❖♥ ❛ ●❛♣ ❜❡t✇❡❡♥ ❚r✉t❤ ❛♥❞ Pr♦✈❛❜✐❧✐t②✿ ❙♦♠❡ ♦❢ ❚♦t❛❧
❋✉♥❝t✐♦♥s ❆r❡ ◆♦t ❵Pr♦✈❛❜❧②✬ ❚♦t❛❧✳
❙❡♠✐♥❛r ▲♦❣✐❝ ❛♥❞ ❆♥❛❧②s✐s
❻✳ ❋r✐❞❛② ✷✺ ▼❛②✱ ✷✵✶✷
◆❛♦❤✐ ❊❣✉❝❤✐
▼❛t❤❡♠❛t✐❝❛❧ ■♥st✐t✉t❡✱ ❚♦❤♦❦✉ ❯♥✐✈❡rs✐t②✱ ❏❛♣❛♥
❡❣✉❝❤✐❅♠❛t❤✳t♦❤♦❦✉✳❛❝✳❥♣
✶ ■♥tr♦❞✉❝t✐♦♥
❼ ❇② ●ö❞❡❧✬s s❡❝♦♥❞ ✐♥❝♦♠♣❧❡t❡♥❡ss t❤❡♦r❡♠✱ ❢♦r ❛♥② ❝♦♥s✐st❡♥t ❛①✐♦♠❛t✐❝ s②st❡♠ T
♦❢ ▼❛t❤❡♠❛t✐❝s✱ t❤❡r❡ ❡①✐sts ❛ st❛t❡♠❡♥t A ❡①♣r❡ss❡❞ ❜② t❤❡ ❧❛♥❣✉❛❣❡ ♦❢ T s✉❝❤ t❤❛t
AN ✐s tr✉❡✱
✇❤❡r❡ AN ❞❡♥♦t❡s t❤❡ r❡s✉❧t ♦❢ ✐♥t❡r♣r❡t✐♥❣ A ♦✈❡r t❤❡ ♥❛t✉r❛❧ ♥✉♠❜❡rs N✱
❜✉t A ✐s ♥♦t ❛ t❤❡♦r❡♠ ♦❢ T ✳
❼ ❍❡♥❝❡✱ ❢♦r ❛♥② ❝♦♥s✐st❡♥t ❛①✐♦♠❛t✐❝ s②st❡♠ T ♦❢ ▼❛t❤❡♠❛t✐❝s✱ t❤❡r❡ ❡①✐sts ❛ st❛t❡✲
♠❡♥t A❼x, y➁ ❡①♣r❡ss❡❞ ❜② t❤❡ ❧❛♥❣✉❛❣❡ ♦❢ T s✉❝❤ t❤❛t
➛m ❃ N✱ ➜!n ❃ N s✳t✳ AN❼m, n➁ ✐s tr✉❡✱
❜✉t ➛x➜!yA❼x, y➁ ✐s ♥♦t ❛ t❤❡♦r❡♠ ♦❢ T ✳
❼ ◆❛♠❡❧②✱ ❢♦r ❛♥② ❝♦♥s✐st❡♥t ❛①✐♦♠❛t✐❝ s②st❡♠ T ♦❢ ▼❛t❤❡♠❛t✐❝s✱ t❤❡r❡ ❡①✐sts ❛ t♦t❛❧
❢✉♥❝t✐♦♥ f ✂ N Ns✉❝❤ t❤❛t
➛x➜!y“f❼x➁ y➐➐ ✐s ♥♦t ❛ t❤❡♦r❡♠ ♦❢ T ✱
✐✳❡✳✱ t♦t❛❧✐t② ♦❢ f ✐s ♥♦t ♣r♦✈❛❜❧❡ ✐♥ T ✳
◗✉❡st✐♦♥✳ ❍♦✇ ❝❛♥ ✇❡ ✜♥❞ t❤❡ ❜♦✉♥❞❛r② ♦❢ t♦t❛❧ ❢✉♥❝t✐♦♥s ♣r♦✈❛❜❧② t♦t❛❧ ✐♥ ❛ ❣✐✈❡♥
❝♦♥s✐st❡♥t ❛①✐♦♠❛t✐❝ s②st❡♠❄
❻❙❡♠✐♥❛r ▲♦❣✐❝ ❛♥❞ ❆♥❛❧②s✐s ✲ ●❤❡♥t ❯♥✐✈❡rs✐t②✿ ❤tt♣✿✴✴❝❛❣❡✳✉❣❡♥t✳❜❡✴③✇❝✴❧♦❣❛♥s❡♠✐♥❛r✳❡♥✳❤t♠❧
✶
✷ ●❡♥❡r❛❧ ✐❞❡❛s
