Lower bounds are not easier over the reals: inside PH

Texte intégral

(1)Lower bounds are not easier over the reals: inside PH Hervé Fournier, Pascal Koiran. To cite this version: Hervé Fournier, Pascal Koiran. Lower bounds are not easier over the reals: inside PH. [Research Report] LIP RR-1999-21, Laboratoire de l’informatique du parallélisme. 1999, 2+p. �hal-02102035�. HAL Id: hal-02102035 https://hal-lara.archives-ouvertes.fr/hal-02102035 Submitted on 17 Apr 2019. 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) Laboratoire de l’Informatique du Parallélisme ´ Ecole Normale Supérieure de Lyon Unité Mixte de Recherche CNRS-INRIA-ENS LYON no 5668. SPI.

(3) . .

(4) .

(5) . ´ Ecole Normale Supérieure de Lyon 46 Allée d’Italie, 69364 Lyon Cedex 07, France Téléphone : +33(0)4.72.72.80.37 Télécopieur : +33(0)4.72.72.80.80 Adresse e´ lectronique :

(6).

(7)

(8)

(9) . .

(10)

(11)

(12)

(13)

(14)

(15)

(16)

(17)

(18) ! " #$

(19)

(20)

(21)

(22)

(23)

(24)

(25)

(26) %

(27)

(28)

(29)

(30)

(31)

(32)

(33)

(34) &

(35) '

(36) &

(37)

(38)

(39)

(40)

(41)

(42) (

(43)

(44)

(45)

(46) ) *

(47) +,, . -

(48)

(49) . /

(50)

(51)

(52) 0

(53) /

(54) . 1 1 / 2 /

(55)

(56) !" #$ /

(57)

(58)

(59) /

(60) .

(61) /

(62)

(63) /)

(64) /

(65)

(66)

(67) &/

(68) 1

(69) " 3

(70) /

(71) /

(72)

(73)

(74)

(75) / (/ .

(76)

(77) /

(78) .

(79) *

(80) / )/ . +,,.

(81)

(82)

(83)

(84) ! "#$ #%#" ! & ' ()$ .

(85)

(86)

(87) . !

(88) " #$

(89) % & $'

(90) $' . 4

(91)

(92) *

(93)

(94)

(95) ) ) *

(96)

(97)

(98)

(99) !

(100) $. 2 )

(101)

(102) )

(103) )

(104)

(105) %

(106)

(107)

(108) '

(109)

(110)

(111)

(112)

(113)

(114) .

(115)

(116) )

(117) ) )

(118) 5 67

(119)

(120) 89

(121) !: " : #$ :

(122) 1

(123)

(124) 6 ;7

(125) 9

(126)

(127)

(128) ! " #$.

(129)

(130)

(131)

(132) )

(133) +,, 2

(134)

(135)

(136)

(137) )

(138)

(139)

(140) '

(141)

(142) *

(143)

(144) 2

(145) 6;7

(146)

(147) < '

(148) 5

(149)

(150) 8 ()

(151)

(152) " ,3= ( 5

(153)

(154) .

(155) . .

(156) >

(157) )

(158) () 8 < '

(159) ,3= 2

(160)

(161) )

(162)

(163) *

(164) -

(165)

(166) )

(167)

(168) " " (

(169)

(170)

(171) < '

(172)

(173) " '

(174)

(175) " ,3= ?

(176)

(177)

(178)

(179)

(180) .

(181) ()

(182)

(183)

(184) 5

(185) )

(186) @ 8 (

(187) 6;7

(188)

(189) "

(190)

(191) " ,3= 5

(192) !

(193) $8 ( *

(194)

(195)

(196) 6A 7 ')

(197)

(198)

(199)

(200) )

(201)

(202)

(203)

(204) *

(205) ' 6 B7 2

(206) ()

(207)

(208)

(209)

(210)

(211)

(212)

(213)

(214) .

