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L_{R} hypertargets and links #556

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Feb 16, 2018
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L_{R} hypertargets and links #556

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L_{R} hypertargets and links
jamesray1 Feb 8, 2018
c1e0e93
Merge branch 'master' into patch-119
jamesray1 Feb 16, 2018
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10 changes: 5 additions & 5 deletions Paper.tex
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Original file line number Diff line number Diff line change
Expand Up @@ -360,18 +360,18 @@ \subsubsection{Transaction Receipt}\linkdest{Transaction_Receipt}{}
R \equiv (R_{\mathrm{u}}, R_{\mathrm{b}}, R_{\mathbf{l}}, R_{\mathrm{z}})
\end{equation}

The function $L_{R}$ trivially prepares a transaction receipt for being transformed into an RLP-serialised byte array:
\hypertarget{transaction_receipt_preparation_function_for_RLP_serialisation}{}\linkdest{L__R}The function $L_{R}$ trivially prepares a transaction receipt for being transformed into an RLP-serialised byte array:
\begin{equation}
L_{R}(R) \equiv (0 \in \mathbb{B}_{256}, R_{\mathrm{u}}, R_{\mathrm{b}}, R_{\mathbf{l}})
\end{equation}
where $0 \in \mathbb{B}_{256}$ replaces the pre-transaction state root that existed in previous versions of the protocol.

We assert that the status code $R_{\mathrm{z}}$ is an integer.
\linkdest{R__z_assert}We assert that the status code $R_{\mathrm{z}}$ is an integer.
\begin{equation}
R_{\mathrm{z}} \in \mathbb{P}
\end{equation}

We assert $R_{\mathrm{u}}$, the cumulative gas used is a positive integer and that the logs Bloom, $R_{\mathrm{b}}$, is a hash of size 2048 bits (256 bytes):
\linkdest{R__u_assert}We assert $R_{\mathrm{u}}$, the cumulative gas used is a positive integer and that the logs Bloom, $R_{\mathrm{b}}$, is a hash of size 2048 bits (256 bytes):
\begin{equation}
R_{\mathrm{u}} \in \mathbb{P} \quad \wedge \quad R_{\mathrm{b}} \in \mathbb{B}_{256}
\end{equation}
Expand Down Expand Up @@ -409,7 +409,7 @@ \subsubsection{Holistic Validity}
\linkdest{new_state_H__r}{}H_{\mathrm{r}} &\equiv& \mathtt{\small TRIE}(L_S(\Pi(\boldsymbol{\sigma}, B))) & \wedge \\
\linkdest{Ommer_block_hash_H__o}{}H_{\mathrm{o}} &\equiv& \mathtt{\small KEC}(\mathtt{\small RLP}(L_H^*(B_{\mathbf{U}}))) & \wedge \\
\linkdest{tx_block_hash_H__t}{}H_{\mathrm{t}} &\equiv& \mathtt{\small TRIE}(\{\forall i < \lVert B_{\mathbf{T}} \rVert, i \in \mathbb{P}: p(i, L_{T}(B_{\mathbf{T}}[i]))\}) & \wedge \\
\linkdest{Receipts_Root_H__e}{}H_{\mathrm{e}} &\equiv& \mathtt{\small TRIE}(\{\forall i < \lVert B_{\mathbf{R}} \rVert, i \in \mathbb{P}: p(i, L_{R}(B_{\mathbf{R}}[i]))\}) & \wedge \\
\linkdest{Receipts_Root_H__e}{}H_{\mathrm{e}} &\equiv& \mathtt{\small TRIE}(\{\forall i < \lVert B_{\mathbf{R}} \rVert, i \in \mathbb{P}: p(i, \hyperlink{transaction_receipt_preparation_function_for_RLP_serialisation}{L_{R}}(B_{\mathbf{R}}[i]))\}) & \wedge \\
\linkdest{logs_Bloom_filter_H__b}{}H_{\mathrm{b}} &\equiv& \bigvee_{\mathbf{r} \in B_{\mathbf{R}}} \big( \mathbf{r}_{\mathrm{b}} \big)
\end{array}
\end{equation}
Expand All @@ -429,7 +429,7 @@ \subsubsection{Holistic Validity}

\subsubsection{Serialisation}

\hypertarget{block_preparation_function_for_RLP_serialization_L__B}{}\hypertarget{block_preparation_function_for_RLP_serialization_L__H}{}The function $L_{B}$ and $L_{H}$ are the preparation functions for a block and block header respectively. Much like the transaction receipt preparation function $L_{R}$, we assert the types and order of the structure for when the RLP transformation is required:
\hypertarget{block_preparation_function_for_RLP_serialization_L__B}{}\linkdest{L__B}\hypertarget{block_preparation_function_for_RLP_serialization_L__H}{}\linkdest{L__B}The function $L_{B}$ and $L_{H}$ are the preparation functions for a block and block header respectively. Much like the \hyperlink{transaction_receipt_preparation_function_for_RLP_serialisation}{transaction receipt preparation function $L_{R}$}, we assert the types and order of the structure for when the RLP transformation is required:
\begin{eqnarray}
\quad L_{H}(H) & \equiv & (\begin{array}[t]{l}H_{\mathrm{p}}, H_{\mathrm{o}}, H_{\mathrm{c}}, H_{\mathrm{r}}, H_{\mathrm{t}}, H_{\mathrm{e}}, H_{\mathrm{b}}, H_{\mathrm{d}},\\ H_{\mathrm{i}}, H_{\mathrm{l}}, H_{\mathrm{g}}, H_{\mathrm{s}}, H_{\mathrm{x}}, H_{\mathrm{m}}, H_{\mathrm{n}} \; )\end{array} \\
\quad L_{B}(B) & \equiv & \big( L_{H}(B_{H}), L_{T}^*(B_{\mathbf{T}}), L_{H}^*(\hyperlink{ommer_block_headers_B__U}{B_{\mathbf{U}}}) \big)
Expand Down