Gerard Renardel de Lavalette, Barteld Kooi*. Rineke Verbruggeâ . Abstract. Propositional dynamic logic (PDL) is complete but not compact. As a consequence ...
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1 1 1 1 ! "7 0 & ' &0 & & & &0 1 0 & & & 1 & * i{ in gf gilh l{ # ) hloxn ulfsfn ! }! / . }! / . ! iffipgve $ ifpvloxjpilf 45 i{ pqsf }! fsjsnnipgpilf % i{ pqsf ) rsgsfifm & i{ gfo ) pqsf £wnsvts pqs {lvhxkgpilf l{ ! ip jlfpgifn £hsmg psvgpilf tig iysy pqs tsvnilf ripq {lvhxkgs }! / . ) }! / . ! y p in flp qgvo pl tsvi{e pqgp pqsns vxksn gvs nlxfo ripq vsnusjp pl pqs nshgfpijn l{ iysy ihukisn pqgp y ' s nqgkk nqlr if nsj pilf y~ pqgp pqs nenpsh 7 in gknl npvlfmke jlhuksps ripq vsnusjp pl pqsns nshgfpijny )lv pqin rs nqgkk xns nlhs osvitso vxksn pqgp rs ifpvloxjs flry 2
zqs hlnp ihulvpgfp l{ pqsns gvs 0 &01 46 gfo && 45 rqiks pqs lpqsv vxksn gvs lfke xnso pl uvlts 46 gfo 45 y 46 rikk ws xnso if pqs + ifosfwgxh kshhg gfo wlpq 46 gfo 45 if pqs zvxpq kshhgy ' s vshgv pqgp pqs {gjp pqgp 45 qlkon rqiks lfke 45 in ugvp l{ pqs uvll{ nenpsh in snnsfpigk if lxv uvll{ l{ npvlfm jlhukspsfsnny 5 5 1 1 4 ! "7 & % & & & 0 & % &0 1 & * & i{ gfo ¥ ) pqsf ) ¥ npvlfm jxp 4 i{ pqsf ) ospgjqhsfp & 6 i{ ) pqsf ) jlfoipilfgkingpilf 46 i{ ) pqsf osoxjpilf 45 i{ pqsf }! }! npvlfm fsjsnnipgpilf 1 lpijs pqgp pqs npvxjpxvs l{ pqs uvll{ wsklr in gf iffipgve ifoxjpilf ltsv osvitgpilfn / pqin in flp g uvlwksh wsjgxns l{ pqs rskk {lxfosofsnn l{ osvitgpilfny & in sgne pl uvlts xnifm % gfo & y 4 in gknl sgne ripq # gfo & y |vll{ we ifoxjpilf ltsv g osvitgpilf l{ ) y e % rs hge gnnxhs ripqlxp klnn l{ msfsvgkipe pqgp > y zqs jgnsn wsklr gvs fghso we pqs vxks guukiso kgnp { in gf gilh pqsf / gknl {lv ip in g pgxplklmey e # rs qgts ) y lr guuke & prijs pl lwpgif y
zqsvs gvs {lxv jgnsn jlvvsnulfoifm ripq s¤xgkn > ) ) ) lv ) / pqse gkk {lkklr tig pgxplklmisn gfo 4 y ! ' s qgts pl nqlr }! / . ) }! / . }! / . ! {lv ) ripq - ) > y e rs qgts } . / ! / . } . / ! / . ! nl we & gfo ip nx jsn pl nqlr }! / . ) }! / . } . / ! / . gfo pqin in jlvvsjp i{ }! / . } . / ! / . {lv gfo }! / . } . / ! / . {lv y lpq gvs sgnike lwpgifso xnifm
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gne guukijgpilf l{ pqs ifoxjpilf qeulpqsniny lr pqsvs in g ripq ) gfo ) ) y ' ipq pqs ifoxjpilf qeulpqsnin rs msp ) gfo ) ) nl ripq & rs lwpgif ) y ! i{ ) pqsf we & 6 ) / ripq & rs vshlts
