Pr Priced iced Ti Time med Au Automata mata and Ti Time med - - PowerPoint PPT Presentation

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Pr Priced iced Ti Time med Au Automata mata and Ti Time med - - PowerPoint PPT Presentation

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Pr Priced iced Ti Time med Au Automata mata and Ti Time med Ga Game mes Ki Kim m G. . La Lars rsen Aa Aalborg org Unive versity rsity, , DENMAR NMARK Sc Sche heduling uling Pric iced Tim imed Automa mata and Sy Synt

slide-1
SLIDE 1

Pr Priced iced Ti Time med Au Automata mata and Ti Time med Ga Game mes

Ki Kim m G. . La Lars rsen Aa Aalborg

  • rg Unive

versity rsity, , DENMAR NMARK

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SLIDE 2

Sc Sche heduling uling Pric

iced Tim imed Automa mata

and Sy Synt nthe hesis sis Tim

imed Ga Games es

Ki Kim m G. . La Lars rsen Aa Aalborg

  • rg Unive

versity rsity, , DENMAR NMARK

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SLIDE 3

Ov Overview view

  • Timed

med Automata & UPPAAL

  • Symb

mboli

  • lic Verification &

UPPAAL Engine, Options

  • Priced

iced Timed Automata and Timed Game ames

  • Stochastic

chastic Timed Automata Statist tistical ical Model Checking (Lecture+Exercise)4

TRON

CLASSIC

TIGA

CORA

ECDAR SMC

VTSA Summer r School, l, 2013 2013. Kim Larse sen [3]

slide-4
SLIDE 4

Resourc

  • urces

s & Ta Tasks sks

Resource Task

Shared variable Synchronization

VTSA Summer r School, l, 2013 2013. Kim Larse sen [4]

slide-5
SLIDE 5

Task sk Graph aph Sched heduling uling – Example mple + * + * + *

3ps

*

2ps

+

7ps

*

5ps

+

time

P1 P2

5 10 15 20 25 2 3 6 4 5 1

1 2 3 6 5 4

Compute : (D * ( C * ( A + B )) + (( A + B ) + ( C * D )) using 2 processors

P1 (fast)‏ P2 (slow)‏

A B C D C D 4

VTSA Summer r School, l, 2013 2013. Kim Larse sen [5]

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SLIDE 6

Task sk Graph aph Sched heduling uling – Example mple + * + * + *

3ps

*

2ps

+

7ps

*

5ps

+

time

P1 P2

5 10 15 20 25 2 3 6 4 5 1

Compute : (D * ( C * ( A + B )) + (( A + B ) + ( C * D )) using 2 processors

P1 (fast)‏ P2 (slow)‏

A B C D C D

1 2 3 6 5 4

VTSA Summer r School, l, 2013 2013. Kim Larse sen [6]

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SLIDE 7

Task sk Graph aph Sched heduling uling – Example mple + * + * + *

3ps

*

2ps

+

7ps

*

5ps

+

time

P1 P2

5 10 15 20 25 2 3 6 4 5 1

Compute : (D * ( C * ( A + B )) + (( A + B ) + ( C * D )) using 2 processors

P1 (fast)‏ P2 (slow)‏

A B C D C D

1 2 3 6 5 4

VTSA Summer r School, l, 2013 2013. Kim Larse sen [7]

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SLIDE 8

Task sk Graph aph Sched heduling uling – Example mple + * + * + *

3ps

*

2ps

+

7ps

*

5ps

+

time

P1 P2

5 10 15 20 25 2 3 6 4 5 1

Compute : (D * ( C * ( A + B )) + (( A + B ) + ( C * D )) using 2 processors

P1 (fast)‏ P2 (slow)‏

A B C D C D

1 2 3 6 5 4 E<> (Task1.End‏and‏…‏and‏Task6.End)

VTSA Summer r School, l, 2013 2013. Kim Larse sen [8]

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SLIDE 9

Experimenta perimental l Results ults

Abdeddaïm, Kerbaa, Maler

Symbolic A* Branch-&-Bound 60 sec

VTSA Summer r School, l, 2013 2013. Kim Larse sen [9]

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SLIDE 10

Jo Jobshop hop Sched heduling uling

[TACAS’2001]

