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理論計算機科学における 数学の応用: 擬似ランダムネス
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Nobutaka Shimizu
August 16, 2024
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理論計算機科学における 数学の応用: 擬似ランダムネス
東北大学数学科で2024年5月27日に行った談話会での発表資料
keyword: 擬似ランダムネス, エクスパンダーグラフ
Nobutaka Shimizu
August 16, 2024
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Transcript
ཧܭࢉػՊֶʹ͓͚Δ ֶͷԠ༻: ٖࣅϥϯμϜωε ਗ਼ਫ ৳ߴ (౦ژۀେֶ) ஊձ (20245݄27@౦େֶ)
•ཧܭࢉػՊֶͰ(૾Ҏ্ʹ)ֶͷ֓೦͕෯͘ొ •ֶͱཧܭࢉػՊֶͷڞ௨෦ͷҰͭ: ٖࣅϥϯμϜੑ ൃදͷ֓ཁ 2
•ཧܭࢉػՊֶͰ(૾Ҏ্ʹ)ֶͷ֓೦͕෯͘ొ •ֶͱཧܭࢉػՊֶͷڞ௨෦ͷҰͭ: ٖࣅϥϯμϜੑ •ʮ͜Ε͕͜Μͳͱ͜Ζʹڞ௨͕͋Δͷ͔ʂ(ڻ)ʯͱࢥͬͯ΄͍͠ ൃදͷ֓ཁ 3
•ཧܭࢉػՊֶͰ(૾Ҏ্ʹ)ֶͷ֓೦͕෯͘ొ •ֶͱཧܭࢉػՊֶͷڞ௨෦ͷҰͭ: ٖࣅϥϯμϜੑ •ʮ͜Ε͕͜Μͳͱ͜Ζʹڞ௨͕͋Δͷ͔ʂ(ڻ)ʯͱࢥͬͯ΄͍͠ •ʮงғؾʯΛհ ‣ ݫີͳఆٛূ໌ׂѪ ൃදͷ֓ཁ 4
ৗੜ׆ʹ͓͚Δܭࢉػͷར༻ 5
•ܭࢉػͷཧతͳೳྗͦͷݶքΛֶΛͬͯղ໌ (Ԡ༻ֶ) ‣ ࠷దԽΞϧΰϦζϜ ‣ ࠔੑ (ܭࢉྔԼք; ༧) ‣ άϥϑΞϧΰϦζϜ
‣ ҉߸, ֶशཧ ‣ Ϛϧίϑ࿈ ‣ ܭࢉ ‣ ྔࢠΞϧΰϦζϜ ‣ ࢄΞϧΰϦζϜ ‣ σʔλߏ ‣ etc 𝖯 ≠ 𝖭 𝖯 ཧܭࢉػՊֶ (Theoretical Computer Science) 6
TCSͱ(७ਮ)ֶͷܨ͕Γ 7 ରιϘϨϑෆࣜ ϥϯμϜΥʔΫͷղੳ Green—Taoͷఆཧ ऑֶशثͷϒʔεςΟϯά Kazhdanͷੑ࣭ (T) ཚԽ ޡΓగਖ਼ූ߸
