Elementary Languages - Semantics

'5$)7
6HQGFRPPHQWVWR
IS[[#LILVSDQZDZSO
5\V]DUG:yMFLFNL
7UDQVODWHGIURP3ROLVKE\3DWULFN7URPSL]DQG.RQUDG7DOPRQW.DPLQVNL
$Q,QWURGXFWLRQWR/RJLF
:LWK(OHPHQWVRIWKH7KHRU\RI.QRZOHGJH
‹5\V]DUG:yMFLFNL
:DUV]DZD
&KDSWHU(OHPHQWDU\/DQJXDJHV±6HPDQWLFV3DUW
,QWKLVFKDSWHUZHFRQWLQXHWKHSUHVHQWDWLRQRIWKHSRVWXODWHVZKLFKGHILQHWKH PHDQLQJRIORJLFDO
FRQVWDQWV.H\FRQFHSWVWDXWRORJ\ORJLFDOO\WUXHVHQWHQFHODZRIVHQWHQWLDOFDOFXOXV
/RJLFDO&RQQHFWLYHV±1HJDWLRQ&RQMXQFWLRQDQG'LVMXQFWLRQ
:HGLVFXVVHGWKHPHDQLQJRIWKHQHJDWLRQFRQQHFWLYHÔLQ7KHGHILQL
WLRQZHVDZWKHUHQRZUHFHLYHVWKHIRUPRIWKHQH[WSRVWXODWH
Ô
8QGHU DQ DUELWUDU\ JLYHQ LQWHUSUHWDWLRQ Y D VHQWHQFH RI WKH IRUP Ôa LV
WUXHLIDQRQO\LIaLVIDOVHXQGHUWKLVLQWHUSUHWDWLRQ
%\WKHQHJDWLRQRIVHQWHQFHaZHVKDOOPHDQDQ\VHQWHQFHbZKRVHWUXWK
YDOXHLVGHILQHGE\WKHVDPHFRQGLWLRQDVÔa7KHVHQWHQFH$GDPLVQRWWKHIDWKHU
RI$JDWKDLVWKHQHJDWLRQRIWKHVHQWHQFH$GDPLVWKHIDWKHURI$JDWKDEHFDXVHWKLV
VHQWHQFHLVWUXHMXVWLQFDVHWKHVHQWHQFH$GDPLVWKHIDWKHURI$JDWKDLVIDOVH,I±
DVZHVKDOOEHV\VWHPDWLFDOO\GRLQJ±ZHUHVWULFWRXUDQDO\VHVWRXWWHUDQFHVZKLFK
DUHSXUHO\GHVFULSWLYHH[SODQDWRU\WKHQZHFDQVDIHO\DVVXPHWKDWÔPHDQVH[DFWO\
WKHVDPHDVWKHSKUDVHQRWWUXHWKDW6RQRWWUXHWKDWaLVQRWRQO\DZD\RIUHDGLQJ
Ôa LW LV DOVR DQ DOWHUQDWLYH ZD\ RI H[SUHVVLQJ WKH WKRXJKW H[SUHVVHG E\ Ôa DV
IROORZVIURPÔ
7KH PHDQLQJ RI WKH FRQMXQFWLRQ FRQQHFWLYH à LV GHILQHG LQ WKH IROORZLQJ
SRVWXODWH
à
8QGHU D JLYHQ DUELWUDU\ LQWHUSUHWDWLRQ Y D VHQWHQFH RI WKH IRUPaàb LV
WUXH LI DQG RQO\ LI ERWK FRQVWLWXHQW VHQWHQFHV a DQG b DUH WUXH XQGHU WKLV
LQWHUSUHWDWLRQ
7KLVSRVWXODWHLVDOVRXQFRQWURYHUVLDOWKRXJKLQWKLVFDVHWKHPHDQLQJRIàLVQRW
WKH VDPH DV WKH HYHU\GD\ ZRUG DQG 7KH ODWWHU LQ VRPH FRQWH[WV FDQ KDYH WKH
PHDQLQJ RI µUHVXOW¶ 7KH VHQWHQFH +H PHW 'RURWK\ DQG IHOO LQ ORYH PHDQV VRPH
WKLQJTXLWHGLIIHUHQWIURP+HIHOOLQORYHDQGPHW'RURWK\7KLVLVFOHDUO\WKHFDVH
EHFDXVHLQLQWHUSUHWLQJWKHWZRFRQMXQFWLRQVZHVXUPLVHFRUUHFWO\±EHFDXVHUXOHV
RIVSRNHQFRPPXQLFDWLRQGHPDQGWKLVPRUHRQWKLVLQWKDWWKHLUFRQMXQFWV
DUHFRQQHFWHGE\PRUHWKDQWKHIDFWWKDWWKH\DUHERWKWUXH
$VLQWKHFDVHRIQHJDWLRQWKHQDPHFRQMXQFWLRQRIVHQWHQFHVaDQGbZLOO
EHDSSOLHGQRWRQO\WRVHQWHQFHVRIWKHIRUPaàbEXWWRDQ\VHQWHQFHgZKRVH
WUXWKYDOXHLVJLYHQE\WKHVDPHFRQGLWLRQVDVWKRVHGHWHUPLQLQJWKHWUXWKYDOXHRI
aàb,QDFFRUGDQFHZLWKWKLVGHILQLWLRQFRQMXQFWLRQQHHGQRW FRQWDLQHLWKHUWKH
VLJQàRUWKHZRUGDQG$OORIWKHIROORZLQJDUHFRQMXQFWLRQV7RGD\LV7XHV
GD\WK'HFHPEHU-RKQLVWKHKXVEDQGRI$QQHZKHUHDVKHLVWKHEURWKHURI
$JDWKD 7KH FDSLWDO RI 3RODQG LV FXUUHQWO\ :DUVDZ EXW RQFH LW ZDV .UDNyZ
:HPRYHRQWRWKHGLVMXQFWLRQFRQQHFWLYH
£
8QGHU DJLYHQ DUELWUDU\ LQWHUSUHWDWLRQ Y D VHQWHQFH RI WKH IRUP a£b LV
IDOVH LI DQG RQO\ LI ERWK LWV FRQVWLWXHQW VHQWHQFHV a DQG b DUH IDOVH XQGHU
WKLVLQWHUSUHWDWLRQ
7KHSRVWXODWH£GRHVQRWVWDWHGLUHFWO\ZKHQDVHQWHQFHRIWKHIRUPa£b
LVWUXHEXWHYHU\LQWHUSUHWDWLRQYJLYHVWKHVHQWHQFHHLWKHUWKHYDOXHWUXWKRU
IDOVLW\6Ra£bLVXQGHUWKHJLYHQLQWHUSUHWDWLRQWUXHLILWLVQRWIDOVHXQGHUWKLV
LQWHUSUHWDWLRQ 6HQWHQFHV RI WKH IRUP a £ b DQG DOO RWKHU VHQWHQFHV ZKRVH WUXWK
FRQGLWLRQVDUHWKHVDPHDVVHQWHQFHVRIWKLVIRUPDUHFDOOHGGLVMXQFWLRQVRIaDQG
b
127( 6RIRUH[DPSOHZHPD\WDNHWKHIROORZLQJDVDFDVHRIGLVMXQFWLRQ7R
GD\LVDOUHDG\7XHVGD\ DQG SHUKDSV HYHQ :HGQHVGD\7KH VHPDQWLF FRQWHQW RI WKLV VHQ
WHQFH ZH RPLW WKH SUDJPDWLF LQIRUPDWLRQ FRQWDLQHG LQ LW UHGXFHV WR WKH µXVXDO¶ GLVMXQF
WLRQ7RGD\LV7XHVGD\RUWRGD\LV:HGQHVGD\7KHVHSDUDELOLW\RIWKLVGLVMXQFWLRQLVDFRQ
VHTXHQFHRIWKHPHDQLQJRIWKHZRUGV7XHVGD\DQG:HGQHVGD\DQGQRWIURPWKHPHDQLQJ
RIRU7KHVHQWHQFH2QHFDQJHWWRWKH*UDQG&HQWUDOVWDWLRQRQWUDPQRRUQHHGQRW
EH D GLVMXQFWLRQ 7KH LQWHUSUHWDWLRQ DV D GLVMXQFWLRQ LV FRUUHFW LQ D FRQWH[W WKDW LQGLFDWHV
