Various typos fixed thanks to 看K线图.

Paper.tex

@@ -443,7 +443,7 @@ \subsubsection{Block Header Validity}
 \begin{eqnarray}
 & & H_l < {P(H)_H}_l + \left\lfloor\frac{{P(H)_H}_l}{1024}\right\rfloor \quad \wedge \\
 & & H_l > {P(H)_H}_l - \left\lfloor\frac{{P(H)_H}_l}{1024}\right\rfloor \quad \wedge \\
-& & H_l \geqslant 5000
+& & H_l \geqslant 125000
 \end{eqnarray}
 
 $H_s$ is the timestamp of block $H$ and must fulfil the relation:
@@ -550,7 +550,7 @@ \subsection{Execution}
 
 Note the final condition; the sum of the transaction's gas limit, $T_g$, and the gas utilised in this block prior, given by $\ell(B_\mathbf{R})_u$, must be no greater than the block's \textbf{gasLimit}, ${B_H}_l$. 
 
-The execution of a valid transaction begins with an irrevocable change made to the state: the nonce of the account of the sender, $S(T)$, is incremented by one and the balance is reduced by the up-front cost, $v_0$. The gas available for the proceeding computation, $g$, is defined as $T_g - g_0$. The computation, whether contract creation or a message call, results in an eventual state (which may legally be equivalent to the current state), the change to which is deterministic and never invalid: there can be no invalid transactions from this point.
+The execution of a valid transaction begins with an irrevocable change made to the state: the nonce of the account of the sender, $S(T)$, is incremented by one and the balance is reduced by part of the up-front cost, $T_gT_p$. The gas available for the proceeding computation, $g$, is defined as $T_g - g_0$. The computation, whether contract creation or a message call, results in an eventual state (which may legally be equivalent to the current state), the change to which is deterministic and never invalid: there can be no invalid transactions from this point.
 
 We define the checkpoint state $\boldsymbol{\sigma}_0$:
 \begin{eqnarray}
@@ -631,7 +631,7 @@ \section{Contract Creation} \label{ch:create}
 \end{equation}
 \begin{eqnarray}
 \boldsymbol{\sigma}^*[a] &\equiv& \big( 0, v + v', \mathtt{\tiny TRIE}(\varnothing), \mathtt{\tiny KEC}\big(()\big) \big) \\
-\boldsymbol{\sigma}^*[s]_b &\equiv& \boldsymbol{\sigma}^*[s]_b - v
+\boldsymbol{\sigma}^*[s]_b &\equiv& \boldsymbol{\sigma}[s]_b - v
 \end{eqnarray}
 
 where $v'$ is the account's pre-existing value, in the event it was previously in existence:
@@ -728,15 +728,14 @@ \section{Message Call} \label{ch:call}
 \boldsymbol{\sigma}^{**} & \text{otherwise}
 \end{cases} \\
 (\boldsymbol{\sigma}^{**}, g', \mathbf{s}, \mathbf{o}) & \equiv & \begin{cases}
-\Xi_{\mathtt{ECREC}}(\boldsymbol{\sigma}_1, g, I) & \text{if} \quad a = 1 \\
-\Xi_{\mathtt{SHA256}}(\boldsymbol{\sigma}_1, g, I) & \text{if} \quad a = 2 \\
-\Xi_{\mathtt{RIP160}}(\boldsymbol{\sigma}_1, g, I) & \text{if} \quad a = 3 \\
-\Xi_{\mathtt{ID}}(\boldsymbol{\sigma}_1, g, I) & \text{if} \quad a = 4 \\
+\Xi_{\mathtt{ECREC}}(\boldsymbol{\sigma}_1, g, I) & \text{if} \quad r = 1 \\
+\Xi_{\mathtt{SHA256}}(\boldsymbol{\sigma}_1, g, I) & \text{if} \quad r = 2 \\
+\Xi_{\mathtt{RIP160}}(\boldsymbol{\sigma}_1, g, I) & \text{if} \quad r = 3 \\
+\Xi_{\mathtt{ID}}(\boldsymbol{\sigma}_1, g, I) & \text{if} \quad r = 4 \\
 \Xi(\boldsymbol{\sigma}_1, g, I) & \text{otherwise} \end{cases} \\
-I_a & \equiv & a \\
+I_a & \equiv & r \\
 I_o & \equiv & o \\
 I_p & \equiv & p \\
-I_d & \equiv & d \\
 I_\mathbf{d} & \equiv & \mathbf{d} \\
 I_s & \equiv & s \\
 I_v & \equiv & v \\
@@ -976,14 +975,14 @@ \subsection{Ommer Validation}
 
