Planning for Reactive Behaviors in Hide and Seek

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1 University of Pennsylvania ScholarlyCommons Center for Human Modeling and Simulation Department of Computer & Information Science May 1995 Planning for Reactive Behaviors in Hide and Seek Michael B. Moore University of Pennsylvania Christopher W. Geib University of Pennsylvania Barry D. Reich University of Pennsylvania Follow this and additional works at: Recommended Citation Moore, M. B., Geib, C. W., & Reich, B. D. (1995). Planning for Reactive Behaviors in Hide and Seek. Retrieved from Postprint version. Published in Proceedings of the Fifth Conference on Computer Generated Forces and Behavioral Representation, May 1995, 10 pages. This paper is posted at ScholarlyCommons. For more information, please contact

2 Planning for Reactive Behaviors in Hide and Seek Abstract We describe the ZAROFF system, a plan-based controller for the players in a game of hide and seek. The system features visually realistic human figure animation including realistic human locomotion. We discuss the planner's interaction with a changing environment to which it has only limited perceptual access. A hierarchical planner translates the game's goals of finding hiding players into locomotion goals, assisted by a special-purpose search planner. We describe a system of parallel finite state machines for controlling the player's locomotion. Neither path-planning nor explicit instructions are used to drive locomotion; agent control and apparent complexity are the result of the interaction of a few relatively simple behaviors with a complex (and changing) environment. Comments Postprint version. Published in Proceedings of the Fifth Conference on Computer Generated Forces and Behavioral Representation, May 1995, 10 pages. This conference paper is available at ScholarlyCommons:

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5 q l ñ \ ó c ö õ c ] ó _ \ a ] b ú ö õ c ƒ J : X J : J8 X A P :4 L A C< A 4 8 L 2 A X 2 X L 6 : : X Y 2 t C A X a : X L f L 2 J 8 ] d X 4 L 2 A X C : L d C X J L 6 : e 2 J t L 8 X 4 : 8 X e 8 X ü < :L AL 6 8 L A P :4 LA C < A 4 8 L 2 A X C : < 8 L 2 Y : L A L 6 : 8 ü : X L O J M A J 2 L 2 A X 8 X e A C 2 : X L 8 L 2 A X 6 2 J J L 8 L : q < ƒ I ƒ K ý ˆ þ ƒ FJ d J I I < I A C ˆ ƒ þ ÿ F 2 J 8 ] d X 4 L 2 A X L 6 8 L a 8 M Je 2 J L 8 X 4 : 8 X e 8 X ü < : L A 8 J L C : J J Y 8 < d : F 6 : C : < A : CY 8 < d : J C : M C : J : X L a A C :e : J 2 C 8 P < : J L 8 L : J û6 : 4 A a P 2 X 8 L 2 A X A ] 8 J : X J A C 8 X e 8 4 A X L C A < P : 6 8 Y 2 A C C : J d < L J 2 J C : ] : C C :e L A 8 J 8 ý ˆþ ƒ û6 : 8 ü : X L d L 2 < 2 g : J 8 J : L A ] P : 6 8 Y 2 A C J2 X2 X L : C 8 4 L 2 X ü 2 L 6 2 L J: X Y 2 C A X a : X L û6 : ] A < < A 2 X ül 6 C : : 4 < 8 J J : J A ] P : t 6 8 Y 2 A C J 8 C : 4 d C C : X L < R 2 X d J : m ô ôõ ] ù ôú ö ó Š q X 8 L L C 8 4 L 2 A X P : 6 8 Y 2 A Ce C 8 J 8 X 8 ü : X L L A 8 C e 8 ü A 8 < : 2 L 6 : C 8 X A P :4 L A C 8 < A 4 8 L 2 A X f ] 8 ü A 8 <A P :4 L a A Y : J F L 6 : M A 2 X L A ] 8 L L C 8 4 L 2 A X a A Y : J 8 M M C A M C 2 8 L : < R û6 : A d L t M d L J L C : J J Y 8 < d : A ] 8 X 8 L L C 8 4 L 2 A X P : 6 8 Y 2 A C 2 J 6 2 ü 6 6 : X L 6 :8 ü : X L2 J ] 8 C ] C A a L 6 : ü A 8 < 8 X e e : 4 C :8 J : J 8 JL 6 : 8 ü : X LX :8 C J L 6 : ü A 8 < mb ö ú ] ó ù \ Š q X 8 Y A 2 e 8 X 4 : P : 6 8 Y 2 A C 2 J d J :e L A 8 Y A 2 e 4 A < < 2 J 2 A X J P : L : : X L 6 : 8 ü : X L8 X e A P :4 L J A C A L 6 : C 8 ü : X L Jû6 : J : X J A C 4 A a M A X : X LA ] 8 X 8 Y A 2 e 8 X 4 : P : 6 8 Y 2 A C 6 8 J 8 J :4 L A C t J 6 8 M : e C : ü 2 A X A ] J : X J 2 L 2 Y 2 L R f ]L 6 : C : 8 C : X A A P : 4 L J 2 X L 6 2 J C : ü 2 A X F L 6 :A d L M d L A ] L 6 : 8 Y A 2 e 8 X 4 : P : 6 8 Y 2 A C 2 J g : C A 8 L 6 : C2 J : L 6 : A d L M d L 2 J M C A M A C L 2 A X 8 < L A L 6 : e 2 J L 8 X 4 : 8 X e J 2 g : A ] L 6 : e : L :4 L : e A P t : 4 L J l Z a \ _ \ a ] b ú ö õ ] ú ø ] ô ú ö ó