pqs pgxplklme gfo lwpgif y ifoxjpilf ltsv g osvitgpilf l{ y gne ripq 45 y
zgs gfo guuke 4 prijsy ! ' s qgts } / ! / . } / ! / . ! / tig gfo & rs lwpgif } }! / . } }! / . ! gne tig gfo & y zqsns jgnsn {lkklr oivsjpke we pqs ifoxjpilf qeulpqsniny f ihhsoigps jlfns¤xsfjs l{ 46 in pqgp i{ ) in 7 ifjlfninpsfp pqsf , y b \_\cb _` _^ f pqin nsjpilf rs jlfniosv g so kgfmxgms y ' s nqgkk uvlts npvlfm jlhukspsfsnn l{ 7 tig pqs sfif jlfnpvxjpilf l{ g jgflfijgk hlosk if nsjpilf y~y zqsvs gvs prl npsun vnp rs nqlr pqgp stsve 7 jlfninpsfp nsp jgf ws spsfoso pl g hgihgk 7 jlfninpsfp nsp pqsf rs jlfnpvxjp g vius hlosk jlfninpifm l{ hgihgk 7 jlfninpsfp nspny f nsjpilf y rs nqlr pqgp pqin hlosk olsn flp ngpin{e uvlmvgh qgvhlfe we mitifm g jlxfpsvhlosky 7
' s vsjgkk pqs lwtilxn {gjp pqgp g jlkksjpilf l{ {lvhxkgs in hgihgk 7 jlfninpsfp i ip in 7 jlfninpsfp iysy = + gfo jlfpgifn sgjpke lfs
{vlh ) , {lv stsve {lvhxkg if pqs kgfmxgms y f pqs vshgifosv l{ pqin nsjpilf rs rikk lhip pqs uvs 7 ws{lvs jlfninpsfp y 5 5 / ! 5 5 5 1 ! "7 " % & & & !& & & 0 &0 & & A
1 + sp ws g jlfninpsfp nsp iysy = + y + sp ws gf sfxhsvgpilf l{ gkk {lvhxkgs if y ' s nqgkk ifoxjpitske osfs gf ifjvsgn ifm ns¤xsfjs l{ {lvhxkg nspn spsfoifm gfo nqlr pqgp in hgihgk jlfninpsfpy - i{ - , i{ = gfo in flp l{ pqs {lvh }! / . ! - , ) , }! / . lpqsvrins rqsvs in pqs ksgnp fgpxvgk fxhwsv nxjq pqgp = }! / . gfo in l{ pqs {lvh }! / . ! ' s lwnsvts pqgp pqs if pqs kgnp jgns gkrgen sinpn {lv i{ }! / . {lv gkk pqsf we ! gfo & }! / . ! jlfpvgoijpifm = y l pqs osfipilf l{ in jlvvsjpy lr rs jkgih pqs {lkklrifm {lv gkk {lvhxkgs ) / {vlh pqsns jkgihn snus jigkke {vlh gfo ip {lkklrn ihhsoigpske pqgp in hgihgk jlfninpsfp ~y stsve in jlfninpsfp / y / y = , / 2 y / *y / y = + y zqs uvll{n l{ 2 gfo gvs gn xnxgky ' s mits pqs uvll{n if pqs prl xfxnxgk jgnsn foxjpilf ltsv y )lv jlfninpsfje l{ in mitsfy lr gnnxhs pqgp in jlfninpsfpy { if pqs osfipilf l{ pqs vnp lv nsjlfo jgns guukisn ip in jksgv pqgp in jlfninpsfpy { pqs kgnp jgns guukisn gfo rsvs ifjlfninpsfp pqsf }! / . ! }! / . tig 46 nl we xnifm pihsn }! / . jlfpvgoijpifm pqs osfipilf l{ y ' s nqgkk uvlts g hlvs msfsvgk npgpshsfp ripq ifoxjpilf ltsv g osvitgpilf l{ ¥ i{ ¥ ¦ gfo ¥ pqsf y < in gf gilh we y < # we 2 y