▲❡t T ❜❡ ❛♥ ❛①✐♦♠❛t✐❝ s②st❡♠ ♦❢ ▼❛t❤❡♠❛t✐❝s ❛♥❞ Thm❼T ➁ ✂ ➌A ❙ A ✐s ❛ t❤❡♦r❡♠ ♦❢ T ➑✳ Pr♦❜❧❡♠✳ ■♥ ❣❡♥❡r❛❧ Thm❼T ➁ ☞ N✳ ❍❡♥❝❡ t❤❡ s✐③❡ ✭♦r t❤❡ ❝❛r❞✐♥❛❧✐t②✮ ♦❢ Thm❼T ➁ ❞♦❡s
♥♦t ❤❡❧♣ t♦ ❦♥♦✇ t❤❡ str❡♥❣t❤ ♦❢ t❤❡ ♣r♦✈✐♥❣ ♣♦✇❡r ♦❢ T ✳
❙♦❧✉t✐♦♥✭❄✮✳ ❊♠♣❧♦② ♦r❞✐♥❛❧ ♥✉♠❜❡rs t♦ ♠❡❛s✉r❡ t❤❡ ♣r♦✈✐♥❣ ♣♦✇❡r ♦❢ T ✳
■♥❢♦r♠❛❧ ✐❞❡❛✳
✶✳ ❉❡✜♥❡ ❛♥ ❡♥✉♠❡r❛t✐♦♥ α ✭ Aα ♦❢ st❛t❡♠❡♥ts ❜② ♦r❞✐♥❛❧ ♥✉♠❜❡rs✱ ❛♥❞
✷✳ ❆ss♦❝✐❛t❡ ❛♥ ♦r❞✐♥❛❧ ♥✉♠❜❡r αT ✇✐t❤ T s✉❝❤ t❤❛t Thm❼T ➁ ❜ ➌Aα❙ α ❅ αT➑✳
■t ❤♦❧❞s t❤❛t αT ❅ ω1 ➥1 ✭t❤❡ ❧❡❛st ✉♥❝♦✉♥t❛❜❧❡ ♦r❞✐♥❛❧✮ s✐♥❝❡ Thm❼T ➁ ✐s ❝♦✉♥t❛❜❧❡✳
✸ ❖r❞✐♥❛❧ ♥✉♠❜❡rs
❖r❞✐♥❛❧ ♥✉♠❜❡rs ❛r❡ ❛♥ ❡①t❡♥s✐♦♥ ♦❢ ♥❛t✉r❛❧ ♥✉♠❜❡rs t♦ ✐♥✜♥✐t❡ ♥✉♠❜❡rs✿ 0, 1, 2, . . . ω, ω✔ 1, . . . ω ✔ ω, . . . ω2, . . . ωω, . . . ωω , . . .
✇❤❡r❡ ω ❞❡♥♦t❡s t❤❡ ❧❡❛st ✐♥✜♥✐t❡ ♦r❞✐♥❛❧ ♥✉♠❜❡r✳
❊①❛♠♣❧❡ ✶✳ ✶✳ ω ☞ ❵0, 1, 2, . . .❡ ☞ ❵1, 2, 3, . . .❡ ☞ ❵0, 2, 4, . . .❡✳
✷✳ ω ✔ 1 ☞ ❵1, 2, 3, . . . 0❡ ❛♥❞ ω ✔ ω ☞ ❵0, 2, 4, . . . 1, 3, 5, . . .❡✳
✸✳ ω2 ω ω supm❅ωω m ❛♥❞ ωω supm❅ωωm✳
✹ ❆ ❜♦✉♥❞✐♥❣ ❢✉♥❝t✐♦♥ G
α❉❡✜♥✐t✐♦♥ ✷✳ ✭❙♦♠❡ ❡①❛♠♣❧❡s ♦❢✮ ✐♥❞✉❝t✐✈❡ ❞❡✜♥✐t✐♦♥s ♦❢ Gα ♦♥ α ❇ ωω✳ G0❼m➁ m✔ 1, Gω❼k✔1➁❼m➁ Gω k✔m✔1❼m➁,
Gk✔1❼m➁ Gk❼m➁ ✔ 1, Gω2❼m➁ Gω❼m✔1➁❼m➁,
Gω❼m➁ Gm✔1❼m➁, Gωk✔1❼m➁ Gωk❼m✔1➁❼m➁, Gω✔k✔1❼m➁ Gω✔k❼m➁ ✔ 1, Gωω❼m➁ Gωm✔1❼m➁.
