2... ∈Words(ϕ)σ = A0 A1 A2... ∈ Words(ϕ) can beextended to an accepting run B0 B1 B2 . . .B0 B1 B2 . . .B0 B1 B2 . . . in GGG
A0A0A0 A1A1A1 A2A2A2 A3A3A3 ......... ∈ Words(ϕ)∈ Words(ϕ)∈Words(ϕ)↓↓↓ ↓↓↓ ↓↓↓ ↓↓↓B0B0B0 B1B1B1 B2B2B2 B3B3B3 ......... accepting run
where Bi ={ψ ∈ cl(ϕ) : AiAi+1Ai+2... |= ψ
}Bi =
{ψ ∈ cl(ϕ) : AiAi+1Ai+2... |= ψ
}Bi =
{ψ ∈ cl(ϕ) : AiAi+1Ai+2... |= ψ
}
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LTL ��� GNBA ltlmc3.2-46a
LTL formula ϕϕϕ��� GNBA GGG for Words(ϕ)Words(ϕ)Words(ϕ)
states of GGG === (certain) sets of subformulas of ϕϕϕs.t. each word σ = A0 A1 A2... ∈ Words(ϕ)σ = A0 A1 A2... ∈Words(ϕ)σ = A0 A1 A2... ∈ Words(ϕ) can beextended to an accepting run B0 B1 B2 . . .B0 B1 B2 . . .B0 B1 B2 . . . in GGG
A0A0A0 A1A1A1 A2A2A2 A3A3A3 ......... ∈ Words(ϕ)∈ Words(ϕ)∈Words(ϕ)↓↓↓ ↓↓↓ ↓↓↓ ↓↓↓B0B0B0 B1B1B1 B2B2B2 B3B3B3 ......... accepting run
where Bi ={ψ ∈ cl(ϕ) : AiAi+1Ai+2... |= ψ
}Bi =
{ψ ∈ cl(ϕ) : AiAi+1Ai+2... |= ψ
}Bi =
{ψ ∈ cl(ϕ) : AiAi+1Ai+2... |= ψ
}���set of subformulas of ϕϕϕ and their negations
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LTL ��� GNBA ltlmc3.2-47
LTL formula ϕϕϕ��� GNBA GGG for Words(ϕ)Words(ϕ)Words(ϕ)
states of GGG === (certain) sets of subformulas of ϕϕϕs.t. each word σ = A0 A1 A2... ∈ Words(ϕ)σ = A0 A1 A2... ∈Words(ϕ)σ = A0 A1 A2... ∈ Words(ϕ) can beextended to an accepting run B0 B1 B2 . . .B0 B1 B2 . . .B0 B1 B2 . . . in GGG
Example: ϕ = a U(¬a ∧ b)ϕ = a U(¬a ∧ b)ϕ = a U(¬a ∧ b)
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LTL ��� GNBA ltlmc3.2-47
LTL formula ϕϕϕ��� GNBA GGG for Words(ϕ)Words(ϕ)Words(ϕ)
states of GGG === (certain) sets of subformulas of ϕϕϕs.t. each word σ = A0 A1 A2... ∈ Words(ϕ)σ = A0 A1 A2... ∈Words(ϕ)σ = A0 A1 A2... ∈ Words(ϕ) can beextended to an accepting run B0 B1 B2 . . .B0 B1 B2 . . .B0 B1 B2 . . . in GGG
Example: ϕ = a U(¬a ∧ b)ϕ = a U(¬a ∧ b)ϕ = a U(¬a ∧ b)
{a} {a} {a, b} {b} ∅ ∅ . . . |= ϕ{a} {a} {a, b} {b} ∅ ∅ . . . |= ϕ{a} {a} {a, b} {b} ∅ ∅ . . . |= ϕ
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LTL ��� GNBA ltlmc3.2-47
LTL formula ϕϕϕ��� GNBA GGG for Words(ϕ)Words(ϕ)Words(ϕ)
states of GGG === (certain) sets of subformulas of ϕϕϕs.t. each word σ = A0 A1 A2... ∈ Words(ϕ)σ = A0 A1 A2... ∈Words(ϕ)σ = A0 A1 A2... ∈ Words(ϕ) can beextended to an accepting run B0 B1 B2 . . .B0 B1 B2 . . .B0 B1 B2 . . . in GGG
Example: ϕ = a U(¬a ∧ b)ϕ = a U(¬a ∧ b)ϕ = a U(¬a ∧ b)
{a} {a} {a, b} {b} ∅ ∅ . . . |= ϕ↓ ↓ ↓ ↓ ↓ ↓B0 B1 B2 B3 B4 B5
{a} {a} {a, b} {b} ∅ ∅ . . . |= ϕ↓ ↓ ↓ ↓ ↓ ↓B0 B1 B2 B3 B4 B5
{a} {a} {a, b} {b} ∅ ∅ . . . |= ϕ↓ ↓ ↓ ↓ ↓ ↓B0 B1 B2 B3 B4 B5
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LTL ��� GNBA ltlmc3.2-47
LTL formula ϕϕϕ��� GNBA GGG for Words(ϕ)Words(ϕ)Words(ϕ)
states of GGG === (certain) sets of subformulas of ϕϕϕs.t. each word σ = A0 A1 A2... ∈ Words(ϕ)σ = A0 A1 A2... ∈Words(ϕ)σ = A0 A1 A2... ∈ Words(ϕ) can beextended to an accepting run B0 B1 B2 . . .B0 B1 B2 . . .B0 B1 B2 . . . in GGG
Example: ϕ = a U(¬a ∧ b)ϕ = a U(¬a ∧ b)ϕ = a U(¬a ∧ b) ψ = ¬a ∧ bψ = ¬a ∧ bψ = ¬a ∧ b
{a} {a} {a, b} {b} ∅ ∅ . . . |= ϕ↓ ↓ ↓ ↓ ↓ ↓B0 B1 B2 B3 B4 B5
{a} {a} {a, b} {b} ∅ ∅ . . . |= ϕ↓ ↓ ↓ ↓ ↓ ↓B0 B1 B2 B3 B4 B5
{a} {a} {a, b} {b} ∅ ∅ . . . |= ϕ↓ ↓ ↓ ↓ ↓ ↓B0 B1 B2 B3 B4 B5
where the BiBiBi ’s are subsets of{a,¬a, b,¬b, ψ,¬ψ, ϕ,¬ϕ}{a,¬a, b,¬b, ψ,¬ψ, ϕ,¬ϕ}{a,¬a, b,¬b, ψ,¬ψ, ϕ,¬ϕ}
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LTL ��� GNBA ltlmc3.2-47
LTL formula ϕϕϕ��� GNBA GGG for Words(ϕ)Words(ϕ)Words(ϕ)
states of GGG === (certain) sets of subformulas of ϕϕϕs.t. each word σ = A0 A1 A2... ∈ Words(ϕ)σ = A0 A1 A2... ∈Words(ϕ)σ = A0 A1 A2... ∈ Words(ϕ) can beextended to an accepting run B0 B1 B2 . . .B0 B1 B2 . . .B0 B1 B2 . . . in GGG
Example: ϕ = a U(¬a ∧ b)ϕ = a U(¬a ∧ b)ϕ = a U(¬a ∧ b) ψ = ¬a ∧ bψ = ¬a ∧ bψ = ¬a ∧ b
{a} {a} {a, b} {b} ∅ ∅ . . . |= ϕ↓a¬b¬ψϕ
{a} {a} {a, b} {b} ∅ ∅ . . . |= ϕ
↓a¬b¬ψϕ
{a} {a} {a, b} {b} ∅ ∅ . . . |= ϕ↓a¬b¬ψϕ
just for better readability:tuple rather than set notation
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LTL ��� GNBA ltlmc3.2-47
LTL formula ϕϕϕ��� GNBA GGG for Words(ϕ)Words(ϕ)Words(ϕ)
states of GGG === (certain) sets of subformulas of ϕϕϕs.t. each word σ = A0 A1 A2... ∈ Words(ϕ)σ = A0 A1 A2... ∈Words(ϕ)σ = A0 A1 A2... ∈ Words(ϕ) can beextended to an accepting run B0 B1 B2 . . .B0 B1 B2 . . .B0 B1 B2 . . . in GGG
Example: ϕ = a U(¬a ∧ b)ϕ = a U(¬a ∧ b)ϕ = a U(¬a ∧ b) ψ = ¬a ∧ bψ = ¬a ∧ bψ = ¬a ∧ b
{a} {a} {a, b} {b} ∅ ∅ . . . |= ϕ↓ ↓a¬b¬ψϕ
a¬b¬ψϕ
{a} {a} {a, b} {b} ∅ ∅ . . . |= ϕ
↓ ↓a¬b¬ψϕ
a¬b¬ψϕ
{a} {a} {a, b} {b} ∅ ∅ . . . |= ϕ↓ ↓a¬b¬ψϕ
a¬b¬ψϕ
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LTL ��� GNBA ltlmc3.2-47
LTL formula ϕϕϕ��� GNBA GGG for Words(ϕ)Words(ϕ)Words(ϕ)
states of GGG === (certain) sets of subformulas of ϕϕϕs.t. each word σ = A0 A1 A2... ∈ Words(ϕ)σ = A0 A1 A2... ∈Words(ϕ)σ = A0 A1 A2... ∈ Words(ϕ) can beextended to an accepting run B0 B1 B2 . . .B0 B1 B2 . . .B0 B1 B2 . . . in GGG
Example: ϕ = a U(¬a ∧ b)ϕ = a U(¬a ∧ b)ϕ = a U(¬a ∧ b) ψ = ¬a ∧ bψ = ¬a ∧ bψ = ¬a ∧ b
{a} {a} {a, b} {b} ∅ ∅ . . . |= ϕ↓ ↓ ↓a¬b¬ψϕ
a¬b¬ψϕ
ab¬ψϕ
{a} {a} {a, b} {b} ∅ ∅ . . . |= ϕ
↓ ↓ ↓a¬b¬ψϕ
a¬b¬ψϕ
ab¬ψϕ
{a} {a} {a, b} {b} ∅ ∅ . . . |= ϕ↓ ↓ ↓a¬b¬ψϕ
a¬b¬ψϕ
ab¬ψϕ
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LTL ��� GNBA ltlmc3.2-47
LTL formula ϕϕϕ��� GNBA GGG for Words(ϕ)Words(ϕ)Words(ϕ)
states of GGG === (certain) sets of subformulas of ϕϕϕs.t. each word σ = A0 A1 A2... ∈ Words(ϕ)σ = A0 A1 A2... ∈Words(ϕ)σ = A0 A1 A2... ∈ Words(ϕ) can beextended to an accepting run B0 B1 B2 . . .B0 B1 B2 . . .B0 B1 B2 . . . in GGG
Example: ϕ = a U(¬a ∧ b)ϕ = a U(¬a ∧ b)ϕ = a U(¬a ∧ b) ψ = ¬a ∧ bψ = ¬a ∧ bψ = ¬a ∧ b
{a} {a} {a, b} {b} ∅ ∅ . . . |= ϕ↓ ↓ ↓ ↓a¬b¬ψϕ
a¬b¬ψϕ
ab¬ψϕ
¬abψϕ
{a} {a} {a, b} {b} ∅ ∅ . . . |= ϕ
↓ ↓ ↓ ↓a¬b¬ψϕ
a¬b¬ψϕ
ab¬ψϕ
¬abψϕ
{a} {a} {a, b} {b} ∅ ∅ . . . |= ϕ↓ ↓ ↓ ↓a¬b¬ψϕ
a¬b¬ψϕ
ab¬ψϕ
¬abψϕ
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LTL ��� GNBA ltlmc3.2-47
LTL formula ϕϕϕ��� GNBA GGG for Words(ϕ)Words(ϕ)Words(ϕ)
states of GGG === (certain) sets of subformulas of ϕϕϕs.t. each word σ = A0 A1 A2... ∈ Words(ϕ)σ = A0 A1 A2... ∈Words(ϕ)σ = A0 A1 A2... ∈ Words(ϕ) can beextended to an accepting run B0 B1 B2 . . .B0 B1 B2 . . .B0 B1 B2 . . . in GGG
Example: ϕ = a U(¬a ∧ b)ϕ = a U(¬a ∧ b)ϕ = a U(¬a ∧ b) ψ = ¬a ∧ bψ = ¬a ∧ bψ = ¬a ∧ b
{a} {a} {a, b} {b} ∅ ∅ . . . |= ϕ↓ ↓ ↓ ↓ ↓a¬b¬ψϕ
a¬b¬ψϕ
ab¬ψϕ
¬abψϕ
¬a¬b¬ψ¬ϕ
{a} {a} {a, b} {b} ∅ ∅ . . . |= ϕ
↓ ↓ ↓ ↓ ↓a¬b¬ψϕ
a¬b¬ψϕ
ab¬ψϕ
¬abψϕ
¬a¬b¬ψ¬ϕ
{a} {a} {a, b} {b} ∅ ∅ . . . |= ϕ↓ ↓ ↓ ↓ ↓a¬b¬ψϕ
a¬b¬ψϕ
ab¬ψϕ
¬abψϕ
¬a¬b¬ψ¬ϕ
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LTL ��� GNBA ltlmc3.2-47
LTL formula ϕϕϕ��� GNBA GGG for Words(ϕ)Words(ϕ)Words(ϕ)
states of GGG === (certain) sets of subformulas of ϕϕϕs.t. each word σ = A0 A1 A2... ∈ Words(ϕ)σ = A0 A1 A2... ∈Words(ϕ)σ = A0 A1 A2... ∈ Words(ϕ) can beextended to an accepting run B0 B1 B2 . . .B0 B1 B2 . . .B0 B1 B2 . . . in GGG
Example: ϕ = a U(¬a ∧ b)ϕ = a U(¬a ∧ b)ϕ = a U(¬a ∧ b) ψ = ¬a ∧ bψ = ¬a ∧ bψ = ¬a ∧ b
{a} {a} {a, b} {b} ∅ ∅ . . . |= ϕ↓ ↓ ↓ ↓ ↓ ↓a¬b¬ψϕ
a¬b¬ψϕ
ab¬ψϕ
¬abψϕ
¬a¬b¬ψ¬ϕ
¬a¬b¬ψ¬ϕ
. . .. . .. . .
{a} {a} {a, b} {b} ∅ ∅ . . . |= ϕ
↓ ↓ ↓ ↓ ↓ ↓a¬b¬ψϕ
a¬b¬ψϕ
ab¬ψϕ
¬abψϕ
¬a¬b¬ψ¬ϕ
¬a¬b¬ψ¬ϕ
. . .. . .. . .
{a} {a} {a, b} {b} ∅ ∅ . . . |= ϕ↓ ↓ ↓ ↓ ↓ ↓a¬b¬ψϕ
a¬b¬ψϕ
ab¬ψϕ
¬abψϕ
¬a¬b¬ψ¬ϕ
¬a¬b¬ψ¬ϕ
. . .. . .. . .
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Closure of LTL formulas ltlmc3.2-48
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Closure of LTL formulas ltlmc3.2-48
Let ϕϕϕ be an LTL formula. Then:
subf (ϕ)subf (ϕ)subf (ϕ)def=def=def= set of all subformulas of ϕϕϕ
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Closure of LTL formulas ltlmc3.2-48
Let ϕϕϕ be an LTL formula. Then:
subf (ϕ)subf (ϕ)subf (ϕ)def=def=def= set of all subformulas of ϕϕϕ
cl(ϕ)cl(ϕ)cl(ϕ)def=def=def= subf (ϕ) ∪ {¬ψ : ψ ∈ subf (ϕ)}subf (ϕ) ∪ {¬ψ : ψ ∈ subf (ϕ)}subf (ϕ) ∪ {¬ψ : ψ ∈ subf (ϕ)}
where ψψψ and ¬¬ψ¬¬ψ¬¬ψ are identified
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Closure of LTL formulas ltlmc3.2-48
Let ϕϕϕ be an LTL formula. Then:
subf (ϕ)subf (ϕ)subf (ϕ)def=def=def= set of all subformulas of ϕϕϕ
cl(ϕ)cl(ϕ)cl(ϕ)def=def=def= subf (ϕ) ∪ {¬ψ : ψ ∈ subf (ϕ)}subf (ϕ) ∪ {¬ψ : ψ ∈ subf (ϕ)}subf (ϕ) ∪ {¬ψ : ψ ∈ subf (ϕ)}
where ψψψ and ¬¬ψ¬¬ψ¬¬ψ are identified
Example: if ϕ = a U(¬a ∧ b)ϕ = a U(¬a ∧ b)ϕ = a U(¬a ∧ b) then
cl(ϕ) = {a, b,¬a ∧ b, ϕ} ∪ {¬a,¬b,¬(¬a ∧ b),¬ϕ}cl(ϕ) = {a, b,¬a ∧ b, ϕ} ∪ {¬a,¬b,¬(¬a ∧ b),¬ϕ}cl(ϕ) = {a, b,¬a ∧ b, ϕ} ∪ {¬a,¬b,¬(¬a ∧ b),¬ϕ}
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Closure of LTL formulas ltlmc3.2-48
Let ϕϕϕ be an LTL formula. Then:
subf (ϕ)subf (ϕ)subf (ϕ)def=def=def= set of all subformulas of ϕϕϕ
cl(ϕ)cl(ϕ)cl(ϕ)def=def=def= subf (ϕ) ∪ {¬ψ : ψ ∈ subf (ϕ)}subf (ϕ) ∪ {¬ψ : ψ ∈ subf (ϕ)}subf (ϕ) ∪ {¬ψ : ψ ∈ subf (ϕ)}
where ψψψ and ¬¬ψ¬¬ψ¬¬ψ are identified
Example: if ϕ = a U(¬a ∧ b)ϕ = a U(¬a ∧ b)ϕ = a U(¬a ∧ b) then
cl(ϕ) = {a, b,¬a ∧ b, ϕ} ∪ {¬a,¬b,¬(¬a ∧ b),¬ϕ}cl(ϕ) = {a, b,¬a ∧ b, ϕ} ∪ {¬a,¬b,¬(¬a ∧ b),¬ϕ}cl(ϕ) = {a, b,¬a ∧ b, ϕ} ∪ {¬a,¬b,¬(¬a ∧ b),¬ϕ}
Example: if ϕ′ = �aϕ′ = �aϕ′ = �a
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Closure of LTL formulas ltlmc3.2-48
Let ϕϕϕ be an LTL formula. Then:
subf (ϕ)subf (ϕ)subf (ϕ)def=def=def= set of all subformulas of ϕϕϕ
cl(ϕ)cl(ϕ)cl(ϕ)def=def=def= subf (ϕ) ∪ {¬ψ : ψ ∈ subf (ϕ)}subf (ϕ) ∪ {¬ψ : ψ ∈ subf (ϕ)}subf (ϕ) ∪ {¬ψ : ψ ∈ subf (ϕ)}
where ψψψ and ¬¬ψ¬¬ψ¬¬ψ are identified
Example: if ϕ = a U(¬a ∧ b)ϕ = a U(¬a ∧ b)ϕ = a U(¬a ∧ b) then
cl(ϕ) = {a, b,¬a ∧ b, ϕ} ∪ {¬a,¬b,¬(¬a ∧ b),¬ϕ}cl(ϕ) = {a, b,¬a ∧ b, ϕ} ∪ {¬a,¬b,¬(¬a ∧ b),¬ϕ}cl(ϕ) = {a, b,¬a ∧ b, ϕ} ∪ {¬a,¬b,¬(¬a ∧ b),¬ϕ}
Example: if ϕ′ = �aϕ′ = �aϕ′ = �a = ¬♦¬a = ¬(true U¬a)= ¬♦¬a = ¬(true U¬a)= ¬♦¬a = ¬(true U¬a)
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Closure of LTL formulas ltlmc3.2-48
Let ϕϕϕ be an LTL formula. Then:
subf (ϕ)subf (ϕ)subf (ϕ)def=def=def= set of all subformulas of ϕϕϕ
cl(ϕ)cl(ϕ)cl(ϕ)def=def=def= subf (ϕ) ∪ {¬ψ : ψ ∈ subf (ϕ)}subf (ϕ) ∪ {¬ψ : ψ ∈ subf (ϕ)}subf (ϕ) ∪ {¬ψ : ψ ∈ subf (ϕ)}
where ψψψ and ¬¬ψ¬¬ψ¬¬ψ are identified
Example: if ϕ = a U(¬a ∧ b)ϕ = a U(¬a ∧ b)ϕ = a U(¬a ∧ b) then
cl(ϕ) = {a, b,¬a ∧ b, ϕ} ∪ {¬a,¬b,¬(¬a ∧ b),¬ϕ}cl(ϕ) = {a, b,¬a ∧ b, ϕ} ∪ {¬a,¬b,¬(¬a ∧ b),¬ϕ}cl(ϕ) = {a, b,¬a ∧ b, ϕ} ∪ {¬a,¬b,¬(¬a ∧ b),¬ϕ}
Example: if ϕ′ = �aϕ′ = �aϕ′ = �a = ¬♦¬a = ¬(true U¬a)= ¬♦¬a = ¬(true U¬a)= ¬♦¬a = ¬(true U¬a) then
cl(ϕ′) = {a,¬a, true,¬true,�a,¬�a}cl(ϕ′) = {a,¬a, true,¬true,�a,¬�a}cl(ϕ′) = {a,¬a, true,¬true,�a,¬�a}127 / 527
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Elementary formula-sets ltlmc3.2-50a
B ⊆ cl(ϕ)B ⊆ cl(ϕ)B ⊆ cl(ϕ) is elementary iff:
(i) BBB is maximal consistent w.r.t. prop. logic,i.e., if ψψψ, ψ1 ∧ ψ2 ∈ cl(ϕ)ψ1 ∧ ψ2 ∈ cl(ϕ)ψ1 ∧ ψ2 ∈ cl(ϕ) then:
ψ �∈ Bψ �∈ Bψ �∈ B iff ¬ψ ∈ B¬ψ ∈ B¬ψ ∈ B
ψ1 ∧ ψ2 ∈ Bψ1 ∧ ψ2 ∈ Bψ1 ∧ ψ2 ∈ B iff ψ1 ∈ Bψ1 ∈ Bψ1 ∈ B and ψ2 ∈ Bψ2 ∈ Bψ2 ∈ B
true ∈ cl(ϕ)true ∈ cl(ϕ)true ∈ cl(ϕ) implies true ∈ Btrue ∈ Btrue ∈ B
(ii) BBB is locally consistent with respect to until UUU,i.e., if ψ1 Uψ2 ∈ cl(ϕ)ψ1 Uψ2 ∈ cl(ϕ)ψ1 Uψ2 ∈ cl(ϕ) then:
if ψ1 Uψ2 ∈ Bψ1 Uψ2 ∈ Bψ1 Uψ2 ∈ B and ψ2 �∈ Bψ2 �∈ Bψ2 �∈ B then ψ1 ∈ Bψ1 ∈ Bψ1 ∈ B
if ψ2 ∈ Bψ2 ∈ Bψ2 ∈ B then ψ1 Uψ2 ∈ Bψ1 Uψ2 ∈ Bψ1 Uψ2 ∈ B138 / 527
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Elementary or not? ltlmc3.2-49
Let ϕ = a U(¬a ∧ b)ϕ = a U(¬a ∧ b)ϕ = a U(¬a ∧ b).
B1 = {a, b,¬a ∧ b, ϕ}B1 = {a, b,¬a ∧ b, ϕ}B1 = {a, b,¬a ∧ b, ϕ}
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Elementary or not? ltlmc3.2-49
Let ϕ = a U(¬a ∧ b)ϕ = a U(¬a ∧ b)ϕ = a U(¬a ∧ b).
B1 = {a, b,¬a ∧ b, ϕ}B1 = {a, b,¬a ∧ b, ϕ}B1 = {a, b,¬a ∧ b, ϕ} not elementarypropositional inconsistent
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Elementary or not? ltlmc3.2-49
Let ϕ = a U(¬a ∧ b)ϕ = a U(¬a ∧ b)ϕ = a U(¬a ∧ b).
B1 = {a, b,¬a ∧ b, ϕ}B1 = {a, b,¬a ∧ b, ϕ}B1 = {a, b,¬a ∧ b, ϕ} not elementarypropositional inconsistent
B2 = {¬a, b, ϕ}B2 = {¬a, b, ϕ}B2 = {¬a, b, ϕ}
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Elementary or not? ltlmc3.2-49
Let ϕ = a U(¬a ∧ b)ϕ = a U(¬a ∧ b)ϕ = a U(¬a ∧ b).
B1 = {a, b,¬a ∧ b, ϕ}B1 = {a, b,¬a ∧ b, ϕ}B1 = {a, b,¬a ∧ b, ϕ} not elementarypropositional inconsistent
B2 = {¬a, b, ϕ}B2 = {¬a, b, ϕ}B2 = {¬a, b, ϕ} not elementary, not maximalas ¬a ∧ b �∈ B2¬a ∧ b �∈ B2¬a ∧ b �∈ B2
¬(¬a ∧ b) �∈ B2¬(¬a ∧ b) �∈ B2¬(¬a ∧ b) �∈ B2
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Elementary or not? ltlmc3.2-49
Let ϕ = a U(¬a ∧ b)ϕ = a U(¬a ∧ b)ϕ = a U(¬a ∧ b).
B1 = {a, b,¬a ∧ b, ϕ}B1 = {a, b,¬a ∧ b, ϕ}B1 = {a, b,¬a ∧ b, ϕ} not elementarypropositional inconsistent
B2 = {¬a, b, ϕ}B2 = {¬a, b, ϕ}B2 = {¬a, b, ϕ} not elementary, not maximalas ¬a ∧ b �∈ B2¬a ∧ b �∈ B2¬a ∧ b �∈ B2
¬(¬a ∧ b) �∈ B2¬(¬a ∧ b) �∈ B2¬(¬a ∧ b) �∈ B2
B3 = {¬a, b,¬a ∧ b,¬ϕ}B3 = {¬a, b,¬a ∧ b,¬ϕ}B3 = {¬a, b,¬a ∧ b,¬ϕ}
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Elementary or not? ltlmc3.2-49
Let ϕ = a U(¬a ∧ b)ϕ = a U(¬a ∧ b)ϕ = a U(¬a ∧ b).
B1 = {a, b,¬a ∧ b, ϕ}B1 = {a, b,¬a ∧ b, ϕ}B1 = {a, b,¬a ∧ b, ϕ} not elementarypropositional inconsistent
B2 = {¬a, b, ϕ}B2 = {¬a, b, ϕ}B2 = {¬a, b, ϕ} not elementary, not maximalas ¬a ∧ b �∈ B2¬a ∧ b �∈ B2¬a ∧ b �∈ B2
¬(¬a ∧ b) �∈ B2¬(¬a ∧ b) �∈ B2¬(¬a ∧ b) �∈ B2
B3 = {¬a, b,¬a ∧ b,¬ϕ}B3 = {¬a, b,¬a ∧ b,¬ϕ}B3 = {¬a, b,¬a ∧ b,¬ϕ} not elementarynot locally consistent for UUU
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Elementary or not? ltlmc3.2-49
Let ϕ = a U(¬a ∧ b)ϕ = a U(¬a ∧ b)ϕ = a U(¬a ∧ b).
B1 = {a, b,¬a ∧ b, ϕ}B1 = {a, b,¬a ∧ b, ϕ}B1 = {a, b,¬a ∧ b, ϕ} not elementarypropositional inconsistent
B2 = {¬a, b, ϕ}B2 = {¬a, b, ϕ}B2 = {¬a, b, ϕ} not elementary, not maximalas ¬a ∧ b �∈ B2¬a ∧ b �∈ B2¬a ∧ b �∈ B2
¬(¬a ∧ b) �∈ B2¬(¬a ∧ b) �∈ B2¬(¬a ∧ b) �∈ B2
B3 = {¬a, b,¬a ∧ b,¬ϕ}B3 = {¬a, b,¬a ∧ b,¬ϕ}B3 = {¬a, b,¬a ∧ b,¬ϕ} not elementarynot locally consistent for UUU
B4 = {¬a,¬b,¬(¬a ∧ b),¬ϕ}B4 = {¬a,¬b,¬(¬a ∧ b),¬ϕ}B4 = {¬a,¬b,¬(¬a ∧ b),¬ϕ}
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Elementary or not? ltlmc3.2-49
Let ϕ = a U(¬a ∧ b)ϕ = a U(¬a ∧ b)ϕ = a U(¬a ∧ b).