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¦ y zl nqlr pqgp }! / . ! gnnxhs xnifm jlfpvgulnipilf pqgp pqin in flp pqs jgns pqsf we , }! / . ! y + sp ws pqs ifos ripq }! / . ! pqsf = {lv lpqsvrins we pqs vnp jgns if pqs osfipilf l{ }! / . ! ¦ nl , }! / . ¦ {lv nlhs we pqs kgnp jgns l{ pqs osfipilf l{ y xp gknl we gnnxhupilf }! / . nl , }! / . ) }! / . {lv nlhs jlfpvgoijpifm pqs jlfninpsfje l{ ~y < 45 we y < % oivsjp jlfns¤xsfjs l{ pqs ifoxjpilf qeulpqsniny < & nl ¥ ¥¥ gfo ¥ - ¥¥ {lv nlhs ¥¥y e pqs ifoxjpilf qeulpqsnin rs msp ¥¥ ¦ nl ¥ - ¥¥ ¦ / we guukeifm pqs ifoxjpilf qeulpqsnin gmgif rs lwpgif y lr rs jgf osfs pqs jgflfijgk hlosk fssoso {lv npvlfm jlhukspsfsnny
6 5 & 0 &' & & & 0 & 7 8 ) 9 : # ) ; ! < 8 & < 9 : ) 8 1 } < ; 8 zqs pvxpq kshhg nqlrn pqgp 7 ) spsfon pl gkk {lvhxkgs l{ pqs kgfmxgms 5 5 1 5 5 ( 8 0 1 & $ & & % & 7 ) ¢ 1 foxjpilf ltsv y zqs gplhij gfo uvlulnipilfgk jgnsn gvs npgfogvoy ' s rikk uvlts pqs jgns }! we ifoxjpilf ltsv ! / pqs jgnsn {lv jlhuks uvlmvghn ! l{ pqs {lvhn . . / gfo . - gvs sgne nl rs lfke mits pqs uvll{n l{ pqs vshgififm prl xfxnxgk jgnsny lps pqgp pqs uvll{ gn g rqlks qgn pqs {lvh l{ gf ifoxjpilf ltsv g rskk lvosvifm l{ {lvhxkgs rqsvs }! in jlfniosvso pl ws g nxw{lvhxkg l{ }! ! y ~y ! gplhijy nifm pqs osfipilf l{ pqs pvxpq vskgpilf gfo pqs ifoxjpilf qeulpqsnin 7 ) {lv gkk 8 rs nss pqgp 7 ) } in s¤xitgksfp pl {lv gkk 8 {lv gkk }
p in stiosfp {lkklrn {vlh } y zl nss pqgp ihukisn } gn rskk rs gvmxs tig jlfpvgulnipilfy l gnnxhs } = iysy we hg ihgk jlfninpsfje , } y ' s nqgkk nqlr pqgp pqsvs in g hgihgk jlfninpsfp ripq {lv gkk nxjq pqgp }! gfo , y e pqs + ifosfwgxh kshhg ip nx jsn pl nqlr pqgp } - , in jlfninpsfpy nnxhs ip in flp iysy } - , + pqsf } tig 46 y zqxn ripq 45 } } } y sfjs } gfo } jlfpvgoijpifm pqs gnnxhupilf } = y y ! . ! 7 ) }. ! {lv gkk 7 ) }. }. {lv gkk }. ! xnifm pqs ifoxjpilf qeulpqsnin if pqs nsjlfo npsu gfo ! if pqs kgnp npsuy lps pqgp if pqs pvxpq kshhg rs ol flp uvlts pqs oxgk uvlusvpe {lv uvlmvghn 9 D i {lv gkk nxjq pqgp }! ip qlkon we osfi pilf {lv gplhij uvlmvghn y f nsjpilf y rs skgwlvgps lf pqin kgj l{ {xkk qgvhlfe y 15 1 65 2 44 ! "7 & !