SLIDE 1
r rts rtrst - - PowerPoint PPT Presentation
r rts rtrst - - PowerPoint PPT Presentation
r rts rtrst s t r sttt r rt
SLIDE 2
SLIDE 3
❚♦♣♦❧♦❣✐❝❛❧ ❛♥❞ ❞❡r✐✈❛t✐♦♥❛❧ s❡♠❛♥t✐❝s
❙❡♠❛♥t✐❝s ♦♥ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡s ❝❛♥ ❜❡ ❜✉✐❧t ✉s✐♥❣ ❝❧♦s✉r❡ ♦♣❡r❛t♦r cl ✇❤❡r❡ cl(A) ✐s t❤❡ ❝❧♦s✉r❡ ♦❢ A✳ ❚❤❡ s❡♠❛♥t✐❝s ❞❡✜♥❡❞ ❧✐❦❡ t❤✐s✿ Vcl(♦φ) = cl(Vcl(φ)) ❖r ✉s✐♥❣ ❞❡r✐✈❛t✐✈❡ ♦♣❡r❛t♦r d✱ ✇❤❡r❡ d(A) ✐s t❤❡ s❡t ♦❢ ❛❧❧ ❧✐♠✐t ♣♦✐♥ts ♦❢ A✳ ❚❤❡ s❡♠❛♥t✐❝s ❞❡✜♥❡❞ ❧✐❦❡ t❤✐s✿ Vd(♦φ) = d(Vd(φ)) ❝❧♦s✉r❡ s❡♠❛♥t✐❝s ❞❡r✐✈❛t✐♦♥❛❧ s❡♠❛♥t✐❝s ❛❧❧ s♣❛❝❡s S4 ✭▼❝❑✐♥s❡② ✫ ❚❛rs❦✐✬✶✾✹✹✮ wK4 ✭❊s❛❦✐❛✬✶✾✽✶✮ Q✱ ❈❛♥t♦r s♣❛❝❡ S4 D4 ✭❙❤❡❤t♠❛♥✬✶✾✾✵✮ R S4 D4 + G2 ✭❙❤❡❤t♠❛♥✬✷✵✵✵✮ Rn, n ≥ 2 S4 D4 + G1 ✭❙❤❡❤t♠❛♥✬✶✾✾✵✮ wK4 = K + ♦♦p → ♦p ∨ p D4 = K + ♦♦p → ♦p + ♦⊤
SLIDE 4
❚❤❡ ♣r♦❞✉❝t ♦❢ ❑r✐♣❦❡ ❢r❛♠❡s
❋♦r t✇♦ ❢r❛♠❡s F1 = (W1, R1) ❛♥❞ F2 = (W2, R2) F1 × F2 = (W1 × W2, R∗
1, R∗ 2), ✇❤❡r❡ (a1, a2)R∗ 1(b1, b2) ⇔ a1R1b1 & a2 = b2
(a1, a2)R∗
2(b1, b2) ⇔ a1 = b1 & a2R2b2
❋♦r t✇♦ ❧♦❣✐❝s L1 ❛♥❞ L2 L1 × L2 = Log({F1 × F2 | F1 | = L1 & F2 | = L2}) ✭❙❤❡❤t♠❛♥✱ ✶✾✼✽✮ ❋♦r t✇♦ ❝❧❛ss❡s ♦❢ ❢r❛♠❡s F1 ❛♥❞ F2 Log({F1 × F2 | F1 ∈ F1 & F2 ∈ F2}) ⊇ Log(F1) ∗ Log(F2)+ +12p ↔ 12p + ♦12p → 2♦1p✳ K × K = K ∗ K + 12p ↔ 12p + ♦12p → 2♦1p S4 × S4 = S4 ∗ S4 + 12p ↔ 12p + ♦12p → 2♦1p ✳ ✳ ✳
SLIDE 5
❚❤❡ ♣r♦❞✉❝t ♦❢ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡s
