Binary representations Sample Clauses

Binary representations and 𝑘 𝑖=1 b𝑖 · 2𝑘 −𝑖 = 𝑣. For ℓ ≥ 𝑘, we additionally define bitsℓ (𝑣) as the ℓ-bit string obtained b𝑖 · 2𝑘 −𝑖 = 𝑣. For any ℓ ≥ bits(𝑣) , val(bitsℓ (𝑣)) = 𝑣. We also include the remark reverse opera.tion will be val(bits): given a bitstri ng bits := b1b2 . . . b𝑘 (where every b𝑖 ∈ {0, 1}), . We need to establish a few notations and implicit remarks. For a value 𝑣 ∈ N, we define its binary representation bits(𝑣) := b1b2 . . . b𝑘 such that 2𝑘 −1 ≤ 𝑣 < 2𝑘 , b𝑖 ∈ {0, 1} for every 1 ≤ 𝑖 ≤ 𝑘, by prepending ℓ − 𝑘 zeroes to bits(𝑣). We denote the length of a bitstring bits by bits . The val(bits) := 𝑘