& & & & & 1 & A & 1 & 0 & 1 & $ & A 1 e jlfpvgulnipilfy xuulns = pqsf - , in jlfninpsfpy e kshhg - , in spsfoso pl g hgihgk jlfninpsfp nsp ripq , y lr we kshhg rs jlfjkxos pqgp if pqs jgflfijgk vius hlosk 7 ) = gn osnivsoy lps pqgp pqs jlhukspsfsnn uvll{ ihhsoigpske mitsn g jgflfijgk npgfogvo hlosk jlfpvgve pl pqs sgvke uvll{n l{ rsg jlhukspsfsnn {lv gn pqse guusgv if } * rqijq xns flfnpgfogvo hloskny
lhsrqgp nxvuvinifmke pqs 1 uvlusvpe 7 ) uvltso if kshhg {lv pqs jgflfijgk hlosk in flp hgpjqso we iysy pqs uvlusvpe 9 D }! ¦ {lv gkk uvlmvghn ! gfo gkk hgihgk jlfninpsfp nspn ) y )lv gplhij uvl mvghn pqin qlkon we osfipilfy zqs uvlmvgh jlfnpvxjplvn / ) - uvsnsvts gfo ip gknl qlkon {lv psnp uvlmvghny p {gikn qlrstsv {lv uvlmvghn ripq ipsvgpilfny f } uvlmvgh oinqgvhlfe iysy {gikxvs l{ uvlmvgh qgvhlfe {lv fips jgflfijgk hloskn {lv rgn jkgihso ripqlxp uvll{y ' s nqgkk nspjq g jlxfpsvsghuks {lv ripq pqs uvlmvgh !y zqs hg ihgk jlfninpsfp nspn rikk ws osfso we 7 ) ) 7 ) %
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~ uijpxvs l{ pqs jlxfpsvhlosk 7 y zqs nlkio kifsn ifoijgps pqs gjjsnniwikipe vskgpilf l{ pqs gplhij uvlmvgh pqs ognqso kifsn {lv y rqsvs % ) gvs prl rlvkon if nlhs hlosk 7 pl ws jlfnpvxjpsoy lvs uvsjinske rs nqgkk nqlr 7 ) } ! 7 ) % 7 ) } 7 ) % , 2 lr ihhsoigpske ihukisn } ! ¦ gfo 2 ihukisn 9 § rqsvs 9 § 9 ! in mitsf we pqs jgflfijgk hlosk nss osfipilf * y ' s osfs 7 if prl npsun vnp rs npxoe pqs hlosk ) pqsf rs spsfo pl 7 y f rs npgvp ripq ifpsvuvspifm lfke pqs kgfmxgms : fl uvlulnipilfgk gplhn wsniosn + gfo gn pqs lfke gplhij uvlmvgh / rqsf mlifm pl 7 rs goo pqs gplhij uvlmvgh y zqs spsfnilf l{ 9: in pqs npvijp lvosv vskgpilf lf y ' s rvips } {lv ) gfo } ! {lv 9 D ¦ y ss ) imxvs ~y ' s nge pqgp ¦ in & & & ! & i } {lv nlhs y zqs jvxjigk uvlusvpe l{ in {lv stsve if : ) } in fips lv jlfips * ¦ in jlfips i in fipsy l pqsvs in fl {lvhxkg ripq symy } y zqs qgvo ugvp if uvltifm * kisn if foifm pqs jlvvsnulfoifm uvlusvpe {lv uvlmvghn rqijq vskgpilfn lf gvs vsuvsnsfpgwks gn } ! {lv nlhs uvlmvgh ! if : . ' s nqgkk osfs pqin uvlusvpe xnifm pqs jlkksjpilf l{ ! & nxwnspn l{ i } ) % ¦
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p in vgpqsv lwtilxn pqgp in jklnso xfosv - gfo y lvs nxvuvinifm in pqgp in jklnso xfosv ' & jlf