c ô \ = j j M C A Y 2 e : J ü : X : C 8 < < A 4 A a A L 2 A XA ] A P :4 L J2 X : < = :4 : L ( = 8 e < : C / r r o F8 X e 2 J d J :e 2 X L A ü : X : C 8 L : 6 d a 8 X < A 4 A a A L 2 A X û6 : 4 : X L C 8 < 4 A X t L C A < a : X 2 J a A ] = j j 2 J8 M : C 4 : M L 2 A X 4 A X L C A < F 8 X e 8 4 L 2 A X < A A M S d C 2 X ü L 6 : M : C 4 : M L 2 A X M 6 8 J :L 6 : P : t 6 8 Y 2 A CA d L M d L J 8 C : e : L : C a 2 X :e F e d C 2 X ü L 6 : 4 A X L C A < M 6 8 J : L 6 : X : L ] A A L M A J 2 L 2 A X 2 J J : < :4 L :e F 8 X e e d C 2 X ü L 6 : 8 4 L 2 A X M 6 8 J : L 6 : J L : M 2 JL 8 : X l l _ \ a ] b ú ö õ ù a \ ø ú ó ^ y A C 4 A X L C A < A ] L 6 : 8 ü : X L J O P : 6 8 Y 2 A C J : d J : 8 J : L A ] f X 2 L :J L 8 L : ] ù aú ó \ c C d X 2 X M 8 C 8 < < : < 2 L 6 L 6 : J 2 a d < 8 L 2 A X û6 : J : a X : J 8 C : C : J M A X J 2 P < : ] A C P : 6 8 Y 2 A C J 4 6 :e d < 2 X ü 8 X e 8 4 L 8 J ü 6 < : Y : < 2 X L : C ] 8 4 : L A L 6 : P : 6 8 Y 2 A C J û6 : Ra 8 R 2 X J L 8 X L 2 t 8 L : A L 6 : C a X : J 8 X e : 2 L 6 : CC d X 2 X M 8 C 8 < < : < 2 L 6 L 6 : a A C 8 2 L ] A C L 6 : a L A : 2 L H A a a d X L 2 A X 8 a A X ü J L 8 L : a X : J 2 J8 < J A M A J J 2 P < : + : a 8 : d J :A ] L 6 2 J 4 A a a d X L 2 A X 8 a A X ü a X : J L A 2 a t M < : a : X L 2 X L : C 8 4 L 2 A X P : L : : X M < 8 R : C J j :4 L 2 A X s + 6 : X L 6 : M < 8 X X : C 4 A a a 2 L J L A8 X 8 4 L 2 A X 2 L 2 X Y A : J L 6 : q 4 L 2 A X p :4 d L 2 A X a A e d < : û6 : q 4 L 2 A X p : t 4 d L 2 A X a A e d < :2 X J L 8 X L 2 8 L : J 8 X e C d X JL 6 : a X : J X :4 : J J 8 C R L A 4 8 C C R A d LL 6 8 L 8 4 L 2 A X p a X : 2 JC : J M A X J 2 P < :] A CL 6 :J 4 6 :e d < 2 X ü8 X e 4 A X L C A < A ] L 6 : 8 M M C A M C 2 8 L : P : 6 8 Y 2 A C J Ž ú \ ö ú \ Š q f : < e t A ] t Y 2 : P : 6 8 Y 2 A C d J : J 8 J : X J A CL A e : L : C a 2 X : 6 : L 6 : CA C X A L L 6 : 8 ü : X L 2 J Y 2 J 2 P < : L A 8 X R A L 6 : C 8 ü : X L J û6 : A d L M d L 2 J M C A M A C L 2 A X 8 < L A L 6 : X d a P : C A ] 8 ü : X L J O f : < e J t A ] t Y 2 :2 L 2 J 2 X 8 X e 2 X Y : C J : < R M C A M A C L 2 A X 8 <L A L 6 : e 2 J L 8 X 4 : JL AL 6 : J : 8 ü : X L J û6 2 J P : 6 8 Y 2 A C 2 J M C 2 a 8 C 2 < R d J :e L A J d M M A C L 6 2 e : C J 2 X

6 State 1 State 2 Turning Attraction EXIT State 1 Avoidance EXIT Monitor Monitor o o / y 2 ü d C : / jû6 : ql L C 8 4 L X : ql L C 8 4 L 2 A X q X I I < I < ý þ 4 A X J 2 J L J A ] L A J L 8 L : J y 2 ü t d C : / f X c ô ] ô \ñ L 6 : 8 ü : X L L d C X J L A ] 8 4 : L 6 : A P :4 LA C M A 2 X L A ] 8 L L C 8 4 L 2 A X f X c ô ] ô \ Z 8 X8 L L C 8 4 t L 2 A X P : 6 8 Y 2 A C2 J P A d X e L A L 6 : 8 ü : X L û6 : 8 ü : X L 8 < JL A L 6 : ü A 8 <A P : 4 L A C < A 4 8 L 2 A X + 6 : X L 6 : 8 ü : X L 8 C C 2 Y : J F L 6 2 J P : 6 8 Y 2 A C 2 J d X P A d X e 8 X e L 6 : a X :: 2 L J q X 8 L L C 8 4 L a X : A M L 2 A X 8 < < R 2 X 4 < d e : J 8 a A X 2 t L A C 2 X ü M C A 4 : J J :4 JF 2 X M 8 C 8 < < : < F ] A C8 X 8 C t P 2 L C 8 C R 4 A X e 2 L 2 A X 8 z f j ^ : M C : J J 2 A X M 8 J J :e 8 J 8 X 8 C ü d a : X L 6 : X L 6 : a X : 2 J4 C :8 L :e L A P : L C d : f ] L 6 2 J4 A X e 2 L 2 A X : Y 8 < d 8 L : J L A L C d : 8 L 8 X R L 2 a : F L 6 : 8 L L C 8 4 L 2 A X P : 6 8 Y 2 A C 2 J d X P A d X e 8 X e L 6 : a X : : 2 L J û6 : Y 8 < d :C : L d C X :e P R L 6 : a X : 6 : X 2 L : 2 L J2 X e L : JL 6 : 4 8 d J : A ] L 6 : : 2 L + : d J : L 6 : a A X 2 L A C 2 X ü M C A 4 : J J M C 2 a 8 C 2 < R L A 4 A X t L C A < L 6 : P : 6 8 Y 2 A C A ] L 6 : J : : : C q J L 6 : J : : : C : t M < A C : J L 6 : : X Y 2 C A X a : X LF a A Y 2 X ü] C A a A X : < A 4 8 L 2 A X L A 8 X A L 6 : C L 6 C A d ü 6 L 6 : P 2 X e 2 X ü 8 X e d X P 2 X e 2 X ü A ] 8 L L C 8 4 L 2 A X P : 6 8 Y 2 A C JF 2 L JM C A ü C : J J 2 J2 X L : C C d M L :e 2 ] 8 L 8 X R L 2 a :8 6 2 e : C P :4 A a : J Y 2 J 2 P < : û6 2 J 8 < < A J L 6 : J : : : C L A J L A M 8 X e ] A C a d < 8 L : 8 X : M < 8 X F L 8 t 2 X ü 8 e Y 8 X L 8 ü : A ]L 6 : A M M A C L d X 