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\section{Verilog}
Verilog is a hardware description language commonly used to design hardware. A Verilog design can then be synthesised into more basic logic which describes how different gates connect to each other, called a netlist. This representation can then be put onto either a field-programmable gate array (FPGA) or turned into an application-specific integrated circuit (ASIC) to implement the design that was described in Verilog. The Verilog standard is quite large though, and not all Verilog features are needed to be able to describe hardware. Many Verilog features are only useful for simulation and do not affect the actual hardware itself, which means that these features do not have to be modelled in the semantics. In addition to that, as the HLS algorithm dictates which Verilog constructs are generated, meaning the Verilog subset that has to be modelled by the semantics can be reduced even further to only support the constructs that are needed. Only supporting a smaller subset in the semantics also means that there is less chance that the standard is misunderstood, and that the semantics actually model how the Verilog is simulated.
The Verilog semantics are based on the semantics proposed by \citet{loow19_verif_compil_verif_proces}, which were used to create a formal translation from HOL logic into a Verilog circuit. These semantics are quite practical as they restrict themselves to a small subset of Verilog, which can nonetheless be used to model all hardware constructs one would want to design. An abstraction of the Verilog syntax that is generated is shown below:
\begin{align*}
v\quad ::=&\; \mathit{sz} \yhkeyword{'d} n\\
\textit{op}\quad ::=&\; \yhkeyword{+ } | \yhkeywordsp{- } | \yhkeywordsp{* } \cdots \\
e\quad ::=&\; v\;\; |\;\; x\;\; |\;\; e \yhkeyword{[} e \yhkeyword{]}\;\; |\;\; e\ \mathit{op}\ e\;\; |\;\; \yhkeyword{!} e\;\; |\;\; \yhkeyword{\textasciitilde} e\;\; |\;\; e \yhkeywordsp{? } e \yhkeywordsp{: } e\\
s\quad ::=&\; s\ s\ |\ \epsilon\\[-2pt]
|&\; \yhkeyword{if(} e \yhkeyword{) } s \yhkeywordsp{else } s\\[-2pt]
|&\; \yhkeyword{case(} e \yhkeyword{) } e : s\ \{\ e : s\ \}\ [\ s\ ] \yhkeywordsp{endcase}\\[-2pt]
|&\; e = e \yhkeyword{;}\\[-2pt]
|&\; e \Leftarrow e \yhkeyword{;}\\
d\quad ::=&\; \yhkeyword{[n-1:0] } r\ |\ \yhkeyword{[n-1:0] } r \yhkeywordsp{[m-1:0]}\\
m\quad ::=&\; \yhkeyword{reg } d \yhkeyword{;}\ |\ \yhkeyword{input wire } d \yhkeyword{;}\ |\ \yhkeyword{output reg } d \yhkeyword{;}\\
|&\; \yhkeywordsp{always @(posedge clk) } s \\
m \text{ list}\quad ::=&\; \{ m \}
\end{align*}
The main addition to the Verilog syntax is the explicit declaration of inputs and outputs, as well as variables and arrays. This means that the declarations have to be handled in the semantics as well, adding to the safety that all the registers are declared properly with the right size, as this affects how the Verilog module is synthesised and simulated. In addition to that, literal values are not represented by a list of nested Boolean values, but instead they are represented by a size and its value, meaning a Boolean is represented as a value with size one. Finally, the last difference is that the syntax supports two dimensional arrays in Verilog explicitly which model memory so that we can reason about array loads and stores properly. \JW{In the $m$ category, should it be `reg d; m' rather than just `reg d;'?}\YH{Yes that's actually completely true, I added a $\vec{m}$ rule for now, but can also add the $m$ afterwards.}
\subsection{Semantics}
Existing operational semantics~\cite{loow19_verif_compil_verif_proces} were adapted for the semantics of the language that CoqUp eventually targets. This semantics is a small-step operational semantics at the clock cycle level, as hardware typically does not terminate in any way, however, within each clock cycle the semantics are constructed in a big-step style semantics. This style of semantics matches the small-step operational semantics of CompCert's register transfer language (RTL) quite well.
At the top-level, always blocks describe logic which is run every time some event occurs. The only event that is supported by these semantics is detecting the rising rising edge of the clock, so that we can implement synchronous logic. As soon as an event occurs, the hardware will be executed, meaning if there are multiple always blocks that get triggered by the event, these will run in parallel. However, as the semantics should be deterministic, we impose an order on the always blocks and execute them sequentially. However, to preserve the fact that the statements inside of the always block are executed in parallel, nonblocking assignments to variables need to be kept in a different association map compared to blocking assignments to variables. This preserves the behaviour that blocking assignments change the value of the variable inside of the clock cycle, whereas the nonblocking assignments only take place at the end of the clock cycle, and in parallel. We can denote these two association maps as $s = (\Gamma_{\rm r}, \Gamma_{\rm a}, \Delta_{\rm r}, \Delta_{\rm a})$, where $\Gamma_{\rm r}$ is the current value of the registers, $\Gamma_{\rm a}$ is the current value of the array, and $\Delta_{\rm r}$ and $\Delta_{\rm a}$ are the values of the variables and arrays when the clock cycle ends.