xfjpilfny zqin in g jlfns¤xsfjs l{ pqs {gjp pqgp npvijp ifjkxnilf in g rskklvosvifm lf qsfjs stsve iffips jlf
xfjpilf " " l{ npgwks nspn in s¤xgk pl nlhs fips nxwjlf
xfjpilf " - ¢ ¢ ¢ - " y ' s spsfo npgwikipe pl vskgpilfn in osfso we 9 i 9 $ 9 gfo jkgih in jklnso xfosv fips jlf
xfjpilfn gfo gvwipvgve oin
xfj pilfn gfo xfosv G jlhulnipilfy lr rs jgf jqgvgjpsvins pqs vsuvsnsfpgwks vskgpilfn 9 } ! {lv nlhs uvlmvgh ! if : i 9 9 - 9 - 9 rqsvs {lv nlhs 9 > lv 9 ) 9 > lv 9 ) ) {lv nlhs gfo 9 y zqs lfke i{ , ugvp in uvltso plmspqsv ripq * if nihxkpgfslxn ifoxjpilf ltsv gfo ! y ' s spsfo pl 7 # ) 9 : ) 9 rqsvs # - % ) - ) $ 9 : ) - % ) 9 ) - ) ) ~ ) ) )
- ) ) ) ~ ) ) $ $ ss ) imxvs ~y £wnsvts pqgp 9 : ) pqs spsfnilf l{ 9 : if y ' s jkgih {lv gkk if : 7 ) % i } in jlfipsy vsjgkk pqgp } osflpsn ifpsvuvspgpilf if / rs nqgkk xns 9 D pl osflps pqs ifpsvuvspgpilf l{ ! if 7 y zqin in uvltso tig nihxkpgfslxn ifoxjpilf ltsv gfo ! if : plmspqsv ripq pqs npgpshsfpn % ) 9 D ) } ! jlfips gfo % ) % 9 D ) ) } ! jlfipsy knl {lv gkk 7 ) 7 ) ) ( rqsvs in osfso gn ripq gkk ljjxvvsfjsn l{ vsukgjso we + . y zqs vnp s¤xitgksfjs if ( gknl qlkon {lv % y n g jlfns¤xsfjs l{ gfo ( rs qgts {lv gkk if : 7 ) % 7 ) gfo pqin sfpgikn {lv i{ 7 ) } ! pqsf 7 ) {lv stsve qsfjs 7 ) {lv stsve nl 7 ) % gfo 7 ) % y f lvosv pl tsvi{e 2 rs uxp } + pqsf 7 ) ) gfo ) ) # 7 ) ) nl ifosso 7 ) } gfo 7 ) % , xnifm y zqsvs{lvs wlpq gfo 2 qlko gfo uvlmvgh oinqgvhlfe in oshlfnpvgpsoy
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f pqin nsjpilf rs spsfo pqs npvlfm jlhukspsfsnn vsnxkp {lv 7 pl suin pshij klmij ripq jlhhlf flrksomsy ) ivnp rs vsusgp pqs osfipilfn kgf mxgms gfo nshgfpijny )lv hlvs lf suinpshij klmij nss }3 / rs kllnske wgns pqs gilhn wsklr lf pqs tsvnilf l{ }yy
245 6 6 1 & & & ! &0 1 ! & & " 1 & 0 ' & & & $ 0 % & ! & & 1 & * ,
0 123 45 64 1 245 6 6 & 0 & 7 ) B 1 7 8 ) 9 : ) ; 0 * < 8 = >? & & ! & & 0 ? < 9 : ¦ 8 8 ? & ! & $ & & ? < ; " @ ? & & & % ! & A zqs pvxpq osfipilf in gn xnxgk / rs lfke mits pqs jkgxnsn {lv pqs hlogk lusvgplvny 7 ) B i 7 ) C {lv gkk C nxjq pqgp B 9 :C 7 ) B i 7 ) C {lv gkk C nxjq pqgp B 9 :C {lv gfe 7 ) B i 7 ) C {lv gkk C nxjq pqgp C in vsgjqgwks {vlh B svs C in vsgjqgwks {vlh B i pqsvs in g ugpq l{ ksfmpq ~ if pqs vius hlosk {vlh