B1 = {a, b,¬a ∧ b, ϕ}B1 = {a, b,¬a ∧ b, ϕ}B1 = {a, b,¬a ∧ b, ϕ} not elementarypropositional inconsistent
B2 = {¬a, b, ϕ}B2 = {¬a, b, ϕ}B2 = {¬a, b, ϕ} not elementary, not maximalas ¬a ∧ b �∈ B2¬a ∧ b �∈ B2¬a ∧ b �∈ B2
¬(¬a ∧ b) �∈ B2¬(¬a ∧ b) �∈ B2¬(¬a ∧ b) �∈ B2
B3 = {¬a, b,¬a ∧ b,¬ϕ}B3 = {¬a, b,¬a ∧ b,¬ϕ}B3 = {¬a, b,¬a ∧ b,¬ϕ} not elementarynot locally consistent for UUU
B4 = {¬a,¬b,¬(¬a ∧ b),¬ϕ}B4 = {¬a,¬b,¬(¬a ∧ b),¬ϕ}B4 = {¬a,¬b,¬(¬a ∧ b),¬ϕ} elementary
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Example: elementary formula-sets ltlmc3.2-51
closure cl(ϕ)cl(ϕ)cl(ϕ):• set of all subformulas of ϕϕϕ and their negations• ψψψ and ¬¬ψ¬¬ψ¬¬ψ are identified
elementary formula-sets: subsets BBB of cl(ϕ)cl(ϕ)cl(ϕ)• maximal consistent w.r.t. propositional logic• locally consistent w.r.t. UUU
For ϕ = a U(¬a ∧ b)ϕ = a U(¬a ∧ b)ϕ = a U(¬a ∧ b), the elementary sets are:
{ a, b,¬(¬a ∧ b), ϕ}{ a, b,¬(¬a ∧ b), ϕ}{ a, b,¬(¬a ∧ b), ϕ}{ a,¬b,¬(¬a ∧ b), ϕ}{ a,¬b,¬(¬a ∧ b), ϕ}{ a,¬b,¬(¬a ∧ b), ϕ}{¬a, b, ¬a ∧ b , ϕ}{¬a, b, ¬a ∧ b , ϕ}{¬a, b, ¬a ∧ b , ϕ}
{ a, b,¬(¬a ∧ b),¬ϕ}{ a, b,¬(¬a ∧ b),¬ϕ}{ a, b,¬(¬a ∧ b),¬ϕ}{ a,¬b,¬(¬a ∧ b),¬ϕ}{ a,¬b,¬(¬a ∧ b),¬ϕ}{ a,¬b,¬(¬a ∧ b),¬ϕ}{¬a,¬b,¬(¬a ∧ b),¬ϕ}{¬a,¬b,¬(¬a ∧ b),¬ϕ}{¬a,¬b,¬(¬a ∧ b),¬ϕ}
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Encoding of LTL semantics in a GNBA ltlmc3.2-39-copy
idea: encode the semantics of the operators appearingin ϕϕϕ by appropriate components of the GNBA GGG:
semantics of ... encoding
propositional logictruetruetrue, ¬¬¬, ∧∧∧ in the states
next©©© in the transition relation
until UUU expansion law, least fixed point
ψ1 Uψ2ψ1 Uψ2ψ1 Uψ2 ≡≡≡ ψ2 ∨ (ψ1 ∧©(ψ1 Uψ2))ψ2 ∨ (ψ1 ∧©(ψ1 Uψ2))ψ2 ∨ (ψ1 ∧©(ψ1 Uψ2))���
↖↖↖↗↗↗ ↑↑↑encoded inthe states
encoded in thetransition relation
acceptancecondition
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Encoding of LTL semantics in a GNBA ltlmc3.2-39-copy
idea: encode the semantics of the operators appearingin ϕϕϕ by appropriate components of the GNBA GGG:
semantics of ... encoding
propositional logictruetruetrue, ¬¬¬, ∧∧∧ in the states ←−←−←− elementary
formula sets
next©©© in the transition relation
until UUU expansion law, least fixed point
ψ1 Uψ2ψ1 Uψ2ψ1 Uψ2 ≡≡≡ ψ2 ∨ (ψ1 ∧©(ψ1 Uψ2))ψ2 ∨ (ψ1 ∧©(ψ1 Uψ2))ψ2 ∨ (ψ1 ∧©(ψ1 Uψ2))���
↖↖↖↗↗↗ ↑↑↑elementaryformula sets
encoded in thetransition relation
acceptancecondition
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GNBA for LTL-formula ϕϕϕ ltlmc3.2-57
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GNBA for LTL-formula ϕϕϕ ltlmc3.2-57
G = (Q, 2AP , δ,Q0,F)G = (Q, 2AP , δ,Q0,F)G = (Q, 2AP , δ,Q0,F)
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GNBA for LTL-formula ϕϕϕ ltlmc3.2-57
G = (Q, 2AP , δ,Q0,F)G = (Q, 2AP , δ,Q0,F)G = (Q, 2AP , δ,Q0,F)
state space: QQQ ==={B ⊆ cl(ϕ) : B
{B ⊆ cl(ϕ) : B
{B ⊆ cl(ϕ) : B is elementary
}}}
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GNBA for LTL-formula ϕϕϕ ltlmc3.2-57
G = (Q, 2AP , δ,Q0,F)G = (Q, 2AP , δ,Q0,F)G = (Q, 2AP , δ,Q0,F)
state space: QQQ ==={B ⊆ cl(ϕ) : B
{B ⊆ cl(ϕ) : B
{B ⊆ cl(ϕ) : B is elementary
}}}initial states: Q0Q0Q0 ===
{B ∈ Q : ϕ ∈ B
}{B ∈ Q : ϕ ∈ B
}{B ∈ Q : ϕ ∈ B
}
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GNBA for LTL-formula ϕϕϕ ltlmc3.2-57
G = (Q, 2AP , δ,Q0,F)G = (Q, 2AP , δ,Q0,F)G = (Q, 2AP , δ,Q0,F)
state space: QQQ ==={B ⊆ cl(ϕ) : B
{B ⊆ cl(ϕ) : B
{B ⊆ cl(ϕ) : B is elementary
}}}initial states: Q0Q0Q0 ===
{B ∈ Q : ϕ ∈ B
}{B ∈ Q : ϕ ∈ B
}{B ∈ Q : ϕ ∈ B
}transition relation: for B ∈ QB ∈ QB ∈ Q and A ∈ 2APA ∈ 2APA ∈ 2AP :
if A �= B ∩ APA �= B ∩ APA �= B ∩ AP then δ(B,A) = ∅δ(B ,A) = ∅δ(B ,A) = ∅
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GNBA for LTL-formula ϕϕϕ ltlmc3.2-57
G = (Q, 2AP , δ,Q0,F)G = (Q, 2AP , δ,Q0,F)G = (Q, 2AP , δ,Q0,F)
state space: QQQ ==={B ⊆ cl(ϕ) : B
{B ⊆ cl(ϕ) : B
{B ⊆ cl(ϕ) : B is elementary
}}}initial states: Q0Q0Q0 ===
{B ∈ Q : ϕ ∈ B
}{B ∈ Q : ϕ ∈ B
}{B ∈ Q : ϕ ∈ B
}transition relation: for B ∈ QB ∈ QB ∈ Q and A ∈ 2APA ∈ 2APA ∈ 2AP :
if A �= B ∩ APA �= B ∩ APA �= B ∩ AP then δ(B,A) = ∅δ(B ,A) = ∅δ(B ,A) = ∅
if A = B ∩ APA = B ∩ APA = B ∩ AP then δ(B,A) =δ(B ,A) =δ(B ,A) = set of all B ′ ∈ QB ′ ∈ QB ′ ∈ Q s.t.
©ψ ∈ B©ψ ∈ B©ψ ∈ B iff ψ ∈ B ′ψ ∈ B ′ψ ∈ B ′
ψ1 Uψ2 ∈ Bψ1 Uψ2 ∈ Bψ1 Uψ2 ∈ B iff (ψ2 ∈ B) ∨ (ψ1 ∈ B ∧ ψ1 Uψ2 ∈ B ′)(ψ2 ∈ B) ∨ (ψ1 ∈ B ∧ ψ1 Uψ2 ∈ B ′)(ψ2 ∈ B) ∨ (ψ1 ∈ B ∧ ψ1 Uψ2 ∈ B ′)
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GNBA for LTL-formula ϕϕϕ ltlmc3.2-57
G = (Q, 2AP , δ,Q0,F)G = (Q, 2AP , δ,Q0,F)G = (Q, 2AP , δ,Q0,F)
state space: QQQ ==={B ⊆ cl(ϕ) : B
{B ⊆ cl(ϕ) : B
{B ⊆ cl(ϕ) : B is elementary
}}}initial states: Q0Q0Q0 ===
{B ∈ Q : ϕ ∈ B
}{B ∈ Q : ϕ ∈ B
}{B ∈ Q : ϕ ∈ B
}transition relation: for B ∈ QB ∈ QB ∈ Q and A ∈ 2APA ∈ 2APA ∈ 2AP :
if A �= B ∩ APA �= B ∩ APA �= B ∩ AP then δ(B,A) = ∅δ(B ,A) = ∅δ(B ,A) = ∅
if A = B ∩ APA = B ∩ APA = B ∩ AP then δ(B,A) =δ(B ,A) =δ(B ,A) = set of all B ′ ∈ QB ′ ∈ QB ′ ∈ Q s.t.
©ψ ∈ B©ψ ∈ B©ψ ∈ B iff ψ ∈ B ′ψ ∈ B ′ψ ∈ B ′
ψ1 Uψ2 ∈ Bψ1 Uψ2 ∈ Bψ1 Uψ2 ∈ B iff (ψ2 ∈ B) ∨ (ψ1 ∈ B ∧ ψ1 Uψ2 ∈ B ′)(ψ2 ∈ B) ∨ (ψ1 ∈ B ∧ ψ1 Uψ2 ∈ B ′)(ψ2 ∈ B) ∨ (ψ1 ∈ B ∧ ψ1 Uψ2 ∈ B ′)
acceptance set F ={Fψ1 Uψ2
: ψ1 Uψ2 ∈ cl(ϕ)}
F ={F ψ1 Uψ2 : ψ1 Uψ2 ∈ cl(ϕ)
}F =
{F ψ1 Uψ2
: ψ1 Uψ2 ∈ cl(ϕ)}
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GNBA for LTL-formula ϕϕϕ ltlmc3.2-57
G = (Q, 2AP , δ,Q0,F)G = (Q, 2AP , δ,Q0,F)G = (Q, 2AP , δ,Q0,F)
state space: QQQ ==={B ⊆ cl(ϕ) : B
{B ⊆ cl(ϕ) : B
{B ⊆ cl(ϕ) : B is elementary
}}}initial states: Q0Q0Q0 ===
{B ∈ Q : ϕ ∈ B
}{B ∈ Q : ϕ ∈ B
}{B ∈ Q : ϕ ∈ B
}transition relation: for B ∈ QB ∈ QB ∈ Q and A ∈ 2APA ∈ 2APA ∈ 2AP :
if A �= B ∩ APA �= B ∩ APA �= B ∩ AP then δ(B,A) = ∅δ(B ,A) = ∅δ(B ,A) = ∅
if A = B ∩ APA = B ∩ APA = B ∩ AP then δ(B,A) =δ(B ,A) =δ(B ,A) = set of all B ′ ∈ QB ′ ∈ QB ′ ∈ Q s.t.
©ψ ∈ B©ψ ∈ B©ψ ∈ B iff ψ ∈ B ′ψ ∈ B ′ψ ∈ B ′
ψ1 Uψ2 ∈ Bψ1 Uψ2 ∈ Bψ1 Uψ2 ∈ B iff (ψ2 ∈ B) ∨ (ψ1 ∈ B ∧ ψ1 Uψ2 ∈ B ′)(ψ2 ∈ B) ∨ (ψ1 ∈ B ∧ ψ1 Uψ2 ∈ B ′)(ψ2 ∈ B) ∨ (ψ1 ∈ B ∧ ψ1 Uψ2 ∈ B ′)
acceptance set F ={Fψ1 Uψ2
: ψ1 Uψ2 ∈ cl(ϕ)}
F ={F ψ1 Uψ2 : ψ1 Uψ2 ∈ cl(ϕ)
}F =
{F ψ1 Uψ2
: ψ1 Uψ2 ∈ cl(ϕ)}
where Fψ1 Uψ2=
{B ∈ Q : ψ1 Uψ2 /∈ B ∨ ψ2 ∈ B
}F ψ1 Uψ2
={B ∈ Q : ψ1 Uψ2 /∈ B ∨ ψ2 ∈ B
}Fψ1 Uψ2
={B ∈ Q : ψ1 Uψ2 /∈ B ∨ ψ2 ∈ B
}157 / 527
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Example: GNBA for ϕ =©aϕ =©aϕ =©a ltlmc3.2-52
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Example: GNBA for ϕ =©aϕ =©aϕ =©a ltlmc3.2-52
a,©aa,©aa,©a a,¬©aa,¬©aa,¬©a
¬a,©a¬a,©a¬a,©a ¬a,¬©a¬a,¬©a¬a,¬©a
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Example: GNBA for ϕ =©aϕ =©aϕ =©a ltlmc3.2-52
a,©aa,©aa,©a a,¬©aa,¬©aa,¬©a
¬a,©a¬a,©a¬a,©a ¬a,¬©a¬a,¬©a¬a,¬©a
initial states: formula-sets BBB with©a ∈ B©a ∈ B©a ∈ B
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Example: GNBA for ϕ =©aϕ =©aϕ =©a ltlmc3.2-52
a,©aa,©aa,©a a,¬©aa,¬©aa,¬©a
¬a,©a¬a,©a¬a,©a ¬a,¬©a¬a,¬©a¬a,¬©a
initial states: formula-sets BBB with©a ∈ B©a ∈ B©a ∈ B
transition relation:
if©a ∈ B©a ∈ B©a ∈ B then δ(B ,B ∩ {a}) = {B ′ : a ∈ B ′}δ(B ,B ∩ {a}) = {B ′ : a ∈ B ′}δ(B,B ∩ {a}) = {B ′ : a ∈ B ′}
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Example: GNBA for ϕ =©aϕ =©aϕ =©a ltlmc3.2-52
a,©aa,©aa,©a a,¬©aa,¬©aa,¬©a
¬a,©a¬a,©a¬a,©a ¬a,¬©a¬a,¬©a¬a,¬©a
aaaaaa
initial states: formula-sets BBB with©a ∈ B©a ∈ B©a ∈ B
transition relation:
if©a ∈ B©a ∈ B©a ∈ B then δ(B ,B ∩ {a}) = {B ′ : a ∈ B ′}δ(B ,B ∩ {a}) = {B ′ : a ∈ B ′}δ(B,B ∩ {a}) = {B ′ : a ∈ B ′}
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Example: GNBA for ϕ =©aϕ =©aϕ =©a ltlmc3.2-52
a,©aa,©aa,©a a,¬©aa,¬©aa,¬©a
¬a,©a¬a,©a¬a,©a ¬a,¬©a¬a,¬©a¬a,¬©a
aaaaaa
¬a¬a¬a
¬a¬a¬a
initial states: formula-sets BBB with©a ∈ B©a ∈ B©a ∈ B
transition relation:
if©a ∈ B©a ∈ B©a ∈ B then δ(B ,B ∩ {a}) = {B ′ : a ∈ B ′}δ(B ,B ∩ {a}) = {B ′ : a ∈ B ′}δ(B,B ∩ {a}) = {B ′ : a ∈ B ′}
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Example: GNBA for ϕ =©aϕ =©aϕ =©a ltlmc3.2-52
a,©aa,©aa,©a a,¬©aa,¬©aa,¬©a
¬a,©a¬a,©a¬a,©a ¬a,¬©a¬a,¬©a¬a,¬©a
aaaaaa
¬a¬a¬a
¬a¬a¬a
initial states: formula-sets BBB with©a ∈ B©a ∈ B©a ∈ B
transition relation:
if©a ∈ B©a ∈ B©a ∈ B then δ(B ,B ∩ {a}) = {B ′ : a ∈ B ′}δ(B ,B ∩ {a}) = {B ′ : a ∈ B ′}δ(B,B ∩ {a}) = {B ′ : a ∈ B ′}if©a /∈ B©a /∈ B©a /∈ B then δ(B ,B ∩ {a}) = {B ′ : a �∈ B ′}δ(B ,B ∩ {a}) = {B ′ : a �∈ B ′}δ(B,B ∩ {a}) = {B ′ : a �∈ B ′}
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Example: GNBA for ϕ =©aϕ =©aϕ =©a ltlmc3.2-52
a,©aa,©aa,©a a,¬©aa,¬©aa,¬©a
¬a,©a¬a,©a¬a,©a ¬a,¬©a¬a,¬©a¬a,¬©a
aaaaaa
aaa¬a¬a¬a
¬a¬a¬a
aaa
initial states: formula-sets BBB with©a ∈ B©a ∈ B©a ∈ B
transition relation:
if©a ∈ B©a ∈ B©a ∈ B then δ(B ,B ∩ {a}) = {B ′ : a ∈ B ′}δ(B ,B ∩ {a}) = {B ′ : a ∈ B ′}δ(B,B ∩ {a}) = {B ′ : a ∈ B ′}if©a /∈ B©a /∈ B©a /∈ B then δ(B ,B ∩ {a}) = {B ′ : a �∈ B ′}δ(B ,B ∩ {a}) = {B ′ : a �∈ B ′}δ(B,B ∩ {a}) = {B ′ : a �∈ B ′}
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Example: GNBA for ϕ =©aϕ =©aϕ =©a ltlmc3.2-52
a,©aa,©aa,©a a,¬©aa,¬©aa,¬©a
¬a,©a¬a,©a¬a,©a ¬a,¬©a¬a,¬©a¬a,¬©a
aaa
¬a¬a¬a
aaa
aaa
¬a¬a¬a
¬a¬a¬a
¬a¬a¬a
aaa
initial states: formula-sets BBB with©a ∈ B©a ∈ B©a ∈ B
transition relation:
if©a ∈ B©a ∈ B©a ∈ B then δ(B ,B ∩ {a}) = {B ′ : a ∈ B ′}δ(B ,B ∩ {a}) = {B ′ : a ∈ B ′}δ(B,B ∩ {a}) = {B ′ : a ∈ B ′}if©a /∈ B©a /∈ B©a /∈ B then δ(B ,B ∩ {a}) = {B ′ : a �∈ B ′}δ(B ,B ∩ {a}) = {B ′ : a �∈ B ′}δ(B,B ∩ {a}) = {B ′ : a �∈ B ′}
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Example: GNBA for ϕ =©aϕ =©aϕ =©a ltlmc3.2-53
a,©aa,©aa,©a a,¬©aa,¬©aa,¬©a
¬a,©a¬a,©a¬a,©a ¬a,¬©a¬a,¬©a¬a,¬©a
¬a¬a¬a
aaaaaa
¬a¬a¬a
aaa
¬a¬a¬a
¬a¬a¬a
aaa
set of acceptance sets:
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Example: GNBA for ϕ =©aϕ =©aϕ =©a ltlmc3.2-53
a,©aa,©aa,©a a,¬©aa,¬©aa,¬©a
¬a,©a¬a,©a¬a,©a ¬a,¬©a¬a,¬©a¬a,¬©a
¬a¬a¬a
aaaaaa
¬a¬a¬a
aaa
¬a¬a¬a
¬a¬a¬a
aaa
set of acceptance sets: F = ∅F = ∅F = ∅
hence: all words having an infinite run are accepted
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Example: GNBA for ϕ =©aϕ =©aϕ =©a ltlmc3.2-53
a,©aa,©aa,©a a,¬©aa,¬©aa,¬©a
¬a,©a¬a,©a¬a,©a ¬a,¬©a¬a,¬©a¬a,¬©a
¬a¬a¬a
aaaaaa
¬a¬a¬a
aaa
¬a¬a¬a
¬a¬a¬a
aaa
set of acceptance sets: F = ∅F = ∅F = ∅
∅∅∅ {a}{a}{a} {a}{a}{a} ∅∅∅ ∅∅∅ . . .. . .. . . |= ©a|= ©a|= ©a
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Example: GNBA for ϕ =©aϕ =©aϕ =©a ltlmc3.2-53
a,©aa,©aa,©a a,¬©aa,¬©aa,¬©a
¬a,©a¬a,©a¬a,©a ¬a,¬©a¬a,¬©a¬a,¬©a
¬a¬a¬a
aaaaaa
¬a¬a¬a
aaa
¬a¬a¬a
¬a¬a¬a
aaa
set of acceptance sets: F = ∅F = ∅F = ∅
∅∅∅
↓↓↓¬a¬a¬a©a©a©a
{a}{a}{a} {a}{a}{a} ∅∅∅ ∅∅∅ . . .. . .. . . |= ©a|= ©a|= ©a
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Example: GNBA for ϕ =©aϕ =©aϕ =©a ltlmc3.2-53
a,©aa,©aa,©a a,¬©aa,¬©aa,¬©a
¬a,©a¬a,©a¬a,©a ¬a,¬©a¬a,¬©a¬a,¬©a
¬a¬a¬a
aaaaaa
¬a¬a¬a
aaa
¬a¬a¬a
¬a¬a¬a
aaa
set of acceptance sets: F = ∅F = ∅F = ∅
∅∅∅
↓↓↓¬a¬a¬a©a©a©a
{a}{a}{a}↓↓↓aaa
©a©a©a
{a}{a}{a} ∅∅∅ ∅∅∅ . . .. . .. . . |= ©a|= ©a|= ©a
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Example: GNBA for ϕ =©aϕ =©aϕ =©a ltlmc3.2-53
a,©aa,©aa,©a a,¬©aa,¬©aa,¬©a
¬a,©a¬a,©a¬a,©a ¬a,¬©a¬a,¬©a¬a,¬©a
¬a¬a¬a
aaaaaa
¬a¬a¬a
aaa
¬a¬a¬a
¬a¬a¬a
aaa
set of acceptance sets: F = ∅F = ∅F = ∅
∅∅∅
↓↓↓¬a¬a¬a©a©a©a
{a}{a}{a}↓↓↓aaa
©a©a©a
{a}{a}{a} ∅∅∅
↓↓↓aaa
©a©a©a
∅∅∅ . . .. . .. . . |= ©a|= ©a|= ©a
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Example: GNBA for ϕ =©aϕ =©aϕ =©a ltlmc3.2-53
a,©aa,©aa,©a a,¬©aa,¬©aa,¬©a
¬a,©a¬a,©a¬a,©a ¬a,¬©a¬a,¬©a¬a,¬©a
¬a¬a¬a
aaaaaa
¬a¬a¬a
aaa
¬a¬a¬a
¬a¬a¬a
aaa
set of acceptance sets: F = ∅F = ∅F = ∅
∅∅∅
↓↓↓¬a¬a¬a©a©a©a
{a}{a}{a}↓↓↓aaa
©a©a©a
{a}{a}{a} ∅∅∅
↓↓↓aaa
©a©a©a
∅∅∅ . . .. . .. . .↓↓↓¬a¬a¬a
©a©a©a
|= ©a|= ©a|= ©a
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Example: GNBA for ϕ =©aϕ =©aϕ =©a ltlmc3.2-53
a,©aa,©aa,©a a,¬©aa,¬©aa,¬©a
¬a,©a¬a,©a¬a,©a ¬a,¬©a¬a,¬©a¬a,¬©a
¬a¬a¬a
aaaaaa
¬a¬a¬a
aaa
¬a¬a¬a
¬a¬a¬a
aaa
set of acceptance sets: F = ∅F = ∅F = ∅
∅∅∅
↓↓↓¬a¬a¬a©a©a©a
{a}{a}{a}↓↓↓aaa
©a©a©a
{a}{a}{a} ∅∅∅
↓↓↓aaa
©a©a©a
∅∅∅ . . .. . .. . .↓↓↓¬a¬a¬a
©a©a©a. . .. . .. . .
|= ©a|= ©a|= ©a
acceptingrun
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Soundness of the GNBA for ϕ =©aϕ =©aϕ =©a ltlmc3.2-53a
a,©aa,©aa,©a a,¬©aa,¬©aa,¬©a
¬a,©a¬a,©a¬a,©a ¬a,¬©a¬a,¬©a¬a,¬©a
{a}{a}{a}
∅∅∅
{a}{a}{a}
{a}{a}{a}
∅∅∅
∅∅∅
∅∅∅
{a}{a}{a}
for all words σ = A0 A1 A2 A3 . . . ∈ Lω(G)σ = A0 A1 A2 A3 . . . ∈ Lω(G)σ = A0 A1 A2 A3 . . . ∈ Lω(G): A1 = {a}A1 = {a}A1 = {a}
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Soundness of the GNBA for ϕ =©aϕ =©aϕ =©a ltlmc3.2-53a
a,©aa,©aa,©a a,¬©aa,¬©aa,¬©a
¬a,©a¬a,©a¬a,©a ¬a,¬©a¬a,¬©a¬a,¬©a
{a}{a}{a}
∅∅∅
{a}{a}{a}
{a}{a}{a}
∅∅∅
∅∅∅
∅∅∅
{a}{a}{a}
for all words σ = A0 A1 A2 A3 . . . ∈ Lω(G)σ = A0 A1 A2 A3 . . . ∈ Lω(G)σ = A0 A1 A2 A3 . . . ∈ Lω(G): A1 = {a}A1 = {a}A1 = {a}proof:
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Soundness of the GNBA for ϕ =©aϕ =©aϕ =©a ltlmc3.2-53a
a,©aa,©aa,©a a,¬©aa,¬©aa,¬©a
¬a,©a¬a,©a¬a,©a ¬a,¬©a¬a,¬©a¬a,¬©a
{a}{a}{a}
∅∅∅
{a}{a}{a}
{a}{a}{a}
∅∅∅
∅∅∅
∅∅∅
{a}{a}{a}
for all words σ = A0 A1 A2 A3 . . . ∈ Lω(G)σ = A0 A1 A2 A3 . . . ∈ Lω(G)σ = A0 A1 A2 A3 . . . ∈ Lω(G): A1 = {a}A1 = {a}A1 = {a}proof: Let B0 B1 B2 . . .B0 B1 B2 . . .B0 B1 B2 . . . be an accepting run for σσσ.
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Soundness of the GNBA for ϕ =©aϕ =©aϕ =©a ltlmc3.2-53a
a,©aa,©aa,©a a,¬©aa,¬©aa,¬©a
¬a,©a¬a,©a¬a,©a ¬a,¬©a¬a,¬©a¬a,¬©a
{a}{a}{a}
∅∅∅
{a}{a}{a}
{a}{a}{a}
∅∅∅
∅∅∅
∅∅∅
{a}{a}{a}
for all words σ = A0 A1 A2 A3 . . . ∈ Lω(G)σ = A0 A1 A2 A3 . . . ∈ Lω(G)σ = A0 A1 A2 A3 . . . ∈ Lω(G): A1 = {a}A1 = {a}A1 = {a}proof: Let B0 B1 B2 . . .B0 B1 B2 . . .B0 B1 B2 . . . be an accepting run for σσσ.
=⇒=⇒=⇒ ©a ∈ B0©a ∈ B0©a ∈ B0
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Soundness of the GNBA for ϕ =©aϕ =©aϕ =©a ltlmc3.2-53a
a,©aa,©aa,©a a,¬©aa,¬©aa,¬©a
¬a,©a¬a,©a¬a,©a ¬a,¬©a¬a,¬©a¬a,¬©a
{a}{a}{a}
∅∅∅
{a}{a}{a}
{a}{a}{a}
∅∅∅
∅∅∅
∅∅∅
{a}{a}{a}
for all words σ = A0 A1 A2 A3 . . . ∈ Lω(G)σ = A0 A1 A2 A3 . . . ∈ Lω(G)σ = A0 A1 A2 A3 . . . ∈ Lω(G): A1 = {a}A1 = {a}A1 = {a}proof: Let B0 B1 B2 . . .B0 B1 B2 . . .B0 B1 B2 . . . be an accepting run for σσσ.
=⇒=⇒=⇒ ©a ∈ B0©a ∈ B0©a ∈ B0 and therefore a ∈ B1a ∈ B1a ∈ B1
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Soundness of the GNBA for ϕ =©aϕ =©aϕ =©a ltlmc3.2-53a
a,©aa,©aa,©a a,¬©aa,¬©aa,¬©a
¬a,©a¬a,©a¬a,©a ¬a,¬©a¬a,¬©a¬a,¬©a
{a}{a}{a}
∅∅∅
{a}{a}{a}
{a}{a}{a}
∅∅∅
∅∅∅
∅∅∅
{a}{a}{a}
for all words σ = A0 A1 A2 A3 . . . ∈ Lω(G)σ = A0 A1 A2 A3 . . . ∈ Lω(G)σ = A0 A1 A2 A3 . . . ∈ Lω(G): A1 = {a}A1 = {a}A1 = {a}proof: Let B0 B1 B2 . . .B0 B1 B2 . . .B0 B1 B2 . . . be an accepting run for σσσ.