Sport Economy Local News Comic Stip

Kim

  • 2. 5 min
  • 4. 1 min
  • 3. 3 min
  • 1. 10 min

Jüri

  • 1. 10 min
  • 2. 20 min
  • 3. 1 min
  • 4. 1 min

Jan

  • 4. 1 min
  • 1. 13 min
  • 3. 11 min
  • 2. 11 min

Wang

  • 1. 1 min
  • 2. 1 min
  • 3. 1 min
  • 4. 1 min

Problem: compute the minimal MAKESPAN

NP-hard

Simulated annealing Shiffted bottleneck Branch-and-Bound Gentic Algorithms

VTSA Summer r School, l, 2013. Kim Larse sen [10 10]

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SLIDE 11

Jo Jobshop hop Sched heduling uling in n UPPAAL AAL

VTSA Summer r School, l, 2013 2013. Kim Larse sen [11 11]

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SLIDE 12

Pr Pric iced ed Tim imed ed Aut Autom

  • mata

ta

slide-13
SLIDE 13

Tas ask Gra raph ph Scheduling heduling – Revis visited ited

+ * + * + *

3ps

*

2ps

+

7ps

*

5ps

+

time

P1 P2

5 10 15 20 25 2 3 6 4 5 1

Compute : (D * ( C * ( A + B )) + (( A + B ) + ( C * D )) using 2 processors

P1 (fast) P2 (slow)

A B C D C D

1 2 3 6 5 4

90W

In use

1oW

Idle

30W

In use

20W

Idle

ENERGY:

VTSA Summer r School, l, 2013 2013. Kim Larse sen [13 13]

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SLIDE 14

Tas ask Gra raph ph Scheduling heduling – Revis visited ited

+ * + * + *

3ps

*

2ps

+

7ps

*

5ps

+

time

P1 P2

5 10 15 20 25 2 3 6 4 5 1

Compute : (D * ( C * ( A + B )) + (( A + B ) + ( C * D )) using 2 processors

P1 (fast)‏ P2 (slow)‏

A B C D C D

90W

In use

10W

Idle

30W

In use

20W

Idle

ENERGY:

1 2 3 6 5 4

VTSA Summer r School, l, 2013 2013. Kim Larse sen [14 14]

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SLIDE 15

Tas ask Gra raph ph Scheduling heduling – Revis visited ited

+ * + * + *

3ps

*

2ps

+

7ps

*

5ps

+

time

P1 P2

5 10 15 20 25 2 3 6 4 5 1

Compute : (D * ( C * ( A + B )) + (( A + B ) + ( C * D )) using 2 processors

P1 (fast)‏ P2 (slow)‏

A B C D C D

90W

In use

10W

Idle

30W

In use

20W

Idle

ENERGY:

1 2 3 6 5 4

VTSA Summer r School, l, 2013 2013. Kim Larse sen [15 15]

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SLIDE 16

A si simple mple examp mple le

VTSA Summer r School, l, 2013 2013. Kim Larse sen [16 16]

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SLIDE 17

A si simple mple examp mple le

Q: What is cheapest cost for reaching ?

VTSA Summer r School, l, 2013 2013. Kim Larse sen [17 17]

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SLIDE 18

Cor

  • rner

ner Poi

  • int

nt Regi gions

  • ns

3 3 THM [Behrmann, Fehnker ..01] [Alur,Torre,Pappas 01] Optimal reachability is decidable for PTA THM [Bouyer, Brojaue, Briuere, Raskin 07] Optimal reachability is PSPACE-complete for PTA

VTSA Summer r School, l, 2013 2013. Kim Larse sen [18 18]

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SLIDE 19

Priced iced Zo Zone nes

A cost function C C(x,y)= 2¢x - 1¢y + 3 A zone Z: 1· x · 2 Æ 0· y · 2 Æ x - y ¸ 0

[CAV01 01]

VTSA Summer r School, l, 2013 2013. Kim Larse sen [19 19]

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SLIDE 20

Priced iced Zo Zone nes – Reset

A cost function C C(x,y) = 2¢x - 1¢y + 3 A zone Z: 1· x · 2 Æ 0· y · 2 Æ x - y ¸ 0 Z[x=0]: x=0 Æ 0· y · 2 C = 1¢y + 3 C= -1¢y + 5

[CAV01 01]

VTSA Summer r School, l, 2013 2013. Kim Larse sen [20 20]

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SLIDE 21

Sym ymbolic bolic Bra ranch nch & & Bound

  • und Al

Algo gorithm rithm

Z’ is bigger & cheaper than Z · is a well-quasi

  • rdering which

guarantees termination!

