diff --git a/Paper.tex b/Paper.tex
index 2d121963..8f5190a9 100644
--- a/Paper.tex
+++ b/Paper.tex
@@ -595,7 +595,7 @@ \subsection{Execution}
 Evaluating $\boldsymbol{\sigma}_P$ from $\boldsymbol{\sigma}_0$ depends on the transaction type; either contract creation or message call; we define the tuple of post-execution provisional state $\boldsymbol{\sigma}_P$, remaining gas $g'$ and substate $A$:
 \begin{equation}
 (\boldsymbol{\sigma}_P, g', A) \equiv \begin{cases}
-\Lambda(\boldsymbol{\sigma}_0, S(T), T_o, &\\ \quad\quad g, T_p, T_v, T_\mathbf{i}, 0, \top) & \text{if} \quad T_t = \varnothing \\
+\Lambda_{3}(\boldsymbol{\sigma}_0, S(T), T_o, &\\ \quad\quad g, T_p, T_v, T_\mathbf{i}, 0, \top) & \text{if} \quad T_t = \varnothing \\
 \Theta_{3}(\boldsymbol{\sigma}_0, S(T), T_o, &\\ \quad\quad T_t, T_t, g, T_p, T_v, T_v, T_\mathbf{d}, 0, \top) & \text{otherwise}
 \end{cases}
 \end{equation}
@@ -606,7 +606,7 @@ \subsection{Execution}
 \end{equation}
 and $T_o$ is the original transactor, which can differ from the sender in the case of a message call or contract creation not directly triggered by a transaction but coming from the execution of EVM-code.
 
-Note we use $\Theta_{3}$ to denote the fact that only the first three components of the function's value are taken; the final represents the message-call's output value (a byte array) and is unused in the context of transaction evaluation.
+Note we use $\Theta_{3}$ and $\Lambda_{3}$ to denote the fact that only the first three components of the functions' values are taken; the final represents the message-call's output value (a byte array) and is unused in the context of transaction evaluation.
 
 After the message call or contract creation is processed, the state is finalised by determining the amount to be refunded, $g^*$ from the remaining gas, $g'$, plus some allowance from the refund counter, to the sender at the original rate.
 \begin{equation}
@@ -642,9 +642,9 @@ \section{Contract Creation} \label{ch:create}
 
 There are a number of intrinsic parameters used when creating an account: sender ($s$), original transactor ($o$), available gas ($g$), gas price ($p$), endowment ($v$) together with an arbitrary length byte array, $\mathbf{i}$, the initialisation EVM code and finally the present depth of the message-call/contract-creation stack ($e$).
 
-We define the creation function formally as the function $\Lambda$, which evaluates from these values, together with the state $\boldsymbol{\sigma}$ to the tuple containing the new state, remaining gas and accrued transaction substate $(\boldsymbol{\sigma}', g', A)$, as in section \ref{ch:transactions}:
+We define the creation function formally as the function $\Lambda$, which evaluates from these values, together with the state $\boldsymbol{\sigma}$ to the tuple containing the new state, remaining gas, accrued transaction substate and an error message $(\boldsymbol{\sigma}', g', A, \mathbf{o})$, as in section \ref{ch:transactions}:
 \begin{equation}
-(\boldsymbol{\sigma}', g', A) \equiv \Lambda(\boldsymbol{\sigma}, s, o, g, p, v, \mathbf{i}, e, w)
+(\boldsymbol{\sigma}', g', A, \mathbf{o}) \equiv \Lambda(\boldsymbol{\sigma}, s, o, g, p, v, \mathbf{i}, e, w)
 \end{equation}
 
 The address of the new account is defined as being the rightmost 160 bits of the Keccak hash of the RLP encoding of the structure containing only the sender and the nonce. Thus we define the resultant address for the new account $a$:
@@ -723,7 +723,7 @@ \section{Contract Creation} \label{ch:create}
 \quad\boldsymbol{\sigma}'[a]_c = \texttt{\small KEC}(\mathbf{o}) & \text{otherwise}
 \end{cases} \\
 \nonumber \text{where} \\
-F &\equiv \big(\boldsymbol{\sigma}^{**} = \varnothing \ \vee\  g^{**} < c \ \vee\  |\mathbf{o}| > 24576\big)
+F &\equiv \big((\boldsymbol{\sigma}^{**} = \varnothing \ \wedge\ \mathbf{o} = \varnothing) \vee\  g^{**} < c \ \vee\  |\mathbf{o}| > 24576\big)
 \end{align}
 
