Relational interprocedural analysis of concurrent programs

Texte intégral

(1)Relational interprocedural analysis of concurrent programs Bertrand Jeannet. To cite this version: Bertrand Jeannet. Relational interprocedural analysis of concurrent programs. [Research Report] RR-6671, INRIA. 2008, pp.36. �inria-00328045�. HAL Id: inria-00328045 https://hal.inria.fr/inria-00328045 Submitted on 9 Oct 2008. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Relational interprocedural analysis of concurrent programs Bertrand Jeannet. N° 6671 Octobre 2008. ISSN 0249-6399. apport de recherche. ISRN INRIA/RR--6671--FR+ENG. Thème COM.

(3)

(4)

(5) !" !# $% && '()*! + ,.-0/1-3254&6875,.2544,9/ :;=<

(6) >@?A[]BD\ CFEHGJILKNMO<0>@?0KQP

(7) R9>@>SUT=VWP0XYT.MZK SUVW^_?

(8) `badcZRYef?

(9) MQaR9^hgQcOM iQX9Û^5R9cOMjU?(cO?1Pk;U?

(10) ckPk;U?lT5mnon.pJqErBDPMOR9s=cO?(tYuou9vEHw9w^=XYxo?0K {y \z|0}

(11) ~UY}o ?@? LMZ?

(12) T=jMO;U?@cZ?

(13) XMZVRoT=XYXYÛ^=cOR.X9Pk;hMORhVWT.MO?

(14) cZÛcZRJP?0jUSUcZX9 X9T=XYWIJKNVKR9 KO? SU?

(15) T.MZVWX9ÛcZR9xocZX9>@KMZRPRoT=PS=cOcZ?

(16) T.MÛcZR9x9ckXY>KP

(17) R9>@^_R.KN?1jRYXU J?1jT.S=>s5?

(18) cR9 MO;=cO?1X9jUK0TMZ;U?cZ?

(19) XMZVRoT=XY&XYÛ^=cOR.X9Pk;5X{KN? \ SU?

(20) T.MOVXY ÛcZR9x9ckXY>#VWKDXYT_XYWI.0?0jhsLIPRo>` ÛSJMZVT=xKNSU>@>XYcZVW?0KRYÛcZRJP?0jUSUcO?1K

(21) XYT=jsLIÛcZR9^=X9xoXYMOVWTUxcZ?0XoPk;=XYs=VWVMfIVWTJRocO>XMZVRoT S=KOVT=xMZ;U?0KO?KOSU>@>XYcZV?1K

(22) ?xo?

(23) TU?0cZX9VW

(24) ?MO;UVK@XYÛ^=cOR.X9Pk;MORPRoT=PS=cOcZ?

(25) T.M^=cORox9ckXY>K sLI{PRo>^=SJMOVWTUxRoc?0X9Pk;{MO;UcZ?0XojÛcZRLP

(26) ?0jJS=cO?DKNSU>@>XYcZVW?0K]MO;=XYMMkXY9?1K]VWT.MOR@X9P0PR9S=ToMMO;U? ^=X9cZX9W?

(27) ? J?1PSJMZVRoTR9"MZ;U?(RYMZ;U?

(28) cMZ;UcO?1X9jUK0 : ?0Pk;UTUVP

(29) X9WI99?QjJ?

(30) =TU?QRoSUc]X9T=XYWIJKNVK>@?MO;=RLj@sLIVT=KNMOcZSU>@?

(31) T.MZVTUxMZ;U?QRo^_?0cZXYMOVWR9T=X9 KO?

(32) >XYT.MOVP

(33) K@R9ÛcZR9xocZX9>K

(34) MZ;U?

(35) TsLIX9s=KNMOckX9P MZVT=xMZ;U?P

(36) X9` KNMZXoPkJKR9MZ;U?PRoT=PSUcZcZ?

(37) T.M MOXY;=TcO?1?X9£jU?0K0PMO5VWX9¢9T=?jhX9T=X9XYKNWMIJKNsLVIKMOX9;=s=XYKNMMOckXYX9WP%MZX3VT=IJxK]MOMO?

(38) ;UcZ?>@ÛVT_cZR9Xx9MOck?1XYK

(39) >¡ ?

(40) TL¢LVcZR9TU>@?0ToMkKQVWTR9ckjJ?0cMOR?1X9jMOR :;UVK>@?MZ;URJj¤XYWRQKMZR¥? LMZ?

(41) T=j¦MZR§PRoT=PSUcZcZ?

(42) T.MÛcORox9ckXY>K? JVKfMZVTUx§cO?0WXYMOVWR9T=X9 VWToMZ?

(43) cZÛcZRLP

(44) ?0jJS=cZX9UXYT=X9IJKOVWKQ¨©?9 x=WoTLSU>@?0cOVP

(45) X9=¢XYcZVWX9sUW?0KdX9T=XYWIJKNVK

(46) oKN;=X9^_?XYT=X9IJKOVWKkª ? VW>^=?0>?0T.MO?0jVMR9c(ÛcZR9xocZX9>@KVMZ;KOP0XYXYcD¢3X9cOVXYs=?1K

(47) X9T=j%?? J^_?0cOVW>@?

(48) T.MKN?0¢9?0cZX9 PX9KZKOVWP0XYKOILT=Pk;UcZR9TUVKZXMOVWR9T¦ÛcZRYMZRLP

(49) R9K{VWT¤R9ckjJ?0c{MORVWWS=KfMZcZXYMO?MO;U?«ÛcO?1PVKNVWR9T¬R9R9SUc MO?1Pk;UTUV \ SU?9Us=SJMXYKNRMOR{X9T=XYWIL

(50) ?MO;U?X9ÛÛcZR3 JV>XMZVRoT=KVMQ^5?

(51) cOR9cZ>K

(52) ®.¯°O±² ~³ |o ¢9?0cOV_P0XMOVWR9T¦RY(PRoT=PS=cOcZ?

(53) T.M@ÛcORox9ckXY>K0]VWToMZ?

(54) cZÛcZRLP

(55) ?0jJS=cZX9X9T=XYWIJKNVK0 XYs_KfMZcZXoP MVT.MO?0cO^=cO?

(56) MZXMZVRoT. Centre de recherche INRIA Grenoble – Rhône-Alpes 655, avenue de l’Europe, 38334 Montbonnot Saint Ismier Téléphone : +33 4 76 61 52 00 — Télécopie +33 4 76 61 52 52.

(57) !"

(58) & '(&) )*! % && ! | R9S_ K

(59) MO?0T=jJRoT=KXYSU ÛcZR9x9ckXY>@>@?0KDPRoT=PS=cOcZ?

(60) T.MZKD X9ÛÛcZRJPk;U?@cO?0WXYMOVWR9TJ` TU?

(61) W?jJ?D XYT_XYWILKO?VWT.MO?0cOÛcZRJP

(62) 1jJSU\ckXYW?QjJ?^=cORox9ckXY>@>@?0K]K

(63) \ SU?0T.MOVW?

(64) K

(65) X9T=K% XY^=ÛcORJPk;U? cO?0WXYMOVWR9TU?0W?9S=T«ÛcZR9xocZX9>@>?K

(66) SU?

(67) T.MOVW?

(68) ?1KfMX9T=XYWIJK

(69) ?

(70) TP

(71) X9WP

(72) SUXYT.MjU?0Kc

(73) 0KOSU>

(74) 0K(jJ? ÛcORJP

(75) 0jJSUcZ?99?Md ?

(76) TÛcZR9^=X9x9?1XYT.M&?0T=KOSUVMZ??1KVTUR9cZ>@XYMOVWR9T=Kj XoP

(77) P?1KOKOVWsUVWVM

(78) XYS>@RI9?

(79) T@jJ? P?0Kc

(80) 0KOSU>

(81) 0K0 R9S_K\ x

(82)

(83) T

(84)

(85) ckXYWVWKOR9T=KP?

(86) MNMO?(X9ÛÛcZRLPk;=?DX9SJ {ÛcZR9x9\ ckXY>@>@?0K%PRoT=PS=cOcZ?

(87) T.MZK?

(88) T P

(89) XYPS=WX9ToM^5R9SUcPk;=X SU?DMoPk;U?(jJ?0KQc

(90) 1KNSU>

(91) 1KjJ?(ÛcZRLP

(92) 0jJS=cO? SUVÛcZ?

(93) TUT=?

(94) T.MQ?

(95) TPRo>@^JMO? ?

(96) 0P

(97) SJMOVWR9Th\ ^=X9cZX9W<

(98) W?jJ?0KX9SJMOcZ?0K%M 9Pk;U?1KjJShÛcORox9ckXY>@>@?P

(99) R9T=P

(100) SUcOcZ?

(101) T.M1 :"?1Pk;UTUV S=?

(102) >@?

(103) T.M0LTURoS=Kj

(104)

(105) =TUVKOKOR9T=KTUR9MOcZ?>

(106) MZ;URJjJ?(j XYT=X9IJKO??

(107) TVWT=KfMZcOS=>?0T.MZXYT.M WXK

(108) 0>@X9T.MOV \ S=?R9^

(109)

(110) ckXMZVRoTUTU?0W?KNMZXYT_jUXYckj ÛSUVK?

(111) T¤?