WKDWWKHSHUVRQVSHDNLQJLV QRWVXUH ZKLFKWUDP JRHVWRWKHVWDWLRQ±EXW WKH\ DUH VXUH DW
OHDVW RQH RI WKHP GRHV 7KLV LQWHUSUHWDWLRQ LV QRW FRUUHFW LI WKH LQWHQWLRQ RI WKH XWWHUDQFH
ZDVWRLQIRUPWKHDXGLHQFHWKDWERWKDQGJRWRWKHVWDWLRQ,QWKLVLQWHUSUHWDWLRQLWLVD
FRQMXQFWLRQ
&RQQHFWLYHV,PSOLFDWLRQDQG(TXLYDOHQFH
7KHSRVWXODWHGHILQLQJWKHPHDQLQJRIWKHFRQQHFWLYHŽKDVWKHIRUP
Ž
8QGHUDJLYHQDUELWUDU\LQWHUSUHWDWLRQ Y DVHQWHQFHRIWKHIRUP aŽb LV
IDOVHLIDQGRQO\LIXQGHUWKLVLQWHUSUHWDWLRQaLVWUXHZKHUHDVbLVIDOVH
$ VWDWHPHQW WR WKH HIIHFW WKDW b SURYLGHG WKDW a WKDW VRPHWKLQJ WDNHV SODFH RU
ZRXOG WDNH SODFH SURYLGHG WKDW VRPHWKLQJ HOVH GRHV RU GLG RU DQ\ VHQWHQFH WKDW
H[SUHVVHVLWLVFDOOHGDFRQGLWLRQDOZLWKDQWHFHGHQWaDQGFRQVHTXHQWb7KXVLQ
SDUWLFXODUXQGHUWKHFRPPRQO\DFFHSWHGPHDQLQJRILIWKHQDFRQGLWLRQDOLVDQ\
VHQWHQFHRIWKHIRUPLIaWKHQbDVZHOODQ\VHQWHQFHRIWKHIRUPLIWKHUHZDVWKH
FDVHWKDWaWKHQLWZRXOGEHWKHFDVHWKDWb
6HQWHQFHVRIWKHIRUPaŽbWKXVDQ\VHQWHQFHVZKRVHWUXWKFRQGLWLRQVDUH
H[DFWO\ DV HQYLVDJHG E\ WKH SRVWXODWH Ž DUH FRQGLWLRQDOV DUH DOVR FRQGLWLRQDOV
EXWRIVSHFLILFNLQG±WKHLUPHDQLQJPD\GLIIHUIURPWKDWRIERWKLIaWKHQbDQGLI
WKHUHZDVWKHFDVHWKDWaWKHQLWZRXOGEHWKHFDVHWKDWb7KH\DUHFDOOHGWUDGLWLRQ
DOO\PDWHULDOLPSOLFDWLRQV
,Q WKLV FRQWH[W µPDWHULDO¶ DPRXQWV WR µWZRYDOXHG¶ ,I WKH VXEMHFW RI RXU
DQDO\VHVLVH[FOXVLYHO\WZRYDOXHGFRQQHFWLYHVWKHQLWRIFRXUVHEHFRPHVXQQHFHV
VDU\WRFRQVWDQWO\SUHIL[WKHDGMHFWLYHµWZRYDOXHG¶,WEHFRPHVGHVLUDEOHLIEHVLGHV
WZRYDOXHGPDWHULDOFRQQHFWLYHVRWKHUVDUHDOVRDQDO\]HG$VZHOODVLPSOLFDWLRQ
DOVRRWKHUFRQQHFWLYHVHJQHJDWLRQRUFRQMXQFWLRQQHHGQRWLQJHQHUDOEHµPDWH
ULDO
+RZHYHUWKHFDVHRIPDWHULDOLPSOLFDWLRQLVDVSHFLILFRQH7KLVLVQRWRQO\
EHFDXVH PDWHULDO LPSOLFDWLRQ ZDV WKH IUHTXHQW WRSLF RI DWWHPSWV WR VKRZ WKDW LQ
VSLWHRIZKDWRQHPLJKWH[SHFWWKHPHDQLQJRIŽVHWE\SRVWXODWHŽ GRHV QRW
FDSWXUHFRPPRQO\DFFHSWHGPHDQLQJRILI«WKHQVHHDOVRFKDSWHU,WLVDOVR
EHFDXVH PDWHULDO LPSOLFDWLRQ LV XVXDOO\ RSSRVHG WR IRUPDO LPSOLFDWLRQ WKH ODWWHU
EHLQJGHILQHGDVVHQWHQFHVRIWKHIRUP"[f[Žy[DQGDQ\VHQWHQFHVZKRVH
WUXWKFRQGLWLRQVDUHWKHVDPHLQYLUWXHRIWKHSRVWXODWHV"DQGŽ
7KH PRWLYDWLRQ EHKLQG WKH GLVWLQFWLRQ RI PDWHULDO DQG IRUPDO LPSOLFDWLRQ
ZDV WKH FRQYLFWLRQ WKDW ZKHQHYHU LI « WKHQ LV XVHG LQ JHQHUDO VWDWHPHQWV WKH
PHDQLQJ RI WKLV FRQQHFWLYH LV FORVH WR LI QRW LGHQWLFDO ZLWK WKH PHDQLQJ RI WKH
FRQQHFWLYHŽ,QGHHGYDULRXVVWDWHPHQWVRIWKHIRUPLIVRPHWKLQJKDVSURSHUW\f
WKHQLWDOVRKDVSURSHUW\yHJLIVRPHQXPEHULVGLYLVLEOHE\WKHQLW¶VGLYLVLEOH
E\ PLJKWEH DGHTXDWHO\ H[SUHVVHGE\XVLQJŽ LQVWHDG RI LIWKHQ 2Q WKH RWKHU
KDQGDVKDVEHHQGHPRQVWUDWHGZLWKPDQ\H[DPSOHVJHQHUDOO\UHSODFLQJŽZLWKLI
« WKHQ FDQ OHDG WR UDGLFDO FKDQJHV RI VHQWHQFH PHDQLQJ ,Q SDUWLFXODU WKH LQLWLDO
VHQWHQFHaŽbPD\EHWUXHEXWLIaWKHQbPD\WXUQRXWWREHHYLGHQWO\IDOVH7KH
DSSHDUDQFHRIVXFKFDVHVZDVSURPSWHGWKHQDPHSDUDGR[HVRIPDWHULDOLPSOLFD
WLRQ:HZLOOGHYRWHWKHODVWVHFWLRQVRIFKDSWHUWRWKHSUREOHPRIWKHSDUDGR[HV
RILPSOLFDWLRQ)RUQRZOHW¶VWDNHFDUHRIWKHPHDQLQJRIIRUPDOLPSOLFDWLRQVLWLV
QRWIXOO\GHILQHGE\Ž
7R VLPSOLI\ WKH GLVFXVVLRQ OHW¶V WDNH D FRQFUHWH VHQWHQFH RI WKH IRUP
"[f[Žy[DQGDFRUUHVSRQGLQJVHQWHQFHH[SUHVVHGLQRUGLQDU\ODQJXDJH)RU
H[DPSOHWKHVHQWHQFH
>@
"[OLJKWEORQGH[ŽEOXHH\HG[
DQGLWVHYHU\GD\FRXQWHUSDUW
,IVRPHRQHLVOLJKWEORQGWKHQWKH\KDYHEOXHH\HV
%\"IRUVHQWHQFH>@WREHWUXHDOOWKHSDUWLFXODUFDVHVRIWKLVVHQWHQFH
PXVWEHWUXH7KDWLVVHQWHQFHVRIWKHIRUP
>D@
OLJKWEORQGH[ŽEOXHH\HG[>[D@
ZKHUH D LV DQ\ RI WKH REMHFWV LQ WKH FDVH LQ KDQG ± SHUVRQV UHSUHVHQWHG E\ WKH
YDULDEOH[2QWKHRWKHUKDQGDVHQWHQFHRIWKHIRUP>D@LVIDOVHLQRQO\RQHFDVH
ZKHQLWLVWUXHWKDWOLJKWEORQGH[>[D@DQGDWWKHVDPHWLPHLWLVQRWWUXHWKDW