(215) %

(216) C " " - ( ?1

(217)

(218)

(219)

(220)

(221) )

(222) 6D E7 ) ') (

(223)

(224)

(225)

(226) (

(227)

(228)

(229) *

(230)

(231)

(232)

(233) ; 5

(234)

(235)

(236) *

(237)

(238)

(239)

(240)

(241) () , < '

(242) ()

(243) 8 ? ( ; '

(244) C

(245)

(246)

(247) (

(248)

(249)

(250)

(251) F

(252)

(253)

(254)

(255) 2

(256)

(257)

(258) 5

(259)

(260)

(261)

(262)

(263)

(264)

(265)

(266) 8

(267)

(268)

(269)

(270) &

(271)

(272)

(273)

(274)

(275)

(276)

(277) " ) 6G7 2

(278)

(279)

(280) )

(281)

(282)

(283)

(284)

(285)

(286)

(287)

(288) H

(289)

(290)

(291) 5

(292)

(293) ?

(294)

(295) ? 8

(296)

(297)

(298)

(299)

(300)

(301)

(302) C

(303)

(304)

(305)

(306) A

(307)

(308)

(309)

(310)

(311)

(312) )

(313) *

(314)

(315) . . . . . . . . . . . . . . . . . . . . . .

(316).

(317)

(318) . I

(319)

(320) ) )) )

(321) J

(322) "

(323)

(324)

(325)

(326) I %

(327)

(328) 58 " K 58 " 5

(329) 8 5

(330) 8 I 58 " 5 58 588 (

(331) )

(332) 5 8 I

(333)

(334)

(335)

(336) 58 " 5 8 (

(337)

(338) Æ 9 K

(339) *

(340)

(341)

(342) J

(343)

(344)

(345) I

(346)

(347)

(348) ) Æ

(349) I*

(350)

(351) )

(352)

(353)

(354) . 5

(355) )

(356) " 58 8

(357)

(358) 5 8

(359)

(360)

(361) "

(362) 55 88 " KK "" @ " Æ 58

(363)

(364)

(365) ) > Æ

(366)

(367)

(368)

(369)

(370)

(371)

(372) ) 58 % 5 8 6 7 LI)

(373) ! ()

(374) $

(375)

(376)

(377)

(378)

(379)

(380) )

(381) (

(382)

(383) ()

(384)

(385)

(386) !

(387) K$ !

(388) $

(389)

(390)

(391)

(392) (

(393)

(394)

(395) K

(396)

(397)

(398)

(399)

(400)

(401)

(402)

(403)

(404) ()

(405)

(406) + )

(407)

(408)

(409) '

(410)

(411) 5

(412) 8 )

(413)

(414)

(415)

(416) + ) )

(417)

(418)

(419) 5 8

(420) ')

(421)

(422)

(423)

(424)

(425)

(426)

(427)

(428)

(429) @ K 5 8 (

(430)

(431)

(432)

(433) )

(434)

(435) 5)

(436)

(437)

(438) ) Æ

(439) 8 (

(440) I

(441) >

(442) Æ

(443)

(444)

(445) C 5 8 .

(446) . .

(447) . . . . .

(448) .

(449) . .

(450) .

(451) . .

(452) Æ

(453) . . ;. . .

(454) J

(455) 6 7

(456)

(457)

(458) )

(459) I

(460) + )

(461)

(462)

(463) . )

(464) )>) (

(465)

(466) ))

(467) 5 8 (

(468)

(469)

(470)

(471)

(472)

(473) ( ( )

(474)

(475)

(476) ? 6D E7

(477)

(478) )

(479) )

(480)

(481)

(482) (

(483)

(484)

(485) C

(486) )

(487) ) 6 7 ( >

(488)

(489)

(490)

(491) )

(492) >

(493)

(494) )

(495) )

(496)

(497)

(498) ; %

(499) )

(500)

(501)

(502)

(503)

(504) )

(505)

(506) A . . . . . .

(507)

(508)

(509) .

(510) ) " 6 7 5 8 6 7 %

(511)

(512)

(513)

(514)

(515) )

(516) 5 8 >

(517) Æ

(518)

(519)

(520) (

(521)

(522) M

(523)

(524)

(525) &5 8

(526) Æ 5 8 " @ 5 8

(527)

(528)

(529)

(530)

(531) ) !

(532) "

(533) # " " " &5 8 . " 5 8 J

(534) 5 8 (

(535)

(536) 58 5 8 5 8 . . .