2 L R o o l q Y A 2 e 8 X 4 : q X ˆ ƒ þ ÿ < ý þ 4 A X J 2 J L J A ]A X < R A X : J L 8 L : y 2 ü t d C : l f L P 2 X e J 8 X8 Y A 2 e 8 X 4 : P : 6 8 Y 2 A C L A L 6 : 8 ü : X L ] A C : Y A 2 e 8 P < : A P :4 L 2 XL 6 : : X Y 2 C A X t a : X L F 8 X e L 6 : X : 2 L J W X < 2 : 8 L L C 8 4 L 2 A X F 8 Y A 2 e t y 2 ü d C : l jû6 : q Y A 2 e X : State 1 Turning Not Visible EXIT (Unsuccessfully) Visible Visible Arrived State 2 Object Attraction Not Visible State 3 Location Attraction Visible Arrived EXIT (Successfully) Not Visible y 2 ü d C : o j û6 :H 6 8 J : X : 8 X 4 :2 J a 8 2 X L 8 2 X :e L 6 C A d ü 6 A d L L 6 : : X L 2 C : J 2 a d < 8 t L 2 A X f X L 6 : A P :4 L J L A 8 Y A 2 e 8 C : e A A C JF 8 < < JF 8 X e A L 6 : C8 ü : X L Jq X 8 Y A 2 e a X : 2 J d J :e e d C 2 X ü L 6 : 2 X 2 L 2 8 < 2 g 8 L 2 A X A ] L 6 : J 2 a d < 8 L 2 A X F A X 4 : ] A C : ü : X L û6 2 J M C : Y : X L JL 6 : 8 ü : X L J ] C A a 4 A < t < 2 e 2 X ü2 L 6 :8 4 6 A L 6 : C F 8 < < JF 8 X e e A A C J F 8 X e 4 8 X :8 J 2 < R P :: L : X e :e L A 8 X R A L 6 : C A P :4 L J L 6 8 L a 2 ü 6 L P : 8 e e :e L A L 6 : J R J L : a o o o H 6 8 J 2 X ü û6 : 2 X L : C X 8 < J L C d 4 L d C : A ] 8 < ý < ý þ 2 J 8 < A A M 4 A X J 2 J L 2 X ü A ] L 6 C : : J L 8 L : J y 2 ü d C : o f ] L 6 : L 8 C ü : L F L 6 : 8 ü : X L P : 2 X ü J : e F 2 J X A L2 a a :e 2 8 L : < R Y 2 J 2 t P < : FL 6 : a X : : 2 L J d X J d 4 4 : J J ] d < < R j d 4 4 : J J 2 J 2 X e L :e P R L 6 : : 2 L Y 8 < d : + 6 : X L 6 : L 8 C ü : L 2 J Y 2 J 2 P < : L A L 6 : 8 ü : X LFL 6 : a 8 t X : J L 8 C L J 2 X c ô ] ô \ ñ û6 : 8 ü : X L L d C X JL A ] 8 4 : L 6 : L 8 C ü : L 8 X e L 6 : L C 8 X J 2 L 2 A X L A c ô ] ô \ Z 2 J a 8 e :

7 q Human State 2 Move First Foot State 3 Move Other Foot State 1 Calculate Path State 2 Attraction Not At Goal State 1 Calculate Angle Not Done Done At Goal EXIT Not Human State 4 Spin Object EXIT y 2 ü d C : * jû6 :^8 L X : y 2 ü d C : s j û6 : ûd C X X : q X 8 L L C 8 4 L 2 A X L A L 6 : L 8 C ü : L 2 J P A d X e û6 : 8 ü : X L 8 < JL A 8 C e L 6 : L 8 C ü : L d X L 2 <A X : A ] L A L 6 2 X ü J 6 8 M M : X p 2 L 6 : C L 6 : 8 ü : X L 8 C C 2 Y : J 8 L L 6 :L 8 C ü : L 2 X J : L 6 : a X :: 2 L J J d 4 4 : J J ] d < < R FA C L 6 : L 8 C ü : L 4 :8 J : J L A P :Y 2 J 2 P < : L A L 6 : 8 ü : X L û6 : L 8 C ü : L a 8 R 6 8 Y : 8 < :e 8 C A d X e 8 4 A C X : C f X L 6 : < 8 L L : C 4 8 J : L 6 : L C 8 X J 2 L 2 A X L A c ô ] ô \ l 2 J a 8 e : q X 8 L t L C 8 4 L 2 A X L A L 6 : < 8 J L X A X < A 4 8 L 2 A X A ] L 6 : L 8 C ü : L 2 JP A d X e L A L 6 : 8 ü : X L û6 : 8 ü : X L 8 < J L A 8 C e L 6 2 J < A 4 8 L 2 A X d X L 2 < 2 L8 C C 2 Y : J A C L 6 : L 8 C ü : L P :4 A a : J Y 2 J 2 P < : 8 ü 8 2 X f X L 6 : ] A C a : C 4 8 J : L 6 :a X : : 2 L J d X J d 4 4 : J J ] d < < R f X L 6 : < 8 L L : C4 8 J : L 6 : L C 8 X J 2 L 2 A X L A c ô ] ô \ ñ 2 Ja 8 e : M A L : X L 2 8 <2 a M C A Y : a : X L L A L 6 2 J a X : A d < e P : L A 8 e e 8 X p L C 8 M A < 8 L 2 A X j L 8 L : f ] L 6 : a X : 2 J 2 X c ô ] ô \ l 8 X e L 6 : 8 ü : X L 8 C C 2 Y : J 8 L L 6 : < 8 J L X A X < A 4 8 L 2 A X A ] L 6 : L 8 C ü : L2 L 6 A d L J : : 2 X ü L 6 : L 8 C ü : LF 2 X J L : 8 e A ]: 2 L 2 X ü d X J d 4 4 : J J ] d < < R L 6 :8 ü : X L A d < e 8 < 2 X L 6 : e 2 C :4 L 2 A XL 6 :L 8 C ü : L 8 J 8 < 2 X ü P : ] A C : e 2 J 8 M M :8 C 2 X ü û6 2 J A d < e C : d 2 C : a 8 2 X L 8 2 X t 2 X ü L A M 2 :4 : J A ] 2 X ] A C a 8 L 2 A X 8 P A d L L 6 : L 8 C ü : L j 2 L J < 8 J L X A X < A 4 8 L 2 A X 8 X e 2 L JY :4 L A C Y : < A 4 2 L R o o s ûd C X 2 X ü q I < ý þ 2 J d J :e L A C A L 8 L : 8 X A P :4 L A C 6 d a 8 X L A ] 8 4 : 8 XA P :4 L A C < A 4 8 L 2 A X û6 : C A t L 8 L 2 A X a 8 R P : : 2 L 6 : C 4 < A 4 2 J : A C 4 A d X L : C 4 < A 4 t 2 J : F : Y : C 2 J J a 8 < < : C F A C a 8 R P : J M :4 2 f :e q L d C X a X : 4 A X J 2 J L J A ] ] A d CJ L 8 L : J y 2 ü d C : s f X c ô ] ô \ ñ L 6 :8 X ü < : 8 X e e 2 C : 4 L 2 A X A ] C A L 8 L 2 A X 8 