EOXHH\HG[>[D@6XPPDUL]LQJ
>@
6HQWHQFH>@LVWUXHLIDQGRQO\LIDPRQJWKHREMHFWVZKLFKYDULDEOH[UHSUH
VHQWV WKHUH H[LVWV QR REMHFW D VXFK WKDW LW LV WUXH WKDW OLJKW
EORQGH[>[D@ DQG DW WKH VDPH WLPH LW LV QRWWUXH WKDW EOXH
H\HG[>[D@
$UHWKHLPSOLFLWQRUPVZKLFKJXLGHXVLQVHWWOLQJWKHWUXWKYDOXHRIVHQWHQFH
WKHVDPHZKLFKGHILQHWKHWUXWKYDOXHRIVHQWHQFH>@"/HW¶VFRQVLGHU,W¶VTXLWH
REYLRXVWKDWZHWDNHVHQWHQFHWREHIDOVHLQRQO\RQHFDVHLHZKHQWKHUHH[LVWV
VRPHRQH±VRPHFRQFUHWHSHUVRQ11±VXFKWKDW11LVOLJKWEORQGHDQG11GRHV
QRW KDYH EOXH H\HV ,I WKHUH LV QR VXFK 11 WKHQ ZH WDNH WR EH WUXH ,Q RWKHU
ZRUGV
VHQWHQFH LV WUXH LI DQG RQO\ LI DPRQJ WKH REMHFWV ZKLFK WKH FRQWH[WXDO
ZRUG VRPHRQH UHSUHVHQWV WKHUH LV QR 11 VXFK WKDW LW LV WUXH WKDW 11 LV
OLJKWEORQGHDQGDWWKHVDPHWLPHLWLVQRWWUXHWKDW11LVEOXHH\HG
$VIRU >@ DQG LQ WKH FDVH RI >@ DQG WRR WKH GLIIHUHQFH EHWZHHQ WKHP LV
SXUHO\ VW\OLVWLF ± LW GHSHQGV VROHO\ RQ DQ DOWHUQDWLYH YHUEDOL]LQJ RI WKH VDPH
WKRXJKW6RWKHTXHVWLRQZHVHWRXUVHOYHVKDVDQDIILUPDWLYHDQVZHUWKHWUXWKFRQ
GLWLRQVRI>@DQGDUHWKHVDPH6RDUHZHWKXVMXVWLILHGLQVD\LQJWKDWWKHFRQ
QHFWLYHŽDSSHDULQJLQ>@DQGWKHFRQQHFWLYHLI«WRXVHGWRH[SUHVVKDYHWKH
VDPHPHDQLQJ±GRWKH\IXOILOWKHVDPHVHPDQWLFIXQFWLRQV"7KLVGRHVQ¶WVHHPWR
EHVR
,QWKLVUHJDUGRQO\FRQGLWLRQV>@DQGDUHLGHQWLFDOWRWKHOHYHORIVW\OL
]DWLRQ7KHIRUPXODWLRQRI>@KRZHYHUZDVSUHFHGHGE\UHDVRQLQJZKLFKFDQQRW
EH XVHG WR VXSSRUW WKH MXVWLILFDWLRQ RI FRQGLWLRQ 7KH WUXWK RI >@ DV ZH VDLG
REWDLQVRQO\ZKHQDOOVHQWHQFHVRIWKHIRUP>D@DUHWUXH6ROHW¶VFRQVLGHUZKHWKHU
WKH WUXWK FRQGLWLRQ RI LV WKH WUXWK RI DOO VHQWHQFHV ZKLFK DUH VW\OLVWLF FRXQWHU
SDUWVWR>D@LHVHQWHQFHVRIWKHIRUP
11 ,IVRPHRQHLVOLJKWEORQGHWKHQWKH\DUHEOXHH\HG>VRPHRQH11@
+RZZHDQVZHUWKLVTXHVWLRQGHSHQGVVROHO\RQRXUµOLQJXLVWLFLQWXLWLRQ¶±IRUWKH
TXHVWLRQFRQFHUQVLPSOLFLWQRUPVDQGQRWH[SOLFLWQRUPV<HWOHW¶VQRWHWKHIROORZ
LQJ ,I ZH DQVZHU RXU TXHVWLRQ LQ WKH DIILUPDWLYH WKHQ ZH PXVW FRQVLGHU WKH IRO
ORZLQJFRQGLWLRQVDVHTXLYDOHQW
” 7KHUHGRHVQRWH[LVWDQ\FRQFUHWHSHUVRQ11ZKRLVOLJKWEORQGHDQGGRHVQRW
KDYHEOXHH\HV
” (YHU\VHQWHQFHRIWKHIRUP11LVWUXH
7KHTXHVWLRQSUHVHQWVLWVHOIGRWKHLPSOLFLWQRUPVRIQDWXUDOODQJXDJHGHPDQGZH
WUHDWWKHVHWZRFRQGLWLRQVDVHTXLYDOHQW"5DWKHUQRW
/HW¶V LPDJLQH WKH IROORZLQJ VLWXDWLRQ /HW VRPH $GDP EH GDUNEURZQ
KDLUHG:HDVVXPHWKDWVRPHRQHZKRLVDZDUHRIWKLVIDFWVWDWHV,I$GDPLVOLJKW
EORQGHWKHQKHKDVEOXHH\HV:KDWLVWKHWUXWKYDOXHRIWKLVVHQWHQFH",VLWWUXHRU
IDOVH",IZHUHSODFHGLIWKHQZLWKWKHDUURZŽZHJLYHLI«WKHQWKHPHDQLQJRI
ŽWKHDQVZHUZRXOGEHFOHDU%\WKHSRVWXODWHŽZHZRXOGKDYHWRGHFODUHWKH
XWWHUDQFHWUXH%XWRXUTXHVWLRQFRQFHUQVWKHVHQWHQFH,I$GDPLVOLJKWEORQGHWKHQ
KH KDV EOXH H\HV LQ LWV FRPPRQO\ DFFHSWHG XQGHUVWDQGLQJ 7KH TXHVWLRQ RI WKH
WUXWKYDOXHRIWKLVVHQWHQFH±DQGQRWVRPHRURWKHULQWHUSUHWDWLRQ±LVWKHTXHVWLRQ
ZKHWKHU ZKDW LV FRPPXQLFDWHG E\ RQH XWWHULQJ WKLV VHQWHQFH REWDLQV RU QRW 2XU
GLIILFXOW\ UHYROYHV DURXQG WKH IDFW WKDW VLQFH ERWK WKH VSHDNHU DQG ZH NQRZ WKDW
$GDPLVQRWOLJKWEORQGHZHFHDVHWRXQGHUVWDQGZKDWWKHVHQWHQFHLVVXSSRVHGWR
FRPPXQLFDWH $QG LI ZH FHDVH WR XQGHUVWDQG ZKDW WKH VHQWHQFH LV VXSSRVHG WR
FRPPXQLFDWHWKHQZHORVHWKHFDSDFLW\WRDVFULEHLWDWUXWKYDOXH
(VWDEOLVKLQJ ZKDW D VSHDNHU XWWHUV LV VRPHWKLQJ GRQH RQ WKH EDVLV RI OLQ
JXLVWLF QRUPV DFFHSWHG E\ SDUWLFLSDQWV LQ WKH GLVFRXUVH ,Q VRPH FRQWH[WV WKHUH
PD\EHWKH JRYHUQLQJSULQFLSOHWKDWLI«WKHQLVWREHXQGHUVWRRGDVŽ ,QRWKHU
ZRUGVWKHSULQFLSOHWKDWVHQWHQFHVRIWKHIRUPLIaWKHQbDUHWREHWUHDWHGDVPDWH
ULDOLPSOLFDWLRQLVLQIRUFH7KLVSULQFLSOHLVIROORZHGIRULQVWDQFHLQPRVWIRUPDO
GLVFRXUVHV HJ LQ PDWKHPDWLFDO UHDVRQLQJ :H PLJKW LPDJLQH WKDW IURP VRPH
SRLQWV RI YLHZ ZH VKRXOG DSSO\ WKLV SULQFLSOH WR WKH XWWHUDQFH ,I $GDP LV OLJKW
