On the Satisfiability of Metric Temporal Logics over the Reals - - PowerPoint PPT Presentation

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On the Satisfiability of Metric Temporal Logics over the Reals - - PowerPoint PPT Presentation

Nov 05, 2023 252 likes •504 views

On the Satisfiability of Metric Temporal Logics over the Reals Marcello M. Bersani Matteo Rossi Pierluigi San Pietro - Politecnico di Milano Motivations Con$nuous'$me'is'typically'used'when'modeling'hybrid' systems'

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SLIDE 1

On the Satisfiability of Metric Temporal Logics over the Reals

Marcello M. Bersani Matteo Rossi Pierluigi San Pietro

  • Politecnico di Milano
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SLIDE 2

Motivations

  • Con$nuous'$me'is'typically'used'when'modeling'hybrid'

systems'

– computer'systems'that'interact'with'the'physical'world'

  • Also'well'suited'to'capture'asynchrony'in'systems'

– e.g.,'events'that'occur'close'to'each'other,'but'not'at'the'same' $me'

  • Successful'formalisms'and'tools'to'capture'and'analyze'

con$nuous'systems'

– e.g.,'Timed'Automata'(Uppaal)'

  • Con$nuousB$me'temporal'logics'are'useful'to'capture'the'

proper$es'of'systems'

– e.g.,'highBlevel'requirements' – can'also'be'used'to'provide'descrip$ve'models'of'systems'

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SLIDE 3

Motivations

  • Since'‘80'various'aGempts'to'embed'explicit'(real)'

$me'in'LTL'

– Harel,'Pnueli,'Ostroff,'etc.'

  • “A'Really'Temporal'Logic”,'Alur,'Henzinger,'1989'

– (un)Decidability'of'TPTL'over'(dense)'natural'$me'

  • Shortly'aTer,'Metric'Temporal'Logic'(MTL)1'

– (un)Decidability'over'(dense)'natural'$me' – Implicit'use'of'$me'in'parameterized'modali$es'◊<c'

  • Previously'introduced'by'Harel,'Pnueli2'

1'“Real'$me'Logics:'Complexity'and'Expressiveness”','Alur,'Henzinger,'1993'(LICS'1990)' 2'“Applica$ons'of'Temporal'Logic'to'Specifica$on'of'RealB$me'Systems”','Pnueli,'Harel,'1988'

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SLIDE 4

Semantics

φMTL'='p'|'¬φ'|'φ∧φ'|'φUIφ'|'φSIφ' I'='〈a,b〉'or'〈a,∞〉'''''a∈,'b∈

  • Seman$cs'can'be'defined'wrt'Signals)

φU(1,2)ψ'

M:'→ 2AP'

MITL'='fragment'of'MTL's.t.'intervals'I'are'non'punctual'

∃d’∈(1,2),'M,t+d'''ψandM,t’’'φ,∀t’’'∈(t,t+d’)'

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SLIDE 5

Overview1 for SAT

TPTL' MTL3' MITL3' MTL+past' MITL+past'

1'Mainly'from'“On'the'Expressiveness'of'TPTL'and'MTL”','Bouyer,'Chevalier,'Markey,'2009'

MTL0,∞' EXPSPACEBc' PSPACEBc' Undecidable' QMLO=QTL2'

2'“Logics'for'Real'$me:'Decidability'and'Compexity”,'Hirshfeld,'Rabinovich,'2004' 3'“The'Benefits'of'relaxing'punctuality”,'Alur,'Feder,'Henzinger,'1996'

ECL4'

4'“The'Regular'RealB$me'Languages”,''Henzinger,'Raskin,'Schobbens,'1998'

F[1,1]'!'

Explicit'clock'+'Freeze'operator' x.(pU(x<1))' No'clocks'

F[1,1]'"'

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SLIDE 6

QTL (Quantitative Temporal Logic)

φQTL'='p'|'¬φ'|'φ∧φ'|'φUφ'|'φSφ'|'F(0,1)φ'|'P(0,1)φ' '

  • Seman$cs'wrt'Signals)
  • QTL'has'the'same'expressive'power'of'MITL'

M:'→ 2AP'

F(0,1)φ'

∃d’∈(0,1),'M,t+d'''φ'

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SLIDE 7

What we obtained

  • A'new)proof)of'decidability'for'QTL'

'