Pr♦♣♦s✐t✐♦♥ ✸✳ ▲❡t k ❇ m✳
❚❤❡♥ Gk❼m➁ ❅ Gω❼m➁✱ Gω k❼m➁ ❅ Gω2❼m➁✱ ❛♥❞ Gωk❼m➁ ❅ Gωω❼m➁✳ Pr♦♦❢✳ ❇② ✐♥❞✉❝t✐♦♥ ♦♥ k✳
▲❡♠♠❛ ✹✳ ▲❡t 0 ❇ k✳
❚❤❡♥ Gk❼m➁ m ✔ k ✔ 1✱ Gω k❼m➁ k❼m ✔ 1➁✱ ❛♥❞ ❼m ✔ 1➁k❇ Gωk❼m➁✳ Pr♦♦❢✳ ❆❣❛✐♥ ❜② ✐♥❞✉❝t✐♦♥ ♦♥ k✳
✷
✺ ❆①✐♦♠❛t✐❝ s②st❡♠ T
OSR❢♦r ♦r❞❡r❡❞ s❡♠✐✲r✐♥❣s
❉❡✜♥✐t✐♦♥ ✺ ✭❆①✐♦♠❛t✐❝ s②st❡♠ TOSR✮✳
❼ ❚❤❡ ❧❛♥❣✉❛❣❡ ♦❢ TOSR ❝♦♥s✐sts ♦❢ ✔, , 0, 1, ❅✳
❼ ❚❤❡ ❛①✐♦♠s ♦❢ TOSR ❝♦♥s✐st ♦❢ t❤♦s❡ ❛①✐♦♠s ❢♦r ♦r❞❡r❡❞ s❡♠✐✲r✐♥❣s✳
Pr♦♣♦s✐t✐♦♥ ✻✳ ❋♦r ❛♥② f ✂ N N✱ ✐❢ ➛x➜!y“f❼x➁ y➐➐ ✐s ❛ t❤❡♦r❡♠ ♦❢ TOSR✱ t❤❡♥ t❤❡r❡
❡①✐sts ❛♥ ♦r❞✐♥❛❧ ♥✉♠❜❡r α ❅ ωω s✉❝❤ t❤❛t f❼m➁ ❇ Gα❼m➁ ❢♦r ❛❧❧ m✳
▲❡♠♠❛ ✼✳ ■❢ ➛x➜yA❼x, y➁ ✐s ❛ t❤❡♦r❡♠ ♦❢ TOSR✱ t❤❡♥ t❤❡r❡ ❡①✐sts ❛♥ ♦r❞✐♥❛❧ ♥✉♠❜❡r α❅ ωω s✉❝❤ t❤❛t ➛m ❃ N✱ ➜n ❇ Gα❼m➁ s✳t✳ AN❼m, n➁ ✐s tr✉❡✳
Pr♦♦❢✳ ❇② ✐♥❞✉❝t✐♦♥ ♦✈❡r t❤❡ ❢♦r♠❛❧ ♣r♦♦❢ ♦❢ t❤❡ t❤❡♦r❡♠ ➛x➜yA❼x, y➁ ✐♥ TOSR✭❛ss✉♠✐♥❣
♥♦r♠❛❧ ❢♦r♠s ♦❢ TOSR✲♣r♦♦❢s✮✳
✻ ❊①t❡♥❞✐♥❣ T
OSR❜② ❛♥ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥
❉❡✜♥✐t✐♦♥ ✽ ✭❆①✐♦♠❛t✐❝ s②st❡♠ TEXP✮✳
❼ ❚❤❡ ❧❛♥❣✉❛❣❡ ♦❢ TEXP ❝♦♥s✐sts ♦❢ t❤❡ ♦♥❡ ♦❢ TOSR ❛♥❞ 2x✳
❼ ❚❤❡ ❛①✐♦♠s ♦❢ TEXP ❝♦♥s✐st ♦❢ t❤♦s❡ ♦❢ TOSR ❛♥❞ ➛x➜!y❼2x y➁✳
Pr♦♣♦s✐t✐♦♥ ✾✳ ❋♦r ❛♥② f ✂ N N✱ ✐❢ ➛x➜!y“f❼x➁ y➐➐ ✐s ❛ t❤❡♦r❡♠ ♦❢ TEXP✱ t❤❡♥ t❤❡r❡