Bogolyubov—Ruzsaͷิ ࠷ѱ͔࣌Βฏۉ࣌ͷؼண ପԁۂઢ҉߸ ପԁۂઢ ଟ༷ମͷCheegerఆ ނোੑωοτϫʔΫ άϥεϚϯଟ༷ମ 2-to-2༧ Hilbert’s Nullstellensatz Combinatorial Nullstellensatz ΞΠσΞΛഈआ ྫͷߏ ৽ͨͳઃఆ Connes ͷຒΊࠐΈ༧ MIP*=REఆཧ Baum—Connes༧ ηΩϡϦςΟ ฏۉ࣌ܭࢉྔ ੑ࣭ݕࠪ
TCSͱ(७ਮ)ֶͷܨ͕Γ 8 ରιϘϨϑෆࣜ ϥϯμϜΥʔΫͷղੳ Green—Taoͷఆཧ ऑֶशثͷϒʔεςΟϯά ཚԽ ޡΓగਖ਼ූ߸ Bogolyubov—Ruzsaͷิ ࠷ѱ͔࣌Βฏۉ࣌ͷؼண
ପԁۂઢ҉߸ ପԁۂઢ ଟ༷ମͷCheegerఆ ނোੑωοτϫʔΫ άϥεϚϯଟ༷ମ 2-to-2༧ Hilbert’s Nullstellensatz Combinatorial Nullstellensatz ΞΠσΞΛഈआ ྫͷߏ ৽ͨͳઃఆ Connes ͷຒΊࠐΈ༧ MIP*=REఆཧ Baum—Connes༧ ηΩϡϦςΟ ฏۉ࣌ܭࢉྔ Kazhdanͷੑ࣭ (T) ٖࣅϥϯμϜੑ (pseudorandomness)
•ٖࣅཚͷੜ ‣ ཚΛ͏໘ : ϥϯμϜΥʔΫ, MCMC, ֬తޯ๏, etc. ‣ ࣮ࡍͷܭࢉػͰٖࣅཚΛͬͯΔ
- : ԿΒ͔ͷؔ - ͰٖࣅཚΛͨ͘͞Μੜ ( ΛγʔυͱݺͿ) ‣ ༗໊ͳؔ : ϝϧηϯψπΠελ, ઢܗ߹ಉ๏, etc ‣ ྑ࣭ͳ(=࣍ͷग़͕༧ଌͰ͖ͳ͍)ٖࣅཚ͕ཉ͍͠ f: {0,1}32 → {0,1}32 s → f(s) → f(f(s)) → ⋯ s ٖࣅཚ 9
ٖࣅཚ 10 ग़య: e-Gov ๏ྩݕࡧ (https://elaws.e-gov.go.jp/document?lawid=503M62000000001) Χ ジ ϊཧҕһձؔಛఆෳ߹؍ޫࢪઃ۠Ҭඋ๏ࢪߦنଇ ୈ176ผද
(H30੍ఆ)
ٖࣅཚ 11 ग़య: e-Gov ๏ྩݕࡧ (https://elaws.e-gov.go.jp/document?lawid=503M62000000001) Χ ジ ϊཧҕһձؔಛఆෳ߹؍ޫࢪઃ۠Ҭඋ๏ࢪߦنଇ ୈ176ผද
(H30੍ఆ) •࣭͕ѱ͍ٖࣅཚͷࣄྫ (࣮) ‣ 2006ʹൃച͞ΕͨήʔϜιϑτʹͯʮμΠεͷ࣍ͷग़ͷۮح͕ਪଌͰ͖Δʯ ͱ͍͏க໋తͳόά͕ݟ͔ͭΓɺճऩʹࢸͬͨ.
•ٖࣅཚʹཉ͍͠ੑ࣭ ‣ Ұ༷ϥϯμϜͳ ʹର͠, ͕Ұ༷ϥϯμϜ - ݪཧతʹෆՄೳ ( ͕ܾ·Δͱ ܾ·Δ͔Β)
s (s, f(s)) s f(s) ٖࣅϥϯμϜੑ 12
•ٖࣅཚʹཉ͍͠ੑ࣭ ‣ Ұ༷ϥϯμϜͳ ʹର͠, ͕Ұ༷ϥϯμϜ - ݪཧతʹෆՄೳ ( ͕ܾ·Δͱ ܾ·Δ͔Β)