 The exception in the determination of $\boldsymbol{\sigma}'$ dictates that $\mathbf{o}$, the resultant byte sequence from the execution of the initialisation code, specifies the final body code for the newly-created account.
@@ -783,7 +783,7 @@ \section{Message Call} \label{ch:call}
 \boldsymbol{\sigma}^{**} & \text{otherwise}
 \end{cases} \\
 g' & \equiv & \begin{cases}
-0 & \text{if} \quad \boldsymbol{\sigma}^{**} = \varnothing \\
+0 & \text{if} \quad \boldsymbol{\sigma}^{**} = \varnothing \ \wedge \ \mathbf{o} = \varnothing \\
 g^{**} & \text{otherwise}
 \end{cases} \\ \nonumber
 \qquad (\boldsymbol{\sigma}^{**}, g^{**}, A, \mathbf{o}) & \equiv & \begin{cases}
@@ -875,18 +875,21 @@ \subsection{Execution Overview}
 \boldsymbol{\mu}_i & \equiv & 0 \\
 \boldsymbol{\mu}_\mathbf{s} & \equiv & ()
 \end{eqnarray}
-\begin{equation}
+\begin{equation} \label{eq:X-def}
 X\big( (\boldsymbol{\sigma}, \boldsymbol{\mu}, A, I) \big) \equiv \begin{cases}
-\big(\varnothing, \boldsymbol{\mu}, A^0, I, ()\big) & \text{if} \quad Z(\boldsymbol{\sigma}, \boldsymbol{\mu}, I)\\
-O(\boldsymbol{\sigma}, \boldsymbol{\mu}, A, I) \cdot \mathbf{o} & \text{if} \quad \mathbf{o} \neq \varnothing\\
-X\big(O(\boldsymbol{\sigma}, \boldsymbol{\mu}, A, I)\big) & \text{otherwise}\\
+\big(\varnothing, \boldsymbol{\mu}, A^0, I, \varnothing\big) & \text{if} \quad Z(\boldsymbol{\sigma}, \boldsymbol{\mu}, I) \\
+\big(\varnothing, \boldsymbol{\mu}', A^0, I, \mathbf{o}\big) & \text{if} \quad w = \text{\small REVERT} \\
+O(\boldsymbol{\sigma}, \boldsymbol{\mu}, A, I) \cdot \mathbf{o} & \text{if} \quad \mathbf{o} \neq \varnothing \\
+X\big(O(\boldsymbol{\sigma}, \boldsymbol{\mu}, A, I)\big) & \text{otherwise} \\
 \end{cases}
 \end{equation}
 
 where
 \begin{eqnarray}
 \mathbf{o} & \equiv & H(\boldsymbol{\mu}, I) \\
-(a, b, c, d) \cdot e & \equiv & (a, b, c, d, e)
+(a, b, c, d) \cdot e & \equiv & (a, b, c, d, e) \\
+\boldsymbol{\mu}' & \equiv & \boldsymbol{\mu}\ \text{except:} \\
+\boldsymbol{\mu}'_g & \equiv & \boldsymbol{\mu}_g - C(\boldsymbol{\sigma}, \boldsymbol{\mu}, I)
 \end{eqnarray}
 
 Note that, when we evaluate $\Xi$, we drop the fourth element $I'$ and extract the remaining gas $\boldsymbol{\mu}'_g$ from the resultant machine state $\boldsymbol{\mu}'$.
@@ -964,13 +967,13 @@ \subsubsection{Normal Halting}
 The normal halting function $H$ is defined:
 \begin{equation}
 H(\boldsymbol{\mu}, I) \equiv \begin{cases}
-H_{\text{\tiny RETURN}}(\boldsymbol{\mu}) & \text{if} \quad w = \text{\small RETURN} \\
+H_{\text{\tiny RETURN}}(\boldsymbol{\mu}) & \text{if} \quad w \in \{\text{\small RETURN}, \text{\small REVERT}\} \\
 () & \text{if} \quad w \in \{ \text{\small STOP}, \text{\small SELFDESTRUCT} \} \\
 \varnothing & \text{otherwise}
 \end{cases}
 \end{equation}
 