(112) £5?1P MZS=XYT.MS=TU?«VWT.MO?0cOÛc

(113) MkXMZVRoT XYs=KNMOckXYVMO?djJ?1KÛVWW?0K"j X9Û^_?0WK jJ?0KM oPk;U?0KP

(114) R9T=P

(115) SUcZcO?0ToMZ?0K0 !QTU?VT.MZ?

(116) cZÛc

(117)

(118) MZXYMOVWR9T(XYs=KNMOckXYVMO? ^_RocNMkXYT.M?1KQjJR9T=T"

(119)

(120) ?1K^_?0cO>@?M=T_XYW?

(121) >@?

(122) T.MQ RosJMO?0ToMZVRoTj SUT=?XYT=X9IJKO? \ SUV MZ?

(123) cZ>@VTU?o A?MNMZ?¤>

(124) MZ;URJjJ? ^5?

(125) cZ>@?Mj

(126) MZ?

(127) T=jUcO? XYSJ ÛcORox9ckXY>@>@?0KP

(128) R9T=P

(129) SUcOcZ?

(130) T.MkK?1K§X9T=X` IJKO?0KVT.MO?0cO^=cORJP

(131) 1jJSUckXYW?0K?

(132) JVWKNMZX9ToMZ?0K¨©XYT_XYWILKO?jJ?1K]¢X9cOVXYsUW?0K]TLSU>

(133)

(134) cZV \ SU?1K

(135) LXYT=X9IJKO?jJ? R9cZ>?3ª RoS=K X3¢oR9T=KV>@ÛXYT.M

(136)

(137) ?@^_RoSUcjJ?1KÛcORox9ckXY>@\ >@?0$K #¢X9cOVXYsUW?0KKZP

(138) X9WX9VcZ?0K0?M TUR9S=KX3¢oR9T=K? J^

(139)

(140) cZV>@?0ToM

(141) ÛS_KNVW?

(142) SUckK%ÛcZRYMORJP

(143) R9W?0K%PX9KZKNV SU?0KjJ?(KOI.T_Pk;UcORoTUVKOXYMOVWR9\TLX=T j VWWS=KNMOcZ?

(144) cWX{Ûc

(145) 1PVKNVWR9TjU?TURYMZcO?>%

(146)

(147) MO;URJjJ?o>@X9VWKXYS=KZKOV?1KX9ÛÛcZR3 JV>XMZVRoT=K S ?

(148) W? ?£?0P MZSU?9 & ² }1| ° ' |D ¢

(149) 0cOV_P

(150) XYMOVWR9T(jJ?d^=cORox9ckXY>@>@?0KP

(151) R9T=P

(152) SUcOcZ?

(153) T.MkK

(154) 1XYT=X9IJKO?dVT.MZ?

(155) cZÛcORJP

(156) 0jJSUckXYW?9 VT.MO?0cO^=c

(157)

(158) MZXMZVRoTX9s=KNMOckXYVMO?.

(159)

(160) "!$#% &#' )(* % + ,-.'#. w. " d& T.MO?0cOÛcZRJP?1jJSUckXY_X9T=XYWIJKNVKdR9"KN? \ SU?

(161) T.MOVXY5ÛcZR9xocZX9>@KdVWK%%?

(162) W` SUT=jJ?0cZKNMORLRJj@VWTVMZK%ÛcZVT=P

(163) V` ÛW?0K21 AAp9p3s UGLa\ vUq9 3G 4.t 5XYT=j>@R9cZ?cZ?0P

(164) ?

(165) T.M%P

(166) R9T.MOcZVs=SJMOVWR9T=K%PRoT=P?0cOT_K]>XYVWTUWI{XYWx9R9` cZVMZ;U>@VWP0XY&MO?1Pk;UTUV S=?0KXYT=j 63R9cDXYMO?0cOT=XYMOVW¢9?¢.VW?

(167) QK'1 i87G 4:9J;<3=4>4J iG ?.Cu>9L@?LGJuA5b 7QR%?

(168) ¢o?

(169) c1=MZ;U?@VT.MZ?

(170) cZÛcORJP

(171) ?0jJSUckXY&XYT=X9IJKOVWKRY%PR9T_PSUcZcO?0T.MÛcZR9xocZX9>@KVWKD>S=Pk;;=X9cZjU?

(172) c B VMVWKQ.T=RTMZRs_\ ?SUT=jJ?1PVjUXYsUW?9_?

(173) ¢o?

(174) Th;U?

(175) TX9 jUXMkX@¢3X9cOVXYs=?1KXYcZ?_TUVMZC? 1 iQXY>u9u 5b SUTUWVW 9?VTMZ;U?KN? SU?

(176) T.MOVXYP0X9KO?D1 A%X9S4otE5 ?P

(177) R9T=KOVWjU?

(178) cVWTMZ;UVK^_XY^5?

(179) cMO;U?@cZ?0X9Pk;_XYsUVWVMfIX9T=XYWIJKNVKRY]PRoT=PS=cOcZ?

(180) T.M^=cORox9ckXY>K VMO;XU J?0j«TLSU>s5?

(181) cR9]MZ;UcO?1X9jUK0"cZ?0P

(182) SUckKNVW¢9?ÛcZRLP

(183) ?0jJS=cO?1K

(184) KO;=XYcZ?0j«>@?

(185) >@R9cZIX9T=jVWTJ` MO?0cOW?0X3¢LVWTUxKN?0>XYT.MOVP

(186) K0%GLS_Pk;hX9TXYT=X9IJKOVWK;=XoK (i) MZR{>@RLjU?

(187) MZ;U?ÛcZRJP?0jUSUcO?(P0XYW XYT_j cZ?MOS=cOThKN?0>XYT.MOVP

(188) KVWTh?0XoPk;MZ;UcO?1X9j UX9T=j (ii) MOR@MZX99?VWToMZRX9P

(189) P

(190) R9SUT.MMO;=?>@RLjUV_P0XMZVRoT RY xoRos=XYJ¢XYcZVXYsUW?0K>XojJ?sLIMZ;U?QRYMZ;U?

(191) cdMO;UcZ?0XojUKdjJSUcZVWTUxMO;U?? J?0P

(192) SJMOVWR9TR9_MZ;U?QÛcZRJP?` jJSUcZ?RY_MO;U?MZ;UcO?1X9js_?0VTUxXoP MZV¢o?9" MVKÛcZ?0PVKO?

(193) WI(MO;UVKPRo>sUVWT=XYMOVWR9T;=VWPk;@VKjJGV F{P

(194) SUM MORMZXoPkL>? B VWTMZ;U?P

(195) X9KO?;=?

(196) cZ?MZ;U?\ R9MO;U?0c%MO;UcZ?0XojUKjUR.?1K%TURYM>@RLjUVI{xoRos=XY5¢XYcZVWX9sUW?0K0 PX9KZKOVWP0XY5VT.MO?0cO^=cORJP?1jJSUckXY=MO?0Pk;=TUV SU?1KXYÛÛWI HLVTMO;U?(P

(197) XoKN?;U?

(198) cZ?DTURMZ;UcO?1X9j^_?0cNRocO> ÛcZRJP?0jUSUcO?P

(199) X9K0.RoTU?(P

(200) X9TcZ?0jJS_P?MO;U?P

(201) R9T=P

(202) SUcOcZ?

(203) T.M%ÛcORox9ckXY> MORX@KO? \ SU?

(204) T.MZVWX95RoTU?9JsLI PjJRoRo>@TU?ÛSJVThMOVW>@TUxRJjJMZ;U?

(205) ?` PkVW;=T.?0MO?0PkcOLWV?0T=X3x=¢o ?0jÛcZRLjUS=P MR9MO;U?PR9T.MZcORo` I=R` x9ckXYÛ;=KRY]X9"MO;UcZ?0XojUK0_X9K i?1P?

(206) T.M]%R9cZJKR9WWR¦¢X9cOVWR9S=KXYÛ^=cOR.X9Pk;U?1K

(207) &g¬=ckKfM%XYÛÛcZRoXoPk;@VWK]MO;UcZ?0XojL` >RJjJS=WX9c XYT_XYWILKOVK

(208) 0VT(;UVPk;R9TU?PR9T_KNVjJ?

(209) cX%MZ;UcZ?0X9jDVWT.MO?

(210) ckX9PMOVWTUxVMO;(VMZK?0TL¢.VWcZR9TU>@?

(211) T.M1J 1 KMLDu9w 5 ÛcZR9^5RoKO?0KX(>@?MO;=RLj{MZX9.VWTUxVT.MORX9P

(212) P

(213) R9SUT.MMO;U?ÛcORJP

(214) ?0jJSUcZ?P0XYWWKRY"XMO;UcZ?0Xoj@;UVW?VWTJ` ?

(215) cZcZVTUxVWT^=X9cZX9W?

(216) ?0T.¢LVWcORoTU>@?

(217) T.MX9KZKNSU>@^JMZVRoT=KMO;=XYMXYs=KNMOckX9PMZKMO;U?^5RoKZKOVsUW?KfMZ?

(218) ^=K RYQR9MO;U?0cMZ;UcO?1X9jUK0«:;=?X9jU¢3X9T.MZXYxo?¨©X9T=jMZ;U?>@RYMZV¢XYMOVWR9T_ªRYMO;=VWKXYÛ^=cOR.X9Pk;VK(MOR WV>@VMMZ;U?PRo>sUVWT=XMZR9cZVWX9?