•ٖࣅϥϯμϜωε ‣ Ұ༷ϥϯμϜͳ ʹର͠, ͕Ұ༷ϥϯμϜͬΆ͘ݟ͑Δ ‣ Ұ༷ͱࣝผͰ͖ͳ͍Α͏ͳ s (s, f(s)) s f(s) s (s, f(s)) ٖࣅϥϯμϜੑ 13 ݅Λ؇
ͷࣝผ 14 01010101010101010101 01000111001111001111 ͋Δ ͔Βੜ͞Εͨ20จࣈ 𝒟 ࣝผऀ A ͬͪ͜Ұ༷ϥϯμϜ͡Όͳ͍
ͬͪ͜Ұ༷ϥϯμϜͰ͋Ζ͏ Ұ༷ ͔Βੜ͞Εͨ20จࣈ 𝒰
ͷࣝผ 15 01010101010101010101 01000111001111001111 ͋Δ ͔Βੜ͞Εͨ20จࣈ 𝒟 Ұ༷ ͔Βੜ͞Εͨ20จࣈ 𝒰
ࣝผऀ A ؔ : 20จࣈ 0 or 1 A ↦ ࣝผऀ ͕ ͱ Λ -ࣝผ ͢Δ A 𝒟 𝒰 ε def ⟺ Pr[A( 𝒟 ) = 1] − Pr[A( 𝒰 ) = 1] > ε
• Λ ʮ01010101010ʯʮ10101010101ʯͷͲͪΒ͔͕֬ Ͱग़ݱ •ࣝผऀ : ‣ 0ͱ1͕ަޓͳΒ1, ͦ͏Ͱͳ͍ͳΒ0Λग़ྗ •
0.999-ࣝผ ‣ ‣ 𝒟 1/2 A(s) A Pr[A( 𝒟 ) = 1] = 1 Pr[A( 𝒰 ) = 1] = 2/211 ≈ 0.001 ͷࣝผ (ྫ) 16
•ٖࣅϥϯμϜωε ‣ ੍ݶ͞ΕͨࣝผऀͷΫϥε Λߟ͑Δ (ଟ߲ࣜ࣌ؒΞϧΰϦζϜͳͲ) 𝒜 ٖࣅϥϯμϜੑ 17
ʹରͯ͠ -ٖࣅϥϯμϜ Ͱ͋Δ ҙͷ ͕ ͱ Λ -ࣝผ͠ͳ͍ 𝒟 𝒜 ε def ⟺ A ∈ 𝒜 𝒟 𝒰 ε શશೳͷࣝผऀ ੍ݶ͞Εͨࣝผऀ 011010100 ૉ൪ͷจࣈ͕1ͩʂ 0ͱ1͕ަޓ͡Όͳ͍͔Β Ұ༷ϥϯμϜ͔ͳ͊
•ٖࣅϥϯμϜωε ‣ ੍ݶ͞ΕͨࣝผऀͷΫϥε Λߟ͑Δ (ଟ߲ࣜ࣌ؒΞϧΰϦζϜͳͲ) 𝒜 ٖࣅϥϯμϜੑ 18
ʹରͯ͠ -ٖࣅϥϯμϜ Ͱ͋Δ ҙͷ ͕ ͱ Λ -ࣝผ͠ͳ͍ 𝒟 𝒜 ε def ⟺ A ∈ 𝒜 𝒟 𝒰 ε ✓ ܭࢉྔతٖࣅϥϯμϜੑ = ͕ޮతͳΞϧΰϦζϜͷ (ྫ: ଟ߲ࣜ࣌ؒΞϧΰϦζϜ) ✓ ߹ͤతٖࣅϥϯμϜੑ = ͕߹ͤతʹఆ·Δؔͷ (ྫ: ࣍ଟ߲ࣜ) 𝒜 𝒜 d
ܭࢉྔతٖࣅϥϯμϜੑ ฏۉ࣌ܭࢉྔ
•ܭࢉྔ: ܭࢉͷෳࡶੑ (࣌ؒ, ۭؒetc) ΛਤΔई ‣ ͕༩͑ΒΕͨͱ͖, Λܭࢉ͢ΔखؒͲΕ͘Β͍͔? x f(x)
ฏۉ࣌ࠔੑ 20