 The validation of ommer headers means nothing more than verifying that each ommer header is both a valid header and satisfies the relation of $N$th-generation ommer to the present block where $N \leq 6$. The maximum of ommer headers is two. Formally:
 \begin{equation}
-\lVert B_\mathbf{U} \rVert \leqslant 2 \bigwedge_{U \in B_\mathbf{U}} V(U) \; \wedge \; k(U, P(B_H), 6)
+\lVert B_\mathbf{U} \rVert \leqslant 2 \bigwedge_{U \in B_\mathbf{U}} V(U) \; \wedge \; k(U, P(B_H)_H, 6)
 \end{equation}
 
 where $k$ denotes the ``is-kin'' property:
 \begin{equation}
 k(U, H, n) \equiv \begin{cases} false & \text{if} \quad n = 0 \\ 
 s(U, H) &\\
-\quad \vee \; k(U, P(H), n - 1) & \text{otherwise}
+\quad \vee \; k(U, P(H)_H, n - 1) & \text{otherwise}
 \end{cases}
 \end{equation}
 
@@ -997,7 +996,7 @@ \subsection{Transaction Validation}
 
 %where $s[i]$ equals the root of the state trie immediately after the execution of the transaction $B_\mathbf{T}[i]$, and $g[i]$ the total gas used immediately after said transaction.
 
-The given \textbf{gasUsed} must correspond faithfully to the transactions listed: ${B_H}_u$, the total gas used in the block, must be equal to the accumulated gas used according to the final transaction:
+The given \textbf{gasUsed} must correspond faithfully to the transactions listed: ${B_H}_g$, the total gas used in the block, must be equal to the accumulated gas used according to the final transaction:
 \begin{equation}
 {B_H}_g = \ell(\mathbf{R})_u
 \end{equation}
@@ -1007,7 +1006,7 @@ \subsection{Reward Application}
 The application of rewards to a block involves raising the balance of the accounts of the beneficiary address of the block and each ommer by a certain amount. We raise the block's beneficiary account by $R_b$; for each ommer, we raise the block's beneficiary by an additional $\frac{1}{32}$ of the block reward and the beneficiary of the ommer gets rewarded depending on the block number. Formally we define the function $\Omega$:
 \begin{eqnarray}
 \Omega(B, \boldsymbol{\sigma}) & \equiv & \boldsymbol{\sigma}': \boldsymbol{\sigma}' = \boldsymbol{\sigma} \quad \text{except:} \\
-\boldsymbol{\sigma}'[{B_H}_c]_b & = & \boldsymbol{\sigma}[{B_H}_c]_b + (1 + \frac{|B_\mathbf{U}|}{32})R_b \\
+\boldsymbol{\sigma}'[{B_H}_c]_b & = & \boldsymbol{\sigma}[{B_H}_c]_b + (1 + \frac{\lVert B_\mathbf{U}\rVert}{32})R_b \\
 \forall_{U \in B_\mathbf{U}}: \\ \nonumber
  \boldsymbol{\sigma}'[U_c]_b & = & \boldsymbol{\sigma}[U_c]_b + (1 + \frac{1}{8} (U_i - {B_H}_i)) R_b 
 \end{eqnarray}
@@ -1036,7 +1035,7 @@ \subsection{State \& Nonce Validation}\label{sec:statenoncevalidation}
 \Phi(B) & \equiv & B': \quad B' = B^* \quad \text{except:} \\
 B'_n & = & n: \quad x \leqslant \frac{2^{256}}{H_d} \\
 B'_m & = & m \quad \text{with } (x, m) = \mathtt{PoW}(B^*_{\hcancel{n}}, n, \mathbf{d}) \\
-B^* & \equiv & B \quad \text{except:} \quad B'_r = r(\Pi(\Gamma(B), B))
+B^* & \equiv & B \quad \text{except:} \quad B^*_r = r(\Pi(\Gamma(B), B))
 \end{eqnarray}
 With $\mathbf{d}$ being a dataset as specified in appendix \ref{app:ethash}.
 