EORQGH WKHQ KH KDV EOXHH\HV ZKLFK ZH DUH DQDO\]LQJ DW WKH PRPHQW ,I ZH GLG
DSSO\LWWKHQDVZHKDYHVHHQWKLVXWWHUDQFHZRXOGKDYHWREHFRQVLGHUHGWUXH%XW
WKLV QRW EH VR 7KH FRQWH[W PLJKW VDQFWLRQ JUDQWLQJ WKH XWWHUDQFH WKH IROORZLQJ
PHDQLQJ,I$GDPZHUHOLJKWEORQGHWKHQKHZRXOGKDYHEOXHH\HV:HQRWLFHWKDW
WKLVLQWHUSUHWDWLRQRILI«WKHQDQGWKXVRIWKHZKROHVHQWHQFHEXLOWLQWHUPVRIWKLV
FRQQHFWLYHQHHGQRWQHFHVVDULO\ JXDUDQWHHWKHWUXWKRI WKH VHQWHQFH )RU LW PLJKW
KDSSHQDIWHUDOOWKDWWKHUHLVDOLJKWEORQGHSHUVRQZLWKRXWEOXHH\HV
6RLWWXUQVRXWWKDWDOWKRXJKWKHµILQDO¶WUXWKFRQGLWLRQVDQG>@RIVHQ
WHQFHVDQG>@DUHLGHQWLFDOH[FHSWLQJYHUEDOGLIIHUHQFHVWKLVGRHVQRWPHDQWKDW
LI « WKHQ DSSHDULQJ LQ LV D PDWHULDO LPSOLFDWLRQ FRQQHFWLYH ± WKDW LW KDV WKH
PHDQLQJZKLFKWKHSRVWXODWHŽDVFULEHVWKHFRQQHFWLYHŽ
,WPD\EHHDV\WRYLHZORJLFDODQDO\VHVDVWKHSURYHUELDOµKDLUVSOLWWLQJ¶'R
ZH UHDOO\ KDYH WR JR VR GHHSO\ LQWR WKH VLPLODULWLHV DQG GLIIHUHQFHV EHWZHHQ WKH
FRQQHFWLYHV GLVFXVVHG" 'R WKH UHVXOW RI VXFK DQDO\VLV VHUYH DQ\WKLQJ XVHIXO" $V
DOZD\V ZKHQ D TXHVWLRQ FRQFHUQV WKH H[DFWLWXGH ZLWK ZKLFK VRPHWKLQJ LV WR EH
UHVHDUFKHGGLVFXVVHGPHDVXUHGHWFWKHDQVZHUGHSHQGVRQWKHFRQWH[WWKHTXHV
WLRQUHIHUVWR0HDVXULQJGLVWDQFHVEHWZHHQWKH6XQDQGWKH(DUWKLQPLOOLPHWHUVLV
SRLQWOHVV0HDVXULQJWKHOHQJWKVRIVKHOYHVZHZDQWWRSXWLQDERRNVKHOILVYHU\
GHVLUDEOH LI LW¶V D PLOOLPHWHU WRR VPDOO LW ZLOO IDOO $ PLOOLPHWHU WRR ELJ DQG LW
ZRQ¶WILWEHWZHHQWKHVKHOYHV,QIRUPDOFRQWH[WVWKHGLIIHUHQFHEHWZHHQLI«WKHQ
DQGŽLVµRQWKHZKROH¶XQLPSRUWDQW,WGRHVQRWIROORZDWDOOWKDWLWLVDOZD\VLQ
VLJQLILFDQWDOZD\VLQJHQHUDORUDOZD\VLQWKHVHFRQWH[WV$QGVLQFHWKHVHGLIIHU
HQFHVH[LVWDQGLQFHUWDLQFLUFXPVWDQFHVRXJKWQRWWREHLJQRUHGWKHORJLFLDQ¶VWDVN
LVWRUHYHDOWKHPDQGGUDZDWWHQWLRQWRWKHP
6LQFHZHKDYHDOUHDG\LQWURGXFHGWKHFRQFHSWRIIRUPDOLPSOLFDWLRQZHDGG
WZRPRUH FODULILFDWLRQV :H EHJLQ ZLWK WKH VLPSOHU RQH (YHU\GD\ ODQJXDJH GRHV
QRWFRQWDLQYDULDEOHVEXWLWGRHVFRQWDLQYHUEDOFRQVWUXFWLRQVUHYHDOHGE\YDULDEOHV
:KHQ ZH VWDWH WKDW LI WKH UHG OLJKW LV RQ WKHQ FURVVLQJ WKH URDG LV IRUELGGHQ RU
ZKHQZHVD\WKDWLI3LQRFFKLROLHVKLVQRVHZLOOJHWORQJHUZHGRQ¶WXVHDQ\OHWWHUV
[\RUDQ\RWKHUDVWKHFRXQWHUSDUWWRYDULDEOHV:HGRQ¶WHYHQXVHSURQRXQVOLNH
VRPHRQHRUVRPHWKLQJ±ZKLFKLQQDWXUDOODQJXDJHVHUYHDVWKHFRXQWHUSDUWWRYDUL
DEOHV<HWLW¶VFOHDUWKDWERWKWKHVHVHQWHQFHVUHIHUWRDQDUELWUDU\PRPHQWLQWKH
FDVHRIWKHOLJKWVDQDUELWUDU\SODFHVRWKH\FRQFHDOZLWKLQWKHPVHOYHVDYDULDEOH
UHIHUULQJ WR WLPH 1HLWKHU LV WKHUH DQ\ GRXEW WKDW ERWK WKHVH VHQWHQFHV KDYH WKH
FKDUDFWHURIJHQHUDODVVHUWLRQV$QGILQDOO\WKHUHLVQRFRQWURYHUV\DVWRWKHLUIRU
PDOL]HGYHUVLRQV"[f[Žy[6RWKH\DUHLPSOLFDWLRQVLQWKHIRUPDOVHQVH
$QG WKH VHFRQG FODULILFDWLRQ ,Q QDWXUDO ODQJXDJH FDVHV RI FRQGLWLRQDOV
ZKLFK GR QRW H[SUHVV DQ\ JHQHUDO FRQWHQW GRQ¶W µWUDQVODWH¶ LQWR D IRUPXOD OLNH LI
f[WKHQy[DUHUDWKHUSHFXOLDULWLHV$VDQH[DPSOHZHFRQVLGHUHGWKHVHQWHQFH
,I $GDP LV OLJKWEORQGH WKHQ KH KDV EOXH H\HV 6LQFH ± DV ZH DVVXPHG ± ZH DUH
VSHDNLQJDERXWDVSHFLILF$GDPZKRLVIXUWKHUPRUHGDUNEURZQKDLUHGWKHQWKLV
VHQWHQFHWDNHQOLWHUDOO\GRHVQRWH[SUHVVDQ\ JHQHUDOFRQWHQW %XWLW¶VHQRXJKWR
LPDJLQH WKDW ZH GR QRW NQRZ WKH FRORU RI $GDP¶V KDLU WR LPPHGLDWHO\ PDNH WKH
JHQHUDOLQWHUSUHWDWLRQWKHQDWXUDORQH,I$GDPLVRQHRIWKHOLJKWEORQGHVWKHQKH
KDV EOXH H\HV 7KH WHUP RQH RI FRQFHDOV D YDULDEOH 7KH SUHGLFDWH FRXQWHUSDUW RI
WKLVVHQWHQFHLV
>@
"[[ $GDPàOLJKWEORQGH[ŽEOXHH\HG[
7KLVVHQWHQFHLVLQGLVSXWDEO\DJHQHUDOVHQWHQFHDQGLVREYLRXVO\DIRUPDOLPSOLFD
WLRQ7KHQDWXUDOWHQGHQF\WRLQWHUSUHWDOOVHQWHQFHVRIWKHIRUPLIaWKHQbDVIRU
PDOLPSOLFDWLRQVILQGVH[SUHVVLRQLQWKHWHQGHQF\WRLQWHUSUHWWKHVHQWHQFH,I$GDP
LV D OLJKWEORQGH WKHQ KH KDV EOXH H\HV DV WKH VHQWHQFH ,I $GDP ZHUH D OLJKW