•ܭࢉྔ: ܭࢉͷෳࡶੑ (࣌ؒ, ۭؒetc) ΛਤΔई ‣ ͕༩͑ΒΕͨͱ͖, Λܭࢉ͢ΔखؒͲΕ͘Β͍͔? •࠷ѱ࣌ܭࢉྔ: ࠷ѱͳೖྗʹର͢ΔΞϧΰϦζϜͷڍಈ
‣ ͘͝গͷίʔφʔέʔεʹӨڹ͞Ε͏Δ ‣ “pessimism of worst-case analysis” [Frieze, McDiarmid, 1986] ‣ ༧ x f(x) 𝖯 ≠ 𝖭 𝖯 ฏۉ࣌ࠔੑ 21
•ܭࢉྔ: ܭࢉͷෳࡶੑ (࣌ؒ, ۭؒetc) ΛਤΔई ‣ ͕༩͑ΒΕͨͱ͖, Λܭࢉ͢ΔखؒͲΕ͘Β͍͔? •࠷ѱ࣌ܭࢉྔ: ࠷ѱͳೖྗʹର͢ΔΞϧΰϦζϜͷڍಈ
‣ ͘͝গͷίʔφʔέʔεʹӨڹ͞Ε͏Δ ‣ “pessimism of worst-case analysis” [Frieze, McDiarmid, 1986] ‣ ༧ •ฏۉ࣌ܭࢉྔ: ฏۉతͳೖྗʹର͢ΔΞϧΰϦζϜͷڍಈ ‣ গͳ͍ίʔφʔέʔεʹӨڹ͞Εʹ͍͘ x f(x) 𝖯 ≠ 𝖭 𝖯 ฏۉ࣌ࠔੑ 22
•ؔ ͷܭࢉ͕ฏۉ࣌ࠔ ‣ Ұ༷ϥϯμϜͳ ʹରͯ͠, ͷܭࢉ͕͍͠ ‣ ྫ: 10ܻͷϥϯμϜͳೋͭͷૉͷੵͷૉҼղ͍͠ (RSA҉߸)
f x f(x) ฏۉ࣌ࠔੑ 23
•ؔ ͷܭࢉ͕ฏۉ࣌ࠔ ‣ Ұ༷ϥϯμϜͳ ʹରͯ͠, ͷܭࢉ͕͍͠ ‣ ྫ: 10ܻͷϥϯμϜͳೋͭͷૉͷੵͷૉҼղ͍͠ (RSA҉߸)
•ฏۉ࣌ࠔͳؔ ٖࣅཚੜث ‣ ʮ͍͠ʯͱ͍͏ωΨςΟϒͳੑ࣭ΛϙδςΟϒͳ݁ՌʹԠ༻ ‣ ͕ฏۉ࣌ࠔ ҙͷଟ߲ࣜ࣌ؒΞϧΰϦζϜʹͱٖͬͯࣅϥϯμϜ - Ұ༷ϥϯμϜͳ ʹରͯ͠ ͷܭࢉ͍͔͠Β f x f(x) ⇒ f ⟺ (s, f(s)) s f(s) ฏۉ࣌ࠔੑ 24 [Nisan, Wigderson, 1994]
҉߸ 25 •҉߸ •डͨ͠ୈࡾऀʹ͍͔ͳΔใ࿙Ε͍͚ͯͳ͍ ‣ ҉߸จʹԿΒ͔ͷ౷ܭతಛ͕͋ͬͨΒඇࣗ໌ͳใ͕࿙ΕΔ ‣ ୈࡾऀʹͱͬͯϥϯμϜͳจࣈྻʹݟ͑Δ͖ Apple 0011101000100
҉߸ 26 •҉߸ •डͨ͠ୈࡾऀʹ͍͔ͳΔใ࿙Ε͍͚ͯͳ͍ ‣ ҉߸จʹԿΒ͔ͷ౷ܭతಛ͕͋ͬͨΒඇࣗ໌ͳใ͕࿙ΕΔ ‣ ୈࡾऀʹͱͬͯϥϯμϜͳจࣈྻʹݟ͑Δ͖ - ୈࡾऀ੍ݶ͞ΕͨܭࢉೳྗΛ༗͢ΔͱԾఆ
Apple 0011101000100
҉߸ 27 •҉߸ •༗໊ͳ҉߸ํࣜ ‣ RSA҉߸ (ϥϯμϜͳڊେͳೋͭͷૉͷੵͷૉҼղͷࠔੑΛԾఆ) ‣ ֨ࢠ҉߸ (ϥϯμϜͳ֨ࢠ্Ͱͷ࠷֨ࢠͷࠔੑΛԾఆ)
‣ ڀۃతͳඪ: ͷԾఆͷԼͰ҆શͳ҉߸Λ࡞Δ •ฏۉ࣌ࠔͳ 㱺 ΄ͱΜͲͷೖྗͰਖ਼͘͠ղ͚ͳ͍ 𝖯 ≠ 𝖭 𝖯 Apple 0011101000100