(219) JÛR.KNVWR9T¥MfI.^=VWP0XYRYP

(220) R9T=P

(221) SUcZcO?0T=PIVMO;XYT§VWT.MO?0cOW?0X3¢LVWTUx KO?

(222) >XYT.MOVP

(223) K0D 7QR%?

(224) ¢9?0cDMO;U?@R9sUMZXYVWTU?1j«X9T=XYWIJKNVKDVWKTUR9M¢9?0cOIXoP

(225) PS=cZXYMO?s_?1P

(226) XYS_KN?@RYMO;U? XYs_KfMZcZXoP MOVWR9TRY MO;U??

(227) TL¢LVWcORoTU>@?

(228) T.MMZ;UcZ?0X9j=K

(229) gQT=RYMO;=?

(230) cQXY^=ÛcOR.X9Pk;{VK%MOR@s5??1KOKxo?

(231) TU?0cZX9 R9T§MO;U?hPX9KZKRYP

(232) R9T=KOVWjU?

(233) cZ?0jÛcZR9xocZX9N> BO1PLDiQiu A$5jJ?

(234) =TU?0K@XTURYMZVRoT§RYMOckXYT=KZX9PMOVWR9T XYT_j@MOckXYT=KZX9PMOVWR9T=X9=ÛcZRJP?1jJSUcZ?0K0YR9c%;UVPk;MO;=?

(235) I{KOS=P

(236) P

(237) ?

(238) ?0jMORKOSU>@>XYcZV0?ÛcZRJP?1jJSUcZ?0K0 gQT=RYMO;=?

(239) ccZ?0P?0T.MOWI(? JÛWR9cZ?0jX9ÛÛcZRoXoPk;PRoT=KNVKNMZK&VTRJPS=KOVWTUxR9TUWIR9T? J?0P

(240) SJMOVWR9T_KVMZ; XDs_RoSUT=jJ?1jT.S=>s5?

(241) cRY PRoT.MO? LMdKOVMkPk;U?0QK 1"LDiQ>u 9.> RJS: 3(iQu9$v 5 :;UVWKcZ?0KNMOcZVP MOVWR9T@XYWWRQK MOcZR?0KOcZSU?

(242) S=MkK

(243) KO_?sUMOSJckX9MDjJRYVMO]VWR9PT=RoSUX9JckKNV?(T.MZ>@?

(244) cZRJÛjJcOSURJWP

(245) R?0jJMOSU;=ck?XYPLRoXYT=T_KOXYVWWjJIL?0KOcOV?1K jh>@KN?

(246) SUMOs_;UKNRJ?

(247) jMXYRYT=j? JMOR?1PR9SJsUMZMZVRoXYT=VWT@K BPR9MO>@;=?ÛVWWTJ?MZ?

(248) ?

(249) TUcZcO?1?1KOj K VWT.¢X9cOVXYT.MZKX9cO?TUR9MKNRoSUT=jR9cQX9T.I? J?0P

(250) SJMOVWR9TTKOR9>@?(X3I9.MO;UVKQX9ÛÛcZRoX9Pk;VWKQcO?0>` VWTUVWKZP?0T.MRYKOIL>s5R9WVWP?

(251) J?0PSUMOVWR9TX9KVM@XYWRQK(MORjJVKZPR¢9?0csUSUx.KsUSUMVMP

(252) X9TUTUR9Ms_? S=KO?0jR9c^=cOR¢LVWTUxXÛcZR9^5?

(253) cOMfI9 ?ÛcORo^_R.KN?;U?0cO?X9T@X9ÛÛcZRoXoPk;MO;_XM]XYT_XYWI.0?0KXYWJMZ;UcO?1X9jUKVWT@^=XYckXYWW?

(254) UXYT_jMZcZXoPkJK ?£?0PMOVW¢9?0IÛcZRJP?1jJSUcZ?P

(255) X9KXYT=jcO?

(256) MOSUcZT=KVThXPR9T_PSUcZcO?0T.MPRoToMZ? LM0L?0¢9?

(257) TVTMO;=?DP0X9KO? RY SUTLs5R9SUT_jJ?0jcZ?0PS=cZKOVRoT"T^_XYcOMOVPSUXYcVM]VK]XYs=?MZRx9?0TU?

(258) ckXMZ?Roc]XÛcZRJP?0jUSUcO? P VWT XMO;UcZ?0Xoj X^=cORJP?1jJSUcZ?DKOSU>@>XYcZI MO;_XMMZX99?1KVWT.MORX9P0PRoSUT.MMZ;U?^_R.KOKOVs=?>@R¢9?1K ^5?

(259) cOR9cZ>?1jt sLIMZ;U?(RYMZ;U?

(260) cMZ;UcZ?0X9j=KjJPSUcZVTUxMZ;U?(? J?0P

(261) SJMOVWR9TRY P : ?0Pk;UTUVP

(262) X9WI9RoSUc>@?

(263) MO;URJjcZ?

(264) WVW?0KR9T«XÛcZ?

(265) ¢LVRoS=KRocOT 1 ?LGJ-u A5MO;=XYMDcZ?RocO>SUWXYMO?1K PX9KZKOVWP0XYVWToMZ?

(266) cZÛcZRLP

(267) ?0jJS=cZX9dXYT=X9\ IJKOVWK(XoKXYTXYs_KfMZcZXoP MVWToMZ?

(268) cZÛcZ?MZXYMOVWR9UT 1 AApop$X 5RYMO;U? R9^5?

(269) ckXMZVRoT=XYKN?0>@X9T.MOVP

(270) KR9¨ KN? SU?

(271) T.MOVXY ª@ÛcZR9xocZX9>K

(272) WV%XoKNVP

(273) X9WI9sLIX§KNSUVMZX9sUW?XYsJ` KNMOckX9P MZVRoThRYMO;=?ÛcZR9x9ckXY> KP

(274) X9` KNMZXoPkJK

(275) URoTU??0VW>@VT=XYMO?1KMO;=?KOR9SUckP?(R9VWTJ=T=VMfIjJSU? MOR@KNMZX9PkJK0:;UVWKX9s=KfMZcZXoP MZVRoT{? JÛWR9VMZKMO;=?D? \ S=XYWVMfIs5?Mf%?

(276) ?0TXoP MOS_XY XYT=j{R9cZ>XY^=X` ckXY>@?MZ?

(277) ckK VWT{P

(278) X9`KfMkX9PkJK"VWT@R9ckjJ?

(279) cMZR(>XMZPk;@X9P0PSUckXMZ?

(280) WI(ÛcZRLP

(281) ?0jJS=cO?P

(282) X9KXYT_jcZ?MZSUcZT=K

(283) :;U?0TYV5KOR9>@?%j=XMZX¢XYcZVWX9sUW?0K&XYcZ?VWTJ=TUVMO?o9XjUXMkXX9s=KNMOckX9P MZVRoTP

(284) X9Ts5?XYÛÛWVW?0j¨ KNS_Pk; X9KP

(285) R9TL¢9?

(286) {^_RoIL;U?1jJckXRocTLSU>@?

(287) cZVWP0XY¢XYcZVWX9sU?1KZª]MZRW?0Xoj{MZR{XYT?£?0PMOVW¢9?XYT_XYWILKOVK

(288) /. 0. t. XYX[Z\^]%]_E`.

(289) A. % . TMO;UVK"^_XY^5?

(290) c%?dx9?0TU?

(291) ckXYWVW

(292) ?MO;=VWK KfMkX9PkX9s=KNMOckX9P MZVRoT(XYÛÛcZRoXoPk;MZRPRoT=PS=cOcZ?

(293) T.MÛcZRY` x9ckXY>K

(294) LVWT;=VWPk;?0X9Pk;MO;UcZ?0Xoj;_X9K%VMRTP

(295) XYW` KNMZXoPk5:;U?XYs=KNMOckX9PMOVWR9T?jJ?0KZPcZVWs_? XYs=KNMOckX9PMZKKN?0^=XYckXMZ?

(296) WIDMO;U?KNMZX9PkMkXYVWLRY_?1X9Pk;MO;UcZ?0Xoj sUSJMdVM^=XYVWcZK"MO;=?

(297) VWcdKfMkX9PkMORo^=K0VT R9ckjJ?

(298) c]MORcZ?

(299) XMZ?QMZ;U?WRJP

(300) X9=?


(302) T.MZK]RYMZ;U?jJV£?

(303) cZ?

(304) T.MMZ;UcZ?0X9j=K

(305) :;UVKX9s=KfMZcZXoP MZVRoT cO?1KNSUMZKVWTXYTVWToMZ?

(306) cZÛcZRLP

(307) ?0jJS=cZX9X9T=XYWIJKNVKD>@?MO;=\ RLj«RocPR9T_PSUcZcO?0T.MDÛcZR9xocZX9>KMZ;=XMVK X9Kdxo?

(308) TU?0cZX9=X9KPRocOcZ?0KO^_RoT=jJVWTUx(>@?MZ;URJjUKdRocKO? SU?0ToMZVWX9=ÛcZR9xocZX9>K HYVM%jU?0XYKdRoTUWIVMO; MO;U?(PRoT.MOcZR9^=XYcOMRY"ÛcZR9xocZX9>K

(309) LKNRMO;_XMXYWXYW_MZ;U?LTURTX9s=KfMZcZXoP MZVRoTR9cjUXYMZXP

(310) XYT s_?cO?0S=KN?1j CR9cZ?