EORQGHKHZRXOGKDYHEOXHH\HV
%XWHYHQWKHUHPDUNVZHKDYHPDGHDOORZXVWREHFRPHDZDUHRIWKHIDFW
WKDW PDWHULDO LPSOLFDWLRQ FDQ WDNH RQ VXUSULVLQJ PHDQLQJV ZKHQ LW LV LQWHUSUHWHG
DFFRUGLQJ WR WKH UXOHV HVWDEOLVKLQJ WKH FRPPRQO\ DFFHSWHG PHDQLQJ RI LI WKHQ
:HVKDOOWU\WRVKRZWKDWWKHSKHQRPHQRQRIWKHµSDUDGR[HVRIPDWHULDOLPSOLFD
WLRQ¶KDYHWKHLUURRWV LQ IRUJHWWLQJ WKDW WKH PHDQLQJ RI Ž LV VHW E\ WKH SRVWXODWH
ŽDQGWKDWLQWKLVVHQVHWKHFRQQHFWLYHLVQRWQHFHVVDULO\WKHVDPHDVLI«WKHQ
:HPRYHRQWRWKHHTXLYDOHQFHFRQQHFWLYHÙ,QLWVPHDQLQJLWLVUHODWHGWR
WKHLPSOLFDWLRQFRQQHFWLYHQRWRQO\LQFODVVLFDOORJLFEXWRQWKHEDVLVRIORJLFVLQ
ZKLFK ORJLFDO FRQVWDQWV DUH GHILQHG GLIIHUHQWO\ WKDQ KHUH REOLTXH UHIHUHQFHV WR
QRQFODVVLFDOORJLFVDUHVFDWWHUHGWKURXJKRXWWKLVJXLGH6RWKLVFRQQHFWLYHLVXVX
DOO\ GHILQHG DV µWZRZD\ LPSOLFDWLRQ¶ DQG WKXV HJ WKLV LV RQH SRVVLEOH ZD\ RI
GHILQLQJWKLVFRQQHFWLYHDVIROORZV
>'Ù@
aÙbÙGI aŽbàbŽa
7KH QH[W SRVWXODWH HVWDEOLVKHV WKH PHDQLQJ RI WKH FRQQHFWLYH Ù H[DFWO\ LQ DFFRU
GDQFHZLWKGHILQLWLRQ>'Ù@ZHVKRZWKLVWREHWKHFDVHLQWKHQH[WVHFWLRQ
Ù
8QGHUDJLYHQDUELWUDU\LQWHUSUHWDWLRQ Y VHQWHQFHaÙb LVWUXHLIDQGRQO\
LI ERWK FRQVWLWXHQWV a DQG b KDYH RQ WKLV LQWHUSUHWDWLRQ WKH VDPH WUXWK
YDOXH
7KH GLVWLQFWLRQ EHWZHHQ IRUPDO DQG PDWHULDO LPSOLFDWLRQ FDUULHV IRUZDUG
DXWRPDWLFDOO\ WR HTXLYDOHQFH 7KH SKUDVH LI DQG RQO\ LI ZKLFK LV WKH YHUEDO
HTXLYDOHQWRIÙKDVEHHQXVHGH[WUHPHO\RIWHQ+RZHYHUZHKDYH\HWWRFRQQHFW
WZR DUELWUDU\ VHQWHQFHV ZLWK WKLV SKUDVH ,W DOZD\V DSSHDUHG LQ VRPH IRUPDO FRQ
WH[WLHLQDFRQWH[WRIWKHIRUPf[LIDQGRQO\LIy[2IFRXUVHWKHWUXWKYDOXH
RIDVWDWHPHQWRIWKH IRUPf[ LI DQGRQO\LIy[LQDSHGDQWLFIRUPXODWLRQIRU
HYHU\ [f[LI DQG RQO\ LIy[GHSHQGVRQZKHWKHU DOOSDUWLFXODU FDVHVRI WKLV
VHQWHQFHDUHWUXHLQWKHVHQVHGHPDQGHGE\WKHSRVWXODWHÙ
7KH0HWKRG
,QYRNLQJSRVWXODWHVGHILQLQJWKHPHDQLQJRIORJLFDOFRQQHFWLYHVEHFRPHVHDVLHULI
ZHJLYHWKHPWKHIRUPRIWDEOHVVKRZLQJKRZWKHWUXWKYDOXHRIDFRPSRXQGVHQ
WHQFHGHSHQGVRQLWVFRQVWLWXHQWV7KHGHILQLWLRQRIÔFRUUHVSRQGVWRWKHIROORZ
LQJWDEOH
a
Ôa
7DEOH
6LQFHDWZRSODFHFRQQHFWLYHDOZD\VMRLQVWZRVHQWHQFHV±VRPHVHQWHQFHa
DQG VRPH VHQWHQFH b WKH WDEOH PXVW VKRZ KRZ WKH WUXWKYDOXH RI WKH ZKROH GH
SHQGVRQWKHWUXWKYDOXHRIWKHFRQVWLWXHQWVHQWHQFHV6RZHPXVWWDNHLQWRDFFRXQW
QRWWZRSRVVLELOLWLHVYa DQGYa EXWIRXUYa Yb Ya
Yb Ya Yb Ya Yb 7DEOHVGHILQLQJWKHPHDQLQJ
RIWZRSODFHFRQQHFWLYHVPXVWWKHUHIRUHWDNHDPRUHFRPSOH[IRUP7KHWDEOHEH
ORZLVµFROOHFWLYH¶±LWLQFOXGHVDOOWKHFRQQHFWLYHV
a
b
Ôa
aàb
a£b
aŽb
aÙb
7DEOH
7DEOHRUUDWKHUSRVWXODWHVÔàÙZKLFKWKLVWDEOHSUHVHQWVLQ
VKRUWIRUPDOORZVXVWRFDOFXODWHWKHWUXWKYDOXHRIDQDUELWUDU\VHQWHQFHEXLOWLQ
WHUPVRIORJLFDOFRQQHFWLYHVRQFRQGLWLRQWKDWZHNQRZWKHWUXWKYDOXHRIWKHVHQ
WHQFHVFRQQHFWHG7KLVPHWKRGRI FDOFXODWLQJWKHYDOXHV RI IRUPXODH LV FDOOHG WKH
PHWKRG:HLOOXVWUDWHLWZLWKWKHIROORZLQJH[DPSOH:HFDOFXODWHIRUH[DP
SOHWKHYDOXHVRIWKHIRUPXODaŽbàbŽaLQUHODWLRQWRWKHYDOXHVRILWV
FRQVWLWXHQWV,IVXFKFDOFXODWLRQVDUHFROOHFWHGLQWKHIRUPRIDWDEOHLWZLOOWDNHWKH
IRUP
a
b
aŽb
bŽa aŽbàbŽa
7DEOH
&RPSDULQJ WDEOHV DQG ZH REVHUYH WKDW IRU HYHU\ VHQWHQFHV a DQG b WKH
IRUPXODHaŽbàbŽaDQGaÙbDOZD\VSRVVHVVWKHVDPHYDOXHVMXVWDVZH
ZDQWHG
2QHPRUHH[DPSOH/HW¶VFKHFNWKDWWKHWUXWKYDOXHRIDQDUELWUDU\VHQWHQFH
RIWKHIRUPaŽÔaŽbZLOODOZD\VEHDVHQWHQFHRIWKLVIRUPLVDOZD\VWUXH
UHJDUGOHVV RI WKH YDOXHV WDNHQ E\ LWV FRQVWLWXHQWV +HUH LV WKH WDEOH RI WKH FRUUH
VSRQGLQJFDOFXODWLRQV
a
b
Ôa
ÔaŽb
aŽÔaŽb
7DEOH
:H QRWH WKDW LQ FUHDWLQJ WKLV WDEOH ZH ZHQW WKURXJK TXLWH D ELW RI ZDVWHG
ZRUN:HLQWHQGHGWRVKRZ WKDW UHJDUGOHVV RI WKH WUXWKYDOXHV RI WKH VHQWHQFHV a
DQGbXQGHULQWHUSUHWDWLRQYWKHWUXWKYDOXHRIWKHZKROHPXVWEH:HOODSURRI