҉߸ 28 ग़య: e-Gov ๏ྩݕࡧ (https://elaws.e-gov.go.jp/document?lawid=413M60000418002)
҉߸ 29 ग़య: e-Gov ๏ྩݕࡧ (https://elaws.e-gov.go.jp/document?lawid=413M60000418002) •๏ྩͷจݴʹʮૉҼղʯʮ༗ݶମʯʮପԁۂઢʯ •େਉ͕ೝΊΕOKΒ͍͠
߹ͤతٖࣅϥϯμϜੑ ΤΫεύϯμʔάϥϑ
• : -ਖ਼ଇάϥϑ ‣ શʹଓ͍ͯ͠Δล͕ ຊ • : ୯७ϥϯμϜΥʔΫͷભҠ֬ߦྻ ‣
୯७ϥϯμϜΥʔΫ : Ұ༷ϥϯμϜͳྡʹભҠ ‣ G d d P P(u, v) = { 1 d 0 ΤΫεύϯμʔάϥϑ 31 3-ਖ਼ଇάϥϑ ͕ลΛͳ͢ {u, v} ͦΕҎ֎
• : -ਖ਼ଇάϥϑ ‣ શʹଓ͍ͯ͠Δล͕ ຊ • : ୯७ϥϯμϜΥʔΫͷભҠ֬ߦྻ ‣
୯७ϥϯμϜΥʔΫ : Ұ༷ϥϯμϜͳྡʹભҠ ‣ G d d P P(u, v) = { 1 d 0 ΤΫεύϯμʔάϥϑ 32 3-ਖ਼ଇάϥϑ ͕ลΛͳ͢ {u, v} ͦΕҎ֎ ఆٛ (ΤΫεύϯμʔάϥϑ) ભҠ֬ߦྻ ͷݻ༗ ͕ Λຬͨ͢ͱ͖ -ΤΫεύϯμʔͱ͍͏. P 1 = λ1 ≥ … ≥ λn ≥ − 1 max{|λ2 |, |λn |} ≤ λ λ
• : -ਖ਼ଇάϥϑ ‣ શʹଓ͍ͯ͠Δล͕ ຊ • : ୯७ϥϯμϜΥʔΫͷભҠ֬ߦྻ ‣
୯७ϥϯμϜΥʔΫ : Ұ༷ϥϯμϜͳྡʹભҠ ‣ G d d P P(u, v) = { 1 d 0 ΤΫεύϯμʔάϥϑ 33 3-ਖ਼ଇάϥϑ ͕ลΛͳ͢ {u, v} ͦΕҎ֎ ఆٛ (ΤΫεύϯμʔάϥϑ) ભҠ֬ߦྻ ͷݻ༗ ͕ Λຬͨ͢ͱ͖ -ΤΫεύϯμʔͱ͍͏. P 1 = λ1 ≥ … ≥ λn ≥ − 1 max{|λ2 |, |λn |} ≤ λ λ ؆୯ͷͨΊৗʹਖ਼ଇੑΛԾఆ ( ରশͳͷͰ࣮ݻ༗Λͭ) P
•ϥϯμϜΥʔΫͷऩଋੑ ‣ άϥϑ͕͋Δ݅Λຬͨ͢ͱ ͷ ্ͷҰ༷ʹҰҙʹऩଋ ‣ ऩଋͷ͞ͲΕ͘Β͍͔? Xt V ϥϯμϜΥʔΫͱΤΫεύϯμʔ
34 ೋ෦άϥϑ্Ͱऩଋ͠ͳ͍ ඇ࿈݁ͩͱऩଋઌ͕ҰҙͰͳ͍
•ϥϯμϜΥʔΫͷऩଋੑ ‣ άϥϑ͕͋Δ݅Λຬͨ͢ͱ ͷ ্ͷҰ༷ʹҰҙʹऩଋ ‣ ऩଋͷ͞ͲΕ͘Β͍͔? •ΤΫεύϯμʔάϥϑ ‣ ϥϯμϜΥʔΫͷऩଋ͕͍άϥϑ