(311) R¢o?

(312) c13MO;UVK]>@?

(313) MO;URJj{P0XYTXYKNR?1X9jMORXs=X9PkLXYckjX9T=XYWIJKNVK0MO;=XYMVWTJ` ?

(314) ckKdTU?1P?0KZKZXYcZIPRoT=jJVMOVWR9T=KRoc]cZ?0XoPk;UVT=xKOR9>@?¨MfI.^=VWP0XYWI?

(315) cZcORoTU?

(316) RoS=Kkª P

(317) R9TJ_x9SUckXMZVRoT=K

(318) XYT=j;UVWPk;VWK%¢o?

(319) cZIS_KN?

(320) SU VTPRo>sUVWT=XMZVRoT{VMO;R9cZ%X9cZjX9T=XYWIJKNVK B¨V ª?

(321) VMO;U?0c%MORV>` ^MOU;=cORXY¢oM]?Ds_?1R9PcZRo%>X9?1cZKjKNXYS_T=P

(322) PX9?1IJKOKNKOVWSUEK BdUKNRo;=>@?

(323) T@?MOVVWT.>@MZ?0?

(324) KQckKNX?1P MORo?1cOjXYckVMOj;{X9XDT=XYs=WXoIJPkKNLVK%©X9XYcZVWjWKXYMOT=RX9ÛIJcZKORVW¢9K0?(Ys5X@?0P

(325) ÛX9cZS=R9^5KO??

(326) cOMZMf;UI9? XYÛÛcZR3 JV>XYMOVWR9T=K^_?0cNRocO>@?0js.IhMZ;U?MfRXYT=X9IJKOVWKXYcZ?jJV£5?0cO?0T.EM Hd¨VWV ªRocMZR^5?

(327) cOR9cZ> X OKO?

(328) >XYT.MOVP KNWVWP

(329) VT=xR9&XÛcZR9x9ckXY>hoMO;=XYMQVWKOR9XMZ?0KMO;=?D?

(330) J?0PSUMOVWR9T=K%MZ;=XMQXYcZ?D^_R.KOKOVs=? PR9S=ToMZ?

(331) cO`b?

(332) UXY>@ÛW?0KMORXYThVWTL¢3X9cOVXYT_P?DÛcORo^_?0cNMfIo ² }

(333) ~ © z

(334) } ² | BSUcQP

(335) R9T.MOcZVWsUSJMOVWR9ThP

(336) X9Ts5?KOSU>@>@X9cOVW

(337) ?1jXoK%R9WRQEK B %?jJ?=T=?XYT¦VT.MZ?

(338) cZÛcORJP

(339) ?0jJSUckXYX9T=XYWIJKNVK@>@?MO;=RLjRoc{PR9T_PSUcZcO?0T.M@^=cORox9ckXY>K

(340) ;UVPk;?1X9jMOR@s5RYMO;RocOXYckjX9T=js_X9PkL%X9cZjXYT=X9IJKOVWEK H %?{ÛcZR¢9?VMZKKNRoSUT=jJTU?1KOK( c1 M0hMO;=?PRoT=PcZ?MZ?R9^5?

(341) ckXMZVRoT=XY]KO?

(342) >\ X9ToMZVWP0KR9%MZ;U? P

(343) R9T=KOVWjU?

(344) cZ?0j{^=cORox9ckXY>K%s.IS=KNVWTUxXYs_KfMZcZXoP MVT.MZ?

(345) cZÛcO?

(346) MZXYMOVWR9T{MZ?0Pk;UT=V SU?1K H %?KN;=R¡;UR¡MOR¥PRo>sUVWTU?VMVMO;¬XjUXMkXXYs=KNMOckX9PMOVWR9T¦XYT=j¦%?KfMZS=jJI¥MZ;U? P

(347) R9>@ÛW? JVMfIRY"MZ;U?(cZ?0KOSUMZVT=x@X9T=XYWIJKNVEK H %?dVW>^=?0>?0T.MO?0jR9SUc"MO?0Pk;=TUV \ SU?dR9c"^=cORox9ckXY>KVMZ;(=TUVMO?

(348) ` MfIL^5?X9T=j(TLSU>@?

(349) cZVWP0XY ¢XYcZVXYsUW?0K0XYT=j?? J^_?0cOVW>@?

(350) T.MO?1jKO?

(351) ¢o?

(352) ckXYoP

(353) WXoKOKOVP

(354) XY.KNILT=Pk;=cORoTUVWKZXMZVRoT(ÛcZRYMORJP

(355) R9K

(356) MZ;=XMX9WRQKMZRVWWS=KfMZcZXYMO?MZ;U?QÛcZ?0P

(357) VWKOVWR9T@RY R9S=cMO?1Pk;UTUV \ S=?9.sUSJM%X9WKORMORXYT=X9IL

(358) ? MZ;U?XYÛ^=cOR3 JVW>@XYMOVWR9T=KVMQ^_?0cNRocO>K0 } ' ® GL?0PMOVWR9TtjJ?

(359) =TU?1KQMZ;U?@>RJjJ?0RYMO;=?^=cORox9ckXY>K%?P

(360) R9T=KOVWjU?

(361) cX9T=jxoV¢o?0K MO;U?0VcKfMkXYT=jUX9cZjKO?

(362) >XYT.MZVWP0K

(363) T§GL?0PMOVWR9Tw%?{jJ?=T=?MfRVWT=KNMOcZSU\>@?

(364) T.MO?1jKN?0>XYT.MOVP

(365) K ¨cZ?0KO^QjJ?0jUVWP0XMO?1jMORRocOXYckjhXYT=jhs_X9PkL%X9cZjXYT_XYWILKOVKZª=;UVPk;? SUVW^hMO;U?KfMkXYT=j=XYckj KN?0>@X9T.MOVP

(366) KVMZ;§X9jUjUVMZVRoT=XYVTJRocO>XYMOVWR9TMZ;=XMVWs5?? JÛWR9VMO?0j¥VTMZ;U?KNMZXoPkX9sJ` KfMZcZXoP MOVWR9T"lGJ?0P MZVRoT A¥>@RYMZV¢XMZ?0KXYT=j¬jJ?

(367) =TU?1K{R9S=c % #-U#% >

(368) _ jJ?0KZPcZVs5?0KMZ;U?VWT=jJS_P?0jR9cZXYckjXYT=js=X9PkLXYckjXYs=KNMOckX9PMDKO?

(369) >XYT.MOVP

(370) KMORox9?

(371) MO;U?0cVMO; ÛPR9cZcZXocZP ?0MZP VWP0MZVXYRoLTV>@ÛÛcZR.W?

(372) R9>@©K

(373) ?05ToXYMkT=XjMZVRojJTVWKZX9PT=S_jKOKO%?0K?R9XY^JT=MZX9V>IL

(374) XY?%WVMfVMZIKcZP?0RoKO>@SUÛMZK0?

(375) JVMfIo? jJ VKO?P

(376) S=?

(377) ¢9KZK?0T.VWT«MOS=GLX9?1WP IMOVWjJR9?0TT KZP9cZVWVs_MZ?K VT«GJ?0P MZVRoTn@MO;U?? J^_?0cOVW>@?

(378) T.MZKQMO;=XYM%?^_?0cNRocO>@?1jVMO;RoSUcVW>@ÛW?

(379) >@?

(380) T.MZXYMOVWR9T5XYT=j ?{X9T=XYWIL

(381) ?ÛcZXoP MZVWP0XYWIMZ;U?{KNMOcZ?

(382) TUx9MO;XYT=j«?1XYLTU?1KOKO?0KDRYR9S=cXYT=X9IJKOVWKDVWT«MZ?

(383) cZ> R9 ÛcO?1PVKNVWR9T"GJ?0P MZVRoT=KpXYT_jvcZ?0KO^jJVWKZPS_KOKO?0K%MZ;U?(cO?0WXYMO?1j%R9cZ{XYT=jPRoT=PWS=jJ?1K

(384) '() )* ! & ( ! Q)* %! ?PRoT=KOVWjJ?0cX«KOV>@ÛW?PRoT=PSUcZcZ?

(385) T.MVW>@^_?0cZXYMOVW¢9?ÛcORox9ckXY>@>@VT=xWX9TUx9S_XYx9?VMZ;§MZ;U? R9WRVWTUx?0XYMOSUcZ?0EK B¨©Vª{XÛcZR9x9ckXY> VKP

(386) R9>@^5RoKO?0j¦RYX .R9MZ;UcO?1X9jUK0 VT.MO?0cZXoP MZVTUxsLIMZ;U?D>?1XYTRY"KO;=XYcZ?0jxoRos=XY_¢XYcZVWX9sU?1K

(387) LXYT_j{XKO?MRY"T=R9TJ` TU?0KNMO?1jÛcZRY` P?0jUSUcO?1KVMO;¦X¢XYWSU?^=X9cZX9>@?MO?0c^=XoKOKOVT=x^_RoVPI¬¨©XoKVWT !¤Ro#c "%$.ª ¦¨VWVª?0X9Pk; &('X)&+*.