✭✈❛♥ ❇❡♥t❤❡♠ ❡t ❛❧✱ ✷✵✵✺✮ ❋♦r t✇♦ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ X1 = (X1, τ1) ❛♥❞ X2 = (X2, τ2) X1 × X2 = (X1 × X2, τ ∗
1 , τ ∗ 2 ), ✇❤❡r❡ τ ∗ 1 ❤❛s ❜❛s❡ {U1 × x2 | U1 ∈ τ1 & x2 ∈ X2}
τ ∗
2 ❤❛s ❜❛s❡ {x1 × U2 | x1 ∈ X1 & U2 ∈ τ2}
SLIDE 6
❚❤❡ ♣r♦❞✉❝t ♦❢ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡s
✭✈❛♥ ❇❡♥t❤❡♠ ❡t ❛❧✱ ✷✵✵✺✮ ❋♦r t✇♦ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ X1 = (X1, τ1) ❛♥❞ X2 = (X2, τ2) X1 × X2 = (X1 × X2, τ ∗
1 , τ ∗ 2 ), ✇❤❡r❡ τ ∗ 1 ❤❛s ❜❛s❡ {U1 × x2 | U1 ∈ τ1 & x2 ∈ X2}
τ ∗
2 ❤❛s ❜❛s❡ {x1 × U2 | x1 ∈ X1 & U2 ∈ τ2}
x y
SLIDE 7
❚❤❡ ♣r♦❞✉❝t ♦❢ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡s
✭✈❛♥ ❇❡♥t❤❡♠ ❡t ❛❧✱ ✷✵✵✺✮ ❋♦r t✇♦ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ X1 = (X1, τ1) ❛♥❞ X2 = (X2, τ2) X1 × X2 = (X1 × X2, τ ∗
1 , τ ∗ 2 ), ✇❤❡r❡ τ ∗ 1 ❤❛s ❜❛s❡ {U1 × x2 | U1 ∈ τ1 & x2 ∈ X2}
τ ∗
2 ❤❛s ❜❛s❡ {x1 × U2 | x1 ∈ X1 & U2 ∈ τ2}
❋♦r t✇♦ ❧♦❣✐❝s L1 ❛♥❞ L2 L1 ×t L2 = Log({X1 × X2 | X1 | = L1 & X2 | = L2} S4 ×t S4 = Log(Q × Q) = S4 ∗ S4 ✭✈❛♥ ❇❡♥t❤❡♠ ❡t ❛❧✱ ✷✵✵✺✮ Log(R × R) = S4 ∗ S4 ✭❑r❡♠❡r✱ ✷✵✶✵❄✮ Log(Cantor × Cantor) = S4 ∗ S4 ❞✲❧♦❣✐❝ ♦❢ ♣r♦❞✉❝t ♦❢ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡s ✇❛s ❝♦♥s✐❞❡r❡❞ ❜② ▲✳ ❯r✐❞✐❛ ✭✷✵✶✶✮✳ Log d(Q × Q) = D4 ∗ D4