B pl C gklfm gjjsnniwikipe gvvlrn 9: gnnljigpso ripq hshwsvn l{ y zqs nihikgvipe wsprssf gfo suinpshij klmij ripq jlhhlf flrksoms qgn klfm wssf flpsoy f {gjp npvlfm jlhukspsfsnn l{ pqs kgppsv ihhsoigpske vsoxjsn pl pqs {lvhsv we pqs {lkklrifm shwsooifm ¥y xuulns pqs nsp l{ gmsfpn ) ¢ ¢ ¢ ) gfo osfs : ¥ } " ¥ ¥ } - ¢ ¢ ¢ - ¥ ¥ } - ¢ ¢ ¢ - ! ¥ lrstsv rs jgf uvlts nlhspqifm npvlfmsv i{ rs pgs g oivsjp guuvlgjqy p pxvfn lxp pqgp pqs sfif hspqlo jgf if gooipilf pl gknl ws xnso {lv nenpshn kis gfo ripq jlhhlf flrksomsy f lvosv pl mits pqs oivsjp uvll{ rs pqs gilhn gfo mits gf iffipgve osvitgpilf vskgpilfy
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5 4 1 & & & ! & & & & &0 & & & & *
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gkk ifnpgfpigpilfn l{ uvlulnipilfgk pgxplklmisn
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5 4 1 & & & & & & & & & & &0 & & $ & 0 & & & A 00 $ & ? & 0 ? & $ $ 0 $ 0 &' &0 !& & *
flrksoms : : : ulnipits ifpvlnusjpilf ,: : ,: fsmgpits ifpvlnusjpilf ' s spsfo gkk {lxv gilh nenpshn pl iffipgve uvll{ nenpshn & 7 & 7 & 7 gfo & 7 we gooifm g so nsp l{ osvitgpilf vxksny ' s vnp ifpvloxjs nlhs flpgpilf if lvosv pl osnjviws pqs iffipgve ifpvl oxjpilf vxks {lv y ' s rgfp pqin vxks pl jlfpgif gkk ifnpgfjsn l{ pqs {lvh : ¢ ¢ ¢ ) ~ : ¢ ¢ ¢ ) rqsvs in pqs lwtilxn gwwvstigpilf osfso ifoxjpitske we gfo y zqs fsgp rge pl {lvhxkgps pqs iffipgve vxks in pl ifpvloxjs fips ns¤xsfjsn ) ¢ ¢ ¢ ) rqsvs pqs gvs sipqsv {lvhxkgs lv hlogkipisn : {lv ripq / / : : zqs iffipgve vxks hge pqsf ws {lvhxkgpso gn ) ~ y ' s mits pqs osvitgpilf vxksn {lv pqs iffipgve nenpshny
1 1 1 4 & !& & 7 $ & 7 $ & 7 $ 0 & 7 A 0 &' &0 & & & &0 1 0 & & & 1 & *
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i{ in gf gilh l{ hloxn ulfsfn ) ) ~ iffipgve ifpvloxjpilf 45 i{ pqsf : fsjsnnipgpilf % i{ pqsf ) rsgsfifm & i{ gfo ) pqsf lr pqs vsgosv hge jqsj pqgp {lv pqsns nenpshn pqs osvitso vxksn & 4 & 6 gfo 46 l{ kshhg ~ jgf ws uvltso gn rskk gn pqs {lkklrifm gfg klmxs l{ npvlfm fsjsnnipgpilf 45 45 i{ pqsf : : npvlfm fsjsnnipgpilf {lv flrksoms p in ihhsoigps pqgp gkk {lxv nenpshn gvs nlxfo ripq vsnusjp pl pqs guuvl uvigps nshgfpijn & 7 {lv gkk vius {vghsn & 7 {lv vs I sits lfsn & 7 {lv vs I sits pvgfnipits lfsn gfo & 7 {lv s¤xitgksfjs vskg pilfny 15 / & !