(388)

(389) "!$#% &#' )(* % + ,-.'# T P

(390)

(391)

(392) POt

(393) "! #

(394) T t GVar , g $&%#' %)( * + *'% FP i , fpi , -. *%/!0 *1 *-2! 3 Pi FR i , fri , -. *%/ ! !0 41 -2! 5 3 Pi LVar i , li 67 *%)( * + '%+*8

(395) Pi 79:+%

(396) +; fp

(397) frj < i si , ei =>!??@A

(398) BCD!0E#! 0 Pi G = hK, Ii $&%#' %7F GH; 1 . 9 h(n > 0 )?i. s. hca. ll. r:. =. h(n = 0)?i c1. c2 f( x) i. hx := n−1i c3 hr := f (x)i. ∗ ni hr := r. hr := 1i e. c4. f (x)i hret r :=. : X9sUW?t BP xhR9cMO;U? K=X9PMORocOVXY ¨©KOVT=x9W?`bMO;UcZ?0Xoj=ªÛcORox9ckXY>. &: XYsUW?@q-BdGLILT.MZXoP MOVP(jJRo>@X9VT_K

(399) ÛcZRJP?0jUSUcO?;=X9KDVMZKDRT«KO?MRYRJP

(400) X9 ¢X9cOVXYsUW?0K05R9cZ>XY&VTU^=SJMXYT=jR9SJMZÛSJM(^=XYckXY>@?

(401) ` MO?0cZK0 ?QcZ?

(402) WIRoT{KN;_XYcZ?0jx9WR9s=X9J¢XYcZVWX9sU?1KR9cPR9>@>SUTUVP

(403) XMZVRoTXYT=jKOILT=Pk;UcZR9TUVKOXYMOVWR9T s5?Mf%?

(404) ?

(405) TMO;=cO?1X9jUK0J K&Vx.K

(406) Uw3`nx9VW¢9?? UX9>^=?1KRY RoSUcÛcZR9xocZX9> >@RJjJ?0 :;U?Q>X9VTcZ?0KNMOcZVWPMOVWR9T=K]XYcZ?MO;LS=KMZ;U?X9s=KO?

(407) T=P

(408) ?RY ? UP?0^JMOVWR9T_KdR9c]TUR9TJ` WRLP0XYefS=>^_K

(409) ¢XYcZVWX9sUW?XYWVX9KOVTUxR9TMO;U?(KNMZX9Pk¨ X9K%VM;=X9Û^_?0T=KVMO;cZ??0cO?0T=P?^=XYckXY>@?

(410) MO?

(411) c%^=XoKOKOVT=x.ª ^5R9VWToMZ?

(412) ckK%MOR@ÛcZRJP?0jUSUcO?1KXYT=jÛcZRJP?0jUSUcZX9^=XYckXY>@?MZ?

(413) ckK

(414) IKJ1L. MON/P0Q0NSR3T. U&VWYXER[Z. :;U?@KNILT.MZXoP MZVWPjJR9>XYVWT=K?S=KO?X9cO?KNS=>>X9cOVKN?1jVWT«:&XYsUW?qo jJ?

(415) T=RYMO?¢9?1P MZR9ckK RY¢XYcZVWX9sUW?0K0.;=?

(416) cZ?0X9K GVar , LVar jJ?

(417) T=RYMO?(KO?MkKRY&¢3X9cOVXYs=?1K

(418) g l g ,-. VWK(jJ?

(419) =TU?1j«sLIXhKO?M(RY%x9WR9s=X9¢3X9cOVXYs=?1K

(420) XhKO?M(RY%ÛcORJP

(421) ?0jJSUcZ?0K0"X9T=j XÛcZKORJ?PMD?0jURYSUcOP

(422) ?oR9T=jJP

(423) SU?

(424) TUcZcOR9?0MOTo?1Mj¥MOX9;UK cZ?0XojU K¨;]KN?0X9?Pk;TK&VWxoK0 5w3`n9 ª g ]\ - T VWKDjJ?_TU?0jVWKs.jJIh?=VMZT=K?0>j¥XYsLVWTI VMZK¢9?1P MORocR9¨ R9cZ>XY ª(VWTUPÛSJM@^=X9cZX9>?

(425) MO?0cZK fp ]PR9SJ=MZÛSJhfpM@^=,XYfrckXY>@, l?,MZG?

(426) ckK i fr ]X9T=jWRJP

(427) XY ¢XYcZVWX9sUW?0K l ¨MO;=XYMVWT=PWS=jJ?1KR9cZ>XY ^=X9cZX9>?

(428) MO?0cZKkª UX9T=js.IVMkKVWToMZcZX9ÛcZRLP

(429) ?0jJS=cZX9A K[^ ¨©P

(430) R9T.MOcZR9 I_R¤x9ckXYÛ;5ª G :;UD? _a`b RYX^=cORJP?1jJSUcZ? P VKQXx9ckXYÛ; G = hK, Ii ;U?0cO? K VKMZ;U?%KO?M&RY #]R9 P YPR9T.MkXYVWTUVT=xSUTUV \ SU?]?0T.MOcZIX9T=j(?

(431) LVMPR9T.MZcORo ^_RoVT.MZK s X9T=j e H I : K × K → Inst XYs5?

(432) K]?1jJx9?1KdR9MO;U?Dx9ckXY^=;VMZ;{Mf%RLVT=j=KR9VT_KfMZcOS=PMOVWR9T=EK B VT.MOckXY^=cORJP?1jJSUckXYVWT=KfMZcOS_P MOVWR9T_K hRi X9T=jÛcORJP

(433) ?0jJSUcZ?P

(434) X9K hy := P (x)i ;U?0cO? X9T=j y X9cO?DMO;=?(¢9?0PMORocZK%RYX9PMOS=X9 VTU^=SJMXYT_jRoSJMOÛSUMQ^=XYckXY>@?MZ?

(435) ckK

(436) x ToMZcZX9ÛcZRLP

(437) ?0jJS=cZX9VWT=KfMZcOS_P MOVWR9T_KX9cO?KO^_?1PV=?0j X9KXcO?0WXYMOVWR9T R ⊆ (GEnv × XYWWRVTUxhMZR?

(438) L^=cO?1KOKs_R9MO;Xx9S=X9cZjRoT¥x9WR9s_XYX9T=jWRJP

(439) XY%¢XYcZVWX9sUW?0K0 LEnv ) XYT=jXMOckXYT=KNR9cZ>XMOVWR9TRY MO;UR.KN?(¢XYcZVXYsUW?0K0 ?cO? \ S=VcZ?MO;=? G MZRs5?jJ?

(440) MO?

(441) cZ>@VT=VWKNMOVPRocÛcZRJP?1jJSUcZ?P

(442) X9K

(443) Vb ?oV I(c, c ) VWK@XP

(444) XYW%MO;U?0T§MZ;U?

(445) cZ?h? JVKfMkKT=R c KOS=Pk;¥MZ;=XM I(c, c ) Roc I(c , c ) VKX«P0XYW :;U?hSUT=PMOVWR9T=K call X9T=j ret cO?1PRocZj>XMZPk;=VTUx¥P0XYWQXYT=j¦cZ?MZSUcOTU` KOVMZ?TURJjJ?0EK B XYT=j ret(c ) = c call(c) = c :;U? ,- Y_>`Kb RY&MO;U?^=cORox9ckXY> VKQPRoT=KNMOcZS=P MZ?0jhX9KMZ;U?SUT=VRoTRYVT.MOckXYÛcZRY` P?1jJSUckXYA Kc^ G K0SUGcOMO;U?0c>@RJjJV=?0j§sLIcZ?

(446) ^=WXoPVWTUx?0jUx9?0KWX9s_?0W?0j¥s.IÛcZRJP?1jJSUcZ? P

(447) X9KsLI«X d d #%%?0jUx9?¨©P

(448) R9TUTU?1P MZVTUxhMZ;U?{P

(449) X9`KNVMO?MORhMZ;U??

(450) T.MZcOI^5R9VWT.MRY%MO;U? P

(451) X9W?

(452) ?3ªXYT=j¦XYT > ed fd % ?0jUx9?¨©PRoTUTU?1P MOVWTUxMO;=?? JVM^5R9VWT.M{RYMZ;U?P

(453) X9W?

(454) ?MOR t. t 0. i. i. i. i. i. i. i. i. i. j. 2. 0. 00. 0. i. XYX[Z\^]%]_E`. 0. 00. 00. 0.

(455) n. % v σ r = hc, i Γ hσ, Γi hσ, Γ1 , Γ2 i. ∈ ∈ ∈ ∈ ∈ ∈ ∈ ∈. >( %3*aC ??

(456) ( * + '%+ Value > ;#%#' %)(fD #-2! GEnv = GVar → Value > % %7f(f -2! [

(457) Pi LEnv i =S LVar i → Value > %+ %)(fD #-2! [ *Y @A

(458) LEnv = i LEnv i c ! ( *! #B 1

(459) 9: !1 *

(460) *1

(461) ?-. ! < Act = K × LEnv > !1 9:? < a *! (# 4! + B *1

(462) Act + > !1 *! 8 !

(463) BB?#%+ 4! + t + S = GEnv × Act !1 4! + + a %%/ ; 1 S = GEnv × Act × Act. &: XYs=?(w^B]GJ?

(464) >XYT.MOVPjJR9>X9VT=K MO;U?¥cZ?MZSUcZTJ` KOVMO?1ª(KN?0? K VWx=qY :;LS=KhMZ;U?

(465) cZ?¥XYcZ?MO;UcZ?