Z Z  '

THM [Behrmann, Fehnker ..01] [Alur,Torre,Pappas 01] Optimal reachability is decidable for PTA THM [Bouyer, Brojaue, Briuere, Raskin 07] Optimal reachability is PSPACE-complete for PTA

VTSA Summer r School, l, 2013 2013. Kim Larse sen [21 21]

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SLIDE 22

cost t E L T E earliest landing time T target time L latest time e cost rate for being early l cost rate for being late d fixed cost for being late e*(T-t) d+l*(t-T) Planes have to keep separation distance to avoid turbulences caused by preceding planes

Runway

Example ample: : Aircraft craft Land nding ing

VTSA Summer r School, l, 2013 2013. Kim Larse sen [22 22]

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SLIDE 23

Planes have to keep separation distance to avoid turbulences caused by preceding planes land! x >= 4 x=5 x <= 5 x=5 x <= 5 land! x <= 9 cost+=2 cost’=3 cost’=1 4 earliest landing time 5 target time 9 latest time 3 cost rate for being early 1 cost rate for being late 2 fixed cost for being late

Runway

Example ample: : Aircraft craft Land nding ing

VTSA Summer r School, l, 2013 2013. Kim Larse sen [23 23]

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SLIDE 24

Aircraft craft Land nding ing

Source of examples: Baesley et al’2000

VTSA Summer r School, l, 2013 2013. Kim Larse sen [24 24]

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SLIDE 25

Op Opti timal mal In Infi finite nite Sched hedule ule

VTSA Summer r School, l, 2013 2013. Kim Larse sen [25 25]

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SLIDE 26

Op Opti timal mal In Infi finite nite Sched heduling uling

Maximize throughput: i.e. maximize Reward / Time in the long run!

VTSA Summer r School, l, 2013 2013. Kim Larse sen [26 26]

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SLIDE 27

Op Opti timal mal In Infi finite nite Sched heduling uling

Minimize Energy Consumption: i.e. minimize Cost / Time in the long run

VTSA Summer r School, l, 2013 2013. Kim Larse sen [27 27]

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SLIDE 28

Op Opti timal mal In Infi finite nite Sched heduling uling

Maximize throughput: i.e. maximize Reward / Cost in the long run

VTSA Summer r School, l, 2013 2013. Kim Larse sen [28 28]

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SLIDE 29

Bouyer, Brinksma, Larsen: HSCC04,FMSD07

Mean an Pay ay-Off Off Op Optimality imality

c1 c2 c3 cn r1 r2 r3 rn

s Value of path s: val(s) = limn!1 cn/rn Optimal Schedule s*: val(s*) = infs val(s)

Accumulated cost Accumulated reward

: BAD

VTSA Summer r School, l, 2013 2013. Kim Larse sen [29 29]

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SLIDE 30

Larsen, Fahrenberg: INFINITY’08

Discount count Op Optimality imality

c(t1) c(t2) c(t3) c(tn) t1 t2 t3 tn

s Value of path s: val(s) = Optimal Schedule s*: val(s*) = infs val(s)

Cost of time tn Time of step n

 < 1 : discounting factor

: BAD

VTSA Summer r School, l, 2013 2013. Kim Larse sen [30 30]

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SLIDE 31

Soundness undness of f Corner

  • rner Point

int Ab Abstr traction action

VTSA Summer r School, l, 2013 2013. Kim Larse sen [31 31]

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SLIDE 32

Mu Multiple tiple Ob Obje jective ctive Sched heduling uling

P2 P1 P6 P3 P4 P7 P5

16,10 2,3 2,3 6,6 10,16 2,2 8,2

4W 3W

cost1’==4 cost2’==3 cost1 cost2

Pareto Frontier

VTSA Summer r School, l, 2013 2013. Kim Larse sen [32 32]

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SLIDE 33

Ene Energy rgy Aut Automata

  • mata
slide-34
SLIDE 34

Ma Mana naging ging Resourc

  • urces

VTSA Summer r School, l, 2013 2013. Kim Larse sen [34 34]