RIWKLVIDFWLVDOPRVWLPPHGLDWH)RUWKHIRUPXODaŽÔaŽbWRFRPHRXWIDOVH
LWPXVWEHWKHFDVHWKDWÔaŽbLVIDOVH)RUÔaŽbWREHIDOVHZHPXVWKDYH
Ya DQG Yb %XW LI Ya WKHQ DQWHFHGHQW RI WKH ZKROH IRUPXOD LV
IDOVHDQGVRaŽÔaŽbPXVWEHWUXHYaŽÔaŽb $QHIIHFWRIWKHFDOFXODWLRQVRUFRUUHVSRQGLQJUHDVRQLQJZHKDYHGRQHLV
WRVKRZWKDWQRVHQWHQFHRIWKHIRUPaŽÔaŽbµFDQEH¶IDOVH±LWLVWUXHXQGHU
HYHU\LQWHUSUHWDWLRQY:HVKDOOFDOODQ\VXFKVHQWHQFHORJLFDOO\WUXHRUDWDXWRO
RJ\$VDV\PERORIDWDXWRORJ\ZHZLOOXVHWKHVDPHVLJQDVIRULPSOLFDWLRQLH
_ 7KHIDFWWKDWDOOVHQWHQFHVRIWKHIRUPaŽÔaŽbDUHORJLFDOO\WUXHZKLFK
ZHKDYHSURYHGLVZULWWHQ
_ aŽÔaŽb
6WDWHPHQWVWKDWVHQWHQFHVLQDJLYHQVFKHPDHJaŽÔaŽbDUHORJLFDOO\WUXH
DUHFDOOHGODZVRIORJLF7KRVHODZVRIORJLFZKRVHSURRIVUHTXLUHRQO\WKHSURSHU
WLHV RI ORJLFDO FRQQHFWLYHV DUH FDOOHG WKH ODZV RI VHQWHQFH FDOFXOXV 7KH ODZV RI
ORJLFZKLFKDUHHVWDEOLVKHGE\DQDO\]LQJERWKFRQQHFWLYHDQGTXDQWLILHUSURSHUWLHV
DUHFDOOHGODZVRISUHGLFDWHFDOFXOXV
7KHXVHRIWKHVDPHV\PERO_ IRUERWKLPSOLFDWLRQDQGORJLFDOWUXWKKDVD
VROLGIRUPDOIRXQGDWLRQ7KLVLVWKDWIRUHYHU\VHQWHQFHWKHIROORZLQJUHODWLRQVKLS
LVVDWLVILHG
_ aLIDQGRQO\LIº_ a
6RFDOOHGµ%DVLF/DZVRI/RJLF¶
7KHVHDUHWZRVWDWHPHQWVRIVHQWHQFHFDOFXOXVZKLFKZHUHRQFHFRXQWHGDPRQJWKH
µEDVLFODZVRIORJLF¶7KLVQDPHLVQRORQJHUXVHG7KHODZVZHVKDOOSUHVHQWGRQRW
DSSHDUWREHDQ\PRUHEDVLFWKDQPDQ\RWKHUODZVRIORJLF<HWWKHUHLVVRPHWKLQJ
ZKLFKGLVWLQJXLVKHVWKHP7KH\ZHUHRQFHWKHVXEMHFWVRIQXPHURXVSKLORVRSKLFDO
DQDO\VHV ZKRVH DXWKRUV WULHG WR SURYH WKDW ORJLF QHHGV WR EH UHIRUPHG LQ VXFK D
ZD\DVWKRVHODZVFHDVHGWREHLPSRUWDQW
/DZRIWKH([FOXGHG0LGGOH
/DZRI&RQWUDGLFWLRQ
_ a£Ôa
_ ÔaàÔa
7KHIDFWa£ÔaLVDWDXWRORJ\LVHDVLO\FKHFNHG,IYa WKHQYa£Ôa EHFDXVHWKHILUVWRIWKHWZRGLVMXQFWVLVWUXH,IYa WKHQYÔa DQGWKXV
DJDLQYa£Ôa 7KLVWLPHEHFDXVHWKHVHFRQGGLVMXQFWLVWUXH
7KH/DZRIWKH([FOXGHG0LGGOHLVVRPHWLPHVXQGHUVWRRGDVDODZZKLFK
VWDWHVWKDWHYHU\VHQWHQFHKDVDQHVWDEOLVKHGWUXWKYDOXHHLWKHULWLVWUXHRUIDOVH
&216758&7,9,67 /2*,&6 7KH /DZ RI WKH ([FOXGHG 0LGGOH _ a £ Ôa KDV
EHHQWKHVXEMHFWRIQXPHURXVFULWLFLVPV,QWKHRSLQLRQRI%URXZHUD'XWFKSKLORVRSKHURI
PDWKHPDWLFVWRVKRZWKH IDOVLW\ RI Ôa LV QRW D SURRI RI WKH WUXWK RI a DV KDG EHHQ DV
VXPHGE\ORJLFLDQV7KHFODVVLFDOXQGHUVWDQGLQJRIWUXWKLVIDXOW\7KHUHVXOWRIWKLVFULWL
FLVPZDVWKHDSSHDUDQFHGXHWR/(-%URXZHU$+H\WLQJRIDV\VWHPRILQWXLWLRQLVW
ORJLFLQZKLFKWKHODZRIWKHH[FOXGHGPLGGOHLVQRWDWDXWRORJ\7KLVLVWKHHDUOLHVWV\VWHP
QXPHURXVWRGD\RIVRFDOOHGFRQVWUXFWLYLVWORJLF±LHV\VWHPVZKHUHWUXWKLVXQGHUVWRRG
QRWDVFRUUHVSRQGHQFHZLWKDVWDWHRIDIIDLUVWKHFODVVLFDOFRQFHSWLRQRIWUXWKEXWDV³WKH
H[LVWHQFHRIDQHIIHFWLYHSURRI´
/RRVHO\ H[SUHVVHG WKH ODZ RI FRQWUDGLFWLRQ VWDWHV WKDW WKHUH GRHV QRW H[LVW DQ\
VHQWHQFHaZKLFKLVDWWKHVDPHWLPHWUXHDQGIDOVH
3$5$&216,67(17 /2*,& 7KH ODZ RI FRQWUDGLFWLRQ ZDV TXHVWLRQHG E\ -DQ
àXNDVLZHLF]ZKHQKHDUJXHGWKDWWKHUHFDQH[LVWVHQWHQFHVVXFKWKDWQHLWKHUWKHVHQWHQFH
QRULWVQHJDWLRQFDQEHFRQVLGHUHGIDOVH7KLVFULWLFLVPEHFDPHWKHLQVSLUDWLRQIRUWKH%UD
]LOLDQORJLFLDQ1HZWRQ&GD&RVWDWRFUHDWHDSDUDFRQVLVWHQWORJLFPRUHSUHFLVHO\PDQ\
V\VWHPV RI VXFK ORJLF +H FRQVLGHUV ZH VKRXOG DOORZ IRU WKH H[LVWHQFH RI VHQWHQFHV IRU
ZKLFKERWKWKH\DQGWKHLUQHJDWLRQVDUHWUXH7KLVRIFRXUVHPHDQVWKDWWKHFRQFHSWRIQH
JDWLRQZKLFKLQRXUFDVHKDVEHHQGHVFULEHGDV³WZRYDOXHG´PXVWEHGHILQHGGLIIHUHQWO\
,Q D µSDUDFRQVLVWHQW¶ ZRUOG D ORJLF ZKLFK SHUPLWV WKH H[LVWHQFH RI VHQWHQFHV a
VXFKWKDWaàÔaLVWUXHLVFDOOHGSDUDFRQVLVWHQWORJLFZHVRPHWLPHVDVVXPHWKDWEHVLGHV
µFODVVLFDOWUXWK¶DQGµFODVVLFDOIDOVLW\¶WKHUHDUH\HWRWKHUWUXWKYDOXHV:HDVVXPHIRUH[
DPSOH WKDW EHVLGHV VHQWHQFHV ZKLFK DUH FODVVLFDOO\ WUXH WKHUH DUH DOVR WKRVH ZKLFK DUH
µSDUDFRQVLVWHQWO\WUXH¶±WUXHWRJHWKHUZLWKWKHLUQHJDWLRQV
'LFWXPGH2PQLDQGD)HZ2WKHU$VVHUWLRQV