C : 4 8 < 4 d < 8 L : e f ] L 6 :A P :4 L 2 JX A L 8 6 d a 8 X F 2 L2 J J M d X 8 P A d L 2 L J4 : X L : CA ] a 8 J J 8 X e L 6 : L d C X a X : : t 2 L J Ad a 8 X J 8 C :8 J M :4 2 8 < 4 8 J : û6 : 6 d a 8 X 8 ü : X L C A t L 8 L : J L 6 : f C J L ] A A L L A L 6 : C 2 ü 6 L ] A C 4 < A 4 2 J : F A C L A L 6 : < : ] L ] A C 4 A d X L : C 4 < A 4 2 J : û6 : < : ü 8 X e P A e R ] A < < A F a 8 2 X L 8 2 X 2 X ü L 6 : 8 ü : X L O J P 8 < 8 X 4 : û6 : C A t L 8 L 2 A X 8 X ü < : ] A C L 6 : f C J L ] A A L 2 J L 6 : a 2 X 2 a d a A ] L 6 :ü A 8 < 8 X ü < : 8 X e X 2 X : L R e : ü C : : J û6 : 8 ü : X L L 6 : X ] A < < A J L 6 : f C J L ] A A L2 L 6 2 L J A L 6 : C ] A A L f ] L 6 : ü A 8 < 8 X ü < : 2 Jü C : 8 L : C L 6 8 X X 2 X : L R e : ü C : : JF L 6 2 J 4 R t 4 < : C : M :8 L J d X L 2 < L 6 : ü A 8 < 6 8 JP : : X C : :e q X A M L 2 A X 8 < 8 C ü d a : X L 4 8 X J M :4 2 ] R L 6 8 L L 6 :8 ü : X L X A L ] A < < A 2 L 6 L 6 :A L 6 : C] A A L 2 X L 6 : f X 8 < 4 R 4 < : + 6 : X < A 4 A a A L 2 A X 2 J L A ] A < < A L d C X 2 X ü F 2 L < A A J a A C :X 8 L t d C 8 < 6 : X L 6 : f X 8 < 4 R 4 < : 2 J X A L 4 A a M < : L :e o o * ^8 L 6y A < < A 2 X ü û6 : 2 e : 8 P : 6 2 X e 8 I ý ƒ K K ƒ þ < ý þ 2 J L A ü : X : C 8 L :8 M 8 L 6 ] C A a L 6 :8 ü : X L O J 4 d C C : X L < A 4 8 L 2 A X L A 8 ü A 8 < < A 4 8 L 2 A X J A a :6 : C : 2 X L 6 : A C < e 8 X e 6 8 Y : L 6 : 8 ü : X L ] A < < A L 6 : M 8 L 6 L A L 6 : ü A 8 < qm 8 L 6 ] A < t < A 2 X ü a X :4 A X J 2 J L J A ] L A J L 8 L : J y 2 ü d C : * f X c ô ] ô \ñ L 6 : M 8 L 6 2 J 4 8 < 4 d < 8 L :e 8 J 8 J : C 2 : J A ] 4 A X X :4 L :e J : ü a : X L J û6 : M 8 L J L 6 C : : 2 a M A C t L 8 X L M C A M : C L 2 : J j 2 L 4 A X X : 4 L J L 6 :8 ü : X L L A L 6 : ü A 8 < F 2 L 8 Y A 2 e J A P J L 8 4 < : J P R 8 4 < : 8 C 8 X 4 : J M : 4 2 f :e P R L 6 : d J : CF 8 X e 2 L 2 J 8 M 8 L 6 A ]a 2 X 2 a 8 < < : X ü L 6 ü 2 Y : X L 6 : f C J L L A 4 A X J L C 8 2 X L J f X c ô ] ô \ Z L 6 : 8 ü : X L 2 J 8 L t L C 8 4 L :e L A J d 4 4 : J J 2 Y : Y : C L 2 4 : J8 < A X ü L 6 : M 8 L 6 q J L 6 : 8 ü : X L 8 M M C A : J 8 Y : C L : F L 6 : 8 L L C 8 4 L 2 A X L A

8 L 6 8 L Y : C L : 2 Jd X P A d X e 8 X e 8 X : 8 L L C 8 4 L 2 A X L A L 6 : X : L Y : C L : 8 < A X ü L 6 : M 8 L 6 2 J P A d X e + 6 : X L 6 : 8 ü : X L8 C C 2 Y : J 8 LL 6 : ü A 8 < F L 6 : M 8 L 6 ] A < < A 2 X ü a X :: 2 L J 8 d C M 8 L 6 ] A < < A 2 X ü e 2 : C J ] C A a M 8 L 6 M < 8 X X 2 X ü P : t 4 8 d J : L 6 : 8 ü : X L 2 J X A L 4 A X J L C 8 2 X :e L A L 6 : M 8 L 6 f X J L :8 e F8 X 8 L L C 8 4 L 2 A XL A L 6 : M 8 L 6 2 J P A d X e L A L 6 : 8 ü : X L û6 2 J P : 6 8 Y 2 A C 4 A a M : L : J 2 L 6 8 X R A L 6 : C P A d X e P : 6 8 Y 2 A C ] A C 4 A X L C A < A ] L 6 : 8 ü : X L f ] 8 a A Y t 2 X ü A P : 4 L 4 C A J J :e L 6 : M 8 L 6 F L 6 : M 8 L 6 t ] A < < A 2 X ü 8 ü : X L A d < e 8 Y A 2 e 2 L F 8 X e 2 ] e : C 4 8 a : 2 X L A Y 2 : L 6 : J : : : C A d < e M d C J d : 2 L At Home (Hider entry) State 1 Tag State 7 Synch Chase All Safe See Safe Start State 6 Search State 2 Count done Hide State 5 Seen Count Start ] óó ú ó ^ State 3 Evade Tagged State 4 Go home q M < 8 R : C O J ü A 8 < J X ü : e d C 2 X ü L 6 : 4 A d C J : A ] 8 ü 8 a : A ]6 2 e : 8 X e J : : û6 : J : : : C a d J L f C J L < A t 4 8 L : e : C 8 X e L 6 : X L 8 ü 2 LP : ] A C : 2 L C :8 4 6 : J 6 A a : P 8 J : A2 e : C J f C J L 8 L L : a M L L A < A 4 8 L :8 6 2 e 2 X ü M < 8 4 : + 6 : X A X : 6 8 Y : J d 4 4 : J J ] d < < R 6 2 e e : X F 2 L J L 8 R JL 6 : C : d X L 2 < 2 L 2 J X :4 : J J 8 C R L A a A Y : + 6 : Xª d J 6 :e ] C A a L 6 : 2 C 6 2 e 2 X ü M < 8 4 : P R L 6 : J : : : C F L 6 : R 8 L L : a M L L A a A Y : P 8 4 L A 6 A a : P 8 J : f ]L 6 : R J d 4 4 : J J ] d < < R 8 C t C 2 Y : 6 A a : F L 6 : R 8 2 Ld X L 2 <L 6 : J L 8 C L A ] L 6 : X : L ü 8 a : 6 : X L 6 : 2 C P : 6 8 Y 2 A C C : J : L J f ] A X : 2 J 4 8 d ü 6 LF L 6 : f C J L A X : 4 8 d ü 6 L 8 e A M L J L 6 : P : 6 8 Y 2 A C A ]L 6 : J : : : C û6 : J :6 2 ü 6 t < : Y : <ü A 8 < J 8 C : d 2 4 < R L C 8 X J < 8 L :e 2 X L A J 2 L d 8 L 2 A X 8 < < R 8 M M C A M C 2 8 L : C : 8 4 L 2 Y : P : 6 8 Y 2 A C J P R S «V 0 U O : 2 P / r r l F : C 8 C <M < 8 X X : C S «V 0 U 2 X L : C < : 8 Y : J 6 2 : C 8 C < : M 8 X J 2 A X2 L t L 2 A X : :4 d L 2 A X f Le A : J L 6 2 J P R d J 2 X ü8 X 2 X 4 C : a : X t L 8 << : ] L L A C 2 ü 6 L : M 8 X J 2 A X A ] L 6 : ] C A X L 2 : C A ] L 6 : M < 8 X J L C d 4 L d C : L A J d 4 4 : J J 2 Y : < R < A : C < : Y : < J A ] 8 P t J L C 8 4 L 2 A X û6 d J M < 8 X X 2 X ü A X < R L 8 : J M < 8 4 : L A L 6 : e : ü C : : X :4 : J J 8 C R L A e : L : C a 2 X :L 6 : X : L8 4 L 2 A X L A P : 4 8 C C 2 :e A d L û6 2 J2 J2 a M A C L 8 X L] A CL 6 2 J e A a 8 2 X J 2 X 4 : L 6 : J : : : C e A : J X A L 6 8 Y : 4 A a M < : L : X A < t :e ü : A ] L 6 :e A a 8 2 X 8 X e 2 L J X A < :e ü : 2 J 4 A X J L 8 X L < R X ü 2 X ü q f X 2 L : J L 8 L : a X :2 J d J :e L A X ü : 8 M < 8 R : C O J (Seeker entry) y 2 ü d C : u j ^< 8 R : Cü A 8 < a X : ü A 8 < JA Y : C L 6 : 4 A d C J : A ] L 6 :ü 8 a : y 2 ü d C : u 8 X e L 6 d J 2 X e 2 C : 4 L < R F 6 8 L 8 4 L 2 A X JL 6 : M < 8 R : C L 8 : J 8 L e 2 : C : X L L 2 a : J e d C 2 X ül 6 : ü 8 a : L A : Y : 2 L J ü A 8 < J û6 : < : ] LJ 2 e : A ] y 2 ü d C : u 4 A C C : J M A X e J L A L 6 : 6 2 e : C C A < : 8 X e L 6 : C 2 ü 6 L J 2 e : 4 A C C : J M A X e J L A L 6 : J : : : C C A < : û6 : C : 8 C : L A : X L C R M A 2 X L J L A L 6 2 J X : L A C e : M : X e 2 X ü A X C A < :L 6 : M < 8 R : C 2 < <L 8 : 2 X 2 t L 2 8 < < R F 6 2 e : C JJ L 8 C L 2 X c ô ] ô \ ñ 8 X e J : : : C J J L 8 C L 2 X c ô ] ô \ û6 : 6 2 e : C J L 8 C L J 6 2 e 2 X ü6 : X L 6 : J : : : C P : ü 2 X J L A 4 A d X L û6 2 J 2 J C : M C : J : X L :e2 X L 6 : e 2 t 8 ü C 8 a P R L 6 : e 8 J 6 :e < 2 X : û6 : L C 8 X J 2 L 2 A X 2 X L A c ô ] ô \ Z 2 J 2 X C : J M A X J : L A 4 A a a d X L 2 A X L C 8 X J t a 2 L L :e P R L 6 : J : : : C O J ü A 8 <a X : j 2 a 2 < 8 C < R F L 6 : L C 8 X J 2 L 2 A X ] C A a 6 2 e : C C A < : L A J : : : C C A < : 2 J 2 X C : J M A X J : L A 8 X : L : C X 8 < J 2 ü X 8 < ] C A a L 6 : J : : : C O J a X : F P : 2 X ü L 8 ü ü :e + 6 : X 8 M < 8 R : C < A 4 8 L : J e 2 X ü M < 8 4 : c ô ] ô \ Z F 2 L C : a 8 2 X JL 6 : C :d X L 2 < J : : X P R L 6 : J : : : C 8 X 4 : e 2 J t 4 A Y : C :e FL 6 : M < 8 R : C8 L L : a M L J L A : Y 8 e : L 6 : J : : : C 8 X e C : L d C X J 8 ] : < R L A 6 A a : P 8 J : py 8 e 2 X ü c ô ] ô \l

9 2 J A a M < 2 J 6 :e P R4 A a P 2 X 2 X ü 8 Y A 2 e 8 X 4 : A ] L 6 : J : : : C 8 X e 8 L L C 8 4 L 2 A X L A P 8 J : û6 : J : : : C P : ü 2 X J P R ü A 2 X ül A 6 A a : P 8 J : c ô ] ô \ 8 X e J R X 4 6 C A X 2 g 2 X ü L 6 : J L 8 C L A ]L 6 : ü 8 a : P R ] A C 4 t 2 X ü L 6 : A L 6 : C M < 8 R : C J L A L C 8 X J 2 L 2 A X ] C A a c ô ] ô \ñ L A c ô ] ô \ Z û6 : J : : : C e : < 8 R J L 6 : J L 8 C L A ]2 L J J : 8 C 4 6 P R 4 A d X L 2 X ü c ô ] ô \ L A M : C a 2 L L 6 : 6 2 e : C J L 2 a : L AC :8 4 6 L 6 : 2 C 6 2 e 2 X ü M < 8 4 : Jû6 : X L 6 : J : : : C P : ü 2 X JL A : M < A C : L 6 : : X Y 2 C A X a : X LL A f X e e : C c ô ] ô \ + 6 : X e : C 2 J e 2 J 4 A Y : C :e F 2 L P :4 A a : J L 6 : L 8 C ü : L ] A C J : c ô ] ô \ ± û6 : A d L 4 A a : A ] L 6 : J : e : L : C a 2 X : J 6 : L 6 : C L 6 : M < 8 R : C 2 < < P : e : C X : L ü 8 a : 6 : X e : C 2 JL 8 ü ü :e P : t ] A C :C : X ü P 8 J : A C8 J : : : C 8 ü 8 2 X 6 : X 8 < < L 6 : 6 2 e : C JJ 8 ] : < R : Y 8 e : L 6 : J : : : C f ] 8 J 2 X ü < : 6 2 e : C J d 4 4 : J J ] d < < R : Y 8 e : JL 6 : J : : : CF L 6 : J : : : C 4 A X L 2 X d : J L A J :8 C 4 6 ] A C A L 6 : C 6 2 e : C Jd X L 2 < L 6 : R 8 C : 8 < < J 8 ] : û6 2 Ja 8 R C : J d < L 2 X e 2 : C : X L 6 2 e : C J P : 2 X ü4 6 8 J :e 8 L e 2 : C : X LL 2 a : J q 4 A X J : d : X 4 : A ]< 2 a 2 L :e M : C 4 : M L 2 A X 2 J L 6 : A t J 2 A X 8 < X : :e L Af X e A P :4 L J 8 d C 8 M M C A J L A 2 J A < 8 L : L 6 2 J C : 8 J A X 2 X ü 