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SLIDE 35

Con

  • nsuming

suming & Ha Harve vesting sting Ene nergy gy

Maximize throughput while respecting: 0 · E · MAX

VTSA Summer r School, l, 2013 2013. Kim Larse sen [35 35]

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SLIDE 36

Ene nergy gy Con

  • nstr

strains ains

  • Energy is not only consumed but may also be regained
  • The aim is to continously satisfy some energy constriants

VTSA Summer r School, l, 2013 2013. Kim Larse sen [36 36]

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SLIDE 37

Results ults (so far)

Bouyer, Fahrenberg, Larsen, Markey, Srba: FORMATS 2008

VTSA Summer r School, l, 2013 2013. Kim Larse sen [37 37]

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SLIDE 38

L-Problem roblem fo for 1-Clock Clock Case se

Theorem: The L-problem is decidable in PTIME for 1-clock PTAs

P Bouyer, U Fahrenberg, K Larsen, N Markey,.. . Infinite runs in weighted timed automata with energy constraints. 2008. VTSA Summer r School, l, 2013 2013. Kim Larse sen [38 38]

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SLIDE 39

LU LU-Problem roblem fo for r 1-Clock Clock Ener ergy gy Gam ames

Theorem For 1-clock priced timed games, the existence of a strategy satisfying LU-bounds is undecidable

P Bouyer, U Fahrenberg, K Larsen, N Markey,.. . Infinite runs in weighted timed automata with energy constraints. 2008. VTSA Summer r School, l, 2013 2013. Kim Larse sen [39 39]

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SLIDE 40

Gene neric ric Mo Module ule fo for In Inc/De Dec

VTSA Summer r School, l, 2013 2013. Kim Larse sen [40 40]

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SLIDE 41

1½ ½ Clocks

  • cks =

= Discre screte te Update dates

VTSA Summer r School, l, 2013 2013. Kim Larse sen [41 41]

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SLIDE 42

New w Approac proach: h: Ene nergy gy Fun uncti tions

  • ns
  • Maximize energy

along paths

  • Use this information

to solve general problem

  • P. Bouyer, U. Fahrenberg, K. G. Larsen, N. Markey: Timed automata with observers under energy constraints. HSCC 2010

VTSA Summer r School, l, 2013 2013. Kim Larse sen [42 42]

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SLIDE 43

Ene nergy gy Fun uncti tion

  • n

Gene neral al Strategy ategy Spend just enough time to survive the next negative update

  • P. Bouyer, U. Fahrenberg, K. G. Larsen, N. Markey: Timed automata with observers under energy constraints. HSCC 2010

VTSA Summer r School, l, 2013 2013. Kim Larse sen [43 43]

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SLIDE 44

Expo ponential nential PTA

Gene neral al Strategy ategy Spend just enough time to survive the next negative update so that after next negative update there is a certain positive amount !

Minimal al Fixpoint nt:

  • P. Bouyer, U. Fahrenberg, K. G. Larsen, N. Markey: Timed automata with observers under energy constraints. HSCC 2010

VTSA Summer r School, l, 2013 2013. Kim Larse sen [44 44]

slide-45
SLIDE 45

Expo ponential nential PTA

Energy rgy Functi nction

Thm [BFLM09]: The L-problem is decidable for linear and exponential 1-clock PTAs with negative discrete updates.

  • P. Bouyer, U. Fahrenberg, K. G. Larsen, N. Markey: Timed automata with observers under energy constraints. HSCC 2010

VTSA Summer r School, l, 2013 2013. Kim Larse sen [45 45]

slide-46
SLIDE 46

Mu Multiple tiple Cos

  • sts

ts & Clocks

  • cks
  • LU Problem Un

Undec ecidab idable le

  • 2 clocks + 2 costs [1]
  • 1 clock + 2 costs [2]
  • 2 clocks + 1 cost [3]

(1) Karin Quaas. On the interval-bound problem for weighted timed automata. 2011. (2) Uli Fahrenberg, Line Juhl, Kim G. Larsen, and Jirı Srba. Energy games in multiweighted automata. 2011 (3) Nicolas Markey. Verification of Embedded Systems – Algorithms and Complexity. 2011.