(466) ?LVWT=jUKR9VWT=KNMOcZS=P MZVRoT=KhX` s_?0WVT=x?0jJxo?0KRYx9WR9s_XY A KcÊK BVWToMZcZX9ÛcZRLP

(467) ?0jJS=cZX95VWT=KNMOcZS=P MZVRoT=K hRi _ÛcZRLP

(468) ?0jJS=cO?P

(469) X9K hcall y := jJP?

(470) TU(x)i TURVWT=PRoy>@:=VTUxP?0jJ(x)i xo?0K?XY UT=Pj?0^JÛM%cZP0RJXYPW?1jJ`bMOSUR9cZ` ?{KNMZcOX9?

(471) cNMOMSU?0cZjJT=xoK ?0hret K0 proc(c) R9MO?1Kd MZ;];U?ToMZ¨©cOVIT_jJTU?RJ {jJR9?1 KMO;=;UXo?3Kª ÛcORJP

(472) ?0jJSUcZ?DMO;=XYMPRoToMkXYVWT=K c j. IKJ I. j. aNSR[X

(473) PW R . U cT. R3WYX

(474) . KUR9cMZ;U?KOX99?RYKNVW>@ÛVPVMfI9"cORo>FTUR. Γ2 z }| { { c2 2 n2 n2. oR T«%? X9KZKNSU>@?X¥ÛcORox9ckXY> VMO; R9TUWI§Mf%RMO;=cO?1X9jUK0 :;U?KN?0>@X9T.MOVP«jJRo>XYVWT=KhXYcZ?KOSU>@>XYcZVWKO?0j¬VWT : X9sU?wU g KNMZXYMO? s = (σ, Γ , Γ ) VKjJ?=T=?0j sLI K&Z VxoSUcZZ ?{q-BZ adcZR9xocZX9>` KNMZXYMO?VTKNMZX9TJ` Xx9WR9s=X9?

(475) TL¢LVcZR9T=>?0T.M X9T=j¦MO;=?KfMkX9PkJK RY -

(476) $#RYσMZ;U?tMZ;UcZ?0X9j=K

(477) KOΓ?

(478) ? jUX9cZjKN?0>@X9T.MOVP

(479) K K VWx=q9gT¥X9P MZV¢XYMOVWR9TcZ?0PRocZjVKXh^=X9VcRYQX PR9T.MZcORo^5R9VWToM c XYT=j¬X9T¤WRLP0XY?

(480) TL¢LVcZR9TU>@?0ToM hc , i VWK{MO;U?PSUcZcO?0T.MR9cMOR9^ X9P MZV¢XYMOVWR9TcO?1PR9ckj{R9 MZ;U?MO;=cO?1X9j T ;dTL¢.VWcZR9TU>@?

(481) T.MkK>XY^h¢XYcZVWX9sU?1KMOR¢X9SU?1K

(482) :;U?0IP0XYThs_?PR9T_P

(483) XMZ?

(484) T=XYMO?1jVMO;hMZ;U? 9 R ^5?

(485) ckXMORoc0&XYT=j¥SU^5j=XMO?1jVMO;MZ;U?T=RYMZXYMOVWR9T 7→ v] v, v X9cO?{¢o?0PMOR9ckK(R9 ⊕ ¢3X9cOVXYs=?1K

(486) (v) jJ?

(487) T=RYMO?1KDMO;U?PR9cZcZ?0KO^_RoT=jJVWTUx¢o?0P σ[x MZR9c(R9%¢XYWSU?1K

(488) "X9T=j jJ?0TURYMZ?0K MO;U?: KOSUX9ssL¢9+ ?1AP MO=RockcKfMRY jJv?

(489) =MZTU;=?1XK{M¨jUVWT¥R.?1G=KBTUG.RY`MKfMfP

(490) ILR9WT.?1MZªX9MOVT;U?XYTLKNI{?0>@¢X9XYT.cZVMOXYVP

(491) sUKW?RYVT R9vT=? MZ;UcZ?0X9vj\VWTv VWKOR9XMZVRoT VWT=jJS_P?0jsLIMZ;U?SU ¨ MOckXYT_KNVMOVWR9T{cZ?

(492) XMZVRoT ª:;U?MZcZX9T=KOVMZVRoTcO?0WXYMOVWR9T ÛcORox9ckXY> VWKdMO;U?0TjJ?_→TU?0j{XoKdXKO^_?1PVXY5X9KOI.T_Pk;UcORoTUR9S_→KÛcO⊆RJjJSS_×P MSR9MO;U?MfRMZcZX9T=KNVMOVWR9T cO?0WXYMOVWR9T=K XYT=j dVWT¥;UVPk;¥MO;=?xoRos=XY?

(493) TL¢LVcZR9TU>@?0ToMVWKKO;=XYcZ?0j ?jJ?

(494) =TU? MO;U?VWTUVMOVXY&→KN?

(495) MRYKfMk→XMZ?0KQXoK S = {hσ, hs , i, hs , ii | init(σ)} ;U?0cO? s XYT=j s jJ?

(496) TUR9MO?DMO;=?KNMZX9cNM^5R9VWT.MR9"MZ;U?>@X9VTÛcZRJP?0jUSUcO?DRY ?0X9Pk;MO;UcZ?0Xoj UXYT_j init X9TVWTUVMZVWX9 PR9T_jJVMZVRoThR9Tx9WR9s=X95¢X9cOVXYsUW?0K0 z c1n1 c10. σ. Γ1 }|. . 1n1 10. !"$#%& ( *),+- /. ( 0$1 - ( 2. 1. . c20. 20. $"!#'& 31 - ( 254. 2. t. 76. t nt. t. t nt. 0. 0. 0. t. 1. 2. 0. IKJ38. 9. P: ; <)X

(497) 1W Q. U >T. 1 0. 1. 2 0. 2. 1 0. 2 0. R3WYX

(498) )J. :;U?@R9cZ%X9cZj«PRoW?0PMOVWTUxKN?0>@X9T.MOVP

(499) KDVT_jJS=P?1js.IXMOckXYT=KOVMZVRoTKOIJKfMZ?

(500) > →) Pk;=XYcO` X9P MZ?

(501) cZV0?0KMO;U?KN?

(502) MRYQcZ?0XoPk;=XYsUW?{KNMZXMZ?0KRYXÛcZR9xocZX9>h"cZR9> XKO?MR9=(S,

(503) #% # B X ⊆S ¨Nq1ª reach(X ) = {s | ∃s ∈ X , s → s} 0. >= - . def. Z ?'@ ( . 0. def. E A E+ =. S. i>0. 0. 0. 0. ∗. Ei . &('X)&+*.

(504) p.

(505) "!$#% &#' )(* % + ,-.'# I t (c, c0 ) = hRi R(σ, , σ 0 , 0 ) ˙ ¸ ˙ ¸ σ, Γ·hc, i →t σ 0 , Γ·hc0 , 0 i. I t (c, sj ) = hcall y := Pj (x)i + Ry:=P (x) (σ, , j ) ˙ ¸ j˙ ¸ t σ, Γ·hc, i → σ, Γ·hc, i·hsj , j i. 9 %% <. hσ, Γ1 i →1 hσ 0 , Γ01 i hσ, Γ1 , Γ2 i → hσ 0 , Γ01 , Γ2 i. 9 !?1 <. I t (ej , c) = hret y := Pj (x)i − Ry:=P (σ, , j , σ 0 , 0 ) j (x) ˙ ¸ ˙ ¸ σ, Γ·hcall(c), i·hej , j i →t σ 0 , Γ·hc, 0 i. 9[ <. 9Y5! <. hσ, Γ2 i →2 hσ 0 , Γ02 i hσ, Γ1 , Γ2 i → hσ 0 , Γ1 , Γ02 i. 9[ < 9 <. def. + Ry:=P (σ, , j ) = j (fpj ) = (σ ⊕ )(x) j (x)  0 σ = σ[y 7→ j (frj ) | y ∈ GVar ] − 0 0 def Ry:=Pj (x) (σ, , j , σ , ) = 0 = [y 7→ j (frj ) | y ∈ LVar ]. 9

(506) <. :&XYsUW? ABdG.MZX9T=jUX9cZjB^_?0cZXYMOVWR9T=X9GL?

(507) >XYT.MZVWP0KB&MOckXYT_KNVMOVWR9TcO?0WXYMOVWR9T → R9"MZ;U?MO;UcZ?0Xoj X9T=jMOckXYT=KOVMZVRoTcZ?

(508) XMZVRoT → RY_MO;U?S=U^=cORox9ckXY>h":;U?cZ?

(509) XMOVWR9T_K RoT?0TL¢.VWcZR9TU>@?

(510) T.MkK T X9T=j R jJ?

(511) =TU?(^=X9cZX9>?

(512) MO?0c%^_X9KZKNVWTUx>?1Pk;=XYT=VWKO>K

(513) R M{VK@MO;U?«KfMkXYT=jUX9cZj¦KO?

(514) >XYT.MOVP

(515) KRoc{VWTJ?

(516) cZcOVWTUxVWT.¢X9cOVXYT.MZK0 KURoc %?jJ?

(517) =TU? MO;=?@P

(518) R9T=P

(519) cO?