  • Implemented'tool'deciding'SAT'for'QTL'

– and'all'the'equivalent'logics'(MITL,'ECL,'QMLO)' – unrestricted'(non'Zeno)'signals''

  • To'the'best'of'our'knowledge,'first'tool'that'

can'handle'SAT'for'these'logics'over' con$nuous'$me'

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SLIDE 8

Our solution

QTL)→)CLTLCoverCclocks1'

  • CLTLoc'is'decidable''(PSPACECc)'
  • CLTLoc'formulae'contain'explicit)clocks'
  • Based'on'(PSPACE)'SAT'of'CLTL2'

– A'decision'procedure'is'available'based'on'SMT' (bounded'SAT)3'

3'“Constraint''LTL'Sa$sfiability'Checking'without'Automata”,'Bersani'et'al.,'2012' 2'“An'automata'Theore$c'Approach'to'Constraint'LTL”,'Demri,'D’Souza,'2003' 1'“A'Tool'for'Deciding''Con$nuos'Time'Metric'Temporal'Logic”,'Bersani,'Rossi,'San'Pietro,'2013'

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SLIDE 9

Constraint LTL over clocks

  • Fragment'of'LTL(FO)'B'〈,<,=〉'
  • V'is'a'finite'set'of'clocks,'z∈V)

τ'= c'|'z'|'Xz' α'='p'|'τ1<τ2'|'τ1=τ2' φ'='α'|'¬φ'|'φUφ'|'φSφ'|'Xφ'|'Yφ'

  • Models:'(π,σ)'

z'∈V, c'∈' Standard'LTL'

π:''→2AP' σ:'×V → '

p∈AP'

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SLIDE 10

Constraint LTL over clocks

  • Alur&Dill'clocks'

– Nonnega$ve'' – strongly'monotonic'(except'for'“resets”)' – Clock'progressiveness1'(non'Zeno'signals)'

'z: ' '0.3'' 1.5' 1.8' 2.0' '0'' 0' 1.0' 6.0' 18.4' 0'

Xz >'z' G(φ)'='¬F(¬φ)'='¬(TU'¬φ)

G(z≥0)'∧'G(Xz=0'∨'Xz>z)'∧'(GF(z=0)'∨'FG(z>maxz))'

1'“A'Theory'of'Timed'Automata”,'Alur,'Dill','1994'

'

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SLIDE 11

From signals to CLTLoc models

  • Given'a'QTL'formula'φ'and'θ'one'of'its'subformlae,'

Mθ'is'the'signal'represen$ng'the'changing'points'of'θ'

F(0,1)'a' a' b' b'∧'F(0,1)'a'

M:'→ 2AP'

=1'

Mθ''by'QTL'seman$cs'

=1'

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SLIDE 12

From signals to CLTLoc models

  • Rela$on'from'signals'M'to'CLTLoc'models'(π,σ)''

r(M)'='{(π,σ)i}'

F(0,1)'φ' φ'

(π,σ)' (π0,σ0)' (πi,σi)'

r(M)'' rB1(π,σ)'

Denumerable'subset'of'' …'

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SLIDE 13

From signals to CLTLoc models

  • For'each'subformula'θ'

– Atoms'fθ,'hθ' – Clocks'zθ

0,zθ 1 '

  • Discrete'posi$ons'in'(π,σ)'represent'the'behavior'of'

θ'at'the'corresponding'posi$on'in'Mθ'

F(0,1)'φ' φ' {…}' {…}' {…}' {…}' {…}' {…}' (π,σ)'

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SLIDE 14

From signals to CLTLoc models

  • Each'posi$on'in'π'represents'the'truth'of'θ'at'the'corresponding'

interval'in'Mθ'

– if'atom'fθ'is'true,'θ'holds'in'the'first)point'of'the'current'interval' – if'atom'hθ'is'true,'θ'holds'in'the'rest)of)the)points'of'the'current'interval' θ=F(0,1)'φ' φ' π' {fφ,fθ,' hφ,hθ}' {fφ}' {}' {hθ}' fφ,hφ' fθ,hθ' fφ,hφ' fφ,¬hφ' ¬fφ,¬hφ' ¬fφ,¬hφ' fφ,hφ' …' ¬fθ,hθ' {fφ,fθ,' hφ,hθ}' {fφ,fθ,' hφ,hθ}'