(537) . .

(538) . . . . . . . .

(539) . . . . .

(540) . . .

(541)

(542) .

(543)

(544)

(545) . .

(546) . . . . . . . . . ! ½. A.

(547) I

(548) K

(549)

(550)

(551) " " " J

(552) 2 6E7

(553)

(554)

(555)

(556) ' " ?

(557) )

(558) ' 2 "

(559) 5 8

(560)

(561)

(562) " ,

(563)

(564) " 6 7

(565)

(566)

(567)

(568) 5

(569) ) )

(570)

(571) 8 +

(572)

(573) J

(574)

(575) I

(576)

(577) " " (

(578) C 58 " 58 -

(579) " I

(580) '

(581) )

(582)

(583) 58 F

(584)

(585)

(586)

(587)

(588) 2

(589)

(590)

(591)

(592)

(593) 58

(594)

(595)

(596)

(597)

(598) " 2

(599)

(600)

(601)

(602)

(603)

(604) ) ! ! Æ

(605)

(606) !

(607) ! " 5

(608)

(609)

(610) 8 ,

(611)

(612) "

(613) ! I*

(614) !

(615) ! 2

(616)

(617)

(618)

(619)

(620) !

(621)

(622) " ! (

(623) ) '

(624) -

(625)

(626) *

(627)

(628)

(629) 5 !8 " ! 5 58

(630) ! "

(631) 8 ,

(632) I*

(633) ! " ! % ! "

(634)

(635)

(636)

(637) Æ. 5) I*)

(638) K8 2 "

(639)

(640)

(641) -

(642)

(643)

(644) * %

(645)

(646)

(647)

(648)

(649)

(650) 5

(651) 8

(652)

(653)

(654) J

(655)

(656) Æ

(657) 5

(658)

(659)

(660) " 8 ( J .

(661)

(662) 5 8 C

(663)

(664)

(665) 58

(666) 5 8

(667) 5 8 6 7

(668)

(669)

(670)

(671)

(672)

(673)

(674) ,

(675)

(676) 5 8

(677)

(678)

(679)

(680) 5 8

(681)

(682) 5

(683)

(684)

(685)

(686)

(687)

(688) 8

(689) . . ¾. ½ . ½ . ½ . G. ¾. ¾.

(690) ,

(691) " 5 " 8.

(692)

(693)

(694) '

(695)

(696) " 6 7

(697)

(698)

(699) Æ " # " #

(700)

(701) F

(702)

(703) 5

(704) #

(705)

(706)

(707)

(708)

(709)

(710) 8

(711)

(712)

(713)

(714)

(715)

(716)

(717)

(718)

(719)

(720)

(721)

(722)

(723)

(724) *

(725) 2

(726)

(727)

(728)

(729) )

(730)

(731)

(732) &5 5&5 &5 &5

(733) 8 8 J

(734)

(735)

(736)

(737)

(738)

(739)

(740)

(741)

(742)

(743)

(744)

(745)

(746) . " '

(747)

(748)

(749)

(750)

(751)

(752)

(753)

(754) 2

(755) " F

(756)

(757) 58 -

(758)

(759)

(760)

(761) 58 2 " "

(762) "

(763) F

(764)

(765) 2 "

(766)

(767) #

(768) )

(769)

(770)

(771)

(772) ½. ½. . .

(773) Æ

(774)

(775)

(776)

(777) .

(778)

(779) )

(780)

(781)

(782) # "

(783) )

(784)

(785) '.

(786)

(787)

(788)

(789)

(790)

(791) 5 8

(792)

(793)

(794) Æ

(795)

(796)

(797) ( '

(798)

(799)

(800) )

(801) 5 8

(802) F

(803)

(804) '

(805)

(806)

(807) Æ

(808)

(809)

(810)

(811) > %

(812) Æ

(813) > ) Æ .

(814) $ $ $ :

(815) 5 $ $ 8 5 8

(816)

(817)

(818) 5$ $ 8 " K 2

(819) I

(820) 5 8 5 8

(821)

(822)

(823) (

(824)

(825)

(826) ?

(827) 5) 67 E8 ´½µ. . . . .

(828) . . . . . . . . . . . .

