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ASSUME ADMISSIBLE SYNTHESIS . Francois Raskin Jean Libre de Bruxelles Universit ' as * . . FFIMS Singapore ifip meeting September 2016 REACTIVE SYNTHESIS Classical setting Sys t Enr ? is the winning objective for Sgs

  1. ASSUME ADMISSIBLE SYNTHESIS . Francois Raskin Jean Libre de Bruxelles Universit ' as * . . FFIMS Singapore ifip meeting September 2016

  2. REACTIVE SYNTHESIS Classical setting Sys t ¢ Enr ? is the winning ¢ objective for Sgs adversarial EW is - player 2 sum ⇒ zero game - Correct Sys Tinning strategy =

  3. Sys ¢ Enr ? t completely adversarial Env is → what if Rational ? Ew user = and / be made Eno Sys can or → of several with each components , their objective own . De need richer > setting a

  4. . players . based graph games turn N directed Finite graph z ~ Vertices partitioned are • < - > ( = ^ V=VsoV£ut .tk | verkasofployoei . . vi. > V×V Ee

  5. and Players Strategies objectives ... ,w} :V*.¥ V 0=4,2 oi - , s.t.tt#.vev*.V.. stand for , Players 't .D)eE : ( visit - partofthegskmtodesqn ; - parts of the environment * ma W 2 N "/\ s " ... . . . ¢ ¢2 ¢ ' . serofskakgiesof & Ployoei KEV ... .

  6. outcomes and Strategy profiles of ) { ( % { . .x{w E , oz sx x. as , ... , { - , I - Ei € i profile . ( % = all strategies of the profile broi Qtr , %) ( of , oz No Ns Ne Nn Vi T = = . . . . . . . . . , )=%ftGjD SE N No = : . : if then tj Tcj ) tcjts > , e ° •

  7. Running example 3# tv 2 players : 5 7 % he 4 ¢s=- 12=-17.43 = Players = Player 2

  8. SYNTHESIS RULES

  9. Synthesis rules WIN Ew Sys - W 2 s ... ,or)÷ % } ignored { & 0 . ... . How Voz =) ois : . . . . . arcos ,r ... , inskakord Can be 2- 6 - sum player zero case = any for Players .

  10. ( ~ I Running example win m 3 \ 5 % 4 a =-D 0/2 43 D I > µ 4 ! \ } hf None of the players has winning strategy a Y

  11. Running example win m 3 \ 5 I ( ~ 4 Da ¢s=- 0/2 43 =-D > µ 4 5 Players spoils ¢2±D<>3 By playing 1 -

  12. Running example win m 3 \ 5 I ( ~ 4 Da ¢s=- 0/2 43 D ± > µ 4 ¢s±D<>4 Player 2 spoils By playing 2 2 -

  13. . HYP Synthesis rules WIN , %) Ew Sys - W 2 1 ... Or } . HYP { ok Or . How Foes Voz : . . . . . Our ( os ¢£^ ^¢w Or t → ,q ... , . . . - HYP inskakard Can be % for any Players .

  14. ( ~ I . HYP WIN Running example >[g m 3 \ 5 li 4 Da 0/2=-1343 ¢s=- th , with Pts ok ¢ . -5 a wins - \ > useless solution !

  15. ADMISSIBILITY

  16. Dominated strategy ol for E. if dominated by is

  17. Dominated strategy ol for E. if dominated by is EL tqi as good % Always @ : i . , a) , %) ark. ONCE t ¢ ¢ ' t ⇒ . . .

  18. Dominated strategy ol for E. if dominated by is EL tqi as good % Always @ : i . .oDt¢ , %) ark ONCE ¢ ' t ⇒ . . . , { better Sometimes Z % � 2 � E Ei : i . ON , %) ... ) ONCE .io ¢ 0 , ¥ t ^ oi .

  19. Dominated strategy ol for E. if dominated by is EL tqi as good % Always @ : i . .oDt& , %) ark. ONCE ¢ ' t ⇒ . . , { better Sometimes =) % � 2 � E Ei : i . ON , %) ... ) ONCE .io ¢ 0 , ¥ t ^ % . rational A avoids dominated player strategies →

  20. m 3 \ 5 A ; ¢s= Da 4 , . 4 that takes Any strategy r→5 the dominated 3 4 1→2 strategy by is → , - if not iris even a winning strategy

  21. Admissible strategy . admissible { for ¢ ; E E. is • i E 4. for ¢ dominated not if ' by any is oi E. . . ) set of admissible { , ( oh the Adm c- is i . strategies of Player for ¢i i . the only { Festooned } strategies ! . ) ( oh Adm. = . • ( ¢ ;) t Adm fi = ; . • ,

  22. m 3 \ 5 ( X ; ¢s= Da 4 ✓ . 4 3 4 { 2 EE 1 - → , 9 admissible is indents 335L ' ' ' a

  23. Synthesis rules Admissible ssune W 2 s ... 0w 02 Or FK or ) ,r : , , ... ( ¢ ; ) Adm for E % @ alli.rsisn@VoteAdm.iC4.D ,oI)k¢ Onto : . . admissible strategies of the others Yach all winning against is q .

  24. a 3 \ , X 5 s > YYa3→4 ( ~ I Da 4 ¢s=- 42 43 =-D > y 4 GD n2→2 . Voze e Adm~( oh ) Claim k Hoi Admz ( : : or ) ( Out ¢s^ $2 t or , rule applies ! Admissible Assume ⇒

  25. Assume Admissible Synthesis - profiles Theorem : all for ( , %) AA or a. : , ... ( on ) ¢n¢u .ro/v Our t as , oos , . . , . ser of - profiles Theorem : the rectangular AA is . ST x STW SIX x = . . . , %) taxi } Out ( ( ¢D| , ( $ . D: { Foe e Adm. Sti Adm E oi q . = need for synchronization ! To ⇒ Compositionally ⇒

  26. - C for Theorem : deciding if AA applies Pspaa is Reach ability Muller and Safety objectives . ,

  27. - C for Theorem : deciding if AA applies Pspaa is , Readability Muller and Safety objectives . $2 =-D Safe State Space Value Value Value -2 is o * * tu TKRDD safe KKDD Safe XD Safe A player when playing admissible its value decrease never .

  28. . Cfa Theorem : deciding if AA applies Pspaa is , Readability Muller and Safety objectives . $z=- Dsafe State Space other properties for Value Values Valeo z . Values Win ! # # ⇒ t f + Value Win ⇒ . on ' KRDD safe KKDD Safe ADD Safe Help ! The setofplays compatible ⇒ A player when playing admissible with admissible strategies its value decrease regular is W never . .

  29. Conclusion allows for admissible Assume synthesis • . player ... compositional W Synthesis in zero non . : mm games . : sets of " solutions Rectangular . solution solution Jf then AA gives win gives a a . . ¢z ( (D) - HYP solution and Jf Win gives a • solution . then AA gives a .

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