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SLIDE 15

From signals to CLTLoc models

  • Time'progress'among'posi$ons'is'measured'by'

clocks'

– Clocks'zθ

0,zθ 1'($me'elapsed'since'the'last'two'events)'

F(0,1)'φ' φ' π' {fφ,fθ,' hφ,hθ}' {fφ,fθ,' hφ,hθ}' σ' zφ

0=0)

zφ

1>0'

zφ

0=.3'

zφ

1>0'

{fφ}' zφ

0=1'

zφ

1=0)

{}' zφ

0=2.2'

zφ

1=1.2'

{hθ}' zφ

0=2.4'

zφ

1=1.4'

{fφ,fθ,' hφ,hθ}' zφ

0=0)

zφ

1=2.3'

0.3' 0.7' 1.2' 0.2' 0.9'

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SLIDE 16

Equisatisfiability

  • Given'a'QTL'formula'Φ,'we'build'a'set'of'CLTLoc'

formulae'' {m(θ)'|'θ'subformula'of'Φ}' ' '

M,0'Φ'''''iff'''''(π,σ),0''fΦ'θ'G(m(θ))

(for'all'(π,σ)''r(M))'

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SLIDE 17

Translation for U'

θφU'ψ' φ' ψ'

m(θ):'fθ'⇔'hθ'''''' ∧'' hθ'⇔'hφ'∧'(hψ'∨'X(MφU ('(Mφ'∧'hψ)'∨'fψ)))'

Mφ' Mφ' Mφ' Mφ' Mφ' hψ'

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SLIDE 18

Translation for F(0,1)'

19'

θF(0,1)'φ' φ'

zθ

i=0'

zθ

i=1'

zφ

j=0'

zφ

j>1'

>1' =1'

θ' φ'

m(θ):'''''θ⇔¬fθ'∧'zθ

i=0'∧'X(zθ i>0'U'(''''φ '∧'zθ i=1'∧'zφ j>1'))'

θ'⇔'¬Y(hθ)'∧'hθ'

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SLIDE 19

Translation for F(0,1)'

20'

θF(0,1)'φ' φ'

m(θ):'''''θ⇔'''''ϕ'∧'¬X(¬''ϕU'(''''ϕ'∧'0'<'zϕ

i≤1)'

zφ

i=0'

zϕ

i>1'

>1'

φ' θ' φ'

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SLIDE 20

Complexity

  • The'sa$sfiability'problem'for'QTL'is'known'to'

be'PSPACEBc'

  • CLTLoc'is'PSPACEBc1''
  • The'size'of'formula'θ'G(m(θ))'is'O(|ΦQTL|)'

– PSPACE'complexity'is'preserved'

1'Number'of'subformulae,'max'constant'occurring'(binary'encoding)'

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SLIDE 21

K-bounded SAT

  • Find'a'(infinite)'periodic'model'over'

– Subformulae) – Regions'for'clocks'(not'over'values!!)'

  • with'at.most.K'changing'points''

π' {φ,θ}' Rσ' RlB1' {fφ}' Rl {φ,θ}' {fφ}' Rk. Rk+1'

K=6. 1. 2. 3.

  • 4. 5.

='12(3456)ω'

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SLIDE 22

Example

G(0,100)¬p'→'G(100,200)¬p'' ∧'' p'→'F(0,200)p'' G[0,∞) (

)

∧'p'∧'G(0,100)¬p'' G(0,'∞)('p'→'F(0,1)q'∨'P(0,1)q')' G(0,'∞)('q'→'G(0,100)¬q')' G(0,'∞)('q'→'G(0,100]¬q')'

0' 100'

α=' β=' γ=' δ=' Formula) t) K) periodicity)

α'

10s) 10' B'

α'∧'β'

40s) 10' B'

α'∧'β'∧'γ'

10m) 20' 15m)

α'∧'β'∧'δ'

80m) 30' >12h)

SAT' UNSAT'

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SLIDE 23

Implementation

  • qtlSolver:'hGp://code.google.com/p/qtlsolver/'

– Transla$on'of'QTL'(and'MITL)'to'CLTLoc' – Java'

  • ae2Zot:'arithme$cal'plugin'for'Zot'

– Bounded'SAT'for'CLTL'and'CLTLoc' – SMT'based'

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SLIDE 24

Questions?