=⇒=⇒=⇒ ©a ∈ B0©a ∈ B0©a ∈ B0 and therefore a ∈ B1a ∈ B1a ∈ B1
=⇒=⇒=⇒ the outgoing edges of B1B1B1 have label {a}{a}{a}
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Soundness of the GNBA for ϕ =©aϕ =©aϕ =©a ltlmc3.2-53a
a,©aa,©aa,©a a,¬©aa,¬©aa,¬©a
¬a,©a¬a,©a¬a,©a ¬a,¬©a¬a,¬©a¬a,¬©a
{a}{a}{a}
∅∅∅
{a}{a}{a}
{a}{a}{a}
∅∅∅
∅∅∅
∅∅∅
{a}{a}{a}
for all words σ = A0 A1 A2 A3 . . . ∈ Lω(G)σ = A0 A1 A2 A3 . . . ∈ Lω(G)σ = A0 A1 A2 A3 . . . ∈ Lω(G): A1 = {a}A1 = {a}A1 = {a}proof: Let B0 B1 B2 . . .B0 B1 B2 . . .B0 B1 B2 . . . be an accepting run for σσσ.
=⇒=⇒=⇒ ©a ∈ B0©a ∈ B0©a ∈ B0 and therefore a ∈ B1a ∈ B1a ∈ B1
=⇒=⇒=⇒ the outgoing edges of B1B1B1 have label {a}{a}{a}=⇒=⇒=⇒ {a} = B1 ∩ AP = A1{a} = B1 ∩ AP = A1{a} = B1 ∩ AP = A1
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Example: GNBA for ϕ = a U bϕ = a U bϕ = a U b ltlmc3.2-54
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Example: GNBA for ϕ = a U bϕ = a U bϕ = a U b ltlmc3.2-54
a, b, a U ba, b, a U ba, b, a U b ¬a,¬b,¬(a U b)¬a,¬b,¬(a U b)¬a,¬b,¬(a U b)
¬a, b, a U b¬a, b, a U b¬a, b, a U b
a,¬b,¬(a U b)a,¬b,¬(a U b)a,¬b,¬(a U b)a,¬b, a U ba,¬b, a U ba,¬b, a U b
locally inconsistent: {a, b,¬(a U b)}{a, b,¬(a U b)}{a, b,¬(a U b)}{¬a, b,¬(a U b)}{¬a, b,¬(a U b)}{¬a, b,¬(a U b)}{¬a,¬b, a U b}{¬a,¬b, a U b}{¬a,¬b, a U b}
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Example: GNBA for ϕ = a U bϕ = a U bϕ = a U b ltlmc3.2-54
a, b, a U ba, b, a U ba, b, a U b ¬a,¬b,¬(a U b)¬a,¬b,¬(a U b)¬a,¬b,¬(a U b)
¬a, b, a U b¬a, b, a U b¬a, b, a U b
a,¬b,¬(a U b)a,¬b,¬(a U b)a,¬b,¬(a U b)a,¬b, a U ba,¬b, a U ba,¬b, a U b
initial states: BBB with ϕ = a U b ∈ Bϕ = a U b ∈ Bϕ = a U b ∈ B
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Example: GNBA for ϕ = a U bϕ = a U bϕ = a U b ltlmc3.2-54
a, b, a U ba, b, a U ba, b, a U b ¬a,¬b,¬(a U b)¬a,¬b,¬(a U b)¬a,¬b,¬(a U b)
¬a, b, a U b¬a, b, a U b¬a, b, a U b
a,¬b,¬(a U b)a,¬b,¬(a U b)a,¬b,¬(a U b)a,¬b, a U ba,¬b, a U ba,¬b, a U b
initial states: BBB with ϕ = a U b ∈ Bϕ = a U b ∈ Bϕ = a U b ∈ B
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Example: GNBA for ϕ = a U bϕ = a U bϕ = a U b ltlmc3.2-54
a, b, a U ba, b, a U ba, b, a U b ¬a,¬b,¬(a U b)¬a,¬b,¬(a U b)¬a,¬b,¬(a U b)
¬a, b, a U b¬a, b, a U b¬a, b, a U b
a,¬b,¬(a U b)a,¬b,¬(a U b)a,¬b,¬(a U b)a,¬b, a U ba,¬b, a U ba,¬b, a U b
initial states: BBB with ϕ = a U b ∈ Bϕ = a U b ∈ Bϕ = a U b ∈ B
acceptance condition: just one set of accept states
F =F =F = set of all BBB with ϕ �∈ Bϕ �∈ Bϕ �∈ B or b ∈ Bb ∈ Bb ∈ B
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Example: GNBA for ϕ = a U bϕ = a U bϕ = a U b ←−←−←−NBA ltlmc3.2-54
a, b, a U ba, b, a U ba, b, a U b ¬a,¬b,¬(a U b)¬a,¬b,¬(a U b)¬a,¬b,¬(a U b)
¬a, b, a U b¬a, b, a U b¬a, b, a U b
a,¬b,¬(a U b)a,¬b,¬(a U b)a,¬b,¬(a U b)a,¬b, a U ba,¬b, a U ba,¬b, a U b
initial states: BBB with ϕ = a U b ∈ Bϕ = a U b ∈ Bϕ = a U b ∈ B
acceptance condition: just one set of accept states
F =F =F = set of all BBB with ϕ �∈ Bϕ �∈ Bϕ �∈ B or b ∈ Bb ∈ Bb ∈ B
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Example: (G)NBA for ϕ = a U bϕ = a U bϕ = a U b ltlmc3.2-54
a, b, a U ba, b, a U ba, b, a U b ¬a,¬b,¬(a U b)¬a,¬b,¬(a U b)¬a,¬b,¬(a U b)
¬a, b, a U b¬a, b, a U b¬a, b, a U b
a,¬b,¬(a U b)a,¬b,¬(a U b)a,¬b,¬(a U b)a,¬b, a U ba,¬b, a U ba,¬b, a U b
initial states: BBB with ϕ = a U b ∈ Bϕ = a U b ∈ Bϕ = a U b ∈ B
acceptance condition: just one set of accept states
F =F =F = set of all BBB with ϕ �∈ Bϕ �∈ Bϕ �∈ B or b ∈ Bb ∈ Bb ∈ B
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Example: (G)NBA for ϕ = a U bϕ = a U bϕ = a U b ltlmc3.2-54
a, b, a U ba, b, a U ba, b, a U b ¬a,¬b,¬(a U b)¬a,¬b,¬(a U b)¬a,¬b,¬(a U b)
¬a, b, a U b¬a, b, a U b¬a, b, a U b
a,¬b,¬(a U b)a,¬b,¬(a U b)a,¬b,¬(a U b)a,¬b, a U ba,¬b, a U ba,¬b, a U b
transition relation: B ′ ∈ δ(B,B ∩ AP)B ′ ∈ δ(B ,B ∩ AP)B ′ ∈ δ(B,B ∩ AP) iff
a U b ∈ B ⇐⇒(b ∈ B ∨ ( a ∈ B ∧ a U b ∈ B ′ )
)a U b ∈ B ⇐⇒
(b ∈ B ∨ ( a ∈ B ∧ a U b ∈ B ′ )
)a U b ∈ B ⇐⇒
(b ∈ B ∨ ( a ∈ B ∧ a U b ∈ B ′ )
)189 / 527
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Example: (G)NBA for ϕ = a U bϕ = a U bϕ = a U b ltlmc3.2-54
a, b, a U ba, b, a U ba, b, a U b ¬a,¬b,¬(a U b)¬a,¬b,¬(a U b)¬a,¬b,¬(a U b)
¬a, b, a U b¬a, b, a U b¬a, b, a U b
a,¬b,¬(a U b)a,¬b,¬(a U b)a,¬b,¬(a U b)a,¬b, a U ba,¬b, a U ba,¬b, a U b
¬a ∧ b¬a ∧ b¬a ∧ b
¬a ∧ b¬a ∧ b¬a ∧ b
¬a ∧ b¬a ∧ b¬a ∧ b
transition relation: B ′ ∈ δ(B,B ∩ AP)B ′ ∈ δ(B ,B ∩ AP)B ′ ∈ δ(B,B ∩ AP) iff
a U b ∈ B ⇐⇒(b ∈ B ∨ ( a ∈ B ∧ a U b ∈ B ′ )
)a U b ∈ B ⇐⇒
(b ∈ B ∨ ( a ∈ B ∧ a U b ∈ B ′ )
)a U b ∈ B ⇐⇒
(b ∈ B ∨ ( a ∈ B ∧ a U b ∈ B ′ )
)190 / 527
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Example: (G)NBA for ϕ = a U bϕ = a U bϕ = a U b ltlmc3.2-54
a, b, a U ba, b, a U ba, b, a U b ¬a,¬b,¬(a U b)¬a,¬b,¬(a U b)¬a,¬b,¬(a U b)
¬a, b, a U b¬a, b, a U b¬a, b, a U b
a,¬b,¬(a U b)a,¬b,¬(a U b)a,¬b,¬(a U b)a,¬b, a U ba,¬b, a U ba,¬b, a U b
a ∧ ba ∧ ba ∧ ba ∧ ba ∧ ba ∧ b
transition relation: B ′ ∈ δ(B,B ∩ AP)B ′ ∈ δ(B ,B ∩ AP)B ′ ∈ δ(B,B ∩ AP) iff
a U b ∈ B ⇐⇒(b ∈ B ∨ ( a ∈ B ∧ a U b ∈ B ′ )
)a U b ∈ B ⇐⇒
(b ∈ B ∨ ( a ∈ B ∧ a U b ∈ B ′ )
)a U b ∈ B ⇐⇒
(b ∈ B ∨ ( a ∈ B ∧ a U b ∈ B ′ )
)191 / 527
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Example: (G)NBA for ϕ = a U bϕ = a U bϕ = a U b ltlmc3.2-54
a, b, a U ba, b, a U ba, b, a U b ¬a,¬b,¬(a U b)¬a,¬b,¬(a U b)¬a,¬b,¬(a U b)
¬a, b, a U b¬a, b, a U b¬a, b, a U b
a,¬b,¬(a U b)a,¬b,¬(a U b)a,¬b,¬(a U b)a,¬b, a U ba,¬b, a U ba,¬b, a U ba ∧ ¬ba ∧ ¬ba ∧ ¬b
transition relation: B ′ ∈ δ(B,B ∩ AP)B ′ ∈ δ(B ,B ∩ AP)B ′ ∈ δ(B,B ∩ AP) iff
a U b ∈ B ⇐⇒(b ∈ B ∨ ( a ∈ B ∧ a U b ∈ B ′ )
)a U b ∈ B ⇐⇒
(b ∈ B ∨ ( a ∈ B ∧ a U b ∈ B ′ )
)a U b ∈ B ⇐⇒
(b ∈ B ∨ ( a ∈ B ∧ a U b ∈ B ′ )
)192 / 527
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Example: (G)NBA for ϕ = a U bϕ = a U bϕ = a U b ltlmc3.2-54
a, b, a U ba, b, a U ba, b, a U b ¬a,¬b,¬(a U b)¬a,¬b,¬(a U b)¬a,¬b,¬(a U b)
¬a, b, a U b¬a, b, a U b¬a, b, a U b
a,¬b,¬(a U b)a,¬b,¬(a U b)a,¬b,¬(a U b)a,¬b, a U ba,¬b, a U ba,¬b, a U b
¬a ∧ ¬b¬a ∧ ¬b¬a ∧ ¬b
transition relation: B ′ ∈ δ(B,B ∩ AP)B ′ ∈ δ(B ,B ∩ AP)B ′ ∈ δ(B,B ∩ AP) iff
a U b ∈ B ⇐⇒(b ∈ B ∨ ( a ∈ B ∧ a U b ∈ B ′ )
)a U b ∈ B ⇐⇒
(b ∈ B ∨ ( a ∈ B ∧ a U b ∈ B ′ )
)a U b ∈ B ⇐⇒
(b ∈ B ∨ ( a ∈ B ∧ a U b ∈ B ′ )
)193 / 527
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Example: (G)NBA for ϕ = a U bϕ = a U bϕ = a U b ltlmc3.2-54
a, b, a U ba, b, a U ba, b, a U b ¬a,¬b,¬(a U b)¬a,¬b,¬(a U b)¬a,¬b,¬(a U b)
¬a, b, a U b¬a, b, a U b¬a, b, a U b
a,¬b,¬(a U b)a,¬b,¬(a U b)a,¬b,¬(a U b)a,¬b, a U ba,¬b, a U ba,¬b, a U b
a ∧ ¬ba ∧ ¬ba ∧ ¬b
transition relation: B ′ ∈ δ(B,B ∩ AP)B ′ ∈ δ(B ,B ∩ AP)B ′ ∈ δ(B,B ∩ AP) iff
a U b ∈ B ⇐⇒(b ∈ B ∨ ( a ∈ B ∧ a U b ∈ B ′ )
)a U b ∈ B ⇐⇒
(b ∈ B ∨ ( a ∈ B ∧ a U b ∈ B ′ )
)a U b ∈ B ⇐⇒
(b ∈ B ∨ ( a ∈ B ∧ a U b ∈ B ′ )
)194 / 527
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Example: (G)NBA for ϕ = a U bϕ = a U bϕ = a U b ltlmc3.2-55
a, b, a U ba, b, a U ba, b, a U b ¬a,¬b,¬(a U b)¬a,¬b,¬(a U b)¬a,¬b,¬(a U b)
¬a, b, a U b¬a, b, a U b¬a, b, a U b a,¬b,¬(a U b)a,¬b,¬(a U b)a,¬b,¬(a U b)
a,¬b, a U ba,¬b, a U ba,¬b, a U b
¬a ∧ ¬b¬a ∧ ¬b¬a ∧ ¬ba ∧ ba ∧ ba ∧ b
¬a ∧ b¬a ∧ b¬a ∧ b a ∧ ¬ba ∧ ¬ba ∧ ¬b
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Example: (G)NBA for ϕ = a U bϕ = a U bϕ = a U b ltlmc3.2-55
a, b, a U ba, b, a U ba, b, a U b ¬a,¬b,¬(a U b)¬a,¬b,¬(a U b)¬a,¬b,¬(a U b)
¬a, b, a U b¬a, b, a U b¬a, b, a U b a,¬b,¬(a U b)a,¬b,¬(a U b)a,¬b,¬(a U b)
a,¬b, a U ba,¬b, a U ba,¬b, a U b
¬a ∧ ¬b¬a ∧ ¬b¬a ∧ ¬ba ∧ ba ∧ ba ∧ b
¬a ∧ b¬a ∧ b¬a ∧ b a ∧ ¬ba ∧ ¬ba ∧ ¬b
{a}{a}{a} {a}{a}{a} {a, b}{a, b}{a, b} ∅∅∅ ∅∅∅ ∅∅∅ . . . |= a U b. . . |= a U b. . . |= a U b
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Example: (G)NBA for ϕ = a U bϕ = a U bϕ = a U b ltlmc3.2-55
a, b, a U ba, b, a U ba, b, a U b ¬a,¬b,¬(a U b)¬a,¬b,¬(a U b)¬a,¬b,¬(a U b)
¬a, b, a U b¬a, b, a U b¬a, b, a U b a,¬b,¬(a U b)a,¬b,¬(a U b)a,¬b,¬(a U b)
a,¬b, a U ba,¬b, a U ba,¬b, a U b
¬a ∧ ¬b¬a ∧ ¬b¬a ∧ ¬ba ∧ ba ∧ ba ∧ b
¬a ∧ b¬a ∧ b¬a ∧ b a ∧ ¬ba ∧ ¬ba ∧ ¬b
{a}{a}{a} {a}{a}{a} {a, b}{a, b}{a, b} ∅∅∅ ∅∅∅ ∅∅∅ . . . |= a U b. . . |= a U b. . . |= a U b↓↓↓aaa¬b¬b¬bϕϕϕ
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Example: (G)NBA for ϕ = a U bϕ = a U bϕ = a U b ltlmc3.2-55
a, b, a U ba, b, a U ba, b, a U b ¬a,¬b,¬(a U b)¬a,¬b,¬(a U b)¬a,¬b,¬(a U b)
¬a, b, a U b¬a, b, a U b¬a, b, a U b a,¬b,¬(a U b)a,¬b,¬(a U b)a,¬b,¬(a U b)
a,¬b, a U ba,¬b, a U ba,¬b, a U b
¬a ∧ ¬b¬a ∧ ¬b¬a ∧ ¬ba ∧ ba ∧ ba ∧ b
¬a ∧ b¬a ∧ b¬a ∧ b a ∧ ¬ba ∧ ¬ba ∧ ¬b
{a}{a}{a} {a}{a}{a} {a, b}{a, b}{a, b} ∅∅∅ ∅∅∅ ∅∅∅ . . . |= a U b. . . |= a U b. . . |= a U b↓↓↓aaa¬b¬b¬bϕϕϕ
↓↓↓aaa¬b¬b¬bϕϕϕ
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Example: (G)NBA for ϕ = a U bϕ = a U bϕ = a U b ltlmc3.2-55
a, b, a U ba, b, a U ba, b, a U b ¬a,¬b,¬(a U b)¬a,¬b,¬(a U b)¬a,¬b,¬(a U b)
¬a, b, a U b¬a, b, a U b¬a, b, a U b a,¬b,¬(a U b)a,¬b,¬(a U b)a,¬b,¬(a U b)
a,¬b, a U ba,¬b, a U ba,¬b, a U b
¬a ∧ ¬b¬a ∧ ¬b¬a ∧ ¬ba ∧ ba ∧ ba ∧ b
¬a ∧ b¬a ∧ b¬a ∧ b a ∧ ¬ba ∧ ¬ba ∧ ¬b
{a}{a}{a} {a}{a}{a} {a, b}{a, b}{a, b} ∅∅∅ ∅∅∅ ∅∅∅ . . . |= a U b. . . |= a U b. . . |= a U b↓↓↓aaa¬b¬b¬bϕϕϕ
↓↓↓aaa¬b¬b¬bϕϕϕ
↓↓↓aaabbbϕϕϕ
{ } { } { b} ∅199 / 527
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Example: (G)NBA for ϕ = a U bϕ = a U bϕ = a U b ltlmc3.2-55
a, b, a U ba, b, a U ba, b, a U b ¬a,¬b,¬(a U b)¬a,¬b,¬(a U b)¬a,¬b,¬(a U b)
¬a, b, a U b¬a, b, a U b¬a, b, a U b a,¬b,¬(a U b)a,¬b,¬(a U b)a,¬b,¬(a U b)
a,¬b, a U ba,¬b, a U ba,¬b, a U b
¬a ∧ ¬b¬a ∧ ¬b¬a ∧ ¬ba ∧ ba ∧ ba ∧ b
¬a ∧ b¬a ∧ b¬a ∧ b a ∧ ¬ba ∧ ¬ba ∧ ¬b
{a}{a}{a} {a}{a}{a} {a, b}{a, b}{a, b} ∅∅∅ ∅∅∅ ∅∅∅ . . . |= a U b. . . |= a U b. . . |= a U b↓↓↓aaa¬b¬b¬bϕϕϕ
↓↓↓aaa¬b¬b¬bϕϕϕ
↓↓↓aaabbbϕϕϕ
↓↓↓¬a¬a¬a¬b¬b¬b¬ϕ¬ϕ¬ϕ
{ } { } { b} ∅200 / 527
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Example: (G)NBA for ϕ = a U bϕ = a U bϕ = a U b ltlmc3.2-55
a, b, a U ba, b, a U ba, b, a U b ¬a,¬b,¬(a U b)¬a,¬b,¬(a U b)¬a,¬b,¬(a U b)
¬a, b, a U b¬a, b, a U b¬a, b, a U b a,¬b,¬(a U b)a,¬b,¬(a U b)a,¬b,¬(a U b)
a,¬b, a U ba,¬b, a U ba,¬b, a U b
¬a ∧ ¬b¬a ∧ ¬b¬a ∧ ¬ba ∧ ba ∧ ba ∧ b
¬a ∧ b¬a ∧ b¬a ∧ b a ∧ ¬ba ∧ ¬ba ∧ ¬b
{a}{a}{a} {a}{a}{a} {a, b}{a, b}{a, b} ∅∅∅ ∅∅∅ ∅∅∅ . . . |= a U b. . . |= a U b. . . |= a U b↓↓↓aaa¬b¬b¬bϕϕϕ
↓↓↓aaa¬b¬b¬bϕϕϕ
↓↓↓aaabbbϕϕϕ
↓↓↓¬a¬a¬a¬b¬b¬b¬ϕ¬ϕ¬ϕ
↓↓↓¬a¬a¬a¬b¬b¬b¬ϕ¬ϕ¬ϕ
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Example: (G)NBA for ϕ = a U bϕ = a U bϕ = a U b ltlmc3.2-55
a, b, a U ba, b, a U ba, b, a U b ¬a,¬b,¬(a U b)¬a,¬b,¬(a U b)¬a,¬b,¬(a U b)
¬a, b, a U b¬a, b, a U b¬a, b, a U b a,¬b,¬(a U b)a,¬b,¬(a U b)a,¬b,¬(a U b)
a,¬b, a U ba,¬b, a U ba,¬b, a U b
¬a ∧ ¬b¬a ∧ ¬b¬a ∧ ¬ba ∧ ba ∧ ba ∧ b
¬a ∧ b¬a ∧ b¬a ∧ b a ∧ ¬ba ∧ ¬ba ∧ ¬b
{a}{a}{a} {a}{a}{a} {a, b}{a, b}{a, b} ∅∅∅ ∅∅∅ ∅∅∅ . . . |= a U b. . . |= a U b. . . |= a U b↓↓↓aaa¬b¬b¬bϕϕϕ
↓↓↓aaa¬b¬b¬bϕϕϕ
↓↓↓aaabbbϕϕϕ
↓↓↓¬a¬a¬a¬b¬b¬b¬ϕ¬ϕ¬ϕ
↓↓↓¬a¬a¬a¬b¬b¬b¬ϕ¬ϕ¬ϕ
↓↓↓¬a¬a¬a¬b¬b¬b¬ϕ¬ϕ¬ϕ
{ } { } { b} ∅202 / 527
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Example: (G)NBA for ϕ = a U bϕ = a U bϕ = a U b ltlmc3.2-55
a, b, a U ba, b, a U ba, b, a U b ¬a,¬b,¬(a U b)¬a,¬b,¬(a U b)¬a,¬b,¬(a U b)
¬a, b, a U b¬a, b, a U b¬a, b, a U b a,¬b,¬(a U b)a,¬b,¬(a U b)a,¬b,¬(a U b)
a,¬b, a U ba,¬b, a U ba,¬b, a U b
¬a ∧ ¬b¬a ∧ ¬b¬a ∧ ¬ba ∧ ba ∧ ba ∧ b
¬a ∧ b¬a ∧ b¬a ∧ b a ∧ ¬ba ∧ ¬ba ∧ ¬b
{a}{a}{a} {a}{a}{a} {a, b}{a, b}{a, b} ∅∅∅ ∅∅∅ ∅∅∅ . . . |= a U b. . . |= a U b. . . |= a U b↓↓↓aaa¬b¬b¬bϕϕϕ
↓↓↓aaa¬b¬b¬bϕϕϕ
↓↓↓aaabbbϕϕϕ
↓↓↓¬a¬a¬a¬b¬b¬b¬ϕ¬ϕ¬ϕ
↓↓↓¬a¬a¬a¬b¬b¬b¬ϕ¬ϕ¬ϕ
↓↓↓¬a¬a¬a¬b¬b¬b¬ϕ¬ϕ¬ϕ
acceptingrun
{ } { } { b} ∅203 / 527
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Example: (G)NBA for ϕ = a U bϕ = a U bϕ = a U b ltlmc3.2-56
a, b, a U ba, b, a U ba, b, a U b ¬a,¬b,¬(a U b)¬a,¬b,¬(a U b)¬a,¬b,¬(a U b)
¬a, b, a U b¬a, b, a U b¬a, b, a U b a,¬b,¬(a U b)a,¬b,¬(a U b)a,¬b,¬(a U b)
a,¬b, a U ba,¬b, a U ba,¬b, a U b
¬a ∧ ¬b¬a ∧ ¬b¬a ∧ ¬ba ∧ ba ∧ ba ∧ b
¬a ∧ b¬a ∧ b¬a ∧ b a ∧ ¬ba ∧ ¬ba ∧ ¬b
{a} {a} {a} {a} . . . �|= ϕ{a} {a} {a} {a} . . . �|= ϕ{a} {a} {a} {a} . . . �|= ϕ
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Example: (G)NBA for ϕ = a U bϕ = a U bϕ = a U b ltlmc3.2-56
a, b, a U ba, b, a U ba, b, a U b ¬a,¬b,¬(a U b)¬a,¬b,¬(a U b)¬a,¬b,¬(a U b)
¬a, b, a U b¬a, b, a U b¬a, b, a U b a,¬b,¬(a U b)a,¬b,¬(a U b)a,¬b,¬(a U b)
a,¬b, a U ba,¬b, a U ba,¬b, a U b
q0q0q0
¬a ∧ ¬b¬a ∧ ¬b¬a ∧ ¬ba ∧ ba ∧ ba ∧ b
¬a ∧ b¬a ∧ b¬a ∧ b a ∧ ¬ba ∧ ¬ba ∧ ¬b
{a} {a} {a} {a} . . . �|= ϕ{a} {a} {a} {a} . . . �|= ϕ{a} {a} {a} {a} . . . �|= ϕ
only 111 infinite run: q0 q0 q0 . . .q0 q0 q0 . . .q0 q0 q0 . . .
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Example: (G)NBA for ϕ = a U bϕ = a U bϕ = a U b ltlmc3.2-56
a, b, a U ba, b, a U ba, b, a U b ¬a,¬b,¬(a U b)¬a,¬b,¬(a U b)¬a,¬b,¬(a U b)
¬a, b, a U b¬a, b, a U b¬a, b, a U b a,¬b,¬(a U b)a,¬b,¬(a U b)a,¬b,¬(a U b)
a,¬b, a U ba,¬b, a U ba,¬b, a U b
q0q0q0
{a}{a}{a}
∅∅∅...........................
.........
{a, b}{a, b}{a, b}..................
..................
{b}{b}{b}
..................
.........
.........
{a}{a}{a}
.........
{a} {a} {a} {a} . . . �|= ϕ{a} {a} {a} {a} . . . �|= ϕ{a} {a} {a} {a} . . . �|= ϕ
only 111 infinite run: q0 q0 q0 . . .q0 q0 q0 . . .q0 q0 q0 . . .
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Example: (G)NBA for ϕ = a U bϕ = a U bϕ = a U b ltlmc3.2-56
a, b, a U ba, b, a U ba, b, a U b ¬a,¬b,¬(a U b)¬a,¬b,¬(a U b)¬a,¬b,¬(a U b)
¬a, b, a U b¬a, b, a U b¬a, b, a U b a,¬b,¬(a U b)a,¬b,¬(a U b)a,¬b,¬(a U b)
a,¬b, a U ba,¬b, a U ba,¬b, a U b
q0q0q0
{a}{a}{a}
∅∅∅...........................
.........
{a, b}{a, b}{a, b}..................
..................
{b}{b}{b}
..................
.........
.........
{a}{a}{a}
.........
{a} {a} {a} {a} . . . �|= ϕ{a} {a} {a} {a} . . . �|= ϕ{a} {a} {a} {a} . . . �|= ϕ
only 111 infinite run: q0 q0 q0 . . .q0 q0 q0 . . .q0 q0 q0 . . . not accepting
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GNBA for LTL-formula ϕϕϕ ltlmc3.2-57a
G = (Q, 2AP , δ,Q0,F)G = (Q, 2AP , δ,Q0,F)G = (Q, 2AP , δ,Q0,F)
state space: QQQ ==={B ⊆ cl(ϕ) : B
{B ⊆ cl(ϕ) : B
{B ⊆ cl(ϕ) : B is elementary
}}}initial states: Q0Q0Q0 ===
{B ∈ Q : ϕ ∈ B
}{B ∈ Q : ϕ ∈ B
}{B ∈ Q : ϕ ∈ B
}transition relation: for B ∈ QB ∈ QB ∈ Q and A ∈ 2APA ∈ 2APA ∈ 2AP :
if A �= B ∩ APA �= B ∩ APA �= B ∩ AP then δ(B,A) = ∅δ(B ,A) = ∅δ(B ,A) = ∅
if A = B ∩ APA = B ∩ APA = B ∩ AP then δ(B,A) =δ(B ,A) =δ(B ,A) = set of all B ′ ∈ QB ′ ∈ QB ′ ∈ Q s.t.