:H PRYH RQ WR WKH SUHVHQWDWLRQ RI D IHZ ODZV RI SUHGLFDWH FDOFXOXV :H VKRXOG
H[SODLQWKHIROORZLQJ,QORJLFWH[WERRNVODZVRISUHGLFDWHFDOFXOXVDUHIRUPXODWHG
DVODZVFRQFHUQLQJQRWRQO\VHQWHQFHVEXWVHQWHQWLDOIXQFWLRQVWRR6RRQHDVVXPHV
WKDWIRUH[DPSOHWKHIROORZLQJWKHRUHPFDOOHGGLFWXPGHRPQLLVDODZRIORJLF
$
_ "[f[Žf[
:H GR QRW UHTXLUH WKDW WKH VFKHPDWD "[f[ DQG f[ EH VHQWHQFH VFKHPDWD ,W LV
SHUPLVVLEOHIRUWKHPWREHVFKHPDWDRIVHQWHQWLDOIXQFWLRQV7KLVLVGRQHIRUSXUHO\
WHFKQLFDOUHDVRQV±WKHSUHVHQWDWLRQRIFHUWDLQORJLFDOWKHRUHPVEHFRPHVVLPSOHU,Q
WKHFRQFHSWLRQRIWKLVJXLGHZHVKDOOVWDQGE\WKHSRVLWLRQWKDWWUXWKYDOXHVFDQEH
DVFULEHGRQO\WRVHQWHQFHVDQGQRWWRVHQWHQWLDOIXQFWLRQV7KXVRQO\VHQWHQFHVDQG
QRW VHQWHQWLDO IXQFWLRQV FDQ EH WUXH DQG RQO\ EHWZHHQ VHQWHQFHV FDQ LPSOLFDWLRQ
WDNHSODFH7KHVHFRQYHQWLRQVFRYHUORJLFDOWUXWKDQGORJLFDOLPSOLFDWLRQ%HFDXVH
WKHV\PERO_ VHUYHVERWKDVWKHV\PERORIORJLFDOWUXWKDQGORJLFDOLPSOLFDWLRQWKH
IRUPXODH WR ZKLFK ZH VKDOO DSSO\ LW ZLOO EH WUHDWHG DV VHQWHQFHV 7KLV GHFLVLRQ
GRHVQ¶WDWDOOPHDQWKDWZHPXVWXQGHUWDNH DQ\LPSRUWDQWFKDQJHLQWKHDFFHSWHG
ZD\RISUHVHQWLQJORJLF¶VWKHRUHPVDQGWKHODZVRIORJLFDOGHGXFWLRQ)LWWLQJZKDW
WKH UHDGHU PLJKW KDYH PHW LQ RWKHU WH[WERRNV ZLWK WKHVH DVVXPSWLRQV LV D VLPSOH
PDWWHU
/HW¶V FODULI\ WKH LVVXH ZLWK WKH H[DPSOH RI WKH ODZ GLFWXP GH RPQL /HW¶V
DVVXPHWKDWEHVLGHV[WKHIRUPXODf[LQFOXGHVDVIUHHYDULDEOHV[ [ )RUPXOD
f[ZKLFKZHKDYHXVHGWRH[SUHVVWKHODZGLFWXPGHRPQLLVQRWLQLWVHOIHLWKHUD
VHQWHQWLDOIXQFWLRQRUDVHQWHQFH,WLVDVHQWHQFHVFKHPDDQGPD\EHWUHDWHGDVD
VFKHPDIRUVHQWHQWLDOIXQFWLRQV6RRXUGHFLVLRQDWWKHSULFHRIDOLWWOHFRPSOLFD
WLRQLQH[SUHVVLQJORJLFDOWKHRUHPVGRHVQRWH[FOXGHWKHSRVVLELOLW\RIWUHDWLQJWKH
IRUPXODH "[f[ DQG f[ DV VFKHPDWD RI VHQWHQWLDO IXQFWLRQV :KDW LW GRHV H[
FOXGH ZKLFK LV D GLIIHUHQW PDWWHU LV WUHDWLQJ GLFWXP GH RPQL DV D WKHRUHP DERXW
VHQWHQWLDO IXQFWLRQV ,Q WKH FRQFHSWLRQ ZH KDYH WDNHQ RQ WKH WKHRUHP GLFWXP GH
RPQLFRQFHUQVFRPSOHWHO\DUELWUDU\VHQWHQFHVREWDLQHGE\UHSODFLQJIUHHYDULDEOHV
LQWKHVHQWHQWLDOIXQFWLRQVRIWKHIRUP"[f[DQGf[ZLWKSURSHUQDPHVRIDUEL
WUDU\REMHFWVUHSUHVHQWHGE\WKHYDULDEOHV7KLVUHSODFHPHQWVKRXOGRIFRXUVHWDNH
SODFHLQVXFKDZD\WKDWDQ\RQHYDULDEOHZLOOEHHYHU\ZKHUHUHSODFHGE\WKHVDPH
QDPH
Q
2QH RI WKH SRVVLELOLWLHV IRU LQWURGXFLQJ VXFK D UHSODFHPHQW RSHUDWLRQ GH
SHQGV RQ WUHDWLQJ WKH IUHH YDULDEOHV DV FRQWH[WXDO QDPHV ,I "[f[ DQG f[ DUH
DUELWUDU\FRQFUHWHVHQWHQFHVDQGQRWVFKHPDWDRIVHQWHQFHVWKHQDSSO\LQJWKHVXI
IL[>[D[ D [ D @ZHREWDLQIURP$DWKHRUHPRIWKHIRUP
>$@
Q
Q
_ "[f[Žf[>[D[ D [ D @
Q
Q
ZKHUHWKHV\PEROVDD D DUHWREHWUHDWHGDVµPHWDWKHRUHWLFDO¶V\PEROVQRW
EHORQJLQJWRWKHDQDO\]HGODQJXDJHRIDUELWUDU\REMHFWVIURPWKHVFRSHRIWKHFRU
UHVSRQGLQJYDULDEOHV
7UHDWLQJYDULDEOHVDVFRQWH[WXDOQDPHVLVRQHRIWKHSRVVLELOLWLHVRIGHYHO
RSLQJ$ LQWR D WKHRUHP FRQFHUQLQJ FRQFUHWH VHQWHQFHV 2QO\ LQ WKH FDVH ZKHQ
WKHDQDO\]HGODQJXDJHFRQWDLQVQRSURSHUQDPHVZHDVVXPHGIRUH[DPSOHWKDWLQ
WKHFDVHRIWKHODQJXDJH/
WKLVZLOOEHVRZLOOWKLVEHWKHRQO\SRVVLELOLW\,I
LQWKHODQJXDJHLQTXHVWLRQWKHUHDUHSURSHUQDPHVWKHQWKHYDULDEOHVDSSHDULQJLQ
WKHVFKHPDWDRQWKHULJKWRI$FDQEHUHSODFHGE\WKRVHQDPHVQRWQHFHVVDULO\
FRQWH[WXDOQDPHV:HZLOOIROORZWKHIROORZLQJSULQFLSOH
Q
+202
=
:KHQHYHU WKHUH DUH VFKHPDWD RI VHQWHQWLDO IXQFWLRQV LQ WKH VFRSH RI WKH
V\PERO _ HYHU\ IUHH YDULDEOH DSSHDULQJ LQ WKHVH IXQFWLRQV VKRXOG EH
WUHDWHG DV D V\PERO RI D SURSHU FRQWH[WXDO RU QRW QDPH RI DQ DUELWUDULO\
FKRVHQREMHFWIURPWKHVFRSHRIWKDWYDULDEOH
$IWHUDOOWKHVH H[SODQDWLRQVLW¶VHDV\WRVHHWKDWWKHRUHP$VD\VVRPH
WKLQJFRPSOHWHO\REYLRXV,WVD\VWKDWLIVRPHVHQWHQFHRIWKHIRUP"[f[ LVWUXH
XQGHUWKHDVVXPHGLQWHUSUHWDWLRQYWKHQHYHU\VHQWHQFHRIWKHIRUPf[PXVWDOVR