Xt V ϥϯμϜΥʔΫͱΤΫεύϯμʔ 35 ೋ෦άϥϑ্Ͱऩଋ͠ͳ͍ ඇ࿈݁ͩͱऩଋઌ͕ҰҙͰͳ͍
•ૄͳΤΫεύϯμʔάϥϑ : ૄͳͷʹ࿈݁ੑ͕ڧ͍ ΤΫεύϯμʔͷݟͨ 36 ؆୯ʹஅͰ͖ͦ͏ அ͠ʹ͍͘
‣ ( ) ‣ શͯͷ -ΤΫεύϯμʔ •ఆ ʹରͯ͠
-ΤΫεύϯμʔଘࡏ͢Δ͔? ‣ ϥϯμϜʹ࡞Δͱਖ਼ͷ֬Ͱ (֬తख๏) ‣ ϥϯμϜਖ਼ଇάϥϑ •ϥϯμϜωεΛΘͣʹߏͰ͖Δ͔? (ཚ) → ∞ i → ∞ Gi λ λ < 1 λ λ = 2 d − 1 d + 0.01 ΤΫεύϯμʔάϥϑ 37 -ਖ਼ଇάϥϑͷ -ΤΫεύϯμʔͰ͋Δ d (Gi )i∈ℕ λ def ⟺ [Friedman, 2008]
•తͳߏ ‣ έΠϦʔάϥϑ (܈ͷ࡞༻ΛௐΔॏཁͳಓ۩) ‣ Margulisͷߏ (1973) … ‣ Lubotzky,
Phillips, and Sarnak (1988) ‣ Margulis (1988) ‣ Morgenstern (1994) ‣ ࣍ ͕ಛผͳ߹ͷߏ λ = 5 2 8 < 0.9 d ΤΫεύϯμʔͷߏ 38
•తͳߏ ‣ έΠϦʔάϥϑ (܈ͷ࡞༻ΛௐΔॏཁͳಓ۩) ‣ Margulisͷߏ (1973) … ‣ Lubotzky,
Phillips, and Sarnak (1988) ‣ Margulis (1988) ‣ Morgenstern (1994) ‣ ࣍ ͕ಛผͳ߹ͷߏ λ = 5 2 8 < 0.9 d ΤΫεύϯμʔͷߏ 39 ϥϚψδϟϯάϥϑ (“࠷దͳ”ΤΫεύϯμʔੑΛͭ) λ ≈ 2 d − 1 d
•߹ͤతͳߏ ‣ తͳߏͩͱײ (ͳͥΤΫεύϯμʔੑ͕Γཱͭͷ͔? )͕͍͠ ‣ Reingold, Vadhan, Wigderson (2002)
- δάβάੵ ‣ Marcus, Spielman, Srinivasta (2015) - શͯͷೋ෦ϥϚψδϟϯάϥϑͷߏ - ৫Γࠞͥଟ߲ࣜ (interlacing polynomial) •ະղܾ: ࣍7ͷϥϚψδϟϯάϥϑͷߏ ΤΫεύϯμʔͷߏ (ଓ) 40
•ཁૉ ͷू߹ ʹର͠, ࣍ͷ ্ͷ Λߟ͑Δ ‣ ্ͷ -ਖ਼ଇΤΫεύϯμʔάϥϑ Λߟ͑Δ
‣ ΛҰ༷ϥϯμϜʹબͿ ‣ Λ࢝ͱ͢Δ͞ ͷϥϯμϜΥʔΫͷܦ༝ Λग़ྗ n V Vℓ 𝒟 V d G = (V, E) u1 u1 ℓ − 1 (u1 , …, uℓ ) ٖࣅϥϯμϜੑ 41 u1 u2 uℓ