©ψ ∈ B©ψ ∈ B©ψ ∈ B iff ψ ∈ B ′ψ ∈ B ′ψ ∈ B ′
ψ1 Uψ2 ∈ Bψ1 Uψ2 ∈ Bψ1 Uψ2 ∈ B iff (ψ2 ∈ B) ∨ (ψ1 ∈ B ∧ ψ1 Uψ2 ∈ B ′)(ψ2 ∈ B) ∨ (ψ1 ∈ B ∧ ψ1 Uψ2 ∈ B ′)(ψ2 ∈ B) ∨ (ψ1 ∈ B ∧ ψ1 Uψ2 ∈ B ′)
acceptance set F ={Fψ1 Uψ2
: ψ1 Uψ2 ∈ cl(ϕ)}
F ={F ψ1 Uψ2 : ψ1 Uψ2 ∈ cl(ϕ)
}F =
{F ψ1 Uψ2
: ψ1 Uψ2 ∈ cl(ϕ)}
where Fψ1 Uψ2=
{B ∈ Q : ψ1 Uψ2 /∈ B ∨ ψ2 ∈ B
}F ψ1 Uψ2
={B ∈ Q : ψ1 Uψ2 /∈ B ∨ ψ2 ∈ B
}Fψ1 Uψ2
={B ∈ Q : ψ1 Uψ2 /∈ B ∨ ψ2 ∈ B
}208 / 527
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Soundness ltlmc3.2-soundness-LTL-2-GNBA
.... of the construction LTL formula ϕϕϕ��� GNBA GGG
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Soundness ltlmc3.2-soundness-LTL-2-GNBA
Let ϕϕϕ be an LTL-formula and G = (Q, 2AP , δ,Q0,F)G = (Q, 2AP , δ,Q0,F)G = (Q, 2AP , δ,Q0,F) bethe constructed GNBA.
Claim: Words(ϕ) = Lω(G)Words(ϕ) = Lω(G)Words(ϕ) = Lω(G)
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Soundness ltlmc3.2-soundness-LTL-2-GNBA
Let ϕϕϕ be an LTL-formula and G = (Q, 2AP , δ,Q0,F)G = (Q, 2AP , δ,Q0,F)G = (Q, 2AP , δ,Q0,F) bethe constructed GNBA.
Claim: Words(ϕ) = Lω(G)Words(ϕ) = Lω(G)Words(ϕ) = Lω(G)
“⊆⊆⊆” show: each infinite word A0 A1 A2 ... ∈ (2AP)ωA0 A1 A2 ... ∈ (2AP)ωA0 A1 A2 ... ∈ (2AP)ω
with A0 A1 A2 ... |= ϕA0 A1 A2 ... |= ϕA0 A1 A2 ... |= ϕ
has an accepting run in GGG
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Soundness ltlmc3.2-soundness-LTL-2-GNBA
Let ϕϕϕ be an LTL-formula and G = (Q, 2AP , δ,Q0,F)G = (Q, 2AP , δ,Q0,F)G = (Q, 2AP , δ,Q0,F) bethe constructed GNBA.
Claim: Words(ϕ) = Lω(G)Words(ϕ) = Lω(G)Words(ϕ) = Lω(G)
“⊆⊆⊆” show: each infinite word A0 A1 A2 ... ∈ (2AP)ωA0 A1 A2 ... ∈ (2AP)ωA0 A1 A2 ... ∈ (2AP)ω
with A0 A1 A2 ... |= ϕA0 A1 A2 ... |= ϕA0 A1 A2 ... |= ϕ
has an accepting run in GGG
“⊇⊇⊇” show: for all infinite words A0 A1 A2 ... ∈ Lω(G)A0 A1 A2 ... ∈ Lω(G)A0 A1 A2 ... ∈ Lω(G) :
A0 A1 A2 ... |= ϕA0 A1 A2 ... |= ϕA0 A1 A2 ... |= ϕ
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Soundness ltlmc3.2-soundness-LTL-2-GNBA
Let ϕϕϕ be an LTL-formula and G = (Q, 2AP , δ,Q0,F)G = (Q, 2AP , δ,Q0,F)G = (Q, 2AP , δ,Q0,F) bethe constructed GNBA.
Claim: Words(ϕ) = Lω(G)Words(ϕ) = Lω(G)Words(ϕ) = Lω(G)
“⊆⊆⊆” show: each infinite word A0 A1 A2 ... ∈ (2AP)ωA0 A1 A2 ... ∈ (2AP)ωA0 A1 A2 ... ∈ (2AP)ω
with A0 A1 A2 ... |= ϕA0 A1 A2 ... |= ϕA0 A1 A2 ... |= ϕ
has an accepting run in GGG
“⊇⊇⊇” show: for all infinite words A0 A1 A2 ... ∈ Lω(G)A0 A1 A2 ... ∈ Lω(G)A0 A1 A2 ... ∈ Lω(G) :
A0 A1 A2 ... |= ϕA0 A1 A2 ... |= ϕA0 A1 A2 ... |= ϕ
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Accepting runs for the elements of Words(ϕ)Words(ϕ)Words(ϕ) ltlmc3.2-47-copy
LTL formula ϕϕϕ��� GNBA GGG for Words(ϕ)Words(ϕ)Words(ϕ)
states of GGG === elementary formula-sets B ⊆ cl(ϕ)B ⊆ cl(ϕ)B ⊆ cl(ϕ)
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Accepting runs for the elements of Words(ϕ)Words(ϕ)Words(ϕ) ltlmc3.2-47-copy
LTL formula ϕϕϕ��� GNBA GGG for Words(ϕ)Words(ϕ)Words(ϕ)
states of GGG === elementary formula-sets B ⊆ cl(ϕ)B ⊆ cl(ϕ)B ⊆ cl(ϕ)
s.t. each word σ = A0 A1 A2... ∈Words(ϕ)σ = A0 A1 A2... ∈ Words(ϕ)σ = A0 A1 A2... ∈ Words(ϕ) can beextended to an accepting run B0 B1 B2 . . .B0 B1 B2 . . .B0 B1 B2 . . . in GGG
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Accepting runs for the elements of Words(ϕ)Words(ϕ)Words(ϕ) ltlmc3.2-47-copy
LTL formula ϕϕϕ��� GNBA GGG for Words(ϕ)Words(ϕ)Words(ϕ)
states of GGG === elementary formula-sets B ⊆ cl(ϕ)B ⊆ cl(ϕ)B ⊆ cl(ϕ)
s.t. each word σ = A0 A1 A2... ∈Words(ϕ)σ = A0 A1 A2... ∈ Words(ϕ)σ = A0 A1 A2... ∈ Words(ϕ) can beextended to an accepting run B0 B1 B2 . . .B0 B1 B2 . . .B0 B1 B2 . . . in GGG
Example: ϕ = a U(¬a ∧ b)ϕ = a U(¬a ∧ b)ϕ = a U(¬a ∧ b)
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Accepting runs for the elements of Words(ϕ)Words(ϕ)Words(ϕ) ltlmc3.2-47-copy
LTL formula ϕϕϕ��� GNBA GGG for Words(ϕ)Words(ϕ)Words(ϕ)
states of GGG === elementary formula-sets B ⊆ cl(ϕ)B ⊆ cl(ϕ)B ⊆ cl(ϕ)
s.t. each word σ = A0 A1 A2... ∈Words(ϕ)σ = A0 A1 A2... ∈ Words(ϕ)σ = A0 A1 A2... ∈ Words(ϕ) can beextended to an accepting run B0 B1 B2 . . .B0 B1 B2 . . .B0 B1 B2 . . . in GGG
Example: ϕ = a U(¬a ∧ b)ϕ = a U(¬a ∧ b)ϕ = a U(¬a ∧ b)
{a} {a} {a, b} {b} ∅ ∅ . . . |= ϕ{a} {a} {a, b} {b} ∅ ∅ . . . |= ϕ{a} {a} {a, b} {b} ∅ ∅ . . . |= ϕ
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Accepting runs for the elements of Words(ϕ)Words(ϕ)Words(ϕ) ltlmc3.2-47-copy
LTL formula ϕϕϕ��� GNBA GGG for Words(ϕ)Words(ϕ)Words(ϕ)
states of GGG === elementary formula-sets B ⊆ cl(ϕ)B ⊆ cl(ϕ)B ⊆ cl(ϕ)
s.t. each word σ = A0 A1 A2... ∈Words(ϕ)σ = A0 A1 A2... ∈ Words(ϕ)σ = A0 A1 A2... ∈ Words(ϕ) can beextended to an accepting run B0 B1 B2 . . .B0 B1 B2 . . .B0 B1 B2 . . . in GGG
Example: ϕ = a U(¬a ∧ b)ϕ = a U(¬a ∧ b)ϕ = a U(¬a ∧ b)
{a} {a} {a, b} {b} ∅ ∅ . . . |= ϕ↓ ↓ ↓ ↓ ↓ ↓B0 B1 B2 B3 B4 B5
{a} {a} {a, b} {b} ∅ ∅ . . . |= ϕ↓ ↓ ↓ ↓ ↓ ↓B0 B1 B2 B3 B4 B5
{a} {a} {a, b} {b} ∅ ∅ . . . |= ϕ↓ ↓ ↓ ↓ ↓ ↓B0 B1 B2 B3 B4 B5
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Accepting runs for the elements of Words(ϕ)Words(ϕ)Words(ϕ) ltlmc3.2-47-copy
LTL formula ϕϕϕ��� GNBA GGG for Words(ϕ)Words(ϕ)Words(ϕ)
states of GGG === elementary formula-sets B ⊆ cl(ϕ)B ⊆ cl(ϕ)B ⊆ cl(ϕ)
s.t. each word σ = A0 A1 A2... ∈Words(ϕ)σ = A0 A1 A2... ∈ Words(ϕ)σ = A0 A1 A2... ∈ Words(ϕ) can beextended to an accepting run B0 B1 B2 . . .B0 B1 B2 . . .B0 B1 B2 . . . in GGG
Example: ϕ = a U(¬a ∧ b)ϕ = a U(¬a ∧ b)ϕ = a U(¬a ∧ b) ψ = ¬a ∧ bψ = ¬a ∧ bψ = ¬a ∧ b
{a} {a} {a, b} {b} ∅ ∅ . . . |= ϕ↓ ↓ ↓ ↓ ↓ ↓B0 B1 B2 B3 B4 B5
{a} {a} {a, b} {b} ∅ ∅ . . . |= ϕ↓ ↓ ↓ ↓ ↓ ↓B0 B1 B2 B3 B4 B5
{a} {a} {a, b} {b} ∅ ∅ . . . |= ϕ↓ ↓ ↓ ↓ ↓ ↓B0 B1 B2 B3 B4 B5
where the BiBiBi ’s are states in GGG, i.e., elementarysubsets of {a,¬a, b,¬b, ψ,¬ψ, ϕ,¬ϕ}{a,¬a, b,¬b, ψ,¬ψ, ϕ,¬ϕ}{a,¬a, b,¬b, ψ,¬ψ, ϕ,¬ϕ}
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Accepting runs for the elements of Words(ϕ)Words(ϕ)Words(ϕ) ltlmc3.2-47-copy
LTL formula ϕϕϕ��� GNBA GGG for Words(ϕ)Words(ϕ)Words(ϕ)
states of GGG === elementary formula-sets B ⊆ cl(ϕ)B ⊆ cl(ϕ)B ⊆ cl(ϕ)
s.t. each word σ = A0 A1 A2... ∈Words(ϕ)σ = A0 A1 A2... ∈ Words(ϕ)σ = A0 A1 A2... ∈ Words(ϕ) can beextended to an accepting run B0 B1 B2 . . .B0 B1 B2 . . .B0 B1 B2 . . . in GGG
Example: ϕ = a U(¬a ∧ b)ϕ = a U(¬a ∧ b)ϕ = a U(¬a ∧ b) ψ = ¬a ∧ bψ = ¬a ∧ bψ = ¬a ∧ b
{a} {a} {a, b} {b} ∅ ∅ . . . |= ϕ
↓a¬b¬ψϕ
{a} {a} {a, b} {b} ∅ ∅ . . . |= ϕ↓a¬b¬ψϕ
{a} {a} {a, b} {b} ∅ ∅ . . . |= ϕ
↓a¬b¬ψϕ
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Accepting runs for the elements of Words(ϕ)Words(ϕ)Words(ϕ) ltlmc3.2-47-copy
LTL formula ϕϕϕ��� GNBA GGG for Words(ϕ)Words(ϕ)Words(ϕ)
states of GGG === elementary formula-sets B ⊆ cl(ϕ)B ⊆ cl(ϕ)B ⊆ cl(ϕ)
s.t. each word σ = A0 A1 A2... ∈Words(ϕ)σ = A0 A1 A2... ∈ Words(ϕ)σ = A0 A1 A2... ∈ Words(ϕ) can beextended to an accepting run B0 B1 B2 . . .B0 B1 B2 . . .B0 B1 B2 . . . in GGG
Example: ϕ = a U(¬a ∧ b)ϕ = a U(¬a ∧ b)ϕ = a U(¬a ∧ b) ψ = ¬a ∧ bψ = ¬a ∧ bψ = ¬a ∧ b
{a} {a} {a, b} {b} ∅ ∅ . . . |= ϕ
↓ ↓a¬b¬ψϕ
a¬b¬ψϕ
{a} {a} {a, b} {b} ∅ ∅ . . . |= ϕ↓ ↓a¬b¬ψϕ
a¬b¬ψϕ
{a} {a} {a, b} {b} ∅ ∅ . . . |= ϕ
↓ ↓a¬b¬ψϕ
a¬b¬ψϕ
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Accepting runs for the elements of Words(ϕ)Words(ϕ)Words(ϕ) ltlmc3.2-47-copy
LTL formula ϕϕϕ��� GNBA GGG for Words(ϕ)Words(ϕ)Words(ϕ)
states of GGG === elementary formula-sets B ⊆ cl(ϕ)B ⊆ cl(ϕ)B ⊆ cl(ϕ)
s.t. each word σ = A0 A1 A2... ∈Words(ϕ)σ = A0 A1 A2... ∈ Words(ϕ)σ = A0 A1 A2... ∈ Words(ϕ) can beextended to an accepting run B0 B1 B2 . . .B0 B1 B2 . . .B0 B1 B2 . . . in GGG
Example: ϕ = a U(¬a ∧ b)ϕ = a U(¬a ∧ b)ϕ = a U(¬a ∧ b) ψ = ¬a ∧ bψ = ¬a ∧ bψ = ¬a ∧ b
{a} {a} {a, b} {b} ∅ ∅ . . . |= ϕ
↓ ↓ ↓a¬b¬ψϕ
a¬b¬ψϕ
ab¬ψϕ
{a} {a} {a, b} {b} ∅ ∅ . . . |= ϕ↓ ↓ ↓a¬b¬ψϕ
a¬b¬ψϕ
ab¬ψϕ
{a} {a} {a, b} {b} ∅ ∅ . . . |= ϕ
↓ ↓ ↓a¬b¬ψϕ
a¬b¬ψϕ
ab¬ψϕ
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Accepting runs for the elements of Words(ϕ)Words(ϕ)Words(ϕ) ltlmc3.2-47-copy
LTL formula ϕϕϕ��� GNBA GGG for Words(ϕ)Words(ϕ)Words(ϕ)
states of GGG === elementary formula-sets B ⊆ cl(ϕ)B ⊆ cl(ϕ)B ⊆ cl(ϕ)
s.t. each word σ = A0 A1 A2... ∈Words(ϕ)σ = A0 A1 A2... ∈ Words(ϕ)σ = A0 A1 A2... ∈ Words(ϕ) can beextended to an accepting run B0 B1 B2 . . .B0 B1 B2 . . .B0 B1 B2 . . . in GGG
Example: ϕ = a U(¬a ∧ b)ϕ = a U(¬a ∧ b)ϕ = a U(¬a ∧ b) ψ = ¬a ∧ bψ = ¬a ∧ bψ = ¬a ∧ b
{a} {a} {a, b} {b} ∅ ∅ . . . |= ϕ
↓ ↓ ↓ ↓a¬b¬ψϕ
a¬b¬ψϕ
ab¬ψϕ
¬abψϕ
{a} {a} {a, b} {b} ∅ ∅ . . . |= ϕ↓ ↓ ↓ ↓a¬b¬ψϕ
a¬b¬ψϕ
ab¬ψϕ
¬abψϕ
{a} {a} {a, b} {b} ∅ ∅ . . . |= ϕ
↓ ↓ ↓ ↓a¬b¬ψϕ
a¬b¬ψϕ
ab¬ψϕ
¬abψϕ
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Accepting runs for the elements of Words(ϕ)Words(ϕ)Words(ϕ) ltlmc3.2-47-copy
LTL formula ϕϕϕ��� GNBA GGG for Words(ϕ)Words(ϕ)Words(ϕ)
states of GGG === elementary formula-sets B ⊆ cl(ϕ)B ⊆ cl(ϕ)B ⊆ cl(ϕ)
s.t. each word σ = A0 A1 A2... ∈Words(ϕ)σ = A0 A1 A2... ∈ Words(ϕ)σ = A0 A1 A2... ∈ Words(ϕ) can beextended to an accepting run B0 B1 B2 . . .B0 B1 B2 . . .B0 B1 B2 . . . in GGG
Example: ϕ = a U(¬a ∧ b)ϕ = a U(¬a ∧ b)ϕ = a U(¬a ∧ b) ψ = ¬a ∧ bψ = ¬a ∧ bψ = ¬a ∧ b
{a} {a} {a, b} {b} ∅ ∅ . . . |= ϕ
↓ ↓ ↓ ↓ ↓a¬b¬ψϕ
a¬b¬ψϕ
ab¬ψϕ
¬abψϕ
¬a¬b¬ψ¬ϕ
{a} {a} {a, b} {b} ∅ ∅ . . . |= ϕ↓ ↓ ↓ ↓ ↓a¬b¬ψϕ
a¬b¬ψϕ
ab¬ψϕ
¬abψϕ
¬a¬b¬ψ¬ϕ
{a} {a} {a, b} {b} ∅ ∅ . . . |= ϕ
↓ ↓ ↓ ↓ ↓a¬b¬ψϕ
a¬b¬ψϕ
ab¬ψϕ
¬abψϕ
¬a¬b¬ψ¬ϕ
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Accepting runs for the elements of Words(ϕ)Words(ϕ)Words(ϕ) ltlmc3.2-47-copy
LTL formula ϕϕϕ��� GNBA GGG for Words(ϕ)Words(ϕ)Words(ϕ)
states of GGG === elementary formula-sets B ⊆ cl(ϕ)B ⊆ cl(ϕ)B ⊆ cl(ϕ)
s.t. each word σ = A0 A1 A2... ∈Words(ϕ)σ = A0 A1 A2... ∈ Words(ϕ)σ = A0 A1 A2... ∈ Words(ϕ) can beextended to an accepting run B0 B1 B2 . . .B0 B1 B2 . . .B0 B1 B2 . . . in GGG
Example: ϕ = a U(¬a ∧ b)ϕ = a U(¬a ∧ b)ϕ = a U(¬a ∧ b) ψ = ¬a ∧ bψ = ¬a ∧ bψ = ¬a ∧ b
{a} {a} {a, b} {b} ∅ ∅ . . . |= ϕ
↓ ↓ ↓ ↓ ↓ ↓a¬b¬ψϕ
a¬b¬ψϕ
ab¬ψϕ
¬abψϕ
¬a¬b¬ψ¬ϕ
¬a¬b¬ψ¬ϕ
. . .. . .. . .
{a} {a} {a, b} {b} ∅ ∅ . . . |= ϕ↓ ↓ ↓ ↓ ↓ ↓a¬b¬ψϕ
a¬b¬ψϕ
ab¬ψϕ
¬abψϕ
¬a¬b¬ψ¬ϕ
¬a¬b¬ψ¬ϕ
. . .. . .. . .
{a} {a} {a, b} {b} ∅ ∅ . . . |= ϕ
↓ ↓ ↓ ↓ ↓ ↓a¬b¬ψϕ
a¬b¬ψϕ
ab¬ψϕ
¬abψϕ
¬a¬b¬ψ¬ϕ
¬a¬b¬ψ¬ϕ
. . .. . .. . .
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GNBA for LTL-formula ϕϕϕ ltlmc3.2-57a
G = (Q, 2AP , δ,Q0,F)G = (Q, 2AP , δ,Q0,F)G = (Q, 2AP , δ,Q0,F)
state space: QQQ ==={B ⊆ cl(ϕ) : B
{B ⊆ cl(ϕ) : B
{B ⊆ cl(ϕ) : B is elementary
}}}initial states: Q0Q0Q0 ===
{B ∈ Q : ϕ ∈ B
}{B ∈ Q : ϕ ∈ B
}{B ∈ Q : ϕ ∈ B
}transition relation: for B ∈ QB ∈ QB ∈ Q and A ∈ 2APA ∈ 2APA ∈ 2AP :
if A �= B ∩ APA �= B ∩ APA �= B ∩ AP then δ(B,A) = ∅δ(B ,A) = ∅δ(B ,A) = ∅
if A = B ∩ APA = B ∩ APA = B ∩ AP then δ(B,A) =δ(B ,A) =δ(B ,A) = set of all B ′ ∈ QB ′ ∈ QB ′ ∈ Q s.t.
©ψ ∈ B©ψ ∈ B©ψ ∈ B iff ψ ∈ B ′ψ ∈ B ′ψ ∈ B ′
ψ1 Uψ2 ∈ Bψ1 Uψ2 ∈ Bψ1 Uψ2 ∈ B iff (ψ2 ∈ B) ∨ (ψ1 ∈ B ∧ ψ1 Uψ2 ∈ B ′)(ψ2 ∈ B) ∨ (ψ1 ∈ B ∧ ψ1 Uψ2 ∈ B ′)(ψ2 ∈ B) ∨ (ψ1 ∈ B ∧ ψ1 Uψ2 ∈ B ′)
acceptance set F ={Fψ1 Uψ2
: ψ1 Uψ2 ∈ cl(ϕ)}
F ={F ψ1 Uψ2 : ψ1 Uψ2 ∈ cl(ϕ)
}F =
{F ψ1 Uψ2
: ψ1 Uψ2 ∈ cl(ϕ)}
where Fψ1 Uψ2=
{B ∈ Q : ψ1 Uψ2 /∈ B ∨ ψ2 ∈ B
}F ψ1 Uψ2
={B ∈ Q : ψ1 Uψ2 /∈ B ∨ ψ2 ∈ B
}Fψ1 Uψ2
={B ∈ Q : ψ1 Uψ2 /∈ B ∨ ψ2 ∈ B
}226 / 527
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Elementary formula-sets ltlmc3.2-50a-copy
B ⊆ cl(ϕ)B ⊆ cl(ϕ)B ⊆ cl(ϕ) is elementary iff:
(i) BBB is maximal consistent w.r.t. prop. logic,i.e., if ψψψ, ψ1 ∧ ψ2 ∈ cl(ϕ)ψ1 ∧ ψ2 ∈ cl(ϕ)ψ1 ∧ ψ2 ∈ cl(ϕ) then:
ψ �∈ Bψ �∈ Bψ �∈ B iff ¬ψ ∈ B¬ψ ∈ B¬ψ ∈ B
ψ1 ∧ ψ2 ∈ Bψ1 ∧ ψ2 ∈ Bψ1 ∧ ψ2 ∈ B iff ψ1 ∈ Bψ1 ∈ Bψ1 ∈ B and ψ2 ∈ Bψ2 ∈ Bψ2 ∈ B
true ∈ cl(ϕ)true ∈ cl(ϕ)true ∈ cl(ϕ) implies true ∈ Btrue ∈ Btrue ∈ B
(ii) BBB is locally consistent with respect to until UUU,i.e., if ψ1 Uψ2 ∈ cl(ϕ)ψ1 Uψ2 ∈ cl(ϕ)ψ1 Uψ2 ∈ cl(ϕ) then:
if ψ1 Uψ2 ∈ Bψ1 Uψ2 ∈ Bψ1 Uψ2 ∈ B and ψ2 �∈ Bψ2 �∈ Bψ2 �∈ B then ψ1 ∈ Bψ1 ∈ Bψ1 ∈ B
if ψ2 ∈ Bψ2 ∈ Bψ2 ∈ B then ψ1 Uψ2 ∈ Bψ1 Uψ2 ∈ Bψ1 Uψ2 ∈ B227 / 527
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Soundness ltlmc3.2-soundness-LTL-2-GNBA2
Let ϕϕϕ be an LTL-formula and G = (Q, 2AP , δ,Q0,F)G = (Q, 2AP , δ,Q0,F)G = (Q, 2AP , δ,Q0,F) bethe constructed GNBA.
Claim: Words(ϕ) = Lω(G)Words(ϕ) = Lω(G)Words(ϕ) = Lω(G)
“⊆⊆⊆” show: each infinite word A0 A1 A2 ... ∈ (2AP)ωA0 A1 A2 ... ∈ (2AP)ωA0 A1 A2 ... ∈ (2AP)ω
with A0 A1 A2 ... |= ϕA0 A1 A2 ... |= ϕA0 A1 A2 ... |= ϕ
has an accepting run in GGG
“⊇⊇⊇” show: for all infinite words A0 A1 A2 ... ∈ Lω(G)A0 A1 A2 ... ∈ Lω(G)A0 A1 A2 ... ∈ Lω(G) :
A0 A1 A2 ... |= ϕA0 A1 A2 ... |= ϕA0 A1 A2 ... |= ϕ
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Soundness ltlmc3.2-soundness-LTL-2-GNBA2
Let ϕϕϕ be an LTL-formula and G = (Q, 2AP , δ,Q0,F)G = (Q, 2AP , δ,Q0,F)G = (Q, 2AP , δ,Q0,F) bethe constructed GNBA.
Claim: Words(ϕ) = Lω(G)Words(ϕ) = Lω(G)Words(ϕ) = Lω(G)
“⊆⊆⊆” show: each infinite word A0 A1 A2 ... ∈ (2AP)ωA0 A1 A2 ... ∈ (2AP)ωA0 A1 A2 ... ∈ (2AP)ω
with A0 A1 A2 ... |= ϕA0 A1 A2 ... |= ϕA0 A1 A2 ... |= ϕ
has an accepting run in GGG
“⊇⊇⊇” show: for all infinite words A0 A1 A2 ... ∈ Lω(G)A0 A1 A2 ... ∈ Lω(G)A0 A1 A2 ... ∈ Lω(G) :
A0 A1 A2 ... |= ϕA0 A1 A2 ... |= ϕA0 A1 A2 ... |= ϕ
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Proof of Lω(G) ⊆ Words(ϕ)Lω(G) ⊆ Words(ϕ)Lω(G) ⊆ Words(ϕ) ltlmc3.2-59
Claim: If B0A0→ B1
A1→ B2A2→ ...B0
A0→ B1A1→ B2
A2→ ...B0A0→ B1
A1→ B2A2→ ... is a path in GGG s.t.
∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F
then for all formulas ψ ∈ cl(ϕ)ψ ∈ cl(ϕ)ψ ∈ cl(ϕ):
ψ ∈ B0ψ ∈ B0ψ ∈ B0 iff A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ
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Proof of Lω(G) ⊆ Words(ϕ)Lω(G) ⊆ Words(ϕ)Lω(G) ⊆ Words(ϕ) ltlmc3.2-59
Claim: If B0A0→ B1
A1→ B2A2→ ...B0
A0→ B1A1→ B2
A2→ ...B0A0→ B1
A1→ B2A2→ ... is a path in GGG s.t.
∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F
then for all formulas ψ ∈ cl(ϕ)ψ ∈ cl(ϕ)ψ ∈ cl(ϕ):
ψ ∈ B0ψ ∈ B0ψ ∈ B0 iff A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ
The claim yields that for each σ = A0 A1 A2 . . . ∈ Lω(G)σ = A0 A1 A2 . . . ∈ Lω(G)σ = A0 A1 A2 . . . ∈ Lω(G):
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Proof of Lω(G) ⊆ Words(ϕ)Lω(G) ⊆ Words(ϕ)Lω(G) ⊆ Words(ϕ) ltlmc3.2-59
Claim: If B0A0→ B1
A1→ B2A2→ ...B0
A0→ B1A1→ B2
A2→ ...B0A0→ B1
A1→ B2A2→ ... is a path in GGG s.t.
∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F
then for all formulas ψ ∈ cl(ϕ)ψ ∈ cl(ϕ)ψ ∈ cl(ϕ):
ψ ∈ B0ψ ∈ B0ψ ∈ B0 iff A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ
The claim yields that for each σ = A0 A1 A2 . . . ∈ Lω(G)σ = A0 A1 A2 . . . ∈ Lω(G)σ = A0 A1 A2 . . . ∈ Lω(G):=⇒=⇒=⇒ there is an accepting run B0 B1 B2 . . .B0 B1 B2 . . .B0 B1 B2 . . . for σσσ
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Proof of Lω(G) ⊆ Words(ϕ)Lω(G) ⊆ Words(ϕ)Lω(G) ⊆ Words(ϕ) ltlmc3.2-59
Claim: If B0A0→ B1
A1→ B2A2→ ...B0
A0→ B1A1→ B2
A2→ ...B0A0→ B1
A1→ B2A2→ ... is a path in GGG s.t.
∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F
then for all formulas ψ ∈ cl(ϕ)ψ ∈ cl(ϕ)ψ ∈ cl(ϕ):
ψ ∈ B0ψ ∈ B0ψ ∈ B0 iff A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ
The claim yields that for each σ = A0 A1 A2 . . . ∈ Lω(G)σ = A0 A1 A2 . . . ∈ Lω(G)σ = A0 A1 A2 . . . ∈ Lω(G):=⇒=⇒=⇒ there is an accepting run B0 B1 B2 . . .B0 B1 B2 . . .B0 B1 B2 . . . for σσσ
=⇒=⇒=⇒ B0A0→ B1
A1→ B2A2→ ...B0
A0→ B1A1→ B2
A2→ ...B0A0→ B1
A1→ B2A2→ ... is a path in GGG
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Proof of Lω(G) ⊆ Words(ϕ)Lω(G) ⊆ Words(ϕ)Lω(G) ⊆ Words(ϕ) ltlmc3.2-59
Claim: If B0A0→ B1
A1→ B2A2→ ...B0
A0→ B1A1→ B2
A2→ ...B0A0→ B1
A1→ B2A2→ ... is a path in GGG s.t.
∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F
then for all formulas ψ ∈ cl(ϕ)ψ ∈ cl(ϕ)ψ ∈ cl(ϕ):
ψ ∈ B0ψ ∈ B0ψ ∈ B0 iff A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ
The claim yields that for each σ = A0 A1 A2 . . . ∈ Lω(G)σ = A0 A1 A2 . . . ∈ Lω(G)σ = A0 A1 A2 . . . ∈ Lω(G):=⇒=⇒=⇒ there is an accepting run B0 B1 B2 . . .B0 B1 B2 . . .B0 B1 B2 . . . for σσσ
=⇒=⇒=⇒ B0A0→ B1
A1→ B2A2→ ...B0
A0→ B1A1→ B2
A2→ ...B0A0→ B1
A1→ B2A2→ ... is a path in GGG s.t. ϕ ∈ B0ϕ ∈ B0ϕ ∈ B0
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Proof of Lω(G) ⊆ Words(ϕ)Lω(G) ⊆ Words(ϕ)Lω(G) ⊆ Words(ϕ) ltlmc3.2-59
Claim: If B0A0→ B1
A1→ B2A2→ ...B0
A0→ B1A1→ B2
A2→ ...B0A0→ B1
A1→ B2A2→ ... is a path in GGG s.t.
∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F
then for all formulas ψ ∈ cl(ϕ)ψ ∈ cl(ϕ)ψ ∈ cl(ϕ):
ψ ∈ B0ψ ∈ B0ψ ∈ B0 iff A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ
The claim yields that for each σ = A0 A1 A2 . . . ∈ Lω(G)σ = A0 A1 A2 . . . ∈ Lω(G)σ = A0 A1 A2 . . . ∈ Lω(G):=⇒=⇒=⇒ there is an accepting run B0 B1 B2 . . .B0 B1 B2 . . .B0 B1 B2 . . . for σσσ
=⇒=⇒=⇒ B0A0→ B1
A1→ B2A2→ ...B0
A0→ B1A1→ B2
A2→ ...B0A0→ B1
A1→ B2A2→ ... is a path in GGG s.t. ϕ ∈ B0ϕ ∈ B0ϕ ∈ B0
↑↑↑as B0 ∈ Q0B0 ∈ Q0B0 ∈ Q0
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Proof of Lω(G) ⊆ Words(ϕ)Lω(G) ⊆ Words(ϕ)Lω(G) ⊆ Words(ϕ) ltlmc3.2-59
Claim: If B0A0→ B1
A1→ B2A2→ ...B0
A0→ B1A1→ B2
A2→ ...B0A0→ B1
A1→ B2A2→ ... is a path in GGG s.t.
∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F (*)
then for all formulas ψ ∈ cl(ϕ)ψ ∈ cl(ϕ)ψ ∈ cl(ϕ):
ψ ∈ B0ψ ∈ B0ψ ∈ B0 iff A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ
The claim yields that for each σ = A0 A1 A2 . . . ∈ Lω(G)σ = A0 A1 A2 . . . ∈ Lω(G)σ = A0 A1 A2 . . . ∈ Lω(G):=⇒=⇒=⇒ there is an accepting run B0 B1 B2 . . .B0 B1 B2 . . .B0 B1 B2 . . . for σσσ
=⇒=⇒=⇒ B0A0→ B1
A1→ B2A2→ ...B0
A0→ B1A1→ B2
A2→ ...B0A0→ B1
A1→ B2A2→ ... is a path in GGG s.t. ϕ ∈ B0ϕ ∈ B0ϕ ∈ B0
and (*) holds ↑↑↑as B0 ∈ Q0B0 ∈ Q0B0 ∈ Q0
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Proof of Lω(G) ⊆ Words(ϕ)Lω(G) ⊆ Words(ϕ)Lω(G) ⊆ Words(ϕ) ltlmc3.2-59
Claim: If B0A0→ B1
A1→ B2A2→ ...B0
A0→ B1A1→ B2
A2→ ...B0A0→ B1
A1→ B2A2→ ... is a path in GGG s.t.
∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F (*)
then for all formulas ψ ∈ cl(ϕ)ψ ∈ cl(ϕ)ψ ∈ cl(ϕ):
ψ ∈ B0ψ ∈ B0ψ ∈ B0 iff A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ
The claim yields that for each σ = A0 A1 A2 . . . ∈ Lω(G)σ = A0 A1 A2 . . . ∈ Lω(G)σ = A0 A1 A2 . . . ∈ Lω(G):=⇒=⇒=⇒ there is an accepting run B0 B1 B2 . . .B0 B1 B2 . . .B0 B1 B2 . . . for σσσ
=⇒=⇒=⇒ B0A0→ B1
A1→ B2A2→ ...B0
A0→ B1A1→ B2
A2→ ...B0A0→ B1
A1→ B2A2→ ... is a path in GGG s.t. ϕ ∈ B0ϕ ∈ B0ϕ ∈ B0
and (*) holds ↑↑↑as B0 ∈ Q0B0 ∈ Q0B0 ∈ Q0
=⇒=⇒=⇒ σ = A0 A1 A2 . . . |= ϕσ = A0 A1 A2 . . . |= ϕσ = A0 A1 A2 . . . |= ϕ237 / 527
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Proof of Lω(G) ⊆ Words(ϕ)Lω(G) ⊆ Words(ϕ)Lω(G) ⊆ Words(ϕ) ltlmc3.2-59
Claim: If B0A0→ B1
A1→ B2A2→ ...B0
A0→ B1A1→ B2
A2→ ...B0A0→ B1
A1→ B2A2→ ... is a path in GGG s.t.
∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F (*)
then for all formulas ψ ∈ cl(ϕ)ψ ∈ cl(ϕ)ψ ∈ cl(ϕ):
ψ ∈ B0ψ ∈ B0ψ ∈ B0 iff A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ
The claim yields that for each σ = A0 A1 A2 . . . ∈ Lω(G)σ = A0 A1 A2 . . . ∈ Lω(G)σ = A0 A1 A2 . . . ∈ Lω(G):=⇒=⇒=⇒ there is an accepting run B0 B1 B2 . . .B0 B1 B2 . . .B0 B1 B2 . . . for σσσ
=⇒=⇒=⇒ B0A0→ B1
A1→ B2A2→ ...B0
A0→ B1A1→ B2
A2→ ...B0A0→ B1
A1→ B2A2→ ... is a path in GGG s.t. ϕ ∈ B0ϕ ∈ B0ϕ ∈ B0
and (*) holds ↑↑↑as B0 ∈ Q0B0 ∈ Q0B0 ∈ Q0
=⇒=⇒=⇒ σ = A0 A1 A2 . . . |= ϕσ = A0 A1 A2 . . . |= ϕσ = A0 A1 A2 . . . |= ϕ238 / 527
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Proof of Lω(G) ⊆ Words(ϕ)Lω(G) ⊆ Words(ϕ)Lω(G) ⊆ Words(ϕ) ltlmc3.2-59
Claim: If B0A0→ B1
A1→ B2A2→ ...B0
A0→ B1A1→ B2
A2→ ...B0A0→ B1
A1→ B2A2→ ... is a path in GGG s.t.
∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F (*)
then for all formulas ψ ∈ cl(ϕ)ψ ∈ cl(ϕ)ψ ∈ cl(ϕ):
ψ ∈ B0ψ ∈ B0ψ ∈ B0 iff A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ
Proof by structural induction on ψψψ
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Proof of Lω(G) ⊆ Words(ϕ)Lω(G) ⊆ Words(ϕ)Lω(G) ⊆ Words(ϕ) ltlmc3.2-59
Claim: If B0A0→ B1
A1→ B2A2→ ...B0
A0→ B1A1→ B2
A2→ ...B0A0→ B1
A1→ B2A2→ ... is a path in GGG s.t.
∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F (*)
then for all formulas ψ ∈ cl(ϕ)ψ ∈ cl(ϕ)ψ ∈ cl(ϕ):
ψ ∈ B0ψ ∈ B0ψ ∈ B0 iff A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ
Proof by structural induction on ψψψ
base of induction:ψ = trueψ = trueψ = trueψ = a ∈ APψ = a ∈ APψ = a ∈ AP
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Proof of Lω(G) ⊆ Words(ϕ)Lω(G) ⊆ Words(ϕ)Lω(G) ⊆ Words(ϕ) ltlmc3.2-59
Claim: If B0A0→ B1
A1→ B2A2→ ...B0
A0→ B1A1→ B2
A2→ ...B0A0→ B1
A1→ B2A2→ ... is a path in GGG s.t.
∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F (*)
then for all formulas ψ ∈ cl(ϕ)ψ ∈ cl(ϕ)ψ ∈ cl(ϕ):
ψ ∈ B0ψ ∈ B0ψ ∈ B0 iff A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ
Proof by structural induction on ψψψ
base of induction:ψ = trueψ = trueψ = trueψ = a ∈ APψ = a ∈ APψ = a ∈ AP
induction step:ψ = ¬ψ′ψ = ¬ψ′ψ = ¬ψ′ψ = ψ1 ∧ ψ2ψ = ψ1 ∧ ψ2ψ = ψ1 ∧ ψ2
ψ =©ψ′ψ =©ψ′ψ =©ψ′ψ = ψ1 Uψ2ψ = ψ1 Uψ2ψ = ψ1 Uψ2
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Base of induction ltlmc3.2-60
Claim: If B0A0→ B1B0A0→ B1B0A0→ B1
A1→ B2A2→ ...
A1→ B2A2→ ...
A1→ B2A2→ ... is a path in GGG s.t.
∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F
then for all formulas ψ ∈ cl(ϕ)ψ ∈ cl(ϕ)ψ ∈ cl(ϕ):
ψ ∈ B0ψ ∈ B0ψ ∈ B0 iff A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ
Base of induction:
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Base of induction ltlmc3.2-60
Claim: If B0A0→ B1B0A0→ B1B0A0→ B1
A1→ B2A2→ ...
A1→ B2A2→ ...
A1→ B2A2→ ... is a path in GGG s.t.
∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F
then for all formulas ψ ∈ cl(ϕ)ψ ∈ cl(ϕ)ψ ∈ cl(ϕ):
ψ ∈ B0ψ ∈ B0ψ ∈ B0 iff A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ
Base of induction:
Suppose ψ = true ∈ cl(ϕ)ψ = true ∈ cl(ϕ)ψ = true ∈ cl(ϕ).
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Base of induction ltlmc3.2-60
Claim: If B0A0→ B1B0A0→ B1B0A0→ B1
A1→ B2A2→ ...
A1→ B2A2→ ...
A1→ B2A2→ ... is a path in GGG s.t.
∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F
then for all formulas ψ ∈ cl(ϕ)ψ ∈ cl(ϕ)ψ ∈ cl(ϕ):
ψ ∈ B0ψ ∈ B0ψ ∈ B0 iff A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ
Base of induction:
Suppose ψ = true ∈ cl(ϕ)ψ = true ∈ cl(ϕ)ψ = true ∈ cl(ϕ). Then true ∈ B0true ∈ B0true ∈ B0
note: truetruetrue is contained in all elementary formula-sets
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Base of induction ltlmc3.2-60
Claim: If B0A0→ B1B0A0→ B1B0A0→ B1
A1→ B2A2→ ...
A1→ B2A2→ ...
A1→ B2A2→ ... is a path in GGG s.t.
∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F
then for all formulas ψ ∈ cl(ϕ)ψ ∈ cl(ϕ)ψ ∈ cl(ϕ):
ψ ∈ B0ψ ∈ B0ψ ∈ B0 iff A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ
Base of induction:
Suppose ψ = true ∈ cl(ϕ)ψ = true ∈ cl(ϕ)ψ = true ∈ cl(ϕ). Then true ∈ B0true ∈ B0true ∈ B0 andA0 A1 A2 . . . |= trueA0 A1 A2 . . . |= trueA0 A1 A2 . . . |= true
note: truetruetrue is contained in all elementary formula-setstruetruetrue holds for all paths/traces
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Base of induction ltlmc3.2-60
Claim: If B0A0→ B1B0A0→ B1B0A0→ B1
A1→ B2A2→ ...
A1→ B2A2→ ...
A1→ B2A2→ ... is a path in GGG s.t.
∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F
then for all formulas ψ ∈ cl(ϕ)ψ ∈ cl(ϕ)ψ ∈ cl(ϕ):
ψ ∈ B0ψ ∈ B0ψ ∈ B0 iff A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ
Base of induction:
Suppose ψ = true ∈ cl(ϕ)ψ = true ∈ cl(ϕ)ψ = true ∈ cl(ϕ). Then true ∈ B0true ∈ B0true ∈ B0 andA0 A1 A2 . . . |= trueA0 A1 A2 . . . |= trueA0 A1 A2 . . . |= true
Let ψ = a ∈ APψ = a ∈ APψ = a ∈ AP.
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Base of induction ltlmc3.2-60
Claim: If B0A0→ B1B0A0→ B1B0A0→ B1
A1→ B2A2→ ...
A1→ B2A2→ ...
A1→ B2A2→ ... is a path in GGG s.t.
∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F
then for all formulas ψ ∈ cl(ϕ)ψ ∈ cl(ϕ)ψ ∈ cl(ϕ):
ψ ∈ B0ψ ∈ B0ψ ∈ B0 iff A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ
Base of induction:
Suppose ψ = true ∈ cl(ϕ)ψ = true ∈ cl(ϕ)ψ = true ∈ cl(ϕ). Then true ∈ B0true ∈ B0true ∈ B0 andA0 A1 A2 . . . |= trueA0 A1 A2 . . . |= trueA0 A1 A2 . . . |= true
Let ψ = a ∈ APψ = a ∈ APψ = a ∈ AP. Then:
a ∈ B0a ∈ B0a ∈ B0
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Base of induction ltlmc3.2-60
Claim: If B0A0→ B1B0A0→ B1B0A0→ B1
A1→ B2A2→ ...
A1→ B2A2→ ...
A1→ B2A2→ ... is a path in GGG s.t.
∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F
then for all formulas ψ ∈ cl(ϕ)ψ ∈ cl(ϕ)ψ ∈ cl(ϕ):
ψ ∈ B0ψ ∈ B0ψ ∈ B0 iff A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ
Base of induction:
Suppose ψ = true ∈ cl(ϕ)ψ = true ∈ cl(ϕ)ψ = true ∈ cl(ϕ). Then true ∈ B0true ∈ B0true ∈ B0 andA0 A1 A2 . . . |= trueA0 A1 A2 . . . |= trueA0 A1 A2 . . . |= true
Let ψ = a ∈ APψ = a ∈ APψ = a ∈ AP. Then:
a ∈ B0 ⇐⇒ a ∈ A0a ∈ B0 ⇐⇒ a ∈ A0a ∈ B0 ⇐⇒ a ∈ A0
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Base of induction ltlmc3.2-60
Claim: If B0A0→ B1B0A0→ B1B0A0→ B1
A1→ B2A2→ ...
A1→ B2A2→ ...
A1→ B2A2→ ... is a path in GGG s.t.
∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F
then for all formulas ψ ∈ cl(ϕ)ψ ∈ cl(ϕ)ψ ∈ cl(ϕ):
ψ ∈ B0ψ ∈ B0ψ ∈ B0 iff A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ
Base of induction:
Suppose ψ = true ∈ cl(ϕ)ψ = true ∈ cl(ϕ)ψ = true ∈ cl(ϕ). Then true ∈ B0true ∈ B0true ∈ B0 andA0 A1 A2 . . . |= trueA0 A1 A2 . . . |= trueA0 A1 A2 . . . |= true
Let ψ = a ∈ APψ = a ∈ APψ = a ∈ AP. Then:
a ∈ B0 ⇐⇒ a ∈ A0a ∈ B0 ⇐⇒ a ∈ A0a ∈ B0 ⇐⇒ a ∈ A0
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Base of induction ltlmc3.2-60
Claim: If B0A0→ B1B0A0→ B1B0A0→ B1
A1→ B2A2→ ...
A1→ B2A2→ ...
A1→ B2A2→ ... is a path in GGG s.t.
∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F A0 = B0 ∩ APA0 = B0 ∩ APA0 = B0 ∩ AP
then for all formulas ψ ∈ cl(ϕ)ψ ∈ cl(ϕ)ψ ∈ cl(ϕ):
ψ ∈ B0ψ ∈ B0ψ ∈ B0 iff A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ
Base of induction:
Suppose ψ = true ∈ cl(ϕ)ψ = true ∈ cl(ϕ)ψ = true ∈ cl(ϕ). Then true ∈ B0true ∈ B0true ∈ B0 andA0 A1 A2 . . . |= trueA0 A1 A2 . . . |= trueA0 A1 A2 . . . |= true
Let ψ = a ∈ APψ = a ∈ APψ = a ∈ AP. Then:
a ∈ B0 ⇐⇒ a ∈ A0a ∈ B0 ⇐⇒ a ∈ A0a ∈ B0 ⇐⇒ a ∈ A0
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Base of induction ltlmc3.2-60
Claim: If B0A0→ B1B0A0→ B1B0A0→ B1
A1→ B2A2→ ...
A1→ B2A2→ ...
A1→ B2A2→ ... is a path in GGG s.t.
∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F A0 = B0 ∩ APA0 = B0 ∩ APA0 = B0 ∩ AP
then for all formulas ψ ∈ cl(ϕ)ψ ∈ cl(ϕ)ψ ∈ cl(ϕ):
ψ ∈ B0ψ ∈ B0ψ ∈ B0 iff A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ
Base of induction:
Suppose ψ = true ∈ cl(ϕ)ψ = true ∈ cl(ϕ)ψ = true ∈ cl(ϕ). Then true ∈ B0true ∈ B0true ∈ B0 andA0 A1 A2 . . . |= trueA0 A1 A2 . . . |= trueA0 A1 A2 . . . |= true
Let ψ = a ∈ APψ = a ∈ APψ = a ∈ AP. Then:
a ∈ B0 ⇐⇒ a ∈ A0 ⇐⇒ A0 A1 A2 . . . |= aa ∈ B0 ⇐⇒ a ∈ A0 ⇐⇒ A0 A1 A2 . . . |= aa ∈ B0 ⇐⇒ a ∈ A0 ⇐⇒ A0 A1 A2 . . . |= a
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Induction step: negation ltlmc3.2-61
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Induction step: negation ltlmc3.2-61
Claim: If B0A0→ B1
A1→ B2A2→ ...B0
A0→ B1A1→ B2
A2→ ...B0A0→ B1
A1→ B2A2→ ... is a path in GGG s.t.
∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F
then for all formulas ψ ∈ cl(ϕ)ψ ∈ cl(ϕ)ψ ∈ cl(ϕ):
ψ ∈ B0ψ ∈ B0ψ ∈ B0 iff A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ
Induction step: for ψ = ¬ψ′ψ = ¬ψ′ψ = ¬ψ′:
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Induction step: negation ltlmc3.2-61
Claim: If B0A0→ B1
A1→ B2A2→ ...B0
A0→ B1A1→ B2
A2→ ...B0A0→ B1
A1→ B2A2→ ... is a path in GGG s.t.
∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F
then for all formulas ψ ∈ cl(ϕ)ψ ∈ cl(ϕ)ψ ∈ cl(ϕ):
ψ ∈ B0ψ ∈ B0ψ ∈ B0 iff A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ
Induction step: for ψ = ¬ψ′ψ = ¬ψ′ψ = ¬ψ′:ψ ∈ B0ψ ∈ B0ψ ∈ B0
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Induction step: negation ltlmc3.2-61
Claim: If B0A0→ B1
A1→ B2A2→ ...B0
A0→ B1A1→ B2
A2→ ...B0A0→ B1
A1→ B2A2→ ... is a path in GGG s.t.
∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F
then for all formulas ψ ∈ cl(ϕ)ψ ∈ cl(ϕ)ψ ∈ cl(ϕ):
ψ ∈ B0ψ ∈ B0ψ ∈ B0 iff A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ
Induction step: for ψ = ¬ψ′ψ = ¬ψ′ψ = ¬ψ′:ψ ∈ B0ψ ∈ B0ψ ∈ B0
iff ψ′ �∈ B0ψ′ �∈ B0ψ′ �∈ B0 (maximal consistency)
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Induction step: negation ltlmc3.2-61
Claim: If B0A0→ B1
A1→ B2A2→ ...B0
A0→ B1A1→ B2
A2→ ...B0A0→ B1
A1→ B2A2→ ... is a path in GGG s.t.
∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F
then for all formulas ψ ∈ cl(ϕ)ψ ∈ cl(ϕ)ψ ∈ cl(ϕ):
ψ ∈ B0ψ ∈ B0ψ ∈ B0 iff A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ
Induction step: for ψ = ¬ψ′ψ = ¬ψ′ψ = ¬ψ′:ψ ∈ B0ψ ∈ B0ψ ∈ B0
iff ψ′ �∈ B0ψ′ �∈ B0ψ′ �∈ B0 (maximal consistency)
iff A0 A1 A2 . . . �|= ψ′A0 A1 A2 . . . �|= ψ′A0 A1 A2 . . . �|= ψ′ (induction hypothesis)
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Induction step: negation ltlmc3.2-61
Claim: If B0A0→ B1
A1→ B2A2→ ...B0
A0→ B1A1→ B2
A2→ ...B0A0→ B1
A1→ B2A2→ ... is a path in GGG s.t.
∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F
then for all formulas ψ ∈ cl(ϕ)ψ ∈ cl(ϕ)ψ ∈ cl(ϕ):
ψ ∈ B0ψ ∈ B0ψ ∈ B0 iff A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ
Induction step: for ψ = ¬ψ′ψ = ¬ψ′ψ = ¬ψ′:ψ ∈ B0ψ ∈ B0ψ ∈ B0
iff ψ′ �∈ B0ψ′ �∈ B0ψ′ �∈ B0 (maximal consistency)
iff A0 A1 A2 . . . �|= ψ′A0 A1 A2 . . . �|= ψ′A0 A1 A2 . . . �|= ψ′ (induction hypothesis)
iff A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ (semantics of ¬¬¬)257 / 527
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Elementary formula-sets ltlmc3.2-50a-copy2
B ⊆ cl(ϕ)B ⊆ cl(ϕ)B ⊆ cl(ϕ) is elementary iff:
(i) BBB is maximal consistent w.r.t. prop. logic,i.e., if ψψψ, ψ1 ∧ ψ2 ∈ cl(ϕ)ψ1 ∧ ψ2 ∈ cl(ϕ)ψ1 ∧ ψ2 ∈ cl(ϕ) then:
ψ �∈ Bψ �∈ Bψ �∈ B iff ¬ψ ∈ B¬ψ ∈ B¬ψ ∈ B
ψ1 ∧ ψ2 ∈ Bψ1 ∧ ψ2 ∈ Bψ1 ∧ ψ2 ∈ B iff ψ1 ∈ Bψ1 ∈ Bψ1 ∈ B and ψ2 ∈ Bψ2 ∈ Bψ2 ∈ B
true ∈ cl(ϕ)true ∈ cl(ϕ)true ∈ cl(ϕ) implies true ∈ Btrue ∈ Btrue ∈ B
(ii) BBB is locally consistent with respect to until UUU,i.e., if ψ1 Uψ2 ∈ cl(ϕ)ψ1 Uψ2 ∈ cl(ϕ)ψ1 Uψ2 ∈ cl(ϕ) then:
if ψ1 Uψ2 ∈ Bψ1 Uψ2 ∈ Bψ1 Uψ2 ∈ B and ψ2 �∈ Bψ2 �∈ Bψ2 �∈ B then ψ1 ∈ Bψ1 ∈ Bψ1 ∈ B
if ψ2 ∈ Bψ2 ∈ Bψ2 ∈ B then ψ1 Uψ2 ∈ Bψ1 Uψ2 ∈ Bψ1 Uψ2 ∈ B258 / 527
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Elementary formula-sets ltlmc3.2-50a-copy2
B ⊆ cl(ϕ)B ⊆ cl(ϕ)B ⊆ cl(ϕ) is elementary iff:
(i) BBB is maximal consistent w.r.t. prop. logic,i.e., if ψψψ, ψ1 ∧ ψ2 ∈ cl(ϕ)ψ1 ∧ ψ2 ∈ cl(ϕ)ψ1 ∧ ψ2 ∈ cl(ϕ) then:
ψ �∈ Bψ �∈ Bψ �∈ B iff ¬ψ ∈ B¬ψ ∈ B¬ψ ∈ B
ψ1 ∧ ψ2 ∈ Bψ1 ∧ ψ2 ∈ Bψ1 ∧ ψ2 ∈ B iff ψ1 ∈ Bψ1 ∈ Bψ1 ∈ B and ψ2 ∈ Bψ2 ∈ Bψ2 ∈ B
true ∈ cl(ϕ)true ∈ cl(ϕ)true ∈ cl(ϕ) implies true ∈ Btrue ∈ Btrue ∈ B
(ii) BBB is locally consistent with respect to until UUU,i.e., if ψ1 Uψ2 ∈ cl(ϕ)ψ1 Uψ2 ∈ cl(ϕ)ψ1 Uψ2 ∈ cl(ϕ) then:
if ψ1 Uψ2 ∈ Bψ1 Uψ2 ∈ Bψ1 Uψ2 ∈ B and ψ2 �∈ Bψ2 �∈ Bψ2 �∈ B then ψ1 ∈ Bψ1 ∈ Bψ1 ∈ B
if ψ2 ∈ Bψ2 ∈ Bψ2 ∈ B then ψ1 Uψ2 ∈ Bψ1 Uψ2 ∈ Bψ1 Uψ2 ∈ B259 / 527
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Induction step: conjunction ltlmc3.2-61a
Claim: If B0A0→ B1
A1→ B2A2→ ...B0
A0→ B1A1→ B2
A2→ ...B0A0→ B1
A1→ B2A2→ ... is a path in GGG s.t.
∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F
then for all formulas ψ ∈ cl(ϕ)ψ ∈ cl(ϕ)ψ ∈ cl(ϕ):
ψ ∈ B0ψ ∈ B0ψ ∈ B0 iff A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ
Induction step: for ψ = ψ1 ∧ ψ2ψ = ψ1 ∧ ψ2ψ = ψ1 ∧ ψ2
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Induction step: conjunction ltlmc3.2-61a
Claim: If B0A0→ B1
A1→ B2A2→ ...B0
A0→ B1A1→ B2
A2→ ...B0A0→ B1
A1→ B2A2→ ... is a path in GGG s.t.
∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F
then for all formulas ψ ∈ cl(ϕ)ψ ∈ cl(ϕ)ψ ∈ cl(ϕ):
ψ ∈ B0ψ ∈ B0ψ ∈ B0 iff A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ
Induction step: for ψ = ψ1 ∧ ψ2ψ = ψ1 ∧ ψ2ψ = ψ1 ∧ ψ2
ψ ∈ B0ψ ∈ B0ψ ∈ B0
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Induction step: conjunction ltlmc3.2-61a
Claim: If B0A0→ B1
A1→ B2A2→ ...B0
A0→ B1A1→ B2
A2→ ...B0A0→ B1
A1→ B2A2→ ... is a path in GGG s.t.
∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F
then for all formulas ψ ∈ cl(ϕ)ψ ∈ cl(ϕ)ψ ∈ cl(ϕ):
ψ ∈ B0ψ ∈ B0ψ ∈ B0 iff A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ
Induction step: for ψ = ψ1 ∧ ψ2ψ = ψ1 ∧ ψ2ψ = ψ1 ∧ ψ2
ψ ∈ B0ψ ∈ B0ψ ∈ B0
iff ψ1, ψ2 ∈ B0ψ1, ψ2 ∈ B0ψ1, ψ2 ∈ B0 (maximal consistency)
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Induction step: conjunction ltlmc3.2-61a
Claim: If B0A0→ B1
A1→ B2A2→ ...B0
A0→ B1A1→ B2
A2→ ...B0A0→ B1
A1→ B2A2→ ... is a path in GGG s.t.
∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F
then for all formulas ψ ∈ cl(ϕ)ψ ∈ cl(ϕ)ψ ∈ cl(ϕ):
ψ ∈ B0ψ ∈ B0ψ ∈ B0 iff A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ
Induction step: for ψ = ψ1 ∧ ψ2ψ = ψ1 ∧ ψ2ψ = ψ1 ∧ ψ2
ψ ∈ B0ψ ∈ B0ψ ∈ B0
iff ψ1, ψ2 ∈ B0ψ1, ψ2 ∈ B0ψ1, ψ2 ∈ B0 (maximal consistency)
iff A0 A1 A2 . . . |= ψ1A0 A1 A2 . . . |= ψ1A0 A1 A2 . . . |= ψ1 and A0 A1 A2 . . . |= ψ2A0 A1 A2 . . . |= ψ2A0 A1 A2 . . . |= ψ2 (IH)
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Induction step: conjunction ltlmc3.2-61a
Claim: If B0A0→ B1
A1→ B2A2→ ...B0
A0→ B1A1→ B2
A2→ ...B0A0→ B1
A1→ B2A2→ ... is a path in GGG s.t.
∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F
then for all formulas ψ ∈ cl(ϕ)ψ ∈ cl(ϕ)ψ ∈ cl(ϕ):
ψ ∈ B0ψ ∈ B0ψ ∈ B0 iff A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ
Induction step: for ψ = ψ1 ∧ ψ2ψ = ψ1 ∧ ψ2ψ = ψ1 ∧ ψ2
ψ ∈ B0ψ ∈ B0ψ ∈ B0
iff ψ1, ψ2 ∈ B0ψ1, ψ2 ∈ B0ψ1, ψ2 ∈ B0 (maximal consistency)
iff A0 A1 A2 . . . |= ψ1A0 A1 A2 . . . |= ψ1A0 A1 A2 . . . |= ψ1 and A0 A1 A2 . . . |= ψ2A0 A1 A2 . . . |= ψ2A0 A1 A2 . . . |= ψ2 (IH)
iff A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ (semantics of ∧∧∧)264 / 527
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Induction step: next step ltlmc3.2-57b
Claim: If B0A0→ B1
A1→ B2A2→ ...B0
A0→ B1A1→ B2
A2→ ...B0A0→ B1
A1→ B2A2→ ... is a path in GGG s.t.
∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F
then for all formulas ψ ∈ cl(ϕ)ψ ∈ cl(ϕ)ψ ∈ cl(ϕ):
ψ ∈ B0ψ ∈ B0ψ ∈ B0 iff A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ
Induction step: for ψ =©ψ′ψ =©ψ′ψ =©ψ′:
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GNBA for LTL-formula ϕϕϕ ltlmc3.2-57b
G = (Q, 2AP , δ,Q0,F)G = (Q, 2AP , δ,Q0,F)G = (Q, 2AP , δ,Q0,F)
state space: QQQ ==={B ⊆ cl(ϕ) : B
{B ⊆ cl(ϕ) : B
{B ⊆ cl(ϕ) : B is elementary
}}}initial states: Q0Q0Q0 ===
{B ∈ Q : ϕ ∈ B
}{B ∈ Q : ϕ ∈ B
}{B ∈ Q : ϕ ∈ B
}transition relation: for B ∈ QB ∈ QB ∈ Q and A ∈ 2APA ∈ 2APA ∈ 2AP :
if A �= B ∩ APA �= B ∩ APA �= B ∩ AP then δ(B,A) = ∅δ(B ,A) = ∅δ(B ,A) = ∅
if A = B ∩ APA = B ∩ APA = B ∩ AP then δ(B,A) =δ(B ,A) =δ(B ,A) = set of all B ′ ∈ QB ′ ∈ QB ′ ∈ Q s.t.
©ψ ∈ B©ψ ∈ B©ψ ∈ B iff ψ ∈ B ′ψ ∈ B ′ψ ∈ B ′
ψ1 Uψ2 ∈ Bψ1 Uψ2 ∈ Bψ1 Uψ2 ∈ B iff (ψ2 ∈ B) ∨ (ψ1 ∈ B ∧ ψ1 Uψ2 ∈ B ′)(ψ2 ∈ B) ∨ (ψ1 ∈ B ∧ ψ1 Uψ2 ∈ B ′)(ψ2 ∈ B) ∨ (ψ1 ∈ B ∧ ψ1 Uψ2 ∈ B ′)
acceptance set F ={Fψ1 Uψ2 : ψ1 Uψ2 ∈ cl(ϕ)
}F =
{F ψ1 Uψ2 : ψ1 Uψ2 ∈ cl(ϕ)
}F =
{F ψ1 Uψ2 : ψ1 Uψ2 ∈ cl(ϕ)
}where Fψ1 Uψ2
={B ∈ Q : ψ1 Uψ2 /∈ B ∨ ψ2 ∈ B
}F ψ1 Uψ2
={B ∈ Q : ψ1 Uψ2 /∈ B ∨ ψ2 ∈ B
}Fψ1 Uψ2
={B ∈ Q : ψ1 Uψ2 /∈ B ∨ ψ2 ∈ B
}266 / 527
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Induction step: next step ltlmc3.2-62
Claim: If B0A0→ B1
A1→ B2A2→ ...B0
A0→ B1A1→ B2
A2→ ...B0A0→ B1
A1→ B2A2→ ... is a path in GGG s.t.
∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F
then for all formulas ψ ∈ cl(ϕ)ψ ∈ cl(ϕ)ψ ∈ cl(ϕ):
ψ ∈ B0ψ ∈ B0ψ ∈ B0 iff A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ
Induction step: for ψ =©ψ′ψ =©ψ′ψ =©ψ′:
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Induction step: next step ltlmc3.2-62
Claim: If B0A0→ B1
A1→ B2A2→ ...B0
A0→ B1A1→ B2
A2→ ...B0A0→ B1
A1→ B2A2→ ... is a path in GGG s.t.
∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F
then for all formulas ψ ∈ cl(ϕ)ψ ∈ cl(ϕ)ψ ∈ cl(ϕ):
ψ ∈ B0ψ ∈ B0ψ ∈ B0 iff A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ
Induction step: for ψ =©ψ′ψ =©ψ′ψ =©ψ′:ψ ∈ B0ψ ∈ B0ψ ∈ B0
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Induction step: next step ltlmc3.2-62
Claim: If B0A0→ B1
A1→ B2A2→ ...B0
A0→ B1A1→ B2
A2→ ...B0A0→ B1
A1→ B2A2→ ... is a path in GGG s.t.
∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F B1 ∈ δ(B0,A0)B1 ∈ δ(B0,A0)B1 ∈ δ(B0,A0)
then for all formulas ψ ∈ cl(ϕ)ψ ∈ cl(ϕ)ψ ∈ cl(ϕ):
ψ ∈ B0ψ ∈ B0ψ ∈ B0 iff A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ
Induction step: for ψ =©ψ′ψ =©ψ′ψ =©ψ′:ψ ∈ B0ψ ∈ B0ψ ∈ B0
iff ψ′ ∈ B1ψ′ ∈ B1ψ′ ∈ B1 (definition of δδδ)
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Induction step: next step ltlmc3.2-62
Claim: If B0A0→ B1
A1→ B2A2→ ...B0
A0→ B1A1→ B2
A2→ ...B0A0→ B1
A1→ B2A2→ ... is a path in GGG s.t.
∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F B1 ∈ δ(B0,A0)B1 ∈ δ(B0,A0)B1 ∈ δ(B0,A0)
then for all formulas ψ ∈ cl(ϕ)ψ ∈ cl(ϕ)ψ ∈ cl(ϕ):
ψ ∈ B0ψ ∈ B0ψ ∈ B0 iff A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ
Induction step: for ψ =©ψ′ψ =©ψ′ψ =©ψ′:ψ ∈ B0ψ ∈ B0ψ ∈ B0
iff ψ′ ∈ B1ψ′ ∈ B1ψ′ ∈ B1 (definition of δδδ)
iff A1 A2 A3 . . . |= ψ′A1 A2 A3 . . . |= ψ′A1 A2 A3 . . . |= ψ′ (induction hypothesis)
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Induction step: next step ltlmc3.2-62
Claim: If B0A0→ B1
A1→ B2A2→ ...B0
A0→ B1A1→ B2
A2→ ...B0A0→ B1
A1→ B2A2→ ... is a path in GGG s.t.
∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F B1 ∈ δ(B0,A0)B1 ∈ δ(B0,A0)B1 ∈ δ(B0,A0)
then for all formulas ψ ∈ cl(ϕ)ψ ∈ cl(ϕ)ψ ∈ cl(ϕ):
ψ ∈ B0ψ ∈ B0ψ ∈ B0 iff A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ
Induction step: for ψ =©ψ′ψ =©ψ′ψ =©ψ′:ψ ∈ B0ψ ∈ B0ψ ∈ B0
iff ψ′ ∈ B1ψ′ ∈ B1ψ′ ∈ B1 (definition of δδδ)
iff A1 A2 A3 . . . |= ψ′A1 A2 A3 . . . |= ψ′A1 A2 A3 . . . |= ψ′ (induction hypothesis)
iff A0 A1 A2 A3 . . . |= ψA0 A1 A2 A3 . . . |= ψA0 A1 A2 A3 . . . |= ψ (semantics of©©©)271 / 527
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Induction step: until ltlmc3.2-63
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Recall: elementary formula-sets ltlmc3.2-63
B ⊆ cl(ϕ)B ⊆ cl(ϕ)B ⊆ cl(ϕ) is elementary iff:
(i) BBB is maximal consistent w.r.t. prop. logic,i.e., if ψψψ, ψ1 ∧ ψ2 ∈ cl(ϕ)ψ1 ∧ ψ2 ∈ cl(ϕ)ψ1 ∧ ψ2 ∈ cl(ϕ) then:
ψ �∈ Bψ �∈ Bψ �∈ B iff ¬ψ ∈ B¬ψ ∈ B¬ψ ∈ B
ψ1 ∧ ψ2 ∈ Bψ1 ∧ ψ2 ∈ Bψ1 ∧ ψ2 ∈ B iff ψ1 ∈ Bψ1 ∈ Bψ1 ∈ B and ψ2 ∈ Bψ2 ∈ Bψ2 ∈ B
true ∈ cl(ϕ)true ∈ cl(ϕ)true ∈ cl(ϕ) implies true ∈ Btrue ∈ Btrue ∈ B
(ii) BBB is locally consistent with respect to until UUU,i.e., if ψ1 Uψ2 ∈ cl(ϕ)ψ1 Uψ2 ∈ cl(ϕ)ψ1 Uψ2 ∈ cl(ϕ) then:
if ψ1 Uψ2 ∈ Bψ1 Uψ2 ∈ Bψ1 Uψ2 ∈ B and ψ2 �∈ Bψ2 �∈ Bψ2 �∈ B then ψ1 ∈ Bψ1 ∈ Bψ1 ∈ B
if ψ2 ∈ Bψ2 ∈ Bψ2 ∈ B then ψ1 Uψ2 ∈ Bψ1 Uψ2 ∈ Bψ1 Uψ2 ∈ B
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Recall: GNBA for LTL-formula ϕϕϕ ltlmc3.2-57d
G = (Q, 2AP , δ,Q0,F)G = (Q, 2AP , δ,Q0,F)G = (Q, 2AP , δ,Q0,F)
state space: QQQ ==={B ⊆ cl(ϕ) : B
{B ⊆ cl(ϕ) : B
{B ⊆ cl(ϕ) : B is elementary
}}}initial states: Q0Q0Q0 ===
{B ∈ Q : ϕ ∈ B
}{B ∈ Q : ϕ ∈ B
}{B ∈ Q : ϕ ∈ B
}transition relation: for B ∈ QB ∈ QB ∈ Q and A ∈ 2APA ∈ 2APA ∈ 2AP :
if A �= B ∩ APA �= B ∩ APA �= B ∩ AP then δ(B,A) = ∅δ(B ,A) = ∅δ(B ,A) = ∅
if A = B ∩ APA = B ∩ APA = B ∩ AP then δ(B,A) =δ(B ,A) =δ(B ,A) = set of all B ′ ∈ QB ′ ∈ QB ′ ∈ Q s.t.
©ψ ∈ B©ψ ∈ B©ψ ∈ B iff ψ ∈ B ′ψ ∈ B ′ψ ∈ B ′
ψ1 Uψ2 ∈ Bψ1 Uψ2 ∈ Bψ1 Uψ2 ∈ B iff (ψ2 ∈ B) ∨ (ψ1 ∈ B ∧ ψ1 Uψ2 ∈ B ′)(ψ2 ∈ B) ∨ (ψ1 ∈ B ∧ ψ1 Uψ2 ∈ B ′)(ψ2 ∈ B) ∨ (ψ1 ∈ B ∧ ψ1 Uψ2 ∈ B ′)
acceptance set F ={Fψ1 Uψ2
: ψ1 Uψ2 ∈ cl(ϕ)}
F ={F ψ1 Uψ2
: ψ1 Uψ2 ∈ cl(ϕ)}
F ={F ψ1 Uψ2
: ψ1 Uψ2 ∈ cl(ϕ)}
where Fψ1 Uψ2=
{B ∈ Q : ψ1 Uψ2 /∈ B ∨ ψ2 ∈ B
}F ψ1 Uψ2 =
{B ∈ Q : ψ1 Uψ2 /∈ B ∨ ψ2 ∈ B
}Fψ1 Uψ2
={B ∈ Q : ψ1 Uψ2 /∈ B ∨ ψ2 ∈ B
}274 / 527
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Recall: GNBA for LTL-formula ϕϕϕ ltlmc3.2-57d
G = (Q, 2AP , δ,Q0,F)G = (Q, 2AP , δ,Q0,F)G = (Q, 2AP , δ,Q0,F)
state space: QQQ ==={B ⊆ cl(ϕ) : B
{B ⊆ cl(ϕ) : B
{B ⊆ cl(ϕ) : B is elementary
}}}initial states: Q0Q0Q0 ===
{B ∈ Q : ϕ ∈ B
}{B ∈ Q : ϕ ∈ B
}{B ∈ Q : ϕ ∈ B
}transition relation: for B ∈ QB ∈ QB ∈ Q and A ∈ 2APA ∈ 2APA ∈ 2AP :
if A �= B ∩ APA �= B ∩ APA �= B ∩ AP then δ(B,A) = ∅δ(B ,A) = ∅δ(B ,A) = ∅
if A = B ∩ APA = B ∩ APA = B ∩ AP then δ(B,A) =δ(B ,A) =δ(B ,A) = set of all B ′ ∈ QB ′ ∈ QB ′ ∈ Q s.t.
©ψ ∈ B©ψ ∈ B©ψ ∈ B iff ψ ∈ B ′ψ ∈ B ′ψ ∈ B ′
ψ1 Uψ2 ∈ Bψ1 Uψ2 ∈ Bψ1 Uψ2 ∈ B iff (ψ2 ∈ B) ∨ (ψ1 ∈ B ∧ ψ1 Uψ2 ∈ B ′)(ψ2 ∈ B) ∨ (ψ1 ∈ B ∧ ψ1 Uψ2 ∈ B ′)(ψ2 ∈ B) ∨ (ψ1 ∈ B ∧ ψ1 Uψ2 ∈ B ′)
acceptance set F ={Fψ1 Uψ2
: ψ1 Uψ2 ∈ cl(ϕ)}
F ={F ψ1 Uψ2
: ψ1 Uψ2 ∈ cl(ϕ)}
F ={F ψ1 Uψ2
: ψ1 Uψ2 ∈ cl(ϕ)}
where Fψ1 Uψ2=
{B ∈ Q : ψ1 Uψ2 /∈ B ∨ ψ2 ∈ B
}F ψ1 Uψ2 =
{B ∈ Q : ψ1 Uψ2 /∈ B ∨ ψ2 ∈ B
}Fψ1 Uψ2
={B ∈ Q : ψ1 Uψ2 /∈ B ∨ ψ2 ∈ B
}275 / 527
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Induction step: until ltlmc3.2-63
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Induction step: until ltlmc3.2-63
Claim: If B0A0→ B1
A1→ B2A2→ ...B0
A0→ B1A1→ B2
A2→ ...B0A0→ B1
A1→ B2A2→ ... is a path in GGG s.t.
∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F
then for all ψ ∈ cl(ϕ)ψ ∈ cl(ϕ)ψ ∈ cl(ϕ): ψ ∈ B0ψ ∈ B0ψ ∈ B0 iff A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ
Induction step for ψ = ψ1 Uψ2ψ = ψ1 Uψ2ψ = ψ1 Uψ2:
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Induction step: until (part “⇐=⇐=⇐=”) ltlmc3.2-63
Claim: If B0A0→ B1
A1→ B2A2→ ...B0
A0→ B1A1→ B2
A2→ ...B0A0→ B1
A1→ B2A2→ ... is a path in GGG s.t.
∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F
then for all ψ ∈ cl(ϕ)ψ ∈ cl(ϕ)ψ ∈ cl(ϕ): ψ ∈ B0ψ ∈ B0ψ ∈ B0 iff A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ
Induction step for ψ = ψ1 Uψ2ψ = ψ1 Uψ2ψ = ψ1 Uψ2:
“⇐=⇐=⇐=”: Suppose A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ.
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Induction step: until (part “⇐=⇐=⇐=”) ltlmc3.2-63
Claim: If B0A0→ B1
A1→ B2A2→ ...B0
A0→ B1A1→ B2
A2→ ...B0A0→ B1
A1→ B2A2→ ... is a path in GGG s.t.
∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F
then for all ψ ∈ cl(ϕ)ψ ∈ cl(ϕ)ψ ∈ cl(ϕ): ψ ∈ B0ψ ∈ B0ψ ∈ B0 iff A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ
Induction step for ψ = ψ1 Uψ2ψ = ψ1 Uψ2ψ = ψ1 Uψ2:
“⇐=⇐=⇐=”: Suppose A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ. Let j ≥ 0j ≥ 0j ≥ 0 s.t.
Aj Aj+1 Aj+2 . . .Aj Aj+1 Aj+2 . . .Aj Aj+1 Aj+2 . . . |= ψ2|= ψ2|= ψ2
Aj−1 Aj Aj−1 . . .Aj−1 Aj Aj−1 . . .Aj−1 Aj Aj−1 . . . |= ψ1|= ψ1|= ψ1
Aj−2 Aj−1 Aj . . .Aj−2 Aj−1 Aj . . .Aj−2 Aj−1 Aj . . . |= ψ1|= ψ1|= ψ1.........
A0 A1 A2 A3 . . .A0 A1 A2 A3 . . .A0 A1 A2 A3 . . . |= ψ1|= ψ1|= ψ1
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Induction step: until (part “⇐=⇐=⇐=”) ltlmc3.2-63
Claim: If B0A0→ B1
A1→ B2A2→ ...B0
A0→ B1A1→ B2
A2→ ...B0A0→ B1
A1→ B2A2→ ... is a path in GGG s.t.
∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F
then for all ψ ∈ cl(ϕ)ψ ∈ cl(ϕ)ψ ∈ cl(ϕ): ψ ∈ B0ψ ∈ B0ψ ∈ B0 iff A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ
Induction step for ψ = ψ1 Uψ2ψ = ψ1 Uψ2ψ = ψ1 Uψ2:
“⇐=⇐=⇐=”: Suppose A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ. Let j ≥ 0j ≥ 0j ≥ 0 s.t.
Aj Aj+1 Aj+2 . . .Aj Aj+1 Aj+2 . . .Aj Aj+1 Aj+2 . . . |= ψ2|= ψ2|= ψ2IH⇒⇒⇒ ψ2 ∈ Bjψ2 ∈ Bjψ2 ∈ Bj
Aj−1 Aj Aj−1 . . .Aj−1 Aj Aj−1 . . .Aj−1 Aj Aj−1 . . . |= ψ1|= ψ1|= ψ1 ⇒⇒⇒ ψ1 ∈ Bj−1ψ1 ∈ Bj−1ψ1 ∈ Bj−1
Aj−2 Aj−1 Aj . . .Aj−2 Aj−1 Aj . . .Aj−2 Aj−1 Aj . . . |= ψ1|= ψ1|= ψ1 ⇒⇒⇒ ψ1 ∈ Bj−2ψ1 ∈ Bj−2ψ1 ∈ Bj−2.........
...
...
...A0 A1 A2 A3 . . .A0 A1 A2 A3 . . .A0 A1 A2 A3 . . . |= ψ1|= ψ1|= ψ1 ⇒⇒⇒ ψ1 ∈ B0ψ1 ∈ B0ψ1 ∈ B0
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Induction step: until (part “⇐=⇐=⇐=”) ltlmc3.2-63
Claim: If B0A0→ B1
A1→ B2A2→ ...B0
A0→ B1A1→ B2
A2→ ...B0A0→ B1
A1→ B2A2→ ... is a path in GGG s.t.
∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F BjBjBj is elementary
then for all ψ ∈ cl(ϕ)ψ ∈ cl(ϕ)ψ ∈ cl(ϕ): ψ ∈ B0ψ ∈ B0ψ ∈ B0 iff A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ
Induction step for ψ = ψ1 Uψ2ψ = ψ1 Uψ2ψ = ψ1 Uψ2:
“⇐=⇐=⇐=”: Suppose A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ. Let j ≥ 0j ≥ 0j ≥ 0 s.t.
Aj Aj+1 Aj+2 . . .Aj Aj+1 Aj+2 . . .Aj Aj+1 Aj+2 . . . |= ψ2|= ψ2|= ψ2IH⇒⇒⇒ ψ2 ∈ Bjψ2 ∈ Bjψ2 ∈ Bj ⇒⇒⇒ ψ ∈ Bjψ ∈ Bjψ ∈ Bj
Aj−1 Aj Aj−1 . . .Aj−1 Aj Aj−1 . . .Aj−1 Aj Aj−1 . . . |= ψ1|= ψ1|= ψ1 ⇒⇒⇒ ψ1 ∈ Bj−1ψ1 ∈ Bj−1ψ1 ∈ Bj−1
Aj−2 Aj−1 Aj . . .Aj−2 Aj−1 Aj . . .Aj−2 Aj−1 Aj . . . |= ψ1|= ψ1|= ψ1 ⇒⇒⇒ ψ1 ∈ Bj−2ψ1 ∈ Bj−2ψ1 ∈ Bj−2.........
...
...
...A0 A1 A2 A3 . . .A0 A1 A2 A3 . . .A0 A1 A2 A3 . . . |= ψ1|= ψ1|= ψ1 ⇒⇒⇒ ψ1 ∈ B0ψ1 ∈ B0ψ1 ∈ B0
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Induction step: until (part “⇐=⇐=⇐=”) ltlmc3.2-63
Claim: If B0A0→ B1
A1→ B2A2→ ...B0
A0→ B1A1→ B2
A2→ ...B0A0→ B1
A1→ B2A2→ ... is a path in GGG s.t.
∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F Bj ∈ δ(Bj−1,Aj−1)Bj ∈ δ(Bj−1,Aj−1)Bj ∈ δ(Bj−1,Aj−1)
then for all ψ ∈ cl(ϕ)ψ ∈ cl(ϕ)ψ ∈ cl(ϕ): ψ ∈ B0ψ ∈ B0ψ ∈ B0 iff A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ
Induction step for ψ = ψ1 Uψ2ψ = ψ1 Uψ2ψ = ψ1 Uψ2:
“⇐=⇐=⇐=”: Suppose A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ. Let j ≥ 0j ≥ 0j ≥ 0 s.t.
Aj Aj+1 Aj+2 . . .Aj Aj+1 Aj+2 . . .Aj Aj+1 Aj+2 . . . |= ψ2|= ψ2|= ψ2IH⇒⇒⇒ ψ2 ∈ Bjψ2 ∈ Bjψ2 ∈ Bj ⇒⇒⇒ ψ ∈ Bjψ ∈ Bjψ ∈ Bj
Aj−1 Aj Aj−1 . . .Aj−1 Aj Aj−1 . . .Aj−1 Aj Aj−1 . . . |= ψ1|= ψ1|= ψ1 ⇒⇒⇒ ψ1 ∈ Bj−1ψ1 ∈ Bj−1ψ1 ∈ Bj−1 ∧∧∧ ψ ∈ Bj−1ψ ∈ Bj−1ψ ∈ Bj−1
Aj−2 Aj−1 Aj . . .Aj−2 Aj−1 Aj . . .Aj−2 Aj−1 Aj . . . |= ψ1|= ψ1|= ψ1 ⇒⇒⇒ ψ1 ∈ Bj−2ψ1 ∈ Bj−2ψ1 ∈ Bj−2.........
...
...
...A0 A1 A2 A3 . . .A0 A1 A2 A3 . . .A0 A1 A2 A3 . . . |= ψ1|= ψ1|= ψ1 ⇒⇒⇒ ψ1 ∈ B0ψ1 ∈ B0ψ1 ∈ B0
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Induction step: until (part “⇐=⇐=⇐=”) ltlmc3.2-63
Claim: If B0A0→ B1
A1→ B2A2→ ...B0
A0→ B1A1→ B2
A2→ ...B0A0→ B1
A1→ B2A2→ ... is a path in GGG s.t.
∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F Bj−1 ∈ δ(Bj−2,Aj−2)Bj−1 ∈ δ(Bj−2,Aj−2)Bj−1 ∈ δ(Bj−2,Aj−2)
then for all ψ ∈ cl(ϕ)ψ ∈ cl(ϕ)ψ ∈ cl(ϕ): ψ ∈ B0ψ ∈ B0ψ ∈ B0 iff A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ
Induction step for ψ = ψ1 Uψ2ψ = ψ1 Uψ2ψ = ψ1 Uψ2:
“⇐=⇐=⇐=”: Suppose A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ. Let j ≥ 0j ≥ 0j ≥ 0 s.t.
Aj Aj+1 Aj+2 . . .Aj Aj+1 Aj+2 . . .Aj Aj+1 Aj+2 . . . |= ψ2|= ψ2|= ψ2IH⇒⇒⇒ ψ2 ∈ Bjψ2 ∈ Bjψ2 ∈ Bj ⇒⇒⇒ ψ ∈ Bjψ ∈ Bjψ ∈ Bj
Aj−1 Aj Aj−1 . . .Aj−1 Aj Aj−1 . . .Aj−1 Aj Aj−1 . . . |= ψ1|= ψ1|= ψ1 ⇒⇒⇒ ψ1 ∈ Bj−1ψ1 ∈ Bj−1ψ1 ∈ Bj−1 ∧∧∧ ψ ∈ Bj−1ψ ∈ Bj−1ψ ∈ Bj−1
Aj−2 Aj−1 Aj . . .Aj−2 Aj−1 Aj . . .Aj−2 Aj−1 Aj . . . |= ψ1|= ψ1|= ψ1 ⇒⇒⇒ ψ1 ∈ Bj−2ψ1 ∈ Bj−2ψ1 ∈ Bj−2 ∧∧∧ ψ ∈ Bj−2ψ ∈ Bj−2ψ ∈ Bj−2.........
...
...
...A0 A1 A2 A3 . . .A0 A1 A2 A3 . . .A0 A1 A2 A3 . . . |= ψ1|= ψ1|= ψ1 ⇒⇒⇒ ψ1 ∈ B0ψ1 ∈ B0ψ1 ∈ B0
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Induction step: until (part “⇐=⇐=⇐=”) ltlmc3.2-63
Claim: If B0A0→ B1
A1→ B2A2→ ...B0
A0→ B1A1→ B2
A2→ ...B0A0→ B1
A1→ B2A2→ ... is a path in GGG s.t.
∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F B1 ∈ δ(B0,A0)B1 ∈ δ(B0,A0)B1 ∈ δ(B0,A0)
then for all ψ ∈ cl(ϕ)ψ ∈ cl(ϕ)ψ ∈ cl(ϕ): ψ ∈ B0ψ ∈ B0ψ ∈ B0 iff A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ
Induction step for ψ = ψ1 Uψ2ψ = ψ1 Uψ2ψ = ψ1 Uψ2:
“⇐=⇐=⇐=”: Suppose A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ. Let j ≥ 0j ≥ 0j ≥ 0 s.t.
