ltl to buechi automata
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LTL to Bchi Automata
LTL to Bchi AutomataFlavio Lerda
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LTL to Bchi Automata
LTL Formulas
Subset of CTL* Distinct from CTL
AFG LTL
f CTL ! f " AFG Contains a sin#le universal $uantifier
The ath formula f holds for ever% ath
Commonl%& A is omitted G is relaced b% 'bo( or al)a%s F is relaced b% 'diamond or eventuall%
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LTL to Bchi Automata
+(amles of LTL formulas
Al)a%s eventuall% &
AGF or AG AF
Al)a%s after eventuall% $ ' $
AG ' ,- F $ or AG ' ,- AF $
Fairness ' A ''GF .ot a CTL formula
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LTL to Bchi Automata
LTL Semantics
Derived from the CTL* semantics Given an infinite e(ecution trace/s0s12
3 's03 4 4' 3 3 1 53 13 53 153 13 53 i0 i3 3 i0 i3 3 16 5 i0 i3 5 0 78 i 73 1
i
is the suffi( of startin# at si
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LTL to Bchi Automata
Bchi Automata
Automaton )hich accets infinite traces A Bchi automatonis 9,tuleS: ;:: F
S is a finite set of states ;S is a set of initial states SS is a transition relation
FS is a set of accetin# states
An infinite se$uence of statesis accetediff it contains accetin# states infinitel%often
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LTL to Bchi Automata
+(amle
S0 S1 S5
1/S0S1S5S5S5S52
5/S0S1S5S1S5S12
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LTL to Bchi Automata
LTL and Bchi Automata
LTL formula =eresents a set of infinite traces)hich
satisf% such formula
Bchi AutomatonAccets a set of infinite traces
>e can build an automaton )hich accetsall and onl%the infinite traces reresentedb% an LTL formula
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LTL to Bchi Automata
Labeled Bchi Automata
Given a set of atomic roosition? Define a labelin# function
& S 5?
+ach state is assi#ned a set of roositionsthat must be true
Similar to the labelin# for the model @
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LTL to Bchi Automata
Generatin# Bchi Automata
>e need a rocedureto #enerate a Bchiautomaton #iven an LTL formula +fficientl%
Formulas are usuall% small Bchi automaton exponential in the size of the formula
The cost of model checin# is proportionalto the sie of theautomaton
.on,deterministic Bchi automata are not equivalent
to deterministic Bchi automata Cannotuse automata minimiation al#orithms Findin# the minimal automata is e(onential
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LTL to Bchi Automata
Aroach
Formula rewriting =e)rite the formula in negated normal formAl% re)ritin# rules
Core translation Turns an LTL formula into a generalized Bchi
automaton
Degeneralization Turns a #eneralied Bchi automaton into aBchi automaton
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LTL to Bchi Automata
=e)ritin#
Negate normal form .e#ation aears onlyin front of literals 6se the follo)in# identities
44/ 4G / F 4 4F / G 4 4'6 / '4 '4
4' / '4 6 '4 'sometimes = is the Release oerator
Dual of Until
ALTL BhiA
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LTL to Bchi Automata
=e)ritin#
Additional rewriting rules The% are not guaranteedto %ield smaller
automata
The sie of the automaton is e(onential inthe sie of the formula
+(amles
' 6 ' '6 ' ' ' GF GF GF '
15820ALTL BhiA
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LTL to Bchi Automata
=e)ritin#
The core al#orithm onlyhandles ! ! "! #
6se the follo)in#& F T6 G 4F 44'T6 4 / F
15820ALTLtBhiA t t
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LTL to Bchi Automata
Core Translation
$ea
@ae use of the follo)in#6 ''6 ''
15820ALTLtBhiA t t
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LTL to Bchi Automata
+(amleF
p
(TUp)
Old:{}
New:{TUp}
Next:{}
Old:{TUp}New:{T}
Next:{TUp}
Old:{TUp}New:{p}
Next:{}
Old:{TUp}New:{}
Next:{TUp}
Old:{TUp}Next:{TUp}
Old:{TUp, p}New:{}
Next:{}
Old:{TUp, p}Next:{}
T pp
Old:{}New:{}
Next:{}
Old:{}Next:{}
15820ALTLtBhiA t t
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LTL to Bchi Automata
Core Translation
Noe =eresent a sub,formula Contain information about the past, the
present and the future
%tate =eresents a state in the final automaton The% are the nodes that have full& expane
15820ALTLtoBchiAutomata
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LTL to Bchi Automata
Core Translation
+(ansion Select a formula from the New field ;f it is a literal: add it to the Oldfield
Ether)ise
'.e):.e(tand '.e):.e(t6
'.e):.e)6 and '.e):.e(t
'.e):.e) and '.e)::.e(t
U (X(U ))
V (X(V ))
15820ALTLtoBchiAutomata
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LTL to Bchi Automata
Core Translation
Noes to states ;f a node has no New formulas Create a ne) node )ith all the Nextformulas Create an ed#e bet)een the t)o nodes Chec if there is an% e$uivalent state
>ith the same Next field
>ith the same Old field
15820ALTLtoBchiAutomata
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LTL to Bchi Automata
Core Translation
A''epting states Generalied Bchi automaton
Multipleaccetin# sets Ene for each Until sub,formula '6 Such that
The Oldfield doesnHt contain 6
or The Oldfield does contain
15820ALTLtoBchiAutomata
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LTL to Bchi Automata
De#eneraliation
Turn a #eneralied Bchi automaton into a()'hi automaton
Consider as man& 'opiesof the automaton as
the number of accepting sets =elace incomin# ed#es from accetin# states
)ith ed#es to the next co%
+ach c%cle must #o throu#h everyco% +ach c%cle must contains accetin# states fromeach accepting set
15820ALTLtoBchiAutomata
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LTL to Bchi Automata
+(amle
T
a b
T
a b
T
1
1,2
2
F a F b
15820ALTLtoBchiAutomata
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LTL to Bchi Automata
+(amle
T
a b
T
a
T
T
a b
T
b
T
15-820ALTLtoBchiAutomata
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LTL to Bchi Automata
+(amle
T
a b
T
a
T
T
a b
T
b
T
15-820ALTLtoBchiAutomata
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LTL to Bchi Automata
+(amle
T
a b
T
a
T
T
a b
T
15-820ALTLtoBchiAutomata
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LTL to Bchi Automata
+(amle
T
a b
T
a
T T
15-820ALTLtoBchiAutomata
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LTL to Bchi Automata
+(amle
T
a b
T
a
TT
15-820ALTLtoBchiAutomata
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LTL to Bchi Automata
Etimiations
Can be done at ea'h stage Tr% to minimie
The number of states and transitions
The number of accepting states ;nvolve Stron#l% connected comonents Fair 'bisimulation
Expensivebut The Bchi automaton is usuall% small The savin# durin# verification can be ver% hi#h
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