(829) . .

(830) Æ

(831) . . . . . . . . ¾. . . . . .

(832) Æ . . . . ! .

(833) . . .

(834) Æ

(835) . . . . . . . !

(836) . . . .

(837) Æ

(838) . ¾ . D. . . . .

(839) % 58

(840)

(841)

(842)

(843)

(844) I

(845)

(846)

(847)

(848)

(849)

(850)

(851)

(852) + J ; :

(853) 5 8

(854)

(855) %@ @ % @ % " K

(856) %1

(857) )

(858)

(859)

(860) % @ @ % @ % " K 2 58

(861) " 5 8

(862) )

(863) *

(864)

(865) ) " " 2

(866)

(867)

(868)

(869)

(870) 1

(871) )

(872) 2

(873) 58

(874)

(875)

(876)

(877)

(878)

(879) % @ @ % @ " K

(880) Æ

(881)

(882) M ) '

(883)

(884)

(885)

(886) ¾

(887) )

(888)

(889)

(890)

(891)

(892) Æ Æ

(893) 5 8

(894)

(895) C &5 8

(896) 5

(897)

(898)

(899)

(900) . @

(901) )

(902) 8 !

(903) $ (

(904)

(905)

(906)

(907)

(908)

(909) .

(910)

(911)

(912) C &5 8 ,

(913) " 2 K

(914)

(915) " K

(916)

(917)

(918)

(919)

(920)

(921)

(922) 31 C " 5 & &8 & 5 8 (

(923)

(924) ' C 58 (

(925)

(926)

(927) Æ

(928) " # " # 5

(929) # 8 58 (

(930)

(931)

(932)

(933)

(934) (

(935) " &5

(936) 8

(937)

(938)

(939)

(940)

(941)

(942) 6 7

(943) " # " # 5

(944) # 8 58 (

(945)

(946)

(947)

(948) 58

(949)

(950) 58 58

(951)

(952)

(953)

(954)

(955) Æ .

(956)

(957)

(958)

(959) )

(960)

(961) 5 8 .

(962) 3

(963) + 3

(964)

(965)

(966)

(967)

(968) ) Æ

(969)

(970)

(971) )

(972)

(973) Æ

(974) . ¾. ·½ . ½. ½. ½. ¾. ¾. ¾. ¾. E. ½. . .

(975)

(976)

(977)

(978) Ê . J

(979) N

(980) I

(981)

(982) ' " K '

(983) 1

(984) )

(985) 5 '8 " K N

(986)

(987) )

(988) " '

(989)

(990) N

(991) ' " K 5 8 )

(992) )>

(993)

(994) N

(995)

(996)

(997) ) " - >

(998) N

(999) )

(1000) I

(1001)

(1002)

(1003)

(1004) 6 7

(1005)

(1006) N 5K

(1007) 8 5 N " K

(1008)

(1009) 8 J

(1010)

(1011)

(1012) Æ

(1013) " 5KN8

(1014) ,

(1015) N

(1016)

(1017)

(1018) K N

(1019)

(1020)

(1021) " N

(1022) 9 N N

(1023)

(1024)

(1025) *

(1026)

(1027)

(1028) I

(1029)

(1030)

(1031) ) Æ

(1032)

(1033) M

(1034)

(1035) 5 8 )

(1036) J

(1037)

(1038)

(1039)

(1040) J

(1041) (

(1042) ) () 2

(1043)

(1044)

(1045)

(1046) *

(1047)

(1048)

(1049)

(1050) N '

(1051)

(1052)

(1053) ?

(1054) N

(1055)

(1056) (

(1057) (

(1058) C

(1059)

(1060)

(1061)

(1062) N.

(1063)

(1064)

(1065)

(1066)

(1067)

(1068)

(1069)

(1070)

(1071)

(1072)

(1073)

(1074) &5K 8 ( ) )

(1075) 5 8 (

(1076)

(1077)

(1078) . ·½. . .

(1079)

(1080)

(1081)

(1082)

(1083)

(1084) .

(1085) ( J

(1086) " 5 8

(1087) .

(1088)

(1089)

(1090)

(1091)

(1092)

(1093)

(1094)