- ❡♥❡r❛❧✐③❛t✐♦♥ t♦ ♥❡✐❣❤❜♦r❤♦♦❞ ❢r❛♠❡s ✇❛s ❞♦♥❡ ❜② ❑✳ ❙❛♥♦ ✭✷✵✶✶✮✳
SLIDE 8
❑♥♦✇♥ r❡s✉❧ts
❚❤❡♦r❡♠ ✭✷✵✶✷✮
▲❡t L1 ❛♥❞ L2 ❜❡ ❢r♦♠ t❤❡ s❡t {D, T, D4, S4} t❤❡♥ L1 ×n L2 = L1 ∗ L2. ◆♦t str❛✐❣❤t❢♦r✇❛r❞ ❜✉t st✐❧❧ ❛
❈♦r♦❧❧❛r②
■♥ ❞❡r✐✈❛t✐♦♥❛❧ s❡♠❛♥t✐❝s ✶✳ D4 ×d D4 = D4 ∗ D4✳ ✷✳ ❬❯r✐❞✐❛✬✷✵✶✶❪ Logd(Q × Q) = D4 ∗ D4 ◆♦t❡ t❤❛t ❛❧❧ t❤❡s❡ ❧♦❣✐❝s ✐♥❝❧✉❞❡ s❡r✐❛❧✐t②✿ ¬⊥✳
SLIDE 9
❑♥♦✇♥ r❡s✉❧ts
❚❤❡♦r❡♠ ✭✷✵✶✷✮
▲❡t L1 ❛♥❞ L2 ❜❡ ❢r♦♠ t❤❡ s❡t {D, T, D4, S4} t❤❡♥ L1 ×n L2 = L1 ∗ L2. ◆♦t str❛✐❣❤t❢♦r✇❛r❞ ❜✉t st✐❧❧ ❛
❈♦r♦❧❧❛r②
■♥ ❞❡r✐✈❛t✐♦♥❛❧ s❡♠❛♥t✐❝s ✶✳ D4 ×d D4 = D4 ∗ D4✳ ✷✳ ❬❯r✐❞✐❛✬✷✵✶✶❪ Logd(Q × Q) = D4 ∗ D4 ◆♦t❡ t❤❛t ❛❧❧ t❤❡s❡ ❧♦❣✐❝s ✐♥❝❧✉❞❡ s❡r✐❛❧✐t②✿ ¬⊥✳
SLIDE 10
❲✐t❤♦✉t s❡r✐❛❧✐t②
■t ✐s ♥♦t t❤❡ ❝❛s❡ ❢♦r ❧♦❣✐❝ K✦
▲❡♠♠❛
❋♦r ❛♥② t✇♦ ♥✲❢r❛♠❡s X1 ❛♥❞ X2 X1 × X2 | = 1⊥ → 21⊥. ❆♥❞ ❡✈❡♥ ♠♦r❡✱ ❢♦r ❛♥② ❝❧♦s❡❞ 1✲❢r❡❡ ❢♦r♠✉❧❛ φ ❛♥❞ ❛♥② ❝❧♦s❡❞ 2✲❢r❡❡ ❢♦r♠✉❧❛ ψ X1 × X2 | = φ → 1φ, X1 × X2 | = ψ → 2ψ.
Pr♦♦❢✳
X1 × X2, (x, y) | = 1⊥ ⇐ ⇒ ∅ ∈ τ ′
1(x, y) ⇐
⇒ ∅ ∈ τ1(x) ⇐ ⇒ ∀y′ ∈ X2 (∅ ∈ τ ′
1(x, y′)) ⇐
⇒ ∀y′ ∈ X2 (X1 × X2, (x, y′) | = 1⊥) = ⇒ X1 × X2, (x, y) | = 21⊥. ❍❡♥❝❡✱ X1 × X2 | = 1⊥ → 21⊥✳
SLIDE 11
❲✐t❤♦✉t s❡r✐❛❧✐t②
■t ✐s ♥♦t t❤❡ ❝❛s❡ ❢♦r ❧♦❣✐❝ K✦
▲❡♠♠❛
❋♦r ❛♥② t✇♦ ♥✲❢r❛♠❡s X1 ❛♥❞ X2 X1 × X2 | = 1⊥ → 21⊥. ❆♥❞ ❡✈❡♥ ♠♦r❡✱ ❢♦r ❛♥② ❝❧♦s❡❞ 1✲❢r❡❡ ❢♦r♠✉❧❛ φ ❛♥❞ ❛♥② ❝❧♦s❡❞ 2✲❢r❡❡ ❢♦r♠✉❧❛ ψ X1 × X2 | = φ → 1φ, X1 × X2 | = ψ → 2ψ.