ٖࣅϥϯμϜੑ 42 άϥϑ ͕ -ΤΫεύϯμʔͳΒ, ʹର͠ -ٖࣅϥϯμϜ G λ
𝒟 𝒜 = {AS : S ⊆ V} (λ/4) ෦ू߹ ʹର͠, Λ S ⊆ V AS : Vℓ → {0,1} AS (u1 , …, uℓ ) = { 1 0 {u1 , …, uℓ } ∩ S ≠ ∅ ͦΕҎ֎ ఆཧ u1 u2 uℓ S = ్தͰ ͷΛ௨ա͔ͨ͠Ͳ͏͔ AS (u1 , …, uℓ ) S
• ͷݩʹʮ͋ͨΓʯorʮͣΕʯ͕͋Δ ‣ গͳ͘ͱ ݸͷʮ͋ͨΓʯ͕͋Δ - Ұ༷ϥϯμϜʹҾ͍͕ͨ͋ͨΔ֬ = •ಠཱҰ༷ϥϯμϜʹ ճ͘͡ΛҾ͘
‣ ʮ͋ͨΓʯ͕ҰճҎ্ग़Δ֬ = ‣ ճҾ͚, 99%ͷ֬Ͱ͋ͨΓΛҾ͚Δ ‣ ͜ͷͱ͖, ϏοτͷϥϯμϜωε͕ඞཁ V δn δ ℓ 1 − (1 − δ)ℓ ℓ = 10/δ 10 log2 n δ Ԡ༻ 43
• ͷݩʹʮ͋ͨΓʯorʮͣΕʯ͕͋Δ ‣ গͳ͘ͱ ݸͷʮ͋ͨΓʯ͕͋Δ - Ұ༷ϥϯμϜʹҾ͍͕ͨ͋ͨΔ֬ = • -ΤΫεύϯμʔάϥϑ্Ͱ
ʹैͬͯ ճ͘͡ΛҾ͘ ‣ άϥϑͷ࣍ ʹґଘ͠ͳ͍ఆʹͰ͖Δ ‣ -ٖࣅϥϯμϜͳͷͰ98%ͷ֬Ͱʮ͋ͨΓʯΛҾ͘ ‣ ༻͍ͨϥϯμϜωε V δn δ 0.04 𝒟 ℓ d n 𝒟 0.01 log2 n + 10 log2 d δ Ԡ༻ 44 u1 u2 uℓ ʮ͋ͨΓʯ ಠཱαϯϓϦϯάΑΓগͳ͍ʂ
•PCPఆཧ •ޡΓగਖ਼ූ߸ •ٖࣅཚੜث •ϋογϡؔ •ฏۉ࣌ܭࢉྔ ΤΫεύϯμʔάϥϑͷԠ༻ 45 [Charles, ’09] [Guruswami,
Kabanets, ’08], [Goldreich, Impagliazzo, Levin, Venkatesan, Zuckerman, 90] [Goldreich, 00] [Sipser, Spielman, 96] [Dinur, 07]
•୯ମෳମͷΤΫεύϯμʔੑ ‣ ߴ࣍ݩΤΫεύϯμʔ •ͷະղܾͷղܾͷཱऀ ‣ ϚτϩΠυʹؔ͢ΔMihail—Vazirani༧ ‣ ہॴݕࠪՄೳޡΓగਖ਼ූ߸ͷߏ •ຊߨٛͰհ ۙͷಈ:
ߴ࣍ݩΤΫεύϯμʔ 46
•ٖࣅϥϯμϜωε ‣ ७ਮֶͱTCSͷ྆ํʹݱΕΔ֓೦ ‣ , ܈, زԿֶʹԠ༻͞Ε͍ͯΔ(Β͍͠) •ΤΫεύϯμʔάϥϑ ‣ έΠϦʔάϥϑΛͬͯߏ
‣ ϥϯμϜωεΛʮઅʯͯ͘͠͡ΛҾ͘ํ๏ •ߴ࣍ݩΤΫεύϯμʔ ‣ ۙͷTCSͰϗοτͳ ‣ ߨٛͰΓ·͢ʂ ·ͱΊ 47 [Lubotzky, ’12]