-The data-returning halt operation, \text{\small RETURN}, has a special function $H_{\text{\tiny RETURN}}$, defined in Appendix \ref{app:vm}.
+The data-returning halt operations, \text{\small RETURN} and \text{\small REVERT}, have a special function $H_{\text{\tiny RETURN}}$, defined in Appendix \ref{app:vm}.
 
 \subsection{The Execution Cycle}
 
@@ -1607,7 +1610,7 @@ \subsection{Gas Cost}
 
 with $C_\text{\tiny CALL}$, $C_\text{\tiny SELFDESTRUCT}$ and $C_\text{\tiny SSTORE}$ as specified in the appropriate section below. We define the following subsets of instructions:
 
-$W_{zero}$ = \{{\small STOP}, {\small RETURN}\}
+$W_{zero}$ = \{{\small STOP}, {\small RETURN}, {\small REVERT}\}
 
 $W_{base}$ = \{{\small ADDRESS}, {\small ORIGIN}, {\small CALLER}, {\small CALLVALUE}, {\small CALLDATASIZE}, {\small CODESIZE}, {\small GASPRICE}, {\small COINBASE},\newline \noindent\hspace*{1cm} {\small TIMESTAMP}, {\small NUMBER}, {\small DIFFICULTY}, {\small GASLIMIT}, {\small POP}, {\small PC}, {\small MSIZE}, {\small GAS}\}
 