(520) MO?^_R.KfMkPR9T_jJVMZVRoTRo^_?0cZXYMOR9c post(X) = {s | ∃s ∈ XX : ⊆s →S s } ;UVPk; %?VWW]X9PMOS=X9WIjJ?1PR9>@^5RoKO?@VT.MORhRo^_?0cZXYMOR9ckKDX9KZKNRJPVXMZ?0jMORMO;=?@MOckXYT=KOVMZVRoT=KDRY%MO;U? VWToMZ?

(521) cZÛcZRLP

(522) ?0jJS=cZX9 I_R x9ckXY^=; G B t. t. +. −. def. [. post(X) =. 0. 0. I(c,c0 ). post(c −−−−→ c0 )(X). (c,c0 )∈K×K. VWK?1X9KOVWIjJ?0jUS=P?1jcZR9> MZ;U?{KO?

(523) >XYT.MOVPcOS=?1KR9%:&XYs+AU KUR9c L?DjJ?=T=?MZ;U?RocOXYckjMOckXYT_Kf?0c]SUT_P MOVWR9T F [X ](X) = X ∪ post(X) VWK >@R9TUR9MOR9T=?XYT_jPRoToMZVTLSURoS=K03XoP

(524) P

(525) R9ckjJVT=xMOR23?0?

(526) TU? K"MO;U?0R9cZ?

(527) >8%?;=X3¢o?. I(c,c0 ). post(c −−−−→ c0 )(X) X0 , X ⊆ S F [X0 ]. GLVWT=P?. def. 0. reach(X0 ) = lfp(F [X0 ]) =. [. 0. (F [X0 ])n (∅). V%IjJS=XYWVMfI9.MO;U?s=XoPkL%X9cZjP

(528) R9W?1P MOVWTUxKN?0>@X9T.MOVP

(529) K%Pk;=XYckX9PMO?

(530) cZVW

(531) ?0KdMO;U?KO?MRY KNMZXYMO?0K PRocO?1X9Pk;=X9sUW?cORo> ¨Vb ?o(?1X9jJVWTUx{MZR.ªXKO?MDRY< #%%# MVWKMO;=?T=XYMOSUckXYPk;URoVWP

(532) ? n≥0. R9cVTJ?0cOcZVWTUxR9cPk;U?0PkLVWTUxXTU?0P

(533) ?0KZKOX9cOI«PR9T_jJVMZVRoTR9TXÛcZR9xocZX9> KNMZXMZ?@MORcZ?0XoPk;X =T=X9 UMfI.^=VWP0XYWI?0cOcZR9TU?0R9S=KP

(534) R9TJ=xoSUckXMOVWR9T" ?(x9?

(535) M%MZ;U?R9WWRVTUxjJ?

(536) =TUVMOVWR9T=KEB VMZ; 0 ! ")*

(537) '* ! & ( ! Q)* %! ?(VT.MOcZRJjJS=P

(538) ?DT=R X>@RLjUV=?1jhKN?0>@X9T.MOVP

(539) K0JVTh;UVPk; qox9WR9s=X95¢X9cOVXYsUW?0K%VW"s_?(^=XoKOKO?0jR9cOMO;hXYT_js_X9PkR9TÛcZRJP?0jUSUcO?(P0XYWWKEH. pre(X) = {s | ∃s0 ∈ X : s → s0 } coreach(X0 ) = lfp(G[X0 ]) G[X0 ](X) = X0 ∪ pre(X). XYX[Z\^]%]_E`. .

(540) v. % . Jt ÛcZRJP?1jJSUcZ?0KVW9?0?

(541) ^XcZR9

(542) ?0TP

(543) R9^LIRYR9cZ>@X9^_XYckXY>@?MZ?

(544) ckK¨©VT=P

(545) S_jJVT=x«MZ;U? MZ;URoKO?;UR9jJVWTUxMO;U?(¢XYWSU?(R9 x9WR9s_XY¢XYcZVWX9sU?1KZª :;UVWK]MO?1Pk;UTUV \ SU?QVK]PX9KZKOVWP0XYUVWT{cZ?

(546) XMOVWR9T_XYUVWT.MO?

(547) cZÛcZRJP?0jUSUcZX9UXYT=X9IJKOVWK0YRoc]cZ?

(548) XMZVTUxMZ;U? ^_R.KOKOVs=?KNMZXYMO?1KRYQXhÛcZRJP?1jJSUcZ?{XMVMZKKfMkXYcOM^_RoVT.MVMO;§VMkK^5RoKZKNVWsUW?KfMkXMZ?0KXYMVMZK ? JVM^5R9VWT.M0d:;U?R9T=I{jJV£?

(549) cZ?

(550) T=P

(551) ?DVWTMO;U?(PRoT=PS=cOcZ?

(552) T.M%P0X9KO?VK%MO;_XM?D;=X3¢o?MZR?0T=KNS=cO? MO;=XYM]MO;=?QMO;=cO?1X9jUKXYx9cZ?

(553) ?1KR9T{MO;U?DPSUcZcO?0T.M]¢XYWSU?QRYxoRos=XY=¢3X9cOVXYs=?1K

(554) .X9K?0X9Pk;@R9 MO;U?0> VW;=X3¢o?VMZKRThPRo^LI{RY MO;U?0> ?{VWW%XoP MOS_XYWIjJ?=T=?{Mf%RVT=KNMOcZSU>@?

(555) T.MZ?0jKO?

(556) >XYT.MOVP

(557) K0R9TU?{jU?0jJVP

(558) XYMO?0jMZRRocN` %X9cZjXYT=X9IJKOVWKX9T=jMZ;U?RYMZ;U?

(559) c(RoTU?jJ?0jUVWP0XMO?1j«MORhs=X9PkLXYckjX9T=XYWIJKNVK

(560) @:;U?xoRoXYR9 XYT§VWT=KNMOcZSU>@?

(561) T.MO?1jKO?

(562) >X9ToMZVWP0KVKMfILÛVP

(563) XYWWI«MZRMOckXYT=KNR9cZ> ÛcORo^_?0cNMZV?1KRoT§?

(564) L?1PSJMZVRoT=K VXYT.s=MOKNRMOckÛX9cOPRoM^_KN?0?0cN>MZV?1XYKT.MOR9VT¥P

(565) K0¨©VT_KfT¥MZcORoSUSU>@c?0P0ToX9MZ?0KO?9j=ª%KfMk?{XMZ?0VWK0UMOR9;_sUXMZMXYP0VWT¥XYTKNMOs5cZ?R9TUWxXYMO?0^=ccORo?^_ J?0ÛcNWMZR9VV?1MOK?1jR9T§VT^_X{R.KO>KOVRos=cO?? P

(566) XYW` KNMZXoPkLK0"T@^=X9cNMZVWP

(567) SUXYc1YVMdVWW=ÛcZR¢LVWjJ?XTU?1P?1KOKZXYcZIPRoT=jJVMOVWR9T=KRocdXYTX9PMOVW¢XMOVWR9T cO?1PR9ckjMZRWV?%s5?

(568) WR¥XYTUR9MO;U?0c&XoP MOVW¢XMZVRoTcZ?0PRocZjVWT@P

(569) XYW` KNMZXoPkLK00MO;=XYM&VWLs5?? JÛWR9VMO?0j s.IMO;U?KNMZXoPk{XYs=KNMOckX9PMOVWR9TMO;=XYMQVWs5?(VT.MOcZRJjJS=P

(570) ?0jVTGL?1P MOVWR9T A=. . 8KJ1L. P0N. RN. . 1W%XEN. T. . >WYX . >T. R3WYX ;. TlMO;U?§R9cZ%X9cZj VWT=KNMOcZSU>@?

(571) T.MO?1jlKN?0>@X9T.MOVP

(572) KX9T X9PMOVW¢3XYMOVWR9TlcZ?0P

(573) R9ckj VK«RY@MO;U?§R9cZ> _;U?

(574) cZ? hc, ς, σ, i ς ∈ GFPEnv = GVar ∪ FP → Value 9?0?

(575) ^=KMOckX9Pk8RYMZ;U?¦¢3X9S=?0KRY xoRos=XY%¢XYcZVWX9sU?1KX9T=jRocO>X9^=X9cZX9>@?MO?0cZKXM@MO;=?KNMZX9cNM@^_RoVT.M@RYQMZ;U?hÛcZRLP

(576) ?` jJS=cO? P H XYT=j XYcZ?MZ;U?(x9WR9s=X9 XYT=jWRLP0XY?


(578) T.MQXM^_RoVT.M σ ∈ GEnv GLS=Pk;X9TX9PMOVW¢3XYMOVWR9TcZ?0P

(579) R9∈ckj@LEnv VWTUJKMZ;U?¢X9SU?1KdR9 R9cZ>@X9=^=X9cZX9>@?MO?0cZK]XYT=jxoRos=XY_¢XYcZcV` XYsUW?0KXMDKNMZX9cNM^_RoVT.MQMZR{MO;=?PS=cOcZ?

(580) T.MQ¢XYWSU?0KRYxoRos=XY&XYT=jhWRJP

(581) XY ¢XYcZVWX9sU?1KQXMD^_RoVT.M c ?TUR ;=X3¢o? Act = K × (GFPEnv × GEnv × LEnv ) XYT=j S = Act × ¨VMO;MO;=?P

(582) R9T=KNMOckXYVWT.MMZ;=XMMZ;U?DMOR9^hX9PMOVW¢XMOVWR9TcZ?0PRocZj=KX9x9cZ?

(583) ?RoTMZ;U?¢XYWSU?0KR9 Act x?