@@ -1210,7 +1209,7 @@ \section{Recursive Length Prefix}\label{app:rlp}
 (128 + \lVert \mathbf{x} \rVert) \cdot \mathbf{x} & \text{else if} \quad \lVert \mathbf{x} \rVert < 56 \\
 \big(183 + \big\lVert \mathtt{\tiny BE}(\lVert \mathbf{x} \rVert) \big\rVert \big) \cdot \mathtt{\tiny BE}(\lVert \mathbf{x} \rVert) \cdot \mathbf{x} & \text{otherwise}
 \end{cases} \\
-\mathtt{\tiny BE}(x) & \equiv & (b_0, b_1, ...): b_0 \neq 0 \wedge \sum_{n = 0}^{n < \lVert \mathbf{b} \rVert - 1} b_n \cdot 256^{\lVert \mathbf{b} \rVert - 1 - n} \\
+\mathtt{\tiny BE}(x) & \equiv & (b_0, b_1, ...): b_0 \neq 0 \wedge x = \sum_{n = 0}^{n < \lVert \mathbf{b} \rVert} b_n \cdot 256^{\lVert \mathbf{b} \rVert - 1 - n} \\
 (a) \cdot (b, c) \cdot (d, e) & = & (a, b, c, d, e)
 \end{eqnarray}
 
@@ -2060,15 +2059,15 @@ \subsubsection{Cache}
 \end{equation}
 Where a single round modifies each subset of the cache as follows:
 \begin{equation}
- E_\text{\tiny RMH}(\mathbf{x}) = \lbrace E_{rmh}(\mathbf{x}, 0), E_{rmh}(\mathbf{x}, 1), ... , E_{rmh}(\mathbf{x}, n - 1) \rbrace
+ E_\text{\tiny RMH}(\mathbf{x}) = \big( E_{rmh}(\mathbf{x}, 0), E_{rmh}(\mathbf{x}, 1), ... , E_{rmh}(\mathbf{x}, n - 1) \big)
 \end{equation}
 \begin{multline}
   E_{rmh}(\mathbf{x}, i) = \texttt{KEC512}(\mathbf{x'}[(i - 1 + n) \mod n] \oplus \mathbf{x'}[\mathbf{x'}[i][0] \mod n]) \\
   \text{with} \quad \mathbf{x'} = \mathbf{x} \quad \text{except} \quad \mathbf{x'}[j] = E_{rmh}(\mathbf{x}, j) \quad \forall \quad j < i
 \end{multline}
 
 \subsubsection{Full dataset calculation} \label{dataset} 
-Essentially, we combine data from $J_{parents}$ pseudorandomly selected cache nodes, and hash that to compute the dataset. The entire dataset is then generated by $d_{size}$ items, each $J_{hashbytes}$ bytes in size:
+Essentially, we combine data from $J_{parents}$ pseudorandomly selected cache nodes, and hash that to compute the dataset. The entire dataset is then generated by a number of items, each $J_{hashbytes}$ bytes in size:
 \begin{equation}
  \mathbf{d}[i] = E_{datasetitem}(\mathbf{c}, i) \quad \forall \quad i < \left\lfloor\frac{d_{size}}{J_{hashbytes}}\right\rfloor
 \end{equation}
@@ -2089,7 +2088,7 @@ \subsubsection{Full dataset calculation} \label{dataset}
 \begin{equation}
  E_{mix}(\mathbf{m}, \mathbf{c}, i, p) = \begin{cases}
 \texttt{KEC512}(\mathbf{c}[i \mod c_{size}] \oplus i) & \text{if} \quad p = 0 \\
-E_\text{\tiny FNV}(\mathbf{m}, \mathbf{c}[E_\text{\tiny FNV}(i \oplus p, \mathbf{m}[p \mod \lfloor J_{hashbytes} / J_{wordbytes} \rfloor]) \mod c_{size}] & \text{otherwise}
+E_\text{\tiny FNV}\big(\mathbf{m}, \mathbf{c}[E_\text{\tiny FNV}(i \oplus p, \mathbf{m}[p \mod \lfloor J_{hashbytes} / J_{wordbytes} \rfloor]) \mod c_{size}] \big) & \text{otherwise}
 \end{cases}
 \end{equation}