Update c1

Increment ent n=3 Decrem ement ent n=12 12

Test-Decreme Decrement

VTSA Summer r School, l, 2013 2013. Kim Larse sen [46 46]

slide-47
SLIDE 47

Mu Multiple tiple Clo locks cks & 1 Cos

  • st
  • L problem is undec

ecidab idable le

  • 4 clocks or more

x=x0 C=C0 x=x0 C=x0+C0 x=x0 C=C0 x=x0 C=(1-x0) +C0

  • P. Bouyer, K. G. Larsen, and N. Markey. Lower-bound constrained runs in weighted timed automata. QEST 2012

unde decid cidab able le

VTSA Summer r School, l, 2013 2013. Kim Larse sen [47 47]

slide-48
SLIDE 48

Uli Fahrenberg, Line Juhl, Kim G. Larsen, and Jirı Srba. Energy games in multiweighted automata. 2011

Mu Multiple tiple Cos

  • sts

ts & 0 Cloc

  • cks

ks

VTSA Summer r School, l, 2013 2013. Kim Larse sen [48 48]

slide-49
SLIDE 49

Con

  • nclusion

clusion

  • Priced Timed Automata a uniform framework

for modeling and solving dynamic ressource allocation problems!

  • Not mentioned here:
  • Model Checking Issues (ext. of CTL and LTL).
  • Future work:
  • Zone-based algorithm for optimal infinite runs.
  • Approximate solutions for priced timed games to

circumvent undecidablity issues.

  • Open problems for Energy Automata.
  • Approximate algorithms for optimal reachability

VTSA Summer r School, l, 2013 2013. Kim Larse sen [49 49]

slide-50
SLIDE 50

Ti Timed med Gam Games es

Ki Kim m G. Lar . Larsen – Aalborg University DENMARK

TIGA

slide-51
SLIDE 51

Model del Checking ecking

: Never two trains at

the crossing at the same time

Environment Controller

VTSA Summer r School, l, 2013 2013. Kim Larse sen [51 51]

slide-52
SLIDE 52

Syn ynthesis thesis

: Never two trains at

the crossing at the same time

Environment Controller

?

VTSA Summer r School, l, 2013 2013. Kim Larse sen [52 52]

slide-53
SLIDE 53

Syn ynthesis thesis

: Never two trains at

the crossing at the same time

Controllable Uncontrollable

Find strategy for controllable actions st behaviour satisfies 

Controller Environment

VTSA Summer r School, l, 2013 2013. Kim Larse sen [53 53]

slide-54
SLIDE 54

Time med Au Automata tomata & & Model el Checking ecking

State te (L1, x=0.81) Transit ansition ions (L1 , x=0.81)

  • 2.1 ->

(L1 , x=2.91)

  • >

(goal , x=2.91)

Ehi hi goal ? Ahi hi goal ? A[ ] : L4 ?

VTSA Summer r School, l, 2013 2013. Kim Larse sen [54 54]

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SLIDE 55

Time med Gam ame Au Automata tomata & Syn ynthesis thesis

Problems to be be considered dered:

  • Does there exist a winning strategy?
  • If yes, compute one (as simple as possible)

controllable uncontrollable

VTSA Summer r School, l, 2013 2013. Kim Larse sen [55 55]

slide-56
SLIDE 56

Decidability idability of

  • f Timed

med Games mes

VTSA Summer r School, l, 2013 2013. Kim Larse sen [56 56]

slide-57
SLIDE 57

Comp

  • mputing

uting Winning nning Sta tates tes

VTSA Summer r School, l, 2013 2013. Kim Larse sen [57 57]

slide-58
SLIDE 58

Reachability chability Games mes

Backwards Fixed-Point Computation

Theorem: The set of winning states is obtained as the least fixpoint

  • f the function: X a p(X) [ Goal

cPred(X) = { q2Q | 9 q’2 X. q c q’} uPred(X) = { q2Q | 9 q’2 X. q u q’} Predt(X,Y) = { q2Q | 9 t. qt2X and 8 s·t. qs2YC } p(X) = Predt[ X [ cPred(X) , uPred(XC) ] Definitions

X Y

Predt(X,Y)

VTSA Summer r School, l, 2013 2013. Kim Larse sen [58 58]

slide-59
SLIDE 59

Sym ymbolic bolic On On-the the-fly fly Al Algo gorithms rithms fo for r Time med Gam ames es [CDF+05, BCD+07]

symbolic version of on-the-fly MC algorithm for modal mu-calculus Liu & Smolka 98