❡①✐sts ❛♥ ♦r❞✐♥❛❧ ♥✉♠❜❡r α ❅ ωω s✉❝❤ t❤❛t f❼m➁ ❇ Gα❼m➁ ❢♦r ❛❧❧ m✳
✼ ❆♥♦t❤❡r ❜♦✉♥❞✐♥❣ ❢✉♥❝t✐♦♥ F
α❉❡✜♥✐t✐♦♥ ✶✵✳ ✭❙♦♠❡ ❡①❛♠♣❧❡s ♦❢✮ ✐♥❞✉❝t✐✈❡ ❞❡✜♥✐t✐♦♥s ♦❢ Fα ♦♥ α ❇ ωω✳ F0❼m➁ m✔ 1, Fω❼k✔1➁❼m➁ Fω k✔m✔1❼m➁,
Fk✔1❼m➁ Fk❼Fk❼m➁➁, Fω2❼m➁ Fω❼m✔1➁❼m➁,
Fω❼m➁ Fm✔1❼m➁, Fωk✔1❼m➁ Fωk❼m✔1➁❼m➁, Fω✔k✔1❼m➁ Fω✔k❼Fω✔k❼m➁➁, Fωω❼m➁ Fωm✔1❼m➁.
Pr♦♣♦s✐t✐♦♥ ✶✶✳ ▲❡t k ❇ m✳
❚❤❡♥ Fk❼m➁ ❅ Fω❼m➁✱ Fω k❼m➁ ❅ Fω2❼m➁✱ ❛♥❞ Fωk❼m➁ ❅ Fωω❼m➁✳ Pr♦♦❢✳ ❇② ✐♥❞✉❝t✐♦♥ ♦♥ k✳
▲❡♠♠❛ ✶✷✳ Fk❼m➁ m ✔ 2k✱ Fω❼m➁ m ✔ 2m✔1 ❛♥❞ 2
2
2m↔➝➝
➜➝➝➙ k 2✬s
❅ Fω k❼m➁✳ Pr♦♦❢✳ ❆❣❛✐♥ ❜② ✐♥❞✉❝t✐♦♥ ♦♥ k✳
✸
✽ ❊①t❡♥❞✐♥❣ T
EXP❜② t❤❡ ✐♥❞✉❝t✐♦♥ s❝❤❡♠❛
❉❡✜♥✐t✐♦♥ ✶✸ ✭❆①✐♦♠❛t✐❝ s②st❡♠ TIND✮✳
❼ ❚❤❡ ❧❛♥❣✉❛❣❡ ♦❢ TIND ✐s t❤❡ ♦♥❡ ♦❢ TEXP✳
❼ ❚❤❡ ❛①✐♦♠s ♦❢ TIND ❝♦♥s✐st ♦❢ t❤♦s❡ ♦❢ TEXP ❛♥❞ t❤❡ ✐♥❞✉❝t✐♦♥ s❝❤❡♠❛✿ A❼0➁ & ➛x❼A❼x➁ A❼x ✔ 1➁➁ ➛xA❼x➁
❘❡♠❛r❦ ✶✹✳ TIND ✐s ♠♦r❡ ❝♦♠♠♦♥❧② ❦♥♦✇♥ ❛s P❡❛♥♦ ❛r✐t❤♠❡t✐❝✱ PA✳
Pr♦♣♦s✐t✐♦♥ ✶✺ ✭●❡♥t③❡♥ ✬✸✻✱ ❈✐❝❤♦♥✲❲❛✐♥❡r ✬✽✸✱ ❇✉❝❤❤♦❧③✲❲❛✐♥❡r ✬✽✼✮✳ ❋♦r ❛♥② f ✂ N N✱ ✐❢ ➛x➜!y“f❼x➁ y➐➐ ✐s ❛ t❤❡♦r❡♠ ♦❢ TIND✱ t❤❡♥ t❤❡r❡ ❡①✐sts ❛♥ ♦r❞✐♥❛❧ ♥✉♠❜❡r α❅ ωω s✉❝❤ t❤❛t f❼m➁ ❇ Fα❼m➁ ❢♦r ❛❧❧ m✳
Pr♦♣♦s✐t✐♦♥ ✶✻✳
✶✳ ❋♦r ❛♥② ♦r❞✐♥❛❧ ♥✉♠❜❡r α ❅ ωω ✱ ➛x➜!y“Fα❼x➁ y➐➐ ✐s ❛ t❤❡♦r❡♠ ♦❢ TIND✳
✷✳ ❋♦r ❛♥② ♦r❞✐♥❛❧ ♥✉♠❜❡r α ❈ ωω ✱ ➛x➜!y“Fα❼x➁ y➐➐ ✐s ♥♦t ❛ t❤❡♦r❡♠ ♦❢ TIND✳