2 X 8 J M :4 2 8 < 2 g :e a A e d < : F 8 < ý K L 6 8 L L C 8 X J < 8 L : J 2 X ] A C a 8 L 2 A X 8 4 d 2 t J 2 L 2 A X ü A 8 < J L A 6 2 ü 6 t < : Y : < M 6 R J < ü A 8 < J L A : M < A C : M 8 C L J A ] L 6 : : X Y 2 C A X a : X L 8 d C8 M M C A L A J :8 C 4 6 M < 8 X X 2 X ü C : d 2 C : JL 6 8 L :8 4 6 M < 8 R : C a 8 2 X L 8 2 X 2 X ] A C t a 8 L 2 A X 8 P A d L L 6 : J L 8 L : A ] 8 6 : d C 2 J L 2 4 J :8 C 4 6 A X 8 X 2 X L : C X 8 < a 8 M û6 : 6 : d C 2 J L 2 4 J :8 C J f X e 2 X ü 8 e : J 2 C :e A P :4 L 8 J2 L Jü A 8 < n o ] ³ \ A 8 Y 2 X ü ü 2 Y : X 8 X A Y : C Y 2 :A ] L 6 : J R J L : a O J 4 A a M A t X : X L JF : X A 2 < < d J L C 8 L : L 6 8 X : 8 a M < : e C 8 X ] C A a 8 L A t M < 8 R : C ü 8 a : A ] 6 2 e : 8 X e J : : ûa 2 < < d J L C 8 L : L 6 : 4 A X e d 4 L A ] 8 J :8 C 4 6 F :2 < <d J : 8 X : 8 a M < : : X Y 2 C A X a : X L 2 L 6 L AP d 2 < e 2 X ü J F A X : A ] J L A 2 X L : C X 8 < C A A a J J : M 8 C 8 L :e P R 8e A A C y 2 ü d C : w 8 d C : 8 a M < : P : ü 2 X J 2 X L 6 : a 2 e e < : A ] 8 ü 8 a : F 8 ] L : CL 6 : 6 2 e : C6 8 J6 2 e e : X 8 X e L 6 : J : : : C 6 8 J f X 2 J 6 : e 4 A d X L 2 X ü û6 : 6 2 e : C 2 J 2 X c ô ] ô \ Z F hider door0 y 2 ü d C : w j ^ < 8 X Y 2 : door2 door1 seeker A ]: 8 a M < :: X Y 2 C A X a : X L 8 2 L 2 X ü L A P : J : : X û6 : J : : : C X ü : J L A c ô ] ô \ A ] 2 L J ü A 8 < a X : 8 X e P : ü 2 X J L A J : : û6 : ü A 8 <a X :J L 8 L : J M : 4 2 f : J 8 J : : ü A 8 < L A L 6 : M < 8 X X : C 8 J µ ¹ º 2 L 6 L 6 :8 e e : e 4 A X J L C 8 2 X L L 6 8 L» ¼ ½ ¹ º ¾ À Á Â Ã û6 : S «V 0 U M < 8 X X : C 4 A X J 2 e : C J L 6 : 8 4 L 2 A X µ ¹ º L A P : M C 2 a 2 L 2 Y : P d L d X e : C J M : 4 2 f :e J 2 X 4 : L 6 : Y 8 C 2 8 P < : ¹ 2 JX A L P A d X e L A 8 M 8 C L 2 4 d < 8 CA P :4 L A ] L R M : À Á Â Ã f X A C e : C L A P 2 X e L 6 :Y 8 C 2 8 P < : F L 6 : J : 8 C 4 6 M < 8 X X : C a d J L P : 4 8 < < : e L A ü : X : C 8 L : 8 M < 8 X ] A C< A 4 8 L 2 X ü 8 À Á Â Ã ûa L 6 2 J : X e F S «V 0 U 8 e e J L A L 6 : M < 8 X 8 Å ÿ ƒÿ 8 X e 4 8 < < J L 6 : J :8 C 4 6M < 8 X X : C L A 2 X J L 8 X L 2 8 L : 8 J :8 C 4 6 M < 8 X y 2 ü d C : Æ 8 û6 : J :8 C 4 6 M < 8 X X : C C :8 J A X J ] C A a L 6 2 J X A < :e ü : 8 4 d 2 J 2 L 2 A X ü A 8 < A ] < A 4 8 L 2 X ü 8 À Á Â Ã F L A L 6 : ü A 8 < A ] : M < A C 2 X ü C : ü 2 A X J 6 : C :8 À Á Â Ã a 2 ü 6 L P : j 8 L 2 J ] R t 2 X ü L 6 2 J ü A 8 < C : d 2 C : J M 6 R J < < R J :8 C X ü L 6 C A d ü 6 M A J J 2 P < : C : ü 2 A X J S «V 0 U 8 J J L 6 : J : 8 C 4 6 M < 8 X X : C L A : M 8 X e L 6 : f X e X A e : p L 2 a : 8 f X e X A e : 2 J : M 8 X e : e F L 6 : J :8 C 4 6 M < 8 X X : C f C J L: 8 a 2 X : J L 6 : : < : X Y 2 C A X t a : X L L A e : L : C a 2 X : 2 ] 8 X A P :4 L A ] L 6 : J M :4 2 f :e L R M : 2 J Y 2 J 2 P < : L A L 6 : 8 ü : X L f ] X A LF L 6 : J :8 C 4 6 M < 8 X X : C J : < : 4 L J 8 C : ü 2 A X L A : M < A C : X : LF ü : X : C 8 L : J 8ü A 8 < L A : M < A C : L 6 8 L C : ü 2 A X F 8 X e 8 e e J2 L L A L 6 : M < 8 X y 2 ü t d C : Æ P û6 : 2 X 2 L 2 8 < a 8 M y 2 ü d C : r A ] C : ü 2 A X J 4 A X t L 8 2 X J 8 < < L 6 :C : ü 2 A X J 2 XL 6 :: X Y 2 C A X a : X L: 4 : M L L 6 : 2 X L : C 2 A C C A A a A ] L 6 :P d 2 < e 2 X üa X L 6 : C 2 ü 6 L U : ü 2 A X J L 6 8 L 8 C : 4 A a M < : L : < R Y 2 J 2 P < : 8 C :a 8 C :e 8 J 6 8 Y 2 X ü P : : X : M < A C :e F M 8 C L 2 8 < < R Y 2 J 2 P < : C : ü 2 A X J 8 C :a 8 C :e