EHWUXH:KDWGRHVWKHSURRIRIWKLVORRNOLNH±EHLQJEDVHGRQWKHVHPDQWLFUXOHV
"DQGŽ"
7KHUHDVRQLQJJRHVDVIROORZV:HWDNHDQDUELWUDU\LQWHUSUHWDWLRQY/HWWKH
VFRSHRIWKHYDULDEOH[XQGHULQWHUSUHWDWLRQYEH8/HWDEHDQDUELWUDU\HOHPHQWRI
WKHVHW8:HDVVXPHWKDWXQGHULQWHUSUHWDWLRQYWKHYDULDEOH[UHSUHVHQWVDQDPHRI
WKLVHOHPHQWLHDFRQWH[WXDOQDPH,IWKHVHQWHQFH"[f[Žf[ZHUHIDOVHLH
Y"[f[ Ž f[ WKHQ LQ DFFRUGDQFH ZLWK WKH WDEOH IRU LPSOLFDWLRQ f[
ZRXOGDOVRKDYHWREHIDOVH7KXVLQYLUWXHRI>"@WKHVHQWHQFH"f[ZLOOEHIDOVH
DQGWKLVPHDQVDJDLQLQYLUWXHRIŽWKDW"[f[Žf[>[D@ LVWUXH6LQFHWKH
LQWHUSUHWDWLRQ ZH KDYH FKRVHQ LV D FRPSOHWHO\ DUELWUDU\ LQWHUSUHWDWLRQ ZH KDYH
SURYHGWKDW"[f[Žf[>[D@LVWUXHIRUHYHU\LQWHUSUHWDWLRQYDQGWKXVZHKDYH
SURYHGWKDW_ "[f[Žf[>[D@
7KHTXDQWLILHUV"DQG$DUHGHILQDEOHLQWHUPVRIHDFKRWKHU)RUERWKWKH
IROORZLQJ UHODWLRQVKLSV DUH ODZV RI ORJLF RI FRXUVH ZH DVVXPH WKDW [ LV WKH RQO\
IUHHYDULDEOHLQIRUPXODf[
$
(
_ "[f[ÙÔ$[Ôf[
_ $[f[ÙÔ"[Ôf[
7KHVHFRQGSURRIRI$JLYHVDJHQHUDORXWOLQHRIKRZWRSURYHWKHVH%XW
LQVWHDGRIVXFKDSURRILWVXIILFHVWRXVHWKHIROORZLQJUHDVRQLQJ:HUHVWULFWRXU
VHOYHVWRFDVH$)RUPXODf[VD\VWKDWDOOWKH[¶VKDYHSURSHUW\f$QGWRVD\
WKDWDOOWKH[¶VKDYHSURSHUW\fLVWRVD\WKDWWKHUHGRHVQRWH[LVWDQ\REMHFW[ZLWK
RXW WKLV SURSHUW\ $QG WKLV LV ZKDW IRUPXOD Ô$[Ôf[ VD\V :KLFK ILQLVKHV RXU
SURRI
7KHPXWXDOGHILQDELOLW\RIWKHTXDQWLILHUVODZ$FDQEHWUHDWHGDVDGHIL
QLWLRQRIWKHTXDQWLILHU"LQWHUPVRI$DQGQHJDWLRQZKHUHDV(FDQEHWUHDWHGDV
DGHILQLWLRQRI$LQWHUPVRI"DQGQHJDWLRQPHDQVWKDWIRUHYHU\ODZFRQFHUQLQJ
RQHRIWKHVHTXDQWLILHUVWKHUHH[LVWVDµGXDO¶ODZFRQFHUQLQJWKHRWKHU7KHWHUP
µGXDO¶LQPDWKHPDWLFVKDVDGHILQLWHPHDQLQJZKLFKZHZRQ¶WJRLQWR7KHGXDO
ODZRI$LV
(
_ f[>[D@Ž$[f[
'HGXFWLRQ7KHRUHP
7KHODZVRIORJLFXQGHUVWRRGLQDFFRUGDQFHZLWKWKHGHILQLWLRQZHJDYHDW
WKHHQGRIVWDWHWKDWVHQWHQFHVRIDJLYHQIRUPDUHORJLFDOWUXWKVWDXWRORJLHV
7KHVHVHQWHQFHVDUHQRWLQWKHPVHOYHVDYHU\LQWHUHVWLQJFDWHJRU\:K\ZRXOGDQ\
RQHEHLQWHUHVWHGLQVHQWHQFHVZKLFKDUHFRPSOHWHO\µFRQWHQWOHVV¶")RURQO\VXFK
VHQWHQFHVFDQEHORJLFDOO\WUXH
:KDW LQFOLQHG ORJLFLDQV WR WKH LQYHVWLJDWLRQ RI WDXWRORJLHV ZDV WKH VWULFW
FRQQHFWLRQEHWZHHQWKHFRQFHSWRIORJLFDOWUXWKDQGWKHFRQFHSWRIORJLFDOHQWDLO
PHQW:HDOUHDG\NQRZWKDWZHPD\XVHWKHVHFRQGRIWKHVHFRQFHSWVWRGHILQHWKH
ILUVWFRPSDUHVHQWHQFHaLVDWDXWRORJ\LVORJLFDOO\WUXHLIDQGRQO\LILWLV
HQWDLOHGE\WKHHPSW\VHWRIVHQWHQFHV:HKDYHDOUHDG\UHFDOOHGVHHWKDWWKH
FRQFHSWRIHQWDLOPHQWZDVGHILQHGLQWKHWKLUWLHVRIWKHODVWFHQWXU\E\$OIUHG7DU
VNL6RKRZZDVORJLFDOHQWDLOPHQWXQGHUVWRRGEHIRUH",W¶VDFRQFHSWJRLQJEDFNWR
DQFLHQWWLPHVDIWHUDOO9DULRXVYHUVLRQVRIWKHIROORZLQJGHILQLWLRQZHUHXVHG
_ 6HQWHQFHaLVORJLFDOO\HQWDLOHGE\VHQWHQFHVbbbLIDQGRQO\LIWKH
LPSOLFDWLRQbàbààbŽaLVORJLFDOO\WUXH
6RLIXQWLO7DUVNLWKHUHODWLRQVKLS_ ZDVWUHDWHGDVDGHILQLWLRQDOUHODWLRQVKLS
WKHQDIWHU7DUVNL¶VGHILQLWLRQLQWKHPDQQHURIWKLVUHODWLRQVKLSEHFDPHDQRQ
GHILQLWLRQDOWKHRUHP/HW¶VFKHFNWKDW_ DQGWKHGHILQLWLRQ>_ @IURPDUHLQ
PXWXDOO\DFFRUG
7KDW a IROORZV ORJLFDOO\ IURP VHQWHQFHV bbb XQGHUVWRRG LQ DFFRU
GDQFH ZLWK >_ @ LV HTXLYDOHQW WR WKH VWDWHPHQW WKDW HYHU\ LQWHUSUHWDWLRQ Y ZKLFK
YHULILHV WKH VHQWHQFHV bbb DOVR YHULILHV WKH VHQWHQFH a 7KLV VWDWHPHQW LQ
WXUQLVHTXLYDOHQWWRWKHVWDWHPHQWWKDWHYHU\LQWHUSUHWDWLRQYYHULILHVWKHLPSOLFD
WLRQ bàbààbŽ aLHWKLVLPSOLFDWLRQLVORJLFDOO\WUXH7KLVVKRUWUHDVRQ
LQJ LV D SURRI RI _ DQG WKXV DOVR RI >_ @ RQ WKH EDVLV RI WKH SUHYLRXVO\ DF
FHSWHG GHILQLWLRQ >_ @ RI WKH FRQFHSW RI HQWDLOPHQW 6R 7DUVNL¶V GHILQLWLRQ RI HQ
WDLOPHQWLVLQDJUHHPHQWZLWKWKHKLVWRULFDOXQGHUVWDQGLQJRIWKLVFRQFHSW
7KHWKHRUHP>_ @LVDSDUWLFXODUO\LPSRUWDQWORJLFDOWKHRUHP,WLVNQRZQDV
WKH'HGXFWLRQ7KHRUHP,QDVRPHZKDWGLIIHUHQWIRUP±LQWKHIRUPRIRQHRIWKH
ODZVRIGHGXFWLRQZHVKDOOIRUPXODWHLWDJDLQLQ