(584) 9TLWR9¢Ls=VcZX9R9TU¢>@XYcZ?0VWToX9MksUK?1XoKZKOªKO]RJ: PVRXMZ?0VWx9j@;.MZMOR?0TP

(585) R9TUT.RYMOMkcZXR9MZ5VRo^_T=RoK

(586) VT.=MkK

(587) ?(LVVWMO;&jJMO;U?

(588) T=?(RYPMORo?TLKO¢9VW?

(589) >T.^=MZVRoI T@MO∈;=XYEnv M g MOXY;=T=?j TUfp?

(590) jJ?

(591) TUR9MO?1KMO;=?PRo^.IRYx9WR9s=X9¢XYcZVWX9sUW?0KXYT_jRocO>X9^=XYckXY>@?

(592) MO?

(593) ckK;URoWjJVWTUx@MZ;U?

(594) VWc¢XYWSU? XMKNMZXYcOM^_RoVT.MRY&MO;=?PSUcZcO?0T.M^=cORJP?1jJSUcZ?9 : X9s 9jU?=TU?1K%MO;U?KO?

(595) >X9ToMZVWPcZSU?1K

(596) iS=?1K(¨bAR9T_PYEq Kª%XYT_j«¨ AR9T=P1t KªMZXYo?0KP

(597) X9cO? RYÛcZR9^=X9xoXYMOVWTUxMO;U?DSU^jUXMZ?R9x9WR9s=X9=¢XYcZVWX9sUW?0KdVWT=jJS_P?0j{s.I@RoTU?MO;=cO?1X9j@MZRMO;=?PRoTJ` PSUcZcO?0T.M MZ;UcO?1X9j &gKX9cZ?0XojJI^=cO?1PVKN?1j 3MO;U?%MZR9^@X9P MZV¢XYMOVWR9TcZ?0P

(598) R9ckjUK"R9UMO;=?P

(599) R9T=P

(600) SUcZcO?0ToM KfMkX9PkJKX9X3IJKQXYxocO?0?R9TMZ;U?P

(601) SUcZcO?0ToMD¢XYWSU?R9dx9WR9s=X9"¢XYcZVWX9sUW?0K0 7QR%?

(602) ¢o?

(603) cQVMDVKTURYM TU?0P

(604) ?0KZKOX9cOVWIMZ;U?QP

(605) XoKN?%RocMZXYVWUX9PMOVW¢XMOVWR9TcZ?0PRocZj=K

(606) 3MZ;=XMd>@?

(607) >@RocOVW

(608) ?MO;U?¢X9SU?R95xoRos=XY ¢3X9cOVXYs=?1KXMQP

(609) X9` KOVMO?0K0:;U?TU?

(610) 8KO?MQRY&VT=VMZVWX9"KfMkXMZ?0KVKjJ?=T=?0jhX9K proc(c). i. i. i. i. 0. 0. Si0 = { hs10 , ς, σ, 1 i, hs20 , ς, σ, 2 i | init(σ) ∧ ς = σ}. ~ ² ]® ~3

(611) } ® | ²

(612) |

(613) }1J} ® | } ®

(614) ² ~ ± J~³ |0}

(615) ~ ® } ® ³h| ® ¥ } ©o| TMZ;UVWK&KO?` >@X9T.MOVP

(616) K09cZ?0X9Pk;_XYsUW?P

(617) XYW` KNMZXoPkLKX9cO?T=?0P?1KOKZXYcZVWI<d"( %. VWT@MO;U?QR9WRVWTUx(KN?0T=KO?9 ® #% > ©

(618) } ² ® '' °

(619) ² ~ ® ³¥|

(620) }1U |{ &#³¬%|

(621) ?

(622) }1WJ` } ®Ro|cO>@ ?0Tj (! ( ' ! Γ = hc , i . . . hc , i ∈ Act i < n" 0. 0. n. n. + i. &('X)&+*.

(623)

(624) "!$#% &#' )(* % + ,-.'#. 4. ¨T.MZcZX>Kª. I t (c, c0 ) = hRi R(, ) ∧ (g0 , fp0 ) = 0 (g0 , fp0 ) Γ · hc, i →ti Γ · hc0 , 0 i 0. I t (c, sj ) = hcall y := Pj (x)i + Ry:=P (, j ) j (x). I t (ej , c) = hret y := Pj (x)i − Ry:=P (, j , 0 ) j (x). Γ · hc, i →ti Γ · hc, i · hsj , j i. Γ · hcall(c), i · hej , j i →ti Γ · hc, 0 i. Γ1 →1 Γ01 Γ01 = Γ001 · hc01 , 01 i 0 2 = 2 [g 7→ 01 (g)]. Γ1 , Γ2 · hc2 , 2 i →i Γ01 , Γ2 · hc2 , 02 i. Γ2 →2 Γ02 Γ02 = Γ002 · hc02 , 02 i 0 1 = 1 [g 7→ 02 (g)]. Γ1 · hc1 , 1 i, Γ2 →i Γ1 · hc1 , 01 i, Γ02. ¨ A%XYW Kª. ¨ i?MKª. ¨bAR9T_PYqEKª. ¨ AR9T=P1tKª. def. + Ry:=P (, j ) = j (g0 , fpj0 , g, fpj ) = (g, x, g, x) j (x) 0 (g0 , fp0 , l \ y) = (g0 , fp0 , l \ y) − 0 def Ry:=Pj (x) (, j , ) = 0 (g \ y, y) = j (g \ y, fr). :&XYsUW? 9^BQT=KNMOcZSU>@?

(625) T.MO?1jKN?0>XYT.MOVP

(626) KEBMOckXYT=KOVMOVWR9TcZ?

(627) XMZVRoT MOckXYT_KNVMOVWR9TcO?0WXYMOVWR9T → R9"MZ;U?SUWÛcZR9x9ckXY>h i. . &# +#% <( =:\ Q E ci I(ci , sj ) = hcall y := Pj (x)i. . . Pj. ! . :\. →ti. ¨ iª ¨ i`ª. R9dMO;U?MO;=cO?1X9j T X9T=j t. j = proc(ci+1 ). ". ". G ! < - 8( %.DY Y^. # \ # x) = i+1 (g0 &,# fpj0?0) ` RocO>@?1j ( #%i (g, ( 1 2 1 2 <d"( %. -7 6$

(628) hΓ ,Γ i$# ∈S,-i ' ] \^ % 6$Γ"D )(8,-Γ <6%

(629) # . . !. X9TPMOXVW¢XMO?0VWR9`bTR9cOcZ?1>PR9?1ckj(jKfMkMOXRMZ?9VWY?Ps5Ro?

(630) T=WjJR¤VMOVWR9XYT=T=KRYMO(i)−(ii) ;=?

(631) cX9P MZV¢X9XYs_MORVW¢oR9?]TXYcZcZ?0?]PRoXQcZT=j@?0VWPT?1KOcZKZ?0XYXocZPkI;=PXYR9s=T_?QjJVP0MZXYVRoWT(`KfMkR9X9cPkJXYK0T . <~ ² " ( ² %| . } C² #% ! ^! T

(632) ! J# #% N ^ &# < "( % &. # O<#% "( %. ( #% &# ² ³ &® |1| ²

(633) } ®

(634) ² ~ ± J~³ |0}

(635) ~ ® } ® ³ | ® ¥ } ©o| MKN;URoSUjs5?PW?0XYc MO;_XMMZ;UVKT=?

(636) KO?

(637) >XYT.MOVP

(638) KjJRL?0KTURYMD>@RLjUVIMO;=?s5?

(639) ;=X3¢LVWR9SUcR9cDÛcORox9ckXY>KEB]R9T=?P

(640) X9T d. s ∈ S0i \ ]\. ]\. s0 ∈ Si. +d. s →∗i s0. d. s ∈ Si. XYWX3ILKQ©XYWs=X9PkMZRMZ;U?KfMkXYT=jUX9cZjKN?0>@X9T.MOVP

(641) K0CR9cZ?RocO>X9WI9%?jJ?

(642) =TU?MO;=?ÛcZRY` ef?0PMOVWR9TSUT=PMOVWR9T π : S → S X9K i. π. . hc10 , ς01 , σ01 , 10 i . . . hc1n1 , ςn11 , σn1 1 , 1n1 i, hc20 , ς02 , σ02 , 20 i . . . hc2n2 , ςn22 , σn2 2 , 2n2 i. XYX[Z\^]%]_E`. . =. *. σn1 1 (= σn2 2 ), 1 1 hc0 , 0 i . . . hc1n1 , 1n1 i, hc20 , 20 i . . . hc2n2 , 2n2 i. +.

(643) q0u. % . T@MO;U?KO? \ SU?0 o?VW>@ÛWVWP

(644) VMZIcO?1KfMZcOVP M S MORVMZK]KNSUs_KN?

(645) M]RY?0`bR9cZ>?1j@KfMkXMO?1K

(646) ?QP

(647) XYT TUR¦R9cZ>SUXMZ?MO;U?PRocOcZ?0PMOVWR9T@RY MZ;U?VWT=KNMOcZSU>@?

(648) T.MO?1j{KO?

(649) >XYT.MOVP

(650) K]( c1 M0&MO;U?DKfMkXYT=j=XYckj KN?0>@X9T.MOVP

(651) K0 ~ ² ]²

(652) | ©