& & & & 7 $ & 7 $ & 7 $ 0 & 7 A & & & & && & & & & A 1 436 e g sfif jlfnpvxjpilf l{ g jgflfijgk hlosk gfgklmlxnke gn if nsjpilf y~y zqs uvsnsfjs l{ pqs guuvluvigps gilhn {vlh gfo if pqs hgihgk jlfninpsfp nspn ifoxjsn pqs guuvluvigps uvlusvpisn l{ pqs gjjsnniwikipe vskgpilfn if pqs jgflfijgk hlosky f pqs gfgklmxs l{ pqs + ifosfwgxh kshhg pqs kgnp jkgxns {lv nqlxko ws - , ) , lpqsvrins rqsvs in pqs ksgnp fgpxvgk fxhwsv ~ nxjq pqgp = gfo in l{ pqs {lvh y zqs osfipilf l{ pqs jgflfijgk hlosk in gn xnxgk {lv suinpshij klmijny zqs hgif oi svsfjs {vlh pqs uvll{ l{ pqs zvxpq kshhg in pqs ifoxjpilf npsu {lv lusvgplv rqijq rlvn tsve nhllpqke 7 ) {lv gkk ) ~ 7 )
{lv gkk ) ~ xnifm pqs ifoxjpilf qeulpqsnin if pqs nsjlfo npsu gfo # & if pqs kgnp npsuy p in jksgv {vlh pqs uvll{ nspjq pqgp gkk {lxv suinpshij klmijn ripq jlh hlf flrksoms gvs jgflfijgk lf pqsiv jgflfijgk {vghsn gkk pqsiv gilhn gvs tgkio }~y 5 2 zl nqlr pqgp pqs iffipgve uvll{ nenpshn gvs npvlfmsv pqgf pqs xnxgk rsgke jlhuksps lfsn ksp xn kll gp pqs rskk flrf sghuks l{ pqs egfpifs msfsvgkny lfniosv g nipxgpilf l{ prl gvhe vsmihsfpn lf prl qikkn lf wlpq niosn l{ g tgkkse if rqijq g qlnpiks gvhe in nipxgpsoy { pqs prl vsmihsfpn gppgj nihxkpgfslxnke pqse rikk jlf¤xsv pqs sfshey { lfke lfs l{ pqs vsmihsfpn gppgjn ip rikk ws os{sgpso we pqs sfshey zqsvs in fl # &
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ifipigk wgppks ukgf gtgikgwks rqijq pqs prl vsmihsfpn gmvss xulfy sipqsv msfsvgk l{ pqs vsmihsfpn rlxko osjios pl gppgj ripqlxp flrifm {lv nxvs pqgp pqs lpqsv lfs rikk gknl gppgj {vlh }3 y zqs jlhhgfoifm msfsvgk l{ pqs vnp oitinilf rinqsn pl jllvoifgps g nihxkpgfslxn gppgj gp nlhs pihs pqs fsp ogey zqs msfsvgkn jgf jlhhxfijgps lfke we hsgfn l{ hsnnsfmsvny lvhgkke ip pgsn g hsnnsfmsv lfs qlxv pl msp {vlh lfs sfjghuhsfp pl pqs lpqsvy lrstsv ip in ulnniwks pqgp qs rikk msp klnp if pqs ogv lv rlvns esp ws jgupxvso we pqs sfshey )lvpxfgpske lf pqin ugvpijxkgv fimqp stsvepqifm mlsn nhllpqkey lr klfm rikk ip pgs pqsh pl jllvoifgps gf gppgj . {vlh }y f{lvpxfgpske ip pxvfn lxp pqgp flvhgkke jlhhlf flrksoms ghlfm pqs prl msfsvgkn jgfflp ws vsgjqso if pqin rge gp sgjq ulifp if pihs lfke
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