10 goto(x) find(x) type(x)=human goto(x) find(x) goto(x) find(x) explore(door1) explore(door1) goto(door1) y 2 ü d C : Æ j A < d L 2 A X A ] L 6 < 8 X ü C 8 M 6 (a) (b) (c) py : M open(door1) ] A C ] d L d C : : M < A C 8 L 2 A X û6 : 4 < A J : J L8 Y 8 2 < 8 P < : d X : t M < A C :e C : ü 2 A X 2 J L 6 : f C J L C A A a 2 X L 6 : P d 2 < e 2 X ü A X L 6 : C 2 ü 6 L F 2 L 2 J C :4 A a a : X e :e ] A C : M < A C 8 L 2 A X û6 2 J ü A 8 < 2 J L 6 : X ] d C L 6 : C : M 8 X e :e P RS «V 0 U L A ü A L A È É Ê 8 X e A M : X 2 L y 2 ü d C : Æ 4 j 2 X 4 :8 < < A ]L 6 : 8 C ü d a : X L J2 X L 6 : f C J L8 4 L 2 A X 8 C : P A d X e L A J M :4 2 f 4 A P :4 L J F2 L 4 8 X P : 4 8 C C 2 :e A d L q 4 L 2 A X p :4 d L 2 A X M : C ] A C a J L 6 2 J 8 4 L 2 A X 2 X e 2 C :4 L < R P R P 2 X e 2 X ü 8 X 8 L t L C 8 4 L 2 A X J : X J A C L A L 6 : J : : : C + 6 : X L 6 : J : : : C 8 C C 2 Y : JF È É Ê 2 J A M : X : e e 2 C :4 L < R P R q 4 L 2 A X p : t 4 d L 2 A X q] L : C È É Ê2 J A M : X : e F S «V 0 U d J : J L 6 : J :8 C 4 6 M < 8 X X : C L A : Y 8 < d 8 L : L 6 : M C A ü C : J J A ] L 6 :J :8 C 4 6 P R : 8 a 2 X 2 X ü L 6 : A C < e ] A CA P : 4 L J 6 8 Y 2 X ü L 6 :M C A M t : C L R À Á Â Ã f ] A X : 2 J< A 4 8 L :e F L 6 : J : 8 C J4 A X J 2 e t : C :e J d 4 4 : J J ] d < f ] X A L F L 6 : J :8 C 4 6 M < 8 X X : CJ : < :4 L J8 X : C : ü 2 A X ] A C : M < A C 8 L 2 A X 8 X e L 6 : J :8 C X ü M C A t 4 : J JC : M :8 L J d X L 2 < L 6 : C : 8 C : X Aa A C :C : ü 2 A X J L A : t M < A C : f X L 6 2 J 4 8 J : F A M : X 2 X ü È É Ê e A : J X A L C : Y :8 < 8 À Á Â Ã F P d L e A : J M : C a 2 L L 6 : 8 ü : X L L A J : : 8 X A L 6 : C C : ü 2 A X L 6 8 L 2 J 8 d L A a 8 L < < R 8 e e :e L A L 6 : J :8 C 4 6 M < 8 X X : C O J2 X L : C X 8 <a 8 M q J L 6 2 J X : C : ü 2 A X 2 J L 6 : 4 < A J : J L d X : M < A C :e J M 8 4 : F A XL 6 :X : L 2 L : C 8 L 2 A X L 6 : M < 8 X X : C2 < <M < 8 X L A : M < A C : 2 L 8 M : X 2 X ü È É Ë e A : JX A L C : Y :8 < L 6 : À Á Â Ã F J A L 6 : J :8 C 4 6 M C A 4 : : e J L A L 6 : X : L 4 < A J : J L d X : M < A C :e C : ü 2 A X FL 6 : C 2 ü 6 L J 2 e : A ] L 6 2 JP d 2 < e 2 X ü A: C : : J : :L 6 :8 e Y 8 X L 8 ü : A ] a 8 2 X L 8 2 X 2 X ü 8 a 8 M f a a :e 2 8 L : < R 8 ] L : CA M : X 2 X üè É Ë F L 6 : 8 ü : X L2 J 2 X t J 2 e : A X : P d 2 < e 2 X ü8 X e e : 4 2 e : J L A ü A L A8 X A X t X : 2 ü 6 P A C 2 X ü C : ü 2 A X j 2 X 4 : L 6 2 J e : J L 2 X 8 L 2 A X C : ü 2 A X 2 J X A X ] C A a 6 8 Y 2 X üj : : X 2 L M C : Y 2 A d J < R F : 4 A d < e J 2 a M < R ü A L 6 : C : û6 2 J A d < e C : J d < L 2 X L 6 : 8 ü : X L 8 < 2 X ü e 2 C :4 L < R L A 8 C e L 6 :e : J L 2 X 8 L 2 A X d X t L 2 < J L A M M :e P R L 6 : 8 < < û A 8 Y A 2 e ü : L L 2 X ü 4 8 d ü 6 L2 X L 6 2 J< A 4 8 < a 2 X 2 ad a F L 6 :J :8 C 4 6 M < 8 X X : C d J : J 2 L J 2 X t L : C X 8 < a 8 M y 2 ü d C : / m L AM < 8 X 8 M 8 L 6 L A L 6 : X : L C : ü 2 A X û6 : A X < R X A XM 8 L 6L 6 : C : 2 J L A : 2 L L 6 : 4 d C t C : X L P d 2 < e 2 X ü L 6 C A d ü 6È É Ê û6 : J :8 C 4 6 M < 8 X X : C C : L d C X J L 6 2 J J : d : X 4 : A ] 2 X L : X L 2 A X J L A S «V 0 U F L 6 : X 2 X Y A : J q 4 L 2 A X p : 4 d L 2 A X L A ü : X : C t 8 L : < A 4 A a A L 2 A X 8 < A X ü L 6 2 J M 8 L 6 py : X L d 8 < < R F 8 ] L : C A M : X 2 X ü È É Î F L 6 : J : : : C f X 8 < < R J : : J8 À Á Â Ã8 X e 4 8 X ü AL A 2 L Ï ö ó ù ø c ú ö ó + :6 8 Y : 2 a M < : a : X L :e 8 M < 8 X t P 8 J :e 4 A X L C A < < : C ] A C8 M < 8 R : C 2 XL 6 :ü 8 a : A ] 6 2 e : 8 X e J : : û6 : 4 A a M < : L : J : : : C 2 J2 a M < : a : X L : e F : 8 C : : L : X e 2 X ü A d C 8 C t L : 4 L d C :L A2 a M < : a : X L L 6 : 6 2 e : C 8 d C8 ü : X Le R X 8 a t < < R C :8 4 L J L A X ü : J 2 X L 6 :: X Y 2 C A X a : X L F ] C A a 8 Y A 2 e 2 X ü 4 A < < 2 J 2 A X J 2 L 6 A P J L 8 4 < : J 8 X e A L 6 : CM < 8 R : C J L A : M < A 2 L 2 X ü X ü : J 2 X 2 X ] A C a 8 L 2 A X 8 P A d L 6 : C :

11 ñ L 6 :A L 6 : C M < 8 R : C J a 8 R P : 6 2 e 2 X ü û6 : 2 a M < : a : X t L 8 L 2 A X 4 A a P 2 X : J ü : X : C 8 < M d C M A J : M < 8 X X 2 X ü F J M :4 2 8 < M d C M A J : C :8 J A X 2 X ü8 P A d L 4 A X e d 4 L 2 X ü 8 J :8 C 4 6 F 8 X e C : 8 4 L 2 Y : 4 A X L C A < A ] 6 d a 8 X P : 6 8 Y 2 A C J J8 X : :4 L 2 Y : J R J L : a ] A C8 X 2 a 8 L 2 X ü 6 d t a 8 X J 4 8 C C R 2 X ü A d L L 8 J J L 6 8 L C : d 2 C : < A 4 A a A L 2 A X z 2 a 2 L 2 X ü L 6 : 6 d a 8 X 8 ü : X L O J 8 8 C : X : J J A ] 2 L J : X t Y 2 C A X a : X L P R J 2 a d < 8 L :e M : C 4 : M L 2 A X 2 X 4 C :8 J : J L 6 : C : 8 < 2 J a A ] L 6 : P : 6 8 Y 2 A C ü : X : C 8 L :e ± m ù Ð óö \ ^ \ \ ó ô c + : 2 J 6 L A L 6 8 X A d C 8 e Y 2 J A C J wa C a = 8 e < : C 8 X e = A X X 2 : + : P P : C ] A C L 6 : 2 C J d M M A C L A ] L 6 2 J A C û6 8 X J 8 < J A L A L 6 : a F + : < L A X= : 4 : LF Ñ A X 8 L 6 8 X H C 8 P L C : : F = C : L L S A d Y 2 < < : F 8 X e Ñ : w 2 a : C A ] A C 4 A a a : X L 2 X ü A X e C 8 ] L JA ] L 6 2 JM 8 M : C Ó \ \ õ \ ó ù \ c = 8 e < : C F w F ^6 2 < < 2 M JF H F 8 X e+ : P P : CF = /r r o Ô þ K I þ Õ Ö ƒ I Ø ý þ < ÛÜ þ Ý I þ ƒ ÿ ƒ I ƒ K 8 ] A C e W X 2 Y : C J 2 L R ^C : J J = :4 : L F + F 8 X e = 8 e < : CF w f / r r o f X L : ü C 8 L :e P : 6 8 Y 2 A C 8 < 8 ü : X L 8 C L :4 L d C : f X Þ ƒ < ÿ þ ƒ I ý ß ý þ ÿ ƒ < ƒ ƒ I Ø I ÿ á ƒ < ÿ â ý ˆþ ƒ ã I I þ ƒ F * Æ u w = :4 : L F + 1 /r r s û6 : Ñ 8 4 z f j ^ q ^f û:4 6 t X < U : M A C L 1 j t H f j t r s t m / F WX 2 Y : C J 2 L R A ]^ : X X t J R < Y 8 X 2 8 F ^ 6 2 < 8 e : < M F ^ q O : 2 P F H /r r l f X L : X L 2 A X J 2 X a :8 X J t : X e M < 8 X X 2 X ü û:4 6 X < U: M A C L1 j t H f j t r l t Æ o F S : M 8 C L a : X L A ] H A a M d L : C 8 X e f X ] A C a 8 L 2 A X j 4 2 : X 4 : F WX 2 Y : C J 2 L R A ] ^ : X X J R < Y 8 X 2 8 D A F A FU: F = S F = :4 : LF + F 8 X e = 8 e < : CF w f /r r s û: C C 8 2 X X 8 Y 2 ü 8 L 2 A X J 2 < < J8 X e C : 8 J A X t 2 X ü f X Þ ƒ < ÿ þ ƒ I ý á ƒ I ý ƒ < ƒ ƒ I Ø I ÿ á ƒ < ÿ â ý ˆþ ƒ K ã Ý I I þ ƒ D A F A /r r s ä þ I þ < ÿ æ ç þ < ß < ý Ý þ è ƒ Ü K ç é þ Û Þ ÿ þ < I þ Û ÿ Ü þ I þ Õ ê ƒ < ƒ ƒ I þ ƒ ^6 S S 2 J J : C L 8 L 2 A X F WX 2 Y : C t J 2 L R A ] ^ : X X J R < Y 8 X A A C : F 1 = F O : 2 P F H F 8 X e U : F = S /r r * ^< 8 X X 2 X ü 8 X e L : C C 8 2 X C :8 J A X 2 X ü f X Ü Ü Ü ë Ô þ Ô ç ƒ þ ƒ ë I I ÿ Þ K þ Ü K þ Ý < I þ ƒ 8 < J A W X 2 Y : C J 2 L R A ] ^ : X X J R < Y 8 X 2 8 H f j e : t M 8 C L a : X L û:4 6 X < U: M A C L 1 j t H f j t r s t u o í z fw H z q =l w m U: F = S F D A F A F = :4 : LF + F 8 X e = 8 e < : CF w f /r r s û: C C 8 2 X C : 8 J A X 2 X ü ] A C 6 d a 8 X< A 4 A a A t L 2 A X f X Þ ƒ < ÿ þ ƒ ƒ I Ü þ I þ ƒ ï ð F r r u / m m * O : X : Y 8 F j 2 Lg : C < 8 X e j fp p p H A a M d L : C j A 4 2 : L R ^C : J J _ ú ö ^ õ ] ³ a ú \ c ò ú ù a ] \ _ ò ö ö õ \ 2 J 8 ^6 S 4 8 X e 2 e 8 L : 2 X H A a M d L : C 8 X e f X ] A C a 8 L 2 A X j 4 2 : X 4 : 8 L L 6 : W X 2 Y : C J 2 L R A ] ^ : X X J R < Y 8 X 2 8 A 2 J 4 d C C : X L C : t J :8 C X q C L 2 f < f X L : < < 2 ü : X 4 :2 J M < 8 X X 2 X ü ] A C 2 X ] A C a 8 L 2 A X8 4 d 2 J 2 L 2 A X A : C :4 : 2 Y :e 6 2 J q = e : ü C : : 2 X ^6 2 < A J A M 6 R 2 X /r w u ] C A a L 6 : WX 2 t Y : C J 2 L R A ] 1 8 C R < 8 X e 8 X e 6 2 J 1 j p e : ü C : : 2 X H A a M d L : C 8 X e f X ] A C a 8 L 2 A X j 4 2 : X 4 : 2 X /r r m ] C A a L 6 : WX 2 Y : C J 2 L R A ] ^ : X X J R < Y 8 X 2 8 Ï a õ ú c ôö ³ a \ õ [ \ ú ó 2 J 8 M A J L t e A 4 L A C 8 < C : J :8 C 4 6 ] : < < A 8 L L 6 : W X 2 Y : C J 2 L R A ] = C 2 L 2 J 6 H A < d a P 2 8 A 2 J 4 d C C : X L C : J :8 C X q C L 2 f <f X L : < < 2 ü : X 4 : 2 J M < 8 X X 2 X ü A: C :4 : 2 Y :e 6 2 J ^6 S e : ü C : : 2 X H A a M d L : C 8 X e f X ] A C a 8 L 2 A X j 4 2 : X 4 : ] C A a L 6 : W X 2 Y : C J 2 L R A ] ^ : X X J R < Y 8 X X /r r * _ ] õ õ ô Ó \ ú ù a 2 J 8 ^6 S 4 8 X e 2 e 8 L : 2 X H A a t M d L : C8 X e f X ] A C a 8 L 2 A X j 4 2 : X 4 : 8 L L 6 : WX 2 Y : C t J 2 L R A ] ^ : X X J R < Y 8 X 2 8 A2 J4 d C C : X L C : J : 8 C X t 4 < d e : J M C A Y 2 e 2 X ü 6 2 ü 6 t < : Y : < P : 6 8 Y 2 A C 8 < 4 A X L C A < A Y : C L 6 : 8 X 2 a 8 L 2 A X A ]6 d a 8 X < A 4 A a A L 2 A X A: C :4 : 2 Y : e 6 2 J = j e : ü C : : 2 X 1 8 L 6 : a 8 L 2 4 J 8 X e H A a M d L : C j 4 2 : X 4 :2 X /r w r ] C A al 6 :WX 2 Y : C J 2 L R

12 A ] 1 8 C R < 8 X e 8 X e 6 2 J1 j p e : ü C : : 2 X H A a t M d L : C 8 X e f X ] A C a 8 L 2 A X j 4 2 : X 4 : 2 X /r r / ] C A a L 6 : WX 2 Y : C J 2 L R A ] ^ : X X J R < Y 8 X 2 8 y 2 ü d C : r jf X 2 L 2 8 < a 8 M y 2 ü d C : / m j y 2 X 8 < a 8 M

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