LLL condition::A→(0，+oo) VSSA:s≌Pr[nAESA], HA∈A:Pr[A]≤ MA ∑a T+(A) Indep.IES AEI Recursive bounds: S∈A,BeS: @s\(B)4s +Pr(B)4s\r+(B) S≤A,BtS: HsU(B)us+Pr(B)Usur+(B) Indep.I∈SUr+(B) usu(B)=us uBus\r+(B) us +Pr[B]ur+(B)Hs\r+(B) ∈Γ+(B) I2∈S\Γ+(B) =us+PrB](mdp.(A HA)(Emdep.(AHA)
∀𝑆 ⊆ 𝒜: 𝑞෬𝑆 ≜ Pr[⋂𝐴∈𝑆𝐴ҧ], 𝜇𝑆 ≜ 𝐼𝑛𝑑𝑒𝑝. 𝐼⊆𝑆 ෑ 𝐴∈𝐼 𝜇𝐴 Recursive bounds: 𝑞෬𝑆∖ 𝐵 ≤ 𝑞෬𝑆 + Pr 𝐵 𝑞෬𝑆∖Γ + 𝐵 𝜇𝑆∪{𝐵} ≥ 𝜇𝑆 + Pr(𝐵) 𝜇𝑆∪Γ + ∀𝑆 ⊆ 𝒜, 𝐵 ∉ 𝑆: (𝐵) 𝜇𝑆∪{𝐵} = 𝜇𝑆 + 𝜇𝐵𝜇𝑆∖Γ +(𝐵) ≥ 𝜇𝑆 + Pr[𝐵] 𝜇Γ +(𝐵)𝜇𝑆∖Γ +(𝐵) = 𝜇𝑆 + Pr[𝐵] σ𝐼𝑛𝑑𝑒𝑝. 𝐼1⊆Γ+(𝐵) ∏𝐴∈𝐼1 𝜇𝐴 σ𝐼𝑛𝑑𝑒𝑝. 𝐼2⊆𝑆∖Γ+(𝐵) ∏𝐴∈𝐼2 𝜇𝐴 LLL condition: ∃𝜇: 𝒜 → (0, +∞) ∀𝐴 ∈ 𝒜: Pr[𝐴] ≤ 𝜇𝐴 𝜇Γ+(𝐴) ∀𝑆 ⊆ 𝒜, 𝐵 ∈ 𝑆: 𝐼𝑛𝑑𝑒𝑝. 𝐼 ⊆ 𝑆 ∪ Γ +(𝐵) 𝐼1 ⊆ Γ + 𝐵 𝐼2 ⊆ 𝑆 ∖ Γ +(𝐵）
LLL condition:]:A→(0，+o) ∀SSA:as±Pr[n4esA VA∈A:Pr[A]≤ 八A 4s会 T+(A) Indep.IES AEl Recursive bounds: VS∈L,B∈S: @5\(B)<4s +Pr(B)4s\r+(B) VS≤A,BES:suBy≥4s+Pr(B)sur+(B) Indep.I∈Sur+(B) usu(B)=us+uBus\r+(B) zus Pr[B]ur+(B)Us\r+(B) I1∈T+(B) I2∈S\T+(B) =s+Pr](区ap4erteM几ehA)(Smdep..(Tc） ≥ks+Pr[B](②mdep.1esur+Ae4a） us Pr[B]Hsur+(B)
∀𝑆 ⊆ 𝒜: 𝑞෬𝑆 ≜ Pr[⋂𝐴∈𝑆𝐴ҧ], 𝜇𝑆 ≜ 𝐼𝑛𝑑𝑒𝑝. 𝐼⊆𝑆 ෑ 𝐴∈𝐼 𝜇𝐴 Recursive bounds: 𝑞෬𝑆∖ 𝐵 ≤ 𝑞෬𝑆 + Pr 𝐵 𝑞෬𝑆∖Γ + 𝐵 𝜇𝑆∪{𝐵} ≥ 𝜇𝑆 + Pr(𝐵) 𝜇𝑆∪Γ + ∀𝑆 ⊆ 𝒜, 𝐵 ∉ 𝑆: (𝐵) 𝜇𝑆∪{𝐵} = 𝜇𝑆 + 𝜇𝐵𝜇𝑆∖Γ +(𝐵) ≥ 𝜇𝑆 + Pr[𝐵] 𝜇Γ +(𝐵)𝜇𝑆∖Γ +(𝐵) LLL condition: ∃𝜇: 𝒜 → (0, +∞) ∀𝐴 ∈ 𝒜: Pr[𝐴] ≤ 𝜇𝐴 𝜇Γ+(𝐴) ∀𝑆 ⊆ 𝒜, 𝐵 ∈ 𝑆: ≥ 𝜇𝑆 + Pr[𝐵] σ𝐼𝑛𝑑𝑒𝑝. 𝐼⊆𝑆∪Γ+(𝐵) ∏𝐴∈𝐼 𝜇𝐴 = 𝜇𝑆 + Pr[𝐵] 𝜇𝑆∪Γ +(𝐵) = 𝜇𝑆 + Pr[𝐵] σ𝐼𝑛𝑑𝑒𝑝. 𝐼1⊆Γ+(𝐵) ∏𝐴∈𝐼1 𝜇𝐴 σ𝐼𝑛𝑑𝑒𝑝. 𝐼2⊆𝑆∖Γ+(𝐵) ∏𝐴∈𝐼2 𝜇𝐴 𝐼𝑛𝑑𝑒𝑝. 𝐼 ⊆ 𝑆 ∪ Γ +(𝐵) 𝐼1 ⊆ Γ + 𝐵 𝐼2 ⊆ 𝑆 ∖ Γ +(𝐵）