Pr♦♦❢✳
❙✐♥❝❡ ψ ❞♦❡s ♥♦t ❝♦♥t❛✐♥ ♥❡✐t❤❡r 2✱ ♥♦r ✈❛r✐❛❜❧❡s✱ ✐ts ✈❛❧✉❡ ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ t❤❡ s❡❝♦♥❞ ❝♦♦r❞✐♥❛t❡✳ ▲❡t F = X1 × X2✳ ❙♦ F, (x, y) | = ψ✱ t❤❡♥ ∀y′(F, (x, y′) | = ψ)✱ ❤❡♥❝❡✱ F, (x, y) | = 2ψ✳
SLIDE 12
❲✐t❤♦✉t s❡r✐❛❧✐t②
▲❡♠♠❛
❋♦r ❛♥② t✇♦ ♥✲❢r❛♠❡s X1 ❛♥❞ X2 X1 × X2 | = 1⊥ → 21⊥. ❆♥❞ ❡✈❡♥ ♠♦r❡✱ ❢♦r ❛♥② ❝❧♦s❡❞ 1✲❢r❡❡ ❢♦r♠✉❧❛ φ ❛♥❞ ❛♥② ❝❧♦s❡❞ 2✲❢r❡❡ ❢♦r♠✉❧❛ ψ X1 × X2 | = φ → 1φ, X1 × X2 | = ψ → 2ψ.
❉❡✜♥✐t✐♦♥
❋♦r t✇♦ ✉♥✐♠♦❞❛❧ ❧♦❣✐❝s L1 ❛♥❞ L2✱ ✇❡ ❞❡✜♥❡ ✇❡❛❦ ❝♦♠♠✉t❛t♦r L1, L2 = L1 ∗ L2 + ∆, ✇❤❡r❡ ∆ = {φ → 2φ | φ ✐s ❝❧♦s❡❞ ❛♥❞ 2✲❢r❡❡}∪{ψ → 1ψ | ψ ✐s ❝❧♦s❡❞ ❛♥❞ 1✲❢r❡❡} .
▲❡♠♠❛
❋♦r ❛♥② t✇♦ ♥♦r♠❛❧ ♠♦❞❛❧ ❧♦❣✐❝s L1 ❛♥❞ L2 L1, L2 ⊆ L1 ×n L2✳ ◆♦t❡ t❤❛t ✐❢ ♦⊤ ∈ L1 ∩ L2 t❤❡♥ L1 ∗ L2 | = ∆✳
SLIDE 13
❈♦♠♣❧❡t❡♥❡ss r❡s✉❧ts
❚❤❡♦r❡♠ ✭✷✵✶✹✮
K ×n K = K, K.
❚❤❡♦r❡♠
■❢ ❧♦❣✐❝s L1 ❛♥❞ L2 ❛r❡ ❛①✐♦♠❛t✐③❛❜❧❡ ❜② ❝❧♦s❡❞ ❢♦r♠✉❧❛s ❛♥❞ ❜② ❛①✐♦♠s ❧✐❦❡ p → kp t❤❡♥ L1 ×n L2 = L1, L2.
❈♦r♦❧❧❛r②
K4 ×d K4 = K4, K4.
SLIDE 14
▲♦❣✐❝ ❙✺
❲❡ ♣✉t ∆1 = {φ → 2φ | φ ✐s ❝❧♦s❡❞ ❛♥❞ 2✲❢r❡❡} , com12 = 12p → 21p, com21 = 21p → 12p, chr = ♦12p → 2♦1p.