▲❡♠♠❛ ✶✼ ✭●❡♥t③❡♥ ✬✸✻✮✳ ❋♦r ❛♥② ♦r❞✐♥❛❧ ♥✉♠❜❡r α ❅ ωω ✱ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐s ❛ t❤❡♦r❡♠
♦❢ TIND✿
✏❚❤❡r❡ ✐s ♥♦ ❞❡s❝❡♥❞✐♥❣ ❝❤❛✐♥ α ❆ α1❆ α2❆ ✆✑
✾ ❙✉♠♠❛r②
❼ ❋r♦♠ ●ö❞❡❧✬s s❡❝♦♥❞ ✐♥❝♦♠♣❧❡t❡♥❡ss t❤❡♦r❡♠✱ ❣✐✈❡♥ ❛ ❝♦♥s✐st❡♥t ❛①✐♦♠❛t✐❝ s②st❡♠
♦❢ ▼❛t❤❡♠❛t✐❝s✱ s♦♠❡ ♦❢ t♦t❛❧ ❢✉♥❝t✐♦♥s ❛r❡ ♥♦t ♣r♦✈❛❜❧② t♦t❛❧ ✐♥ t❤❡ s②st❡♠✳
❼ ❆ ❤✐❡r❛r❝❤② ❼Fα➁α❅ω1 ♦❢ t♦t❛❧ ❢✉♥❝t✐♦♥s ✐s ❞❡✜♥❡❞✳
❼ ❲✐t❤ ❛ ❣✐✈❡♥ ❛①✐♦♠❛t✐❝ s②st❡♠ T ✱ ❛♥ ♦r❞✐♥❛❧ ♥✉♠❜❡r αT ❅ ω1 ✐s ❛ss♦❝✐❛t❡❞✱ ✇❤✐❝❤
❡♥❥♦②s t❤❛t Fα ✐s t♦t❛❧ ✐♥ T ✐❢ α ❅ αT✱ ❛♥❞ t❤❛t Fα ✐s ♥♦t t♦t❛❧ ✐♥ T ✐❢ αT ❇ α✳
❼ ❚❤❡ ♦r❞✐♥❛❧ ♥✉♠❜❡r αT ✐♥❞✐❝❛t❡s ❛ ❜♦✉♥❞❛r② ♦❢ t❤❡ ♣r♦✈✐♥❣ ♣♦✇❡r ♦❢ T ✳
✶✵ ❈♦♥❝❧✉❞✐♥❣ r❡♠❛r❦s
▼♦st t❤❡♦r✐❡s ♦❢ ▼❛t❤❡♠❛t✐❝s ❛r❡ ❛①✐♦♠❛t✐③❛❜❧❡ ✐♥ RCA0✱ WKL0✱ ACA0✱ ATR0♦r Π11✏CA0✳
✭❘❡❝❛❧❧ Thm❼RCA0➁ ø Thm❼WKL0➁ ø Thm❼ACA0➁ ø Thm❼ATR0➁ ø Thm❼Π11✏CA0➁✮
❼ αRCA0 αWKL0 ωω ❛♥❞ αACA0 αTIND ωω ✳
❼ αATR0 ❛♥❞ αΠ1
1✏CA0 ❛r❡ ❛❧s♦ ❦♥♦✇♥✳
❼ ❚❡❝❤♥✐❝❛❧❧② αΠ1
2✏CA0 ❤❛s ❜❡❡♥ ❛❝❝❡♣t❡❞ t♦ ❜❡ ❛ ❧✐♠✐t t❤❛t ❝❛♥ ❜❡ ❛ss♦❝✐❛t❡❞ ✇✐t❤
❛①✐♦♠❛t✐❝ s②st❡♠s✳
❼ ❆ss♦❝✐❛t✐♥❣ s✉❝❤ ❛♥ ♦r❞✐♥❛❧ ✇✐t❤ Π1k✏CA0 ❢♦r ❛♥ ❛r❜✐tr❛r② k ✐s ❛ ❜✐❣ ❝❤❛❧❧❡♥❣❡✳
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