@@ -1991,13 +1994,13 @@ \subsection{Instruction Set}
 \textbf{Value} & \textbf{Mnemonic} & $\delta$ & $\alpha$ & \textbf{Description} \vspace{5pt} \\
 0xf0 & {\small CREATE} & 3 & 1 & Create a new account with associated code. \\
 &&&& $\mathbf{i} \equiv \boldsymbol{\mu}_\mathbf{m}[ \boldsymbol{\mu}_\mathbf{s}[1] \dots (\boldsymbol{\mu}_\mathbf{s}[1] + \boldsymbol{\mu}_\mathbf{s}[2] - 1) ]$ \\
-&&&& $(\boldsymbol{\sigma}', \boldsymbol{\mu}'_g, A^+) \equiv \begin{cases}\Lambda(\boldsymbol{\sigma}^*, I_a, I_o, L(\boldsymbol{\mu}_g), I_p, \boldsymbol{\mu}_\mathbf{s}[0], \mathbf{i}, I_e + 1, I_w) & \text{if} \quad \boldsymbol{\mu}_\mathbf{s}[0] \leqslant \boldsymbol{\sigma}[I_a]_b \;\wedge\; I_e < 1024\\ \big(\boldsymbol{\sigma}, \boldsymbol{\mu}_g, \varnothing\big) & \text{otherwise} \end{cases}$ \\
+&&&& $(\boldsymbol{\sigma}', \boldsymbol{\mu}'_g, A^+, \mathbf{o}) \equiv \begin{cases}\Lambda(\boldsymbol{\sigma}^*, I_a, I_o, L(\boldsymbol{\mu}_g), I_p, \boldsymbol{\mu}_\mathbf{s}[0], \mathbf{i}, I_e + 1, I_w) & \text{if} \quad \boldsymbol{\mu}_\mathbf{s}[0] \leqslant \boldsymbol{\sigma}[I_a]_b \;\wedge\; I_e < 1024\\ \big(\boldsymbol{\sigma}, \boldsymbol{\mu}_g, \varnothing\big) & \text{otherwise} \end{cases}$ \\
 &&&& $\boldsymbol{\sigma}^* \equiv \boldsymbol{\sigma} \quad \text{except} \quad \boldsymbol{\sigma}^*[I_a]_n = \boldsymbol{\sigma}[I_a]_n + 1$ \\
 &&&& $A' \equiv A \Cup A^+$ which abbreviates: $A'_\mathbf{s} \equiv A_\mathbf{s} \cup A^+_\mathbf{s} \quad \wedge \quad A'_\mathbf{l} \equiv A_\mathbf{l} \cdot A^+_\mathbf{l} \quad \wedge \quad A'_\mathbf{t} \equiv A_\mathbf{t} \cup A^+_\mathbf{t}$ \\
 &&&& $ \wedge \quad A'_\mathbf{r} \equiv A_\mathbf{r} + A^+_\mathbf{r}$ \\
 &&&& $\boldsymbol{\mu}'_\mathbf{s}[0] \equiv x$ \\
-&&&& where $x=0$ if the code execution for this operation failed due to an exceptional halting \\
-&&&& $Z(\boldsymbol{\sigma}^*, \boldsymbol{\mu}, I) = \top$ or $I_e = 1024$ \\
+&&&& where $x=0$ if the code execution for this operation failed due to an exceptional halting\\
+&&&& (or for a \text{\small REVERT}) $\boldsymbol{\sigma}' = \varnothing$, or $I_e = 1024$ \\
 &&&& (the maximum call depth limit is reached) or $\boldsymbol{\mu}_\mathbf{s}[0] > \boldsymbol{\sigma}[I_a]_b$ (balance of the caller is too \\
 &&&& low to fulfil the value transfer); and otherwise $x=A(I_a, \boldsymbol{\sigma}[I_a]_n)$, the address of the newly \\
 &&&& created account. \\
@@ -2014,7 +2017,7 @@ \subsection{Instruction Set}
 &&&& $A' \equiv A \Cup A^+$ \\
 &&&& $t \equiv \boldsymbol{\mu}_\mathbf{s}[1] \mod 2^{160}$ \\
 &&&& where $x=0$ if the code execution for this operation failed due to an exceptional halting \\
-&&&& $Z(\boldsymbol{\sigma}, \boldsymbol{\mu}, I) = \top$ or if  \\
+&&&& (or for a \text{\small REVERT}) $\boldsymbol{\sigma}' = \varnothing$ or if \\
 &&&& $\boldsymbol{\mu}_\mathbf{s}[2] > \boldsymbol{\sigma}[I_a]_b$ (not enough funds) or $I_e = 1024$ (call depth limit reached); $x=1$ \\
 &&&& otherwise. \\
 &&&& $\boldsymbol{\mu}'_i \equiv M(M(\boldsymbol{\mu}_i, \boldsymbol{\mu}_\mathbf{s}[3], \boldsymbol{\mu}_\mathbf{s}[4]), \boldsymbol{\mu}_\mathbf{s}[5], \boldsymbol{\mu}_\mathbf{s}[6])$ \\
@@ -2072,6 +2075,12 @@ \subsection{Instruction Set}
 &&&& The deeper argument $\boldsymbol{\mu}_\mathbf{s}[3]$, $\boldsymbol{\mu}_\mathbf{s}[4]$, $\boldsymbol{\mu}_\mathbf{s}[5]$ and $\boldsymbol{\mu}_\mathbf{s}[6]$ are respectively replaced with \\
 &&&& $\boldsymbol{\mu}_\mathbf{s}[2]$, $\boldsymbol{\mu}_\mathbf{s}[3]$, $\boldsymbol{\mu}_\mathbf{s}[4]$ and $\boldsymbol{\mu}_\mathbf{s}[5]$. \\
 &&&& The last argument of $\Theta$ is $\bot$. \\
+0xfd & {\small REVERT} & 2 & 0 & Halt execution reverting state changes
+                 but returning data and gas. \\
+0xfd & {\small REVERT} & 2 & 0 & Halt execution reverting state changes but returning data and gas. \\
+&&&& The effect of this operation is described in (\ref{eq:X-def}). \\
+&&&& For the gas calculation, we need to define the memory usage: \\
+&&&& $\boldsymbol{\mu}'_i \equiv M(\boldsymbol{\mu}_i, \boldsymbol{\mu}_\mathbf{s}[0], \boldsymbol{\mu}_\mathbf{s}[1])$ \\
 \midrule
 0xfe & {\small INVALID} & $\varnothing$ & $\varnothing$ & Designated invalid instruction. \\
 \midrule