❚❤❡♦r❡♠
■❢ ❧♦❣✐❝ L ✐s ❛①✐♦♠❛t✐③❛❜❧❡ ❜② ❝❧♦s❡❞ ❢♦r♠✉❧❛s ❛♥❞ ❜② ❛①✐♦♠s ❧✐❦❡ p → kp t❤❡♥ L ×n S5 = L ∗ S5 + ∆1 + com12 + chr. ❋♦r L = S4 ✇❛s ♣r♦✈❡❞ ❜② ❑r❡♠❡r ✐♥ ✷✵✶✶✳
SLIDE 15
❍♦✇ t♦ ♣r♦✈❡
P▲❆◆ ❲❡ ❤❛✈❡ t✇♦ ❧♦❣✐❝s L1 ❛♥❞ L2✳ ▲❡t Γi ❛r❡ ❛❧❧ ❛①✐♦♠s ❢r♦♠ Li ♦❢ ❢♦r♠ p → kp✳ ❈❛♥♦♥✐❝✐t② ♦❢ t❤❡ ❧♦❣✐❝ L1, L2✳ ⇓ ❈♦♥str✉❝t F1 | = L1 ❛♥❞ F2 | = L2✱ ❛♥❞ F1, F2 ։ FL1,L2✱ ❛♥❞ F1, F2 | = ∆✳ ⇓ ❈♦♥str✉❝t Nω
Γ1(F1) × Nω Γ2(F2) ։ N
- F1, F2Γ1∪Γ2
✳ ⇓ ❈❤❡❝❦ t❤❛t Nω
Γ1(F1) |
= L1 ❛♥❞ Nω
Γ2(F2) |
= L2 ❍❡r❡ ·Γ ✐s ❛ s♣❡❝✐❛❧ ♦♣❡r❛t✐♦♥ ✇❤✐❝❤ ♠❛❦❡s s✉r❡ t❤❛t ✐❢ p → kp ∈ Γ t❤❡♥ t❤✐s ❢♦r♠✉❧❛ ✐s ✈❛❧✐❞✳
SLIDE 16
❍♦✇ t♦ ♣r♦✈❡ ❢♦r ❙✺
P▲❆◆ ❲❡ ❤❛✈❡ t✇♦ ❧♦♦❣✐❝ L ❛♥❞ S5 ❈❛♥♦♥✐❝✐t② ♦❢ t❤❡ ❧♦❣✐❝ L, S5]✳ ⇓ ❈♦♥str✉❝t F1 | = L ❛♥❞ F2 | = S5✱ ❛♥❞ F1, F2] ։ FL,S5]✱ ❛♥❞ F1, F2] | = ∆1, com1, chr✳ ⇓ ❈♦♥str✉❝t Nω
Γ(F1) × Nω S5(F2) ։ N
- F1, F2]Γ
✳
SLIDE 17
❚❛❦❡ r♦♦t❡❞ ❢r❛♠❡s F1 = (W1, R1) ❛♥❞ F2 = (W2, R2) s✉❝❤ t❤❛t W1 ∩ W2 = ∅ t❤❡♥ F1F2 = {x1x2 . . . xn | xi ∈ W1 ∪ W2 ❛♥❞ ♣r♦❥❡❝t♦♥ ♦♥ Wi ✐s ❛ ♣❛t❤} ❲❡ ❞❡✜♥❡ ❛ ❙❡♠✐✲❚❤✉❡ s②st❡♠ C12 = {ab → ba | a ∈ W1, b ∈ W2} ❲❡ ❛❧s♦ ❞❡✜♥❡ ❛ ❑r✐♣❦❡ ❢r❛♠❡ F1, F2] = (F1F2, R<
1 , R⊳ 2)
- aR<
1
b ⇐ ⇒ ∃u ∈ W1( b = au)
- aR<
2
b ⇐ ⇒ ∃v ∈ W2( b = av)
- aR⊳
2
b ⇐ ⇒ ∃ b′ ( aR<
2
b′ & b′ = = ⇒
C12
- b)
▲❡♠♠❛
❋♦r F1 ❛♥❞ F2 ❞❡✜♥❡❞ ❛❜♦✈❡ F1, F2] | = com12, chr, ∆1.
SLIDE 18
SLIDE 19