VTSA Summer r School, l, 2013 2013. Kim Larse sen [59 59]

slide-60
SLIDE 60

UPPAAL AAL Tiga ga [CDF+05, BCD+07]

  • Reachability properties:
  • control: A[ p U q ]

until

  • control: Ahi

hi q  control: A[ true U q ]

  • Safety properties:
  • control: A[ p W q ]

weak until

  • control: A[] p  control: A[ p W false ]
  • Time-optimality :
  • control_t*(u,g): A[ p U q ]
  • u is an upper-bound to prune the search
  • g is the time to the goal from the current state

VTSA Summer r School, l, 2013 2013. Kim Larse sen [60 60]

slide-61
SLIDE 61

UPPAAL AAL Tiga ga

DEM EMO

VTSA Summer r School, l, 2013 2013. Kim Larse sen [61 61]

slide-62
SLIDE 62

Mo Model el Che hecki king ng (ex Train Gate) : Never two trains at

the crossing at the same time

Environment Controller

VTSA Summer r School, l, 2013 2013. Kim Larse sen [62 62]

slide-63
SLIDE 63

Synt nthesis hesis (ex Train Gate) : Never two trains at

the crossing at the same time

Environment Controller

?

VTSA Summer r School, l, 2013 2013. Kim Larse sen [63 63]

slide-64
SLIDE 64

Timed med Game mes : Never two trains at

the crossing at the same time

Controllable Uncontrollable

Find strategy for controllable actions st behaviour satisfies 

Controller Environment

VTSA Summer r School, l, 2013 2013. Kim Larse sen [64 64]

slide-65
SLIDE 65

Pla lastic stic Inje jectio ction n Molding lding Mac achine hine

  • Robust and optimal

control

  • Tool Chain
  • Synthesis: UPPAAL

AL TIGA GA

  • Verification: PHAVer

er

  • Performance: SIMUL

MULIN INK

  • 40% improvement of

existing solutions..

Quasiomodo

[CJL+09]

VTSA Summer r School, l, 2013 2013. Kim Larse sen [65 65]

slide-66
SLIDE 66

Oi Oil l Pum ump p Con

  • ntrol

trol Problem

  • blem
  • R1: stay within safe

interval [4.9,25.1]

  • R2: minimize

average/overall oil volume

Quasiomodo

VTSA Summer r School, l, 2013 2013. Kim Larse sen [66 66]

slide-67
SLIDE 67

The he Ma Machine hine (consumption)

  • Infinite cyclic demand

to be satisfied by our control strategy.

  • P: latency 2 s between

state change of pump

  • F: noise 0.1 l/s

Quasiomodo

VTSA Summer r School, l, 2013 2013. Kim Larse sen [67 67]

slide-68
SLIDE 68

Abstract tract Game me Mo Model

  • UPPAAL Tiga
  • ffers games of perfect information
  • Abstract game model such that states only

contain information about:

  • Volume of oil at the beginning of cycle
  • The ideal volume as predicted by the

consumption cycle

  • Current time within the cycle
  • State of the Pump (on/off)
  • Discrete model

D V, V_rate V_acc time

Quasiomodo

VTSA Summer r School, l, 2013 2013. Kim Larse sen [68 68]

slide-69
SLIDE 69

Ma Machine hine (uncontrollable)

Checks whether V under noise gets

  • utside

[Vmin+0.1,Vmax-0.1] Quasiomodo

VTSA Summer r School, l, 2013 2013. Kim Larse sen [69 69]

slide-70
SLIDE 70

Pum ump p (controllable)

Every 1 (one) seconds Quasiomodo

VTSA Summer r School, l, 2013 2013. Kim Larse sen [70 70]

slide-71
SLIDE 71

Too

  • ol

l Cha hain in

Strategy Synthesis TIGA Verification PHAVER Performance Evaluation SIMULINK

Guarante anteed Correctness Robustness with 40% Improvement

Quasiomodo

VTSA Summer r School, l, 2013 2013. Kim Larse sen [71 71]

slide-72
SLIDE 72

LA LAB B Ex Exer ercises cises

www.cs.aau.dk/~kgl/Shanghai2013 Exercise 28 (Jobshop Scheduling Part 1) Exercise 19 (Train Gate Part 1)