Aj Aj+1 Aj+2 . . .Aj Aj+1 Aj+2 . . .Aj Aj+1 Aj+2 . . . |= ψ2|= ψ2|= ψ2IH⇒⇒⇒ ψ2 ∈ Bjψ2 ∈ Bjψ2 ∈ Bj ⇒⇒⇒ ψ ∈ Bjψ ∈ Bjψ ∈ Bj
Aj−1 Aj Aj−1 . . .Aj−1 Aj Aj−1 . . .Aj−1 Aj Aj−1 . . . |= ψ1|= ψ1|= ψ1 ⇒⇒⇒ ψ1 ∈ Bj−1ψ1 ∈ Bj−1ψ1 ∈ Bj−1 ∧∧∧ ψ ∈ Bj−1ψ ∈ Bj−1ψ ∈ Bj−1
Aj−2 Aj−1 Aj . . .Aj−2 Aj−1 Aj . . .Aj−2 Aj−1 Aj . . . |= ψ1|= ψ1|= ψ1 ⇒⇒⇒ ψ1 ∈ Bj−2ψ1 ∈ Bj−2ψ1 ∈ Bj−2 ∧∧∧ ψ ∈ Bj−2ψ ∈ Bj−2ψ ∈ Bj−2.........
...
...
............
A0 A1 A2 A3 . . .A0 A1 A2 A3 . . .A0 A1 A2 A3 . . . |= ψ1|= ψ1|= ψ1 ⇒⇒⇒ ψ1 ∈ B0ψ1 ∈ B0ψ1 ∈ B0 ∧∧∧ ψ ∈ B0ψ ∈ B0ψ ∈ B0
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Induction step: until (part “=⇒=⇒=⇒”) ltlmc3.2-64
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Induction step: until (part “=⇒=⇒=⇒”) ltlmc3.2-64
Claim: If B0A0→ B1
A1→ B2A2→ ...B0
A0→ B1A1→ B2
A2→ ...B0A0→ B1
A1→ B2A2→ ... is a path in GGG s.t.
∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F
then for all ψ ∈ cl(ϕ)ψ ∈ cl(ϕ)ψ ∈ cl(ϕ): ψ ∈ B0ψ ∈ B0ψ ∈ B0 iff A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ
Induction step for ψ = ψ1 Uψ2ψ = ψ1 Uψ2ψ = ψ1 Uψ2:
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Induction step: until (part “=⇒=⇒=⇒”) ltlmc3.2-64
Claim: If B0A0→ B1
A1→ B2A2→ ...B0
A0→ B1A1→ B2
A2→ ...B0A0→ B1
A1→ B2A2→ ... is a path in GGG s.t.
∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F
then for all ψ ∈ cl(ϕ)ψ ∈ cl(ϕ)ψ ∈ cl(ϕ): ψ ∈ B0ψ ∈ B0ψ ∈ B0 iff A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ
Induction step for ψ = ψ1 Uψ2ψ = ψ1 Uψ2ψ = ψ1 Uψ2:
“=⇒=⇒=⇒” Suppose ψ ∈ B0ψ ∈ B0ψ ∈ B0.
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Induction step: until (part “=⇒=⇒=⇒”) ltlmc3.2-64
Claim: If B0A0→ B1
A1→ B2A2→ ...B0
A0→ B1A1→ B2
A2→ ...B0A0→ B1
A1→ B2A2→ ... is a path in GGG s.t.
∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F
then for all ψ ∈ cl(ϕ)ψ ∈ cl(ϕ)ψ ∈ cl(ϕ): ψ ∈ B0ψ ∈ B0ψ ∈ B0 iff A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ
Induction step for ψ = ψ1 Uψ2ψ = ψ1 Uψ2ψ = ψ1 Uψ2:
“=⇒=⇒=⇒” Suppose ψ ∈ B0ψ ∈ B0ψ ∈ B0. There exists j ≥ 0j ≥ 0j ≥ 0 with ψ2 ∈ Bjψ2 ∈ Bjψ2 ∈ Bj ,
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Induction step: until (part “=⇒=⇒=⇒”) ltlmc3.2-64
Claim: If B0A0→ B1
A1→ B2A2→ ...B0
A0→ B1A1→ B2
A2→ ...B0A0→ B1
A1→ B2A2→ ... is a path in GGG s.t.
∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F
then for all ψ ∈ cl(ϕ)ψ ∈ cl(ϕ)ψ ∈ cl(ϕ): ψ ∈ B0ψ ∈ B0ψ ∈ B0 iff A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ
Induction step for ψ = ψ1 Uψ2ψ = ψ1 Uψ2ψ = ψ1 Uψ2:
“=⇒=⇒=⇒” Suppose ψ ∈ B0ψ ∈ B0ψ ∈ B0. There exists j ≥ 0j ≥ 0j ≥ 0 with ψ2 ∈ Bjψ2 ∈ Bjψ2 ∈ Bj ,since otherwise ∀j ≥ 0. ψ2 /∈ Bj∀j ≥ 0. ψ2 /∈ Bj∀j ≥ 0. ψ2 /∈ Bj
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Induction step: until (part “=⇒=⇒=⇒”) ltlmc3.2-64
Claim: If B0A0→ B1
A1→ B2A2→ ...B0
A0→ B1A1→ B2
A2→ ...B0A0→ B1
A1→ B2A2→ ... is a path in GGG s.t.
∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F
then for all ψ ∈ cl(ϕ)ψ ∈ cl(ϕ)ψ ∈ cl(ϕ): ψ ∈ B0ψ ∈ B0ψ ∈ B0 iff A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ
Induction step for ψ = ψ1 Uψ2ψ = ψ1 Uψ2ψ = ψ1 Uψ2:
“=⇒=⇒=⇒” Suppose ψ ∈ B0ψ ∈ B0ψ ∈ B0. There exists j ≥ 0j ≥ 0j ≥ 0 with ψ2 ∈ Bjψ2 ∈ Bjψ2 ∈ Bj ,since otherwise ∀j ≥ 0. ψ2 /∈ Bj∀j ≥ 0. ψ2 /∈ Bj∀j ≥ 0. ψ2 /∈ Bj and therefore:
ψ ∈ B0ψ ∈ B0ψ ∈ B0 ∧∧∧ ψ2 �∈ B0ψ2 �∈ B0ψ2 �∈ B0
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Induction step: until (part “=⇒=⇒=⇒”) ltlmc3.2-64
Claim: If B0A0→ B1
A1→ B2A2→ ...B0
A0→ B1A1→ B2
A2→ ...B0A0→ B1
A1→ B2A2→ ... is a path in GGG s.t.
∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F Bi+1 ∈ δ(Bi ,Ai)Bi+1 ∈ δ(Bi ,Ai)Bi+1 ∈ δ(Bi ,Ai)
then for all ψ ∈ cl(ϕ)ψ ∈ cl(ϕ)ψ ∈ cl(ϕ): ψ ∈ B0ψ ∈ B0ψ ∈ B0 iff A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ
Induction step for ψ = ψ1 Uψ2ψ = ψ1 Uψ2ψ = ψ1 Uψ2:
“=⇒=⇒=⇒” Suppose ψ ∈ B0ψ ∈ B0ψ ∈ B0. There exists j ≥ 0j ≥ 0j ≥ 0 with ψ2 ∈ Bjψ2 ∈ Bjψ2 ∈ Bj ,since otherwise ∀j ≥ 0. ψ2 /∈ Bj∀j ≥ 0. ψ2 /∈ Bj∀j ≥ 0. ψ2 /∈ Bj and therefore:
ψ ∈ B0ψ ∈ B0ψ ∈ B0 ∧∧∧ ψ2 �∈ B0ψ2 �∈ B0ψ2 �∈ B0
⇒⇒⇒ ψ ∈ B1ψ ∈ B1ψ ∈ B1
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Induction step: until (part “=⇒=⇒=⇒”) ltlmc3.2-64
Claim: If B0A0→ B1
A1→ B2A2→ ...B0
A0→ B1A1→ B2
A2→ ...B0A0→ B1
A1→ B2A2→ ... is a path in GGG s.t.
∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F Bi+1 ∈ δ(Bi ,Ai)Bi+1 ∈ δ(Bi ,Ai)Bi+1 ∈ δ(Bi ,Ai)
then for all ψ ∈ cl(ϕ)ψ ∈ cl(ϕ)ψ ∈ cl(ϕ): ψ ∈ B0ψ ∈ B0ψ ∈ B0 iff A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ
Induction step for ψ = ψ1 Uψ2ψ = ψ1 Uψ2ψ = ψ1 Uψ2:
“=⇒=⇒=⇒” Suppose ψ ∈ B0ψ ∈ B0ψ ∈ B0. There exists j ≥ 0j ≥ 0j ≥ 0 with ψ2 ∈ Bjψ2 ∈ Bjψ2 ∈ Bj ,since otherwise ∀j ≥ 0. ψ2 /∈ Bj∀j ≥ 0. ψ2 /∈ Bj∀j ≥ 0. ψ2 /∈ Bj and therefore:
ψ ∈ B0ψ ∈ B0ψ ∈ B0 ∧∧∧ ψ2 �∈ B0ψ2 �∈ B0ψ2 �∈ B0
⇒⇒⇒ ψ ∈ B1ψ ∈ B1ψ ∈ B1 ∧∧∧ ψ2 �∈ B1ψ2 �∈ B1ψ2 �∈ B1
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Induction step: until (part “=⇒=⇒=⇒”) ltlmc3.2-64
Claim: If B0A0→ B1
A1→ B2A2→ ...B0
A0→ B1A1→ B2
A2→ ...B0A0→ B1
A1→ B2A2→ ... is a path in GGG s.t.
∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F Bi+1 ∈ δ(Bi ,Ai)Bi+1 ∈ δ(Bi ,Ai)Bi+1 ∈ δ(Bi ,Ai)
then for all ψ ∈ cl(ϕ)ψ ∈ cl(ϕ)ψ ∈ cl(ϕ): ψ ∈ B0ψ ∈ B0ψ ∈ B0 iff A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ
Induction step for ψ = ψ1 Uψ2ψ = ψ1 Uψ2ψ = ψ1 Uψ2:
“=⇒=⇒=⇒” Suppose ψ ∈ B0ψ ∈ B0ψ ∈ B0. There exists j ≥ 0j ≥ 0j ≥ 0 with ψ2 ∈ Bjψ2 ∈ Bjψ2 ∈ Bj ,since otherwise ∀j ≥ 0. ψ2 /∈ Bj∀j ≥ 0. ψ2 /∈ Bj∀j ≥ 0. ψ2 /∈ Bj and therefore:
ψ ∈ B0ψ ∈ B0ψ ∈ B0 ∧∧∧ ψ2 �∈ B0ψ2 �∈ B0ψ2 �∈ B0
⇒⇒⇒ ψ ∈ B1ψ ∈ B1ψ ∈ B1 ∧∧∧ ψ2 �∈ B1ψ2 �∈ B1ψ2 �∈ B1
⇒⇒⇒ ψ ∈ B2ψ ∈ B2ψ ∈ B2
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Induction step: until (part “=⇒=⇒=⇒”) ltlmc3.2-64
Claim: If B0A0→ B1
A1→ B2A2→ ...B0
A0→ B1A1→ B2
A2→ ...B0A0→ B1
A1→ B2A2→ ... is a path in GGG s.t.
∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F Bi+1 ∈ δ(Bi ,Ai)Bi+1 ∈ δ(Bi ,Ai)Bi+1 ∈ δ(Bi ,Ai)
then for all ψ ∈ cl(ϕ)ψ ∈ cl(ϕ)ψ ∈ cl(ϕ): ψ ∈ B0ψ ∈ B0ψ ∈ B0 iff A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ
Induction step for ψ = ψ1 Uψ2ψ = ψ1 Uψ2ψ = ψ1 Uψ2:
“=⇒=⇒=⇒” Suppose ψ ∈ B0ψ ∈ B0ψ ∈ B0. There exists j ≥ 0j ≥ 0j ≥ 0 with ψ2 ∈ Bjψ2 ∈ Bjψ2 ∈ Bj ,since otherwise ∀j ≥ 0. ψ2 /∈ Bj∀j ≥ 0. ψ2 /∈ Bj∀j ≥ 0. ψ2 /∈ Bj and therefore:
ψ ∈ B0ψ ∈ B0ψ ∈ B0 ∧∧∧ ψ2 �∈ B0ψ2 �∈ B0ψ2 �∈ B0
⇒⇒⇒ ψ ∈ B1ψ ∈ B1ψ ∈ B1 ∧∧∧ ψ2 �∈ B1ψ2 �∈ B1ψ2 �∈ B1
⇒⇒⇒ ψ ∈ B2ψ ∈ B2ψ ∈ B2 ∧∧∧ ψ2 �∈ B2ψ2 �∈ B2ψ2 �∈ B2.........
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Induction step: until (part “=⇒=⇒=⇒”) ltlmc3.2-64
Claim: If B0A0→ B1
A1→ B2A2→ ...B0
A0→ B1A1→ B2
A2→ ...B0A0→ B1
A1→ B2A2→ ... is a path in GGG s.t.
∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F
then for all ψ ∈ cl(ϕ)ψ ∈ cl(ϕ)ψ ∈ cl(ϕ): ψ ∈ B0ψ ∈ B0ψ ∈ B0 iff A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ
Induction step for ψ = ψ1 Uψ2ψ = ψ1 Uψ2ψ = ψ1 Uψ2:
“=⇒=⇒=⇒” Suppose ψ ∈ B0ψ ∈ B0ψ ∈ B0. There exists j ≥ 0j ≥ 0j ≥ 0 with ψ2 ∈ Bjψ2 ∈ Bjψ2 ∈ Bj ,since otherwise ∀j ≥ 0. ψ2 /∈ Bj∀j ≥ 0. ψ2 /∈ Bj∀j ≥ 0. ψ2 /∈ Bj and therefore:
ψ ∈ B0ψ ∈ B0ψ ∈ B0 ∧∧∧ ψ2 �∈ B0ψ2 �∈ B0ψ2 �∈ B0
⇒⇒⇒ ψ ∈ B1ψ ∈ B1ψ ∈ B1 ∧∧∧ ψ2 �∈ B1ψ2 �∈ B1ψ2 �∈ B1
⇒⇒⇒ ψ ∈ B2ψ ∈ B2ψ ∈ B2 ∧∧∧ ψ2 �∈ B2ψ2 �∈ B2ψ2 �∈ B2.........
=⇒ ∀j ≥ 0=⇒ ∀j ≥ 0=⇒ ∀j ≥ 0. Bj �∈ FψBj �∈ FψBj �∈ Fψ where
Fψ = {B : ψ �∈ BFψ = {B : ψ �∈ BFψ = {B : ψ �∈ B or ψ2 ∈ B}ψ2 ∈ B}ψ2 ∈ B}
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Induction step: until (part “=⇒=⇒=⇒”) ltlmc3.2-64
Claim: If B0A0→ B1
A1→ B2A2→ ...B0
A0→ B1A1→ B2
A2→ ...B0A0→ B1
A1→ B2A2→ ... is a path in GGG s.t.
∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F
then for all ψ ∈ cl(ϕ)ψ ∈ cl(ϕ)ψ ∈ cl(ϕ): ψ ∈ B0ψ ∈ B0ψ ∈ B0 iff A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ
Induction step for ψ = ψ1 Uψ2ψ = ψ1 Uψ2ψ = ψ1 Uψ2:
“=⇒=⇒=⇒” Suppose ψ ∈ B0ψ ∈ B0ψ ∈ B0. There exists j ≥ 0j ≥ 0j ≥ 0 with ψ2 ∈ Bjψ2 ∈ Bjψ2 ∈ Bj ,since otherwise ∀j ≥ 0. ψ2 /∈ Bj∀j ≥ 0. ψ2 /∈ Bj∀j ≥ 0. ψ2 /∈ Bj and therefore:
ψ ∈ B0ψ ∈ B0ψ ∈ B0 ∧∧∧ ψ2 �∈ B0ψ2 �∈ B0ψ2 �∈ B0
⇒⇒⇒ ψ ∈ B1ψ ∈ B1ψ ∈ B1 ∧∧∧ ψ2 �∈ B1ψ2 �∈ B1ψ2 �∈ B1
⇒⇒⇒ ψ ∈ B2ψ ∈ B2ψ ∈ B2 ∧∧∧ ψ2 �∈ B2ψ2 �∈ B2ψ2 �∈ B2.........
=⇒ ∀j ≥ 0=⇒ ∀j ≥ 0=⇒ ∀j ≥ 0. Bj �∈ FψBj �∈ FψBj �∈ Fψ where
Fψ = {B : ψ �∈ BFψ = {B : ψ �∈ BFψ = {B : ψ �∈ B or ψ2 ∈ B}ψ2 ∈ B}ψ2 ∈ B}Contradiction!
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Induction step: until (part “=⇒=⇒=⇒”) ltlmc3.2-65
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Induction step: until (part “=⇒=⇒=⇒”) ltlmc3.2-65
Claim: If B0A0→ B1
A1→ B2A2→ ...B0
A0→ B1A1→ B2
A2→ ...B0A0→ B1
A1→ B2A2→ ... is a path in GGG s.t.
∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F
then for all ψ ∈ cl(ϕ)ψ ∈ cl(ϕ)ψ ∈ cl(ϕ): ψ ∈ B0ψ ∈ B0ψ ∈ B0 iff A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ
Induction step for ψ = ψ1 Uψ2ψ = ψ1 Uψ2ψ = ψ1 Uψ2:
Let ψ ∈ B0ψ ∈ B0ψ ∈ B0 and j ≥ 0j ≥ 0j ≥ 0 minimal s.t. ψ2 ∈ Bjψ2 ∈ Bjψ2 ∈ Bj
298 / 527
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Induction step: until (part “=⇒=⇒=⇒”) ltlmc3.2-65
Claim: If B0A0→ B1
A1→ B2A2→ ...B0
A0→ B1A1→ B2
A2→ ...B0A0→ B1
A1→ B2A2→ ... is a path in GGG s.t.
∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F
then for all ψ ∈ cl(ϕ)ψ ∈ cl(ϕ)ψ ∈ cl(ϕ): ψ ∈ B0ψ ∈ B0ψ ∈ B0 iff A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ
Induction step for ψ = ψ1 Uψ2ψ = ψ1 Uψ2ψ = ψ1 Uψ2:
Let ψ ∈ B0ψ ∈ B0ψ ∈ B0 and j ≥ 0j ≥ 0j ≥ 0 minimal s.t. ψ2 ∈ Bjψ2 ∈ Bjψ2 ∈ Bj
IH=⇒=⇒=⇒ Aj Aj+1 . . .Aj Aj+1 . . .Aj Aj+1 . . . |= ψ2|= ψ2|= ψ2
299 / 527
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Induction step: until (part “=⇒=⇒=⇒”) ltlmc3.2-65
Claim: If B0A0→ B1
A1→ B2A2→ ...B0
A0→ B1A1→ B2
A2→ ...B0A0→ B1
A1→ B2A2→ ... is a path in GGG s.t.
∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F
then for all ψ ∈ cl(ϕ)ψ ∈ cl(ϕ)ψ ∈ cl(ϕ): ψ ∈ B0ψ ∈ B0ψ ∈ B0 iff A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ
Induction step for ψ = ψ1 Uψ2ψ = ψ1 Uψ2ψ = ψ1 Uψ2:
Let ψ ∈ B0ψ ∈ B0ψ ∈ B0 and j ≥ 0j ≥ 0j ≥ 0 minimal s.t. ψ2 ∈ Bjψ2 ∈ Bjψ2 ∈ Bj
IH=⇒=⇒=⇒ Aj Aj+1 . . .Aj Aj+1 . . .Aj Aj+1 . . . |= ψ2|= ψ2|= ψ2
¬ψ2 ∈ Bj−1¬ψ2 ∈ Bj−1¬ψ2 ∈ Bj−1
¬ψ2 ∈ Bj−2¬ψ2 ∈ Bj−2¬ψ2 ∈ Bj−2.........
¬ψ2 ∈ B1¬ψ2 ∈ B1¬ψ2 ∈ B1
¬ψ2 ∈ B0¬ψ2 ∈ B0¬ψ2 ∈ B0300 / 527
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Induction step: until (part “=⇒=⇒=⇒”) ltlmc3.2-65
Claim: If B0A0→ B1
A1→ B2A2→ ...B0
A0→ B1A1→ B2
A2→ ...B0A0→ B1
A1→ B2A2→ ... is a path in GGG s.t.
∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F
then for all ψ ∈ cl(ϕ)ψ ∈ cl(ϕ)ψ ∈ cl(ϕ): ψ ∈ B0ψ ∈ B0ψ ∈ B0 iff A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ
Induction step for ψ = ψ1 Uψ2ψ = ψ1 Uψ2ψ = ψ1 Uψ2:
Let ψ ∈ B0ψ ∈ B0ψ ∈ B0 and j ≥ 0j ≥ 0j ≥ 0 minimal s.t. ψ2 ∈ Bjψ2 ∈ Bjψ2 ∈ Bj
IH=⇒=⇒=⇒ Aj Aj+1 . . .Aj Aj+1 . . .Aj Aj+1 . . . |= ψ2|= ψ2|= ψ2
¬ψ2 ∈ Bj−1¬ψ2 ∈ Bj−1¬ψ2 ∈ Bj−1
¬ψ2 ∈ Bj−2¬ψ2 ∈ Bj−2¬ψ2 ∈ Bj−2.........
¬ψ2 ∈ B1¬ψ2 ∈ B1¬ψ2 ∈ B1
¬ψ2, ψ ∈ B0¬ψ2, ψ ∈ B0¬ψ2, ψ ∈ B0 ←−←−←− by assumption301 / 527
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Induction step: until (part “=⇒=⇒=⇒”) ltlmc3.2-65
Claim: If B0A0→ B1
A1→ B2A2→ ...B0
A0→ B1A1→ B2
A2→ ...B0A0→ B1
A1→ B2A2→ ... is a path in GGG s.t.
∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F
then for all ψ ∈ cl(ϕ)ψ ∈ cl(ϕ)ψ ∈ cl(ϕ): ψ ∈ B0ψ ∈ B0ψ ∈ B0 iff A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ
Induction step for ψ = ψ1 Uψ2ψ = ψ1 Uψ2ψ = ψ1 Uψ2:
Let ψ ∈ B0ψ ∈ B0ψ ∈ B0 and j ≥ 0j ≥ 0j ≥ 0 minimal s.t. ψ2 ∈ Bjψ2 ∈ Bjψ2 ∈ Bj
IH=⇒=⇒=⇒ Aj Aj+1 . . .Aj Aj+1 . . .Aj Aj+1 . . . |= ψ2|= ψ2|= ψ2
¬ψ2 ∈ Bj−1¬ψ2 ∈ Bj−1¬ψ2 ∈ Bj−1
¬ψ2 ∈ Bj−2¬ψ2 ∈ Bj−2¬ψ2 ∈ Bj−2.........
¬ψ2 ∈ B1¬ψ2 ∈ B1¬ψ2 ∈ B1
¬ψ2, ψ1, ψ ∈ B0¬ψ2, ψ1, ψ ∈ B0¬ψ2, ψ1, ψ ∈ B0 ←←← local consistency w.r.t. UUU302 / 527
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Induction step: until (part “=⇒=⇒=⇒”) ltlmc3.2-65
Claim: If B0A0→ B1
A1→ B2A2→ ...B0
A0→ B1A1→ B2
A2→ ...B0A0→ B1
A1→ B2A2→ ... is a path in GGG s.t.
∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F Bi+1 ∈ δ(Bi ,Ai)Bi+1 ∈ δ(Bi ,Ai)Bi+1 ∈ δ(Bi ,Ai)
then for all ψ ∈ cl(ϕ)ψ ∈ cl(ϕ)ψ ∈ cl(ϕ): ψ ∈ B0ψ ∈ B0ψ ∈ B0 iff A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ
Induction step for ψ = ψ1 Uψ2ψ = ψ1 Uψ2ψ = ψ1 Uψ2:
Let ψ ∈ B0ψ ∈ B0ψ ∈ B0 and j ≥ 0j ≥ 0j ≥ 0 minimal s.t. ψ2 ∈ Bjψ2 ∈ Bjψ2 ∈ Bj
IH=⇒=⇒=⇒ Aj Aj+1 . . .Aj Aj+1 . . .Aj Aj+1 . . . |= ψ2|= ψ2|= ψ2
¬ψ2 ∈ Bj−1¬ψ2 ∈ Bj−1¬ψ2 ∈ Bj−1
¬ψ2 ∈ Bj−2¬ψ2 ∈ Bj−2¬ψ2 ∈ Bj−2.........
¬ψ2, ψ1, ψ ∈ B1¬ψ2, ψ1, ψ ∈ B1¬ψ2, ψ1, ψ ∈ B1
¬ψ2, ψ1, ψ ∈ B0¬ψ2, ψ1, ψ ∈ B0¬ψ2, ψ1, ψ ∈ B0 ←←← local consistency w.r.t. UUU303 / 527
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Induction step: until (part “=⇒=⇒=⇒”) ltlmc3.2-65
Claim: If B0A0→ B1
A1→ B2A2→ ...B0
A0→ B1A1→ B2
A2→ ...B0A0→ B1
A1→ B2A2→ ... is a path in GGG s.t.
∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F Bi+1 ∈ δ(Bi ,Ai)Bi+1 ∈ δ(Bi ,Ai)Bi+1 ∈ δ(Bi ,Ai)
then for all ψ ∈ cl(ϕ)ψ ∈ cl(ϕ)ψ ∈ cl(ϕ): ψ ∈ B0ψ ∈ B0ψ ∈ B0 iff A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ
Induction step for ψ = ψ1 Uψ2ψ = ψ1 Uψ2ψ = ψ1 Uψ2:
Let ψ ∈ B0ψ ∈ B0ψ ∈ B0 and j ≥ 0j ≥ 0j ≥ 0 minimal s.t. ψ2 ∈ Bjψ2 ∈ Bjψ2 ∈ Bj
IH=⇒=⇒=⇒ Aj Aj+1 . . .Aj Aj+1 . . .Aj Aj+1 . . . |= ψ2|= ψ2|= ψ2
¬ψ2, ψ1, ψ ∈ Bj−1¬ψ2, ψ1, ψ ∈ Bj−1¬ψ2, ψ1, ψ ∈ Bj−1
¬ψ2, ψ1, ψ ∈ Bj−2¬ψ2, ψ1, ψ ∈ Bj−2¬ψ2, ψ1, ψ ∈ Bj−2.........
¬ψ2, ψ1, ψ ∈ B1¬ψ2, ψ1, ψ ∈ B1¬ψ2, ψ1, ψ ∈ B1
¬ψ2, ψ1, ψ ∈ B0¬ψ2, ψ1, ψ ∈ B0¬ψ2, ψ1, ψ ∈ B0 ←←← local consistency w.r.t. UUU304 / 527
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Induction step: until (part “=⇒=⇒=⇒”) ltlmc3.2-65
Claim: If B0A0→ B1
A1→ B2A2→ ...B0
A0→ B1A1→ B2
A2→ ...B0A0→ B1
A1→ B2A2→ ... is a path in GGG s.t.
∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F
then for all ψ ∈ cl(ϕ)ψ ∈ cl(ϕ)ψ ∈ cl(ϕ): ψ ∈ B0ψ ∈ B0ψ ∈ B0 iff A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ
Induction step for ψ = ψ1 Uψ2ψ = ψ1 Uψ2ψ = ψ1 Uψ2:
Let ψ ∈ B0ψ ∈ B0ψ ∈ B0 and j ≥ 0j ≥ 0j ≥ 0 minimal s.t. ψ2 ∈ Bjψ2 ∈ Bjψ2 ∈ Bj
IH=⇒=⇒=⇒ Aj Aj+1 . . .Aj Aj+1 . . .Aj Aj+1 . . . |= ψ2|= ψ2|= ψ2
¬ψ2, ψ1, ψ ∈ Bj−1¬ψ2, ψ1, ψ ∈ Bj−1¬ψ2, ψ1, ψ ∈ Bj−1 =⇒=⇒=⇒ Aj−1 Aj . . .Aj−1 Aj . . .Aj−1 Aj . . . |= ψ1|= ψ1|= ψ1
¬ψ2, ψ1, ψ ∈ Bj−2¬ψ2, ψ1, ψ ∈ Bj−2¬ψ2, ψ1, ψ ∈ Bj−2.........
¬ψ2, ψ1, ψ ∈ B1¬ψ2, ψ1, ψ ∈ B1¬ψ2, ψ1, ψ ∈ B1
¬ψ2, ψ1, ψ ∈ B0¬ψ2, ψ1, ψ ∈ B0¬ψ2, ψ1, ψ ∈ B0 ←←← local consistency w.r.t. UUU305 / 527
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Induction step: until (part “=⇒=⇒=⇒”) ltlmc3.2-65
Claim: If B0A0→ B1
A1→ B2A2→ ...B0
A0→ B1A1→ B2
A2→ ...B0A0→ B1
A1→ B2A2→ ... is a path in GGG s.t.
∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F
then for all ψ ∈ cl(ϕ)ψ ∈ cl(ϕ)ψ ∈ cl(ϕ): ψ ∈ B0ψ ∈ B0ψ ∈ B0 iff A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ
Induction step for ψ = ψ1 Uψ2ψ = ψ1 Uψ2ψ = ψ1 Uψ2:
Let ψ ∈ B0ψ ∈ B0ψ ∈ B0 and j ≥ 0j ≥ 0j ≥ 0 minimal s.t. ψ2 ∈ Bjψ2 ∈ Bjψ2 ∈ Bj
IH=⇒=⇒=⇒ Aj Aj+1 . . .Aj Aj+1 . . .Aj Aj+1 . . . |= ψ2|= ψ2|= ψ2
¬ψ2, ψ1, ψ ∈ Bj−1¬ψ2, ψ1, ψ ∈ Bj−1¬ψ2, ψ1, ψ ∈ Bj−1 =⇒=⇒=⇒ Aj−1 Aj . . .Aj−1 Aj . . .Aj−1 Aj . . . |= ψ1|= ψ1|= ψ1
¬ψ2, ψ1, ψ ∈ Bj−2¬ψ2, ψ1, ψ ∈ Bj−2¬ψ2, ψ1, ψ ∈ Bj−2 =⇒=⇒=⇒ Aj−2 Aj−1 . . .Aj−2 Aj−1 . . .Aj−2 Aj−1 . . . |= ψ1|= ψ1|= ψ1.........
¬ψ2, ψ1, ψ ∈ B1¬ψ2, ψ1, ψ ∈ B1¬ψ2, ψ1, ψ ∈ B1
¬ψ2, ψ1, ψ ∈ B0¬ψ2, ψ1, ψ ∈ B0¬ψ2, ψ1, ψ ∈ B0 ←←← local consistency w.r.t. UUU306 / 527
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Induction step: until (part “=⇒=⇒=⇒”) ltlmc3.2-65
Claim: If B0A0→ B1
A1→ B2A2→ ...B0
A0→ B1A1→ B2
A2→ ...B0A0→ B1
A1→ B2A2→ ... is a path in GGG s.t.
∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F
then for all ψ ∈ cl(ϕ)ψ ∈ cl(ϕ)ψ ∈ cl(ϕ): ψ ∈ B0ψ ∈ B0ψ ∈ B0 iff A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ
Induction step for ψ = ψ1 Uψ2ψ = ψ1 Uψ2ψ = ψ1 Uψ2:
Let ψ ∈ B0ψ ∈ B0ψ ∈ B0 and j ≥ 0j ≥ 0j ≥ 0 minimal s.t. ψ2 ∈ Bjψ2 ∈ Bjψ2 ∈ Bj
IH=⇒=⇒=⇒ Aj Aj+1 . . .Aj Aj+1 . . .Aj Aj+1 . . . |= ψ2|= ψ2|= ψ2
¬ψ2, ψ1, ψ ∈ Bj−1¬ψ2, ψ1, ψ ∈ Bj−1¬ψ2, ψ1, ψ ∈ Bj−1 =⇒=⇒=⇒ Aj−1 Aj . . .Aj−1 Aj . . .Aj−1 Aj . . . |= ψ1|= ψ1|= ψ1
¬ψ2, ψ1, ψ ∈ Bj−2¬ψ2, ψ1, ψ ∈ Bj−2¬ψ2, ψ1, ψ ∈ Bj−2 =⇒=⇒=⇒ Aj−2 Aj−1 . . .Aj−2 Aj−1 . . .Aj−2 Aj−1 . . . |= ψ1|= ψ1|= ψ1.........
...
...
............
...
...
...¬ψ2, ψ1, ψ ∈ B1¬ψ2, ψ1, ψ ∈ B1¬ψ2, ψ1, ψ ∈ B1 =⇒=⇒=⇒ A1 A2 A3 . . .A1 A2 A3 . . .A1 A2 A3 . . . |= ψ1|= ψ1|= ψ1
¬ψ2, ψ1, ψ ∈ B0¬ψ2, ψ1, ψ ∈ B0¬ψ2, ψ1, ψ ∈ B0307 / 527
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Induction step: until (part “=⇒=⇒=⇒”) ltlmc3.2-65
Claim: If B0A0→ B1
A1→ B2A2→ ...B0
A0→ B1A1→ B2
A2→ ...B0A0→ B1
A1→ B2A2→ ... is a path in GGG s.t.
∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F
then for all ψ ∈ cl(ϕ)ψ ∈ cl(ϕ)ψ ∈ cl(ϕ): ψ ∈ B0ψ ∈ B0ψ ∈ B0 iff A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ
Induction step for ψ = ψ1 Uψ2ψ = ψ1 Uψ2ψ = ψ1 Uψ2:
Let ψ ∈ B0ψ ∈ B0ψ ∈ B0 and j ≥ 0j ≥ 0j ≥ 0 minimal s.t. ψ2 ∈ Bjψ2 ∈ Bjψ2 ∈ Bj
IH=⇒=⇒=⇒ Aj Aj+1 . . .Aj Aj+1 . . .Aj Aj+1 . . . |= ψ2|= ψ2|= ψ2
¬ψ2, ψ1, ψ ∈ Bj−1¬ψ2, ψ1, ψ ∈ Bj−1¬ψ2, ψ1, ψ ∈ Bj−1 =⇒=⇒=⇒ Aj−1 Aj . . .Aj−1 Aj . . .Aj−1 Aj . . . |= ψ1|= ψ1|= ψ1
¬ψ2, ψ1, ψ ∈ Bj−2¬ψ2, ψ1, ψ ∈ Bj−2¬ψ2, ψ1, ψ ∈ Bj−2 =⇒=⇒=⇒ Aj−2 Aj−1 . . .Aj−2 Aj−1 . . .Aj−2 Aj−1 . . . |= ψ1|= ψ1|= ψ1.........
...
...
............
...
...
...¬ψ2, ψ1, ψ ∈ B1¬ψ2, ψ1, ψ ∈ B1¬ψ2, ψ1, ψ ∈ B1 =⇒=⇒=⇒ A1 A2 A3 . . .A1 A2 A3 . . .A1 A2 A3 . . . |= ψ1|= ψ1|= ψ1
¬ψ2, ψ1, ψ ∈ B0¬ψ2, ψ1, ψ ∈ B0¬ψ2, ψ1, ψ ∈ B0 =⇒=⇒=⇒ A0 A1 A2 . . .A0 A1 A2 . . .A0 A1 A2 . . . |= ψ1|= ψ1|= ψ1308 / 527
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Induction step: until (part “=⇒=⇒=⇒”) ltlmc3.2-65
Claim: If B0A0→ B1
A1→ B2A2→ ...B0
A0→ B1A1→ B2
A2→ ...B0A0→ B1
A1→ B2A2→ ... is a path in GGG s.t.
∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F
then for all ψ ∈ cl(ϕ)ψ ∈ cl(ϕ)ψ ∈ cl(ϕ): ψ ∈ B0ψ ∈ B0ψ ∈ B0 iff A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ
Induction step for ψ = ψ1 Uψ2ψ = ψ1 Uψ2ψ = ψ1 Uψ2:
Let ψ ∈ B0ψ ∈ B0ψ ∈ B0 and j ≥ 0j ≥ 0j ≥ 0 minimal s.t. ψ2 ∈ Bjψ2 ∈ Bjψ2 ∈ Bj
IH=⇒=⇒=⇒ Aj Aj+1 . . .Aj Aj+1 . . .Aj Aj+1 . . . |= ψ2|= ψ2|= ψ2
¬ψ2, ψ1, ψ ∈ Bj−1¬ψ2, ψ1, ψ ∈ Bj−1¬ψ2, ψ1, ψ ∈ Bj−1 =⇒=⇒=⇒ Aj−1 Aj . . .Aj−1 Aj . . .Aj−1 Aj . . . |= ψ1|= ψ1|= ψ1.........
...
...
............
...
...
...¬ψ2, ψ1, ψ ∈ B0¬ψ2, ψ1, ψ ∈ B0¬ψ2, ψ1, ψ ∈ B0 =⇒=⇒=⇒ A0 A1 A2 . . .A0 A1 A2 . . .A0 A1 A2 . . . |= ψ1|= ψ1|= ψ1
⇓⇓⇓
309 / 527
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Induction step: until (part “=⇒=⇒=⇒”) ltlmc3.2-65
Claim: If B0A0→ B1
A1→ B2A2→ ...B0
A0→ B1A1→ B2
A2→ ...B0A0→ B1
A1→ B2A2→ ... is a path in GGG s.t.
∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F∀F ∈ F∞∃ j ≥ 0.Bj ∈ F
then for all ψ ∈ cl(ϕ)ψ ∈ cl(ϕ)ψ ∈ cl(ϕ): ψ ∈ B0ψ ∈ B0ψ ∈ B0 iff A0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψA0 A1 A2 . . . |= ψ
Induction step for ψ = ψ1 Uψ2ψ = ψ1 Uψ2ψ = ψ1 Uψ2:
Let ψ ∈ B0ψ ∈ B0ψ ∈ B0 and j ≥ 0j ≥ 0j ≥ 0 minimal s.t. ψ2 ∈ Bjψ2 ∈ Bjψ2 ∈ Bj
IH=⇒=⇒=⇒ Aj Aj+1 . . .Aj Aj+1 . . .Aj Aj+1 . . . |= ψ2|= ψ2|= ψ2
¬ψ2, ψ1, ψ ∈ Bj−1¬ψ2, ψ1, ψ ∈ Bj−1¬ψ2, ψ1, ψ ∈ Bj−1 =⇒=⇒=⇒ Aj−1 Aj . . .Aj−1 Aj . . .Aj−1 Aj . . . |= ψ1|= ψ1|= ψ1.........
...
...
............
...
...
...¬ψ2, ψ1, ψ ∈ B0¬ψ2, ψ1, ψ ∈ B0¬ψ2, ψ1, ψ ∈ B0 =⇒=⇒=⇒ A0 A1 A2 . . .A0 A1 A2 . . .A0 A1 A2 . . . |= ψ1|= ψ1|= ψ1
⇓⇓⇓A0 A1 A2 . . . |= ψ = ψ1 Uψ2A0 A1 A2 . . . |= ψ = ψ1 Uψ2A0 A1 A2 . . . |= ψ = ψ1 Uψ2
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Complexity: LTL ��� NBA ltlmc3.2-67
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Complexity: LTL ��� NBA ltlmc3.2-67
For each LTL formula ϕϕϕ, there is an NBA AAA s.t.
Lω(A)Lω(A)Lω(A) === Words(ϕ)Words(ϕ)Words(ϕ)
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Complexity: LTL ��� NBA ltlmc3.2-67
For each LTL formula ϕϕϕ, there is an NBA AAA s.t.
Lω(A)Lω(A)Lω(A) === Words(ϕ)Words(ϕ)Words(ϕ)
LTL formula ϕϕϕ
GNBA GGG
NBA AAA313 / 527
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Complexity: LTL ��� NBA ltlmc3.2-67
For each LTL formula ϕϕϕ, there is an NBA AAA s.t.
Lω(A)Lω(A)Lω(A) === Words(ϕ)Words(ϕ)Words(ϕ)
LTL formula ϕϕϕ
GNBA GGG
NBA AAA size: size(G) · |F|size(G) · |F|size(G) · |F|314 / 527
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Complexity: LTL ��� NBA ltlmc3.2-67
For each LTL formula ϕϕϕ, there is an NBA AAA s.t.
Lω(A)Lω(A)Lω(A) === Words(ϕ)Words(ϕ)Words(ϕ)
LTL formula ϕϕϕ
GNBA GGG
NBA AAA size: size(G) · |F|size(G) · |F|size(G) · |F|
|F||F||F| === number ofacceptancesets in GGG
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Complexity: LTL ��� NBA ltlmc3.2-67
For each LTL formula ϕϕϕ, there is an NBA AAA s.t.
Lω(A)Lω(A)Lω(A) === Words(ϕ)Words(ϕ)Words(ϕ)
LTL formula ϕϕϕ
GNBA GGG
NBA AAA size: size(G) · |F|size(G) · |F|size(G) · |F|
|F||F||F| === number ofacceptancesets in GGG
≤≤≤ |ϕ||ϕ||ϕ|
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Complexity: LTL ��� NBA ltlmc3.2-67
For each LTL formula ϕϕϕ, there is an NBA AAA s.t.
Lω(A)Lω(A)Lω(A) === Words(ϕ)Words(ϕ)Words(ϕ)
LTL formula ϕϕϕ
GNBA GGG
NBA AAA
size: 2|cl(ϕ)|2|cl(ϕ)|2|cl(ϕ)|
size: size(G) · |F|size(G) · |F|size(G) · |F|
|F||F||F| === number ofacceptancesets in GGG
≤≤≤ |ϕ||ϕ||ϕ|
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Complexity: LTL ��� NBA ltlmc3.2-67
For each LTL formula ϕϕϕ, there is an NBA AAA s.t.
Lω(A)Lω(A)Lω(A) === Words(ϕ)Words(ϕ)Words(ϕ) and
size(A)size(A)size(A) ≤≤≤ 2|cl(ϕ)| · |ϕ|2|cl(ϕ)| · |ϕ|2|cl(ϕ)| · |ϕ|
LTL formula ϕϕϕ
GNBA GGG
NBA AAA
size: 2|cl(ϕ)|2|cl(ϕ)|2|cl(ϕ)|
size: size(G) · |F|size(G) · |F|size(G) · |F|
|F||F||F| === number ofacceptancesets in GGG
≤≤≤ |ϕ||ϕ||ϕ|
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Complexity: LTL ��� NBA ltlmc3.2-67
For each LTL formula ϕϕϕ, there is an NBA AAA s.t.
Lω(A)Lω(A)Lω(A) === Words(ϕ)Words(ϕ)Words(ϕ) and
size(A)size(A)size(A) ≤≤≤ 2|cl(ϕ)| · |ϕ|2|cl(ϕ)| · |ϕ|2|cl(ϕ)| · |ϕ| = 2O(|ϕ|)= 2O(|ϕ|)= 2O(|ϕ|)
LTL formula ϕϕϕ
GNBA GGG
NBA AAA
size: 2|cl(ϕ)|2|cl(ϕ)|2|cl(ϕ)|
size: size(G) · |F|size(G) · |F|size(G) · |F|
|F||F||F| === number ofacceptancesets in GGG
≤≤≤ |ϕ||ϕ||ϕ|
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Size of NBA for LTL formulas ltlmc3.2-68
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Size of NBA for LTL formulas ltlmc3.2-68
For the proposed transformation LTL��� NBA:
The constructed NBA for LTL formulas are oftenunnecessarily complicated
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Size of NBA for LTL formulas ltlmc3.2-68
For the proposed transformation LTL��� NBA:
The constructed NBA for LTL formulas are oftenunnecessarily complicated
NBA for©a©a©a
q0q0q0
q1q1q1
q2q2q2 true
true
aaa
constructed GNBA has444 states and 888 edges
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Size of NBA for LTL formulas ltlmc3.2-68
For the proposed transformation LTL��� NBA:
The constructed NBA for LTL formulas are oftenunnecessarily complicated
NBA for a U ba U ba U b
q0q0q0
q1q1q1
aaa
true
bbb
constructed (G)NBA has555 states and 202020 edges
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Size of NBA for LTL formulas ltlmc3.2-68
For the proposed transformation LTL��� NBA:
The constructed NBA for LTL formulas are oftenunnecessarily complicated
... but there exists LTL formulas ϕnϕnϕn such that
• |ϕn| = O(poly(n))|ϕn| = O(poly(n))|ϕn| = O(poly(n))
• each NBA for ϕnϕnϕn has at least 2n2n2n states
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LT-properties that have no “small” NBA ltlmc3.2-69
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LT-properties that have no “small” NBA ltlmc3.2-69
consider the following family of LT-properties (En)n≥1(En)n≥1(En)n≥1:
En =En =En =
{set of all infinite words over 2AP2AP2AP of the form
A1 A2 A3. . .An A1 A2 A3. . .An B1 B2 B3 B4 . . .A1 A2 A3. . .An A1 A2 A3. . .An B1 B2 B3 B4 . . .A1 A2 A3. . .An A1 A2 A3. . .An B1 B2 B3 B4 . . .
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LT-properties that have no “small” NBA ltlmc3.2-69
consider the following family of LT-properties (En)n≥1(En)n≥1(En)n≥1:
En =En =En =
{set of all infinite words over 2AP2AP2AP of the form
A1 A2 A3. . .An A1 A2 A3. . .AnA1 A2 A3. . .An A1 A2 A3. . .AnA1 A2 A3. . .An A1 A2 A3. . .An︸ ︷︷ ︸B1 B2 B3 B4 . . .B1 B2 B3 B4 . . .B1 B2 B3 B4 . . .︸ ︷︷ ︸= xx= xx= xx
for some x ∈(2AP
)∗x ∈
(2AP
)∗x ∈
(2AP
)∗of length nnn
∈(2AP
)ω∈(2AP
)ω∈(2AP
)ωarbitrary
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LT-properties that have no “small” NBA ltlmc3.2-69
consider the following family of LT-properties (En)n≥1(En)n≥1(En)n≥1:
En =En =En =
{set of all infinite words over 2AP2AP2AP of the form
A1 A2 A3. . .An A1 A2 A3. . .AnA1 A2 A3. . .An A1 A2 A3. . .AnA1 A2 A3. . .An A1 A2 A3. . .An︸ ︷︷ ︸B1 B2 B3 B4 . . .B1 B2 B3 B4 . . .B1 B2 B3 B4 . . .︸ ︷︷ ︸= xx= xx= xx
for some x ∈(2AP
)∗x ∈
(2AP
)∗x ∈
(2AP
)∗of length nnn
∈(2AP
)ω∈(2AP
)ω∈(2AP
)ωarbitrary
LTL formula ϕnϕnϕn with Words(ϕn) = EnWords(ϕn) = EnWords(ϕn) = En
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LT-properties that have no “small” NBA ltlmc3.2-69
consider the following family of LT-properties (En)n≥1(En)n≥1(En)n≥1:
En =En =En =
{set of all infinite words over 2AP2AP2AP of the form
A1 A2 A3. . .An A1 A2 A3. . .AnA1 A2 A3. . .An A1 A2 A3. . .AnA1 A2 A3. . .An A1 A2 A3. . .An︸ ︷︷ ︸B1 B2 B3 B4 . . .B1 B2 B3 B4 . . .B1 B2 B3 B4 . . .︸ ︷︷ ︸= xx= xx= xx
for some x ∈(2AP
)∗x ∈
(2AP
)∗x ∈
(2AP
)∗of length nnn
∈(2AP
)ω∈(2AP
)ω∈(2AP
)ωarbitrary
LTL formula ϕnϕnϕn with Words(ϕn) = EnWords(ϕn) = EnWords(ϕn) = En
ϕn =∧
a∈AP
∧0≤i<n
(©ia↔©i+na
)ϕn =
∧a∈AP
∧0≤i<n
(©ia↔©i+na
)ϕn =
∧a∈AP
∧0≤i<n
(©ia↔©i+na
)329 / 527
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LT-properties that have no “small” NBA ltlmc3.2-69
consider the following family of LT-properties (En)n≥1(En)n≥1(En)n≥1:
En =En =En =
{set of all infinite words over 2AP2AP2AP of the form
A1 A2 A3. . .An A1 A2 A3. . .AnA1 A2 A3. . .An A1 A2 A3. . .AnA1 A2 A3. . .An A1 A2 A3. . .An︸ ︷︷ ︸B1 B2 B3 B4 . . .B1 B2 B3 B4 . . .B1 B2 B3 B4 . . .︸ ︷︷ ︸= xx= xx= xx
for some x ∈(2AP
)∗x ∈
(2AP
)∗x ∈
(2AP
)∗of length nnn
∈(2AP
)ω∈(2AP
)ω∈(2AP
)ωarbitrary
LTL formula ϕnϕnϕn with Words(ϕn) = EnWords(ϕn) = EnWords(ϕn) = En
ϕn =∧
a∈AP
∧0≤i<n
(©ia↔©i+na
)ϕn =
∧a∈AP
∧0≤i<n
(©ia↔©i+na
)ϕn =
∧a∈AP
∧0≤i<n
(©ia↔©i+na
)←−←−←− length
O(poly(n))O(poly(n))O(poly(n))
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LT-property EnEnEn for n=1n=1n=1 ltlmc3.2-69a
E1 =E1 =E1 =
{set of all infinite words over 2AP2AP2AP of the form
A A B1 B2 B3 B4 . . .A A B1 B2 B3 B4 . . .A A B1 B2 B3 B4 . . . where AAA,Bj ⊆ APBj ⊆ APBj ⊆ AP for j ≥ 0j ≥ 0j ≥ 0
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LT-property EnEnEn for n=1n=1n=1 ltlmc3.2-69a
E1 =E1 =E1 =
{set of all infinite words over 2AP2AP2AP of the form
A A B1 B2 B3 B4 . . .A A B1 B2 B3 B4 . . .A A B1 B2 B3 B4 . . . where AAA,Bj ⊆ APBj ⊆ APBj ⊆ AP for j ≥ 0j ≥ 0j ≥ 0
NBA for E1E1E1 if AP = {a}AP = {a}AP = {a}:
q0q0q0
q1q1q1
q2q2q2
q2q2q2
trueaaa aaa
¬a¬a¬a ¬a¬a¬a
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LT-property EnEnEn for n=1n=1n=1 ltlmc3.2-69a
E1 =E1 =E1 =
{set of all infinite words over 2AP2AP2AP of the form
A A B1 B2 B3 B4 . . .A A B1 B2 B3 B4 . . .A A B1 B2 B3 B4 . . . where AAA,Bj ⊆ APBj ⊆ APBj ⊆ AP for j ≥ 0j ≥ 0j ≥ 0
NBA for E1E1E1 if AP = {a}AP = {a}AP = {a}:
q0q0q0
q1q1q1
q2q2q2
q2q2q2
trueaaa aaa
¬a¬a¬a ¬a¬a¬a
LTL-formula:a↔©aa↔©aa↔©a
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LT-property EnEnEn for n=1n=1n=1 ltlmc3.2-69a
E1 =E1 =E1 =
{set of all infinite words over 2AP2AP2AP of the form
A A B1 B2 B3 B4 . . .A A B1 B2 B3 B4 . . .A A B1 B2 B3 B4 . . . where AAA,Bj ⊆ APBj ⊆ APBj ⊆ AP for j ≥ 0j ≥ 0j ≥ 0
NBA for E1E1E1 if AP = {a, b}AP = {a, b}AP = {a, b}:
q0q0q0
q1q1q1
q2q2q2
q3q3q3
q4q4q4
q2q2q2
truea ∧ ba ∧ ba ∧ b a ∧ ba ∧ ba ∧ b
a ∧ ¬ba ∧ ¬ba ∧ ¬b a ∧ ¬ba ∧ ¬ba ∧ ¬b
¬a ∧ b¬a ∧ b¬a ∧ b ¬a ∧ b¬a ∧ b¬a ∧ b
¬a ∧ ¬b¬a ∧ ¬b¬a ∧ ¬b ¬a ∧ ¬b¬a ∧ ¬b¬a ∧ ¬b
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LT-property EnEnEn for n=1n=1n=1 ltlmc3.2-69a
E1 =E1 =E1 =
{set of all infinite words over 2AP2AP2AP of the form
A A B1 B2 B3 B4 . . .A A B1 B2 B3 B4 . . .A A B1 B2 B3 B4 . . . where AAA,Bj ⊆ APBj ⊆ APBj ⊆ AP for j ≥ 0j ≥ 0j ≥ 0
NBA for E1E1E1 if AP = {a, b}AP = {a, b}AP = {a, b}:
q0q0q0
q1q1q1
q2q2q2
q3q3q3
q4q4q4
q2q2q2
truea ∧ ba ∧ ba ∧ b a ∧ ba ∧ ba ∧ b
a ∧ ¬ba ∧ ¬ba ∧ ¬b a ∧ ¬ba ∧ ¬ba ∧ ¬b
¬a ∧ b¬a ∧ b¬a ∧ b ¬a ∧ b¬a ∧ b¬a ∧ b
¬a ∧ ¬b¬a ∧ ¬b¬a ∧ ¬b ¬a ∧ ¬b¬a ∧ ¬b¬a ∧ ¬b
LTL-formula:
(a(a(a ↔↔↔ ©a)©a)©a) ∧∧∧(b(b(b ↔↔↔ ©b)©b)©b)
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LT property EnEnEn for n=2n=2n=2 and AP = {a}AP = {a}AP = {a} ltlmc3.2-70
p0p0p0
q1q1q1
q0q0q0
q11q11q11
q10q10q10
q01q01q01
q00q00q00
q111q111q111
q101q101q101
q010q010q010
q000q000q000
qFqFqF
aaa
¬a¬a¬a
aaa
¬a¬a¬a
aaa
¬a¬a¬a
aaa
aaa
¬a¬a¬a
¬a¬a¬a
aaa
¬a¬a¬a
aaa
¬a¬a¬a
true
E2 ={A1A2A1A2σ : A1,A2 ⊆ AP , σ ∈
(2AP)ω
}E2 =
{A1A2A1A2σ : A1,A2 ⊆ AP , σ ∈
(2AP)ω
}E2 =
{A1A2A1A2σ : A1,A2 ⊆ AP, σ ∈
(2AP)ω
}
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LT property EnEnEn for n=2n=2n=2 and AP = {a}AP = {a}AP = {a} ltlmc3.2-70
p0p0p0
q1q1q1
q0q0q0
q11q11q11
q10q10q10
q01q01q01
q00q00q00
q111q111q111
q101q101q101
q010q010q010
q000q000q000
qFqFqF
aaa
¬a¬a¬a
aaa
¬a¬a¬a
aaa
¬a¬a¬a
aaa
aaa
¬a¬a¬a
¬a¬a¬a
aaa
¬a¬a¬a
aaa
¬a¬a¬a
true
E2 ={A1A2A1A2σ : A1,A2 ⊆ AP , σ ∈
(2AP)ω
}E2 =
{A1A2A1A2σ : A1,A2 ⊆ AP , σ ∈
(2AP)ω
}E2 =
{A1A2A1A2σ : A1,A2 ⊆ AP, σ ∈
(2AP)ω
}LTL-formula: (a↔©©a) ∧ (©a↔©©©a)(a↔©©a) ∧ (©a↔©©©a)(a↔©©a) ∧ (©a↔©©©a)
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LT property EnEnEn for n=2n=2n=2 and AP = {a}AP = {a}AP = {a} ltlmc3.2-70
p0p0p0
q1q1q1
q0q0q0
q11q11q11
q10q10q10
q01q01q01
q00q00q00
q111q111q111
q101q101q101
q010q010q010
q000q000q000
qFqFqF
aaa
¬a¬a¬a
aaa
¬a¬a¬a
aaa
¬a¬a¬a
aaa
aaa
¬a¬a¬a
¬a¬a¬a
aaa
¬a¬a¬a
aaa
¬a¬a¬a
true
general case: each NBA for EnEnEn has ≥ 2n≥ 2n≥ 2n states
338 / 527
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LT property EnEnEn for n=2n=2n=2 and AP = {a}AP = {a}AP = {a} ltlmc3.2-70
p0p0p0
q1q1q1
q0q0q0
q11q11q11
q10q10q10
q01q01q01
q00q00q00
q111q111q111
q101q101q101
q010q010q010
q000q000q000
qFqFqF
aaa
¬a¬a¬a
aaa
¬a¬a¬a
aaa
¬a¬a¬a
aaa
aaa
¬a¬a¬a
¬a¬a¬a
aaa
¬a¬a¬a
aaa
¬a¬a¬a
true
general case: each NBA for EnEnEn has ≥ 2n≥ 2n≥ 2n states
En = Words(ϕn)En = Words(ϕn)En = Words(ϕn) where ϕn =∧
a∈AP
∧0≤i<n
(©ia↔©n+ia
)ϕn =
∧a∈AP
∧0≤i<n
(©ia↔©n+ia
)ϕn =
∧a∈AP
∧0≤i<n
(©ia↔©n+ia
)339 / 527
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LT property EnEnEn for n=2n=2n=2 and AP = {a}AP = {a}AP = {a} ltlmc3.2-70
p0p0p0
q1q1q1
q0q0q0
q11q11q11
q10q10q10
q01q01q01
q00q00q00
q111q111q111
q101q101q101
q010q010q010
q000q000q000
qFqFqF
aaa
¬a¬a¬a
aaa
¬a¬a¬a
aaa
¬a¬a¬a
aaa
aaa
¬a¬a¬a
¬a¬a¬a
aaa
¬a¬a¬a
aaa
¬a¬a¬a
true
general case: each NBA for EnEnEn has ≥ 2n≥ 2n≥ 2n states
En = Words(ϕn)En = Words(ϕn)En = Words(ϕn) where ϕn =∧
a∈AP
∧0≤i<n
(©ia↔©n+ia
)ϕn =
∧a∈AP
∧0≤i<n
(©ia↔©n+ia
)ϕn =
∧a∈AP
∧0≤i<n
(©ia↔©n+ia
)340 / 527