We can then define how one step in the semantics looks like. We therefore first need to define the structure of the main module which will contain the logic for the program. In general, functions that are translated to hardware will require basic handshaking signals so that the translated function can be used in hardware. Firstly, they require an input for the clock, so that all the sequential circuits are run at the right time. They then require a start and reset input, so that the hardware generated from the function can be reused multiple times. Finally, they need a finish and return signal, where finish will go high when the result is ready to be read. In addition to that, the function could take an arbitrary number of inputs which act as arguments to the function, so that the function can be called with different arguments. However, in addition to inputs and outputs to the module, we also need to keep track of some internal signals and properties about the module. Firstly, we need to keep track of the internal variables that contain the current state of the module, and the current contents of the stack. Finally, the module will contain the entry point of the module and the list of module items that declare all of the internal registers and contain the encoding of the state machine that behaves in the same way as the function. We can therefore declare it in the following way:
%\JW{I'd be inclined to write `$r~\mathrm{list}$' rather than $\vec{r}$, as it's a little more readable. (Assuming it's more-or-less the same thing?)}
\begin{align*}
\mathit{M} \quad ::= \quad \big\{\ &\mathtt{args} : r \text{ list}\\
&\mathtt{body} : m \text{ list}\\
&\mathtt{entrypoint} : n\\
&\mathtt{st, stk, finish, return, start, reset, clk} : r\\
&\mathtt{stacksize} : n\ \big\}
\end{align*}
The two main evaluation functions are then \textit{erun}, which takes in the current state together with an expression and returns a value, and \textit{srun}, which takes the current state and a statement as input, and returns the updated state. The inductive rules defining \textit{srun} are shown below, where $\sigma_{n} = (\Gamma_{\rm r}^{n}, \Gamma_{\rm a}^{n}, \Delta_{\rm r}^{n}, \Delta_{\rm a}^{n})$:
\begin{gather*}
\label{eq:1}
\inferrule[Skip]{ }{\textit{srun}\ \sigma\ \epsilon\ \sigma}\\
%
\inferrule[Seq]{\textit{srun}\ \sigma_{0}\ \textit{s}_{1}\ \sigma_{1} \\ \textit{srun}\ \sigma_{1}\ \textit{s}_{2}\ \sigma_{2}}{\textit{srun}\ \sigma_{0}\ (\textit{s}_{1}\ \textit{s}_{2})\ \sigma_{2}}\\
%
\inferrule[CondTrue]{\textit{erun}\ \Gamma_{\rm r}^{0}\ \Gamma_{\rm a}^{0}\ c\ v_{c} \\ \yhfunction{valToB}\ v_{c} = \yhconstant{true} \\ \textit{srun}\ \sigma_{0}\ \textit{st}\ \sigma_{1}}{\textit{srun}\ \sigma_{0}\ (\yhkeyword{if(} c \yhkeyword{) } \textit{st} \yhkeywordsp{else } \textit{sf})\ \sigma_{1}}\\
%
\inferrule[CondFalse]{\textit{erun}\ \Gamma_{\rm r}^{0}\ \Gamma_{\rm a}^{0}\ c\ v_{\rm c} \\ \yhfunction{valToB}\ v_{\rm c} = \yhconstant{false} \\ \textit{srun}\ \sigma_{0}\ \textit{sf}\ \sigma_{1}}{\textit{srun}\ \sigma_{0}\ (\yhkeyword{if(} c \yhkeyword{) } \textit{st} \yhkeywordsp{else } \textit{sf})\ \sigma_{1}}\\
%
\inferrule[CaseNoMatch]{\textit{srun}\ \sigma_{0}\ (\yhkeyword{case(} e \yhkeyword{) } cs\ \textit{def} \yhkeywordsp{endcase})\ \sigma_{1} \\ \textit{erun}\ \Gamma_{\rm r}^{0}\ \Gamma_{\rm a}^{0}\ me\ mve \\ \textit{erun}\ \Gamma_{\rm r}^{0}\ \Gamma_{\rm a}^{0}\ e\ ve \\ mve \neq ve}{\textit{srun}\ \sigma_{0}\ (\yhkeyword{case(} e \yhkeyword{) } ((me : sc) :: cs)\ \textit{def} \yhkeywordsp{endcase})\ \sigma_{1}}\\
%
\inferrule[CaseMatch]{\textit{srun}\ \sigma_{0}\ sc\ \sigma_{1} \\ \textit{erun}\ \Gamma_{\rm r}^{0}\ \Gamma_{\rm a}^{0}\ e\ ve \\ \textit{erun}\ \Gamma_{\rm r}^{0}\ \Gamma_{\rm a}^{0}\ me\ mve \\ mve = ve}{\textit{srun}\ \sigma_{0}\ (\yhkeyword{case(} e \yhkeyword{) } ((me : sc) :: cs)\ \textit{def} \yhkeywordsp{endcase})\ \sigma_{1}}\\
%
\inferrule[CaseDefault]{\textit{srun}\ \sigma_{0}\ s\ \sigma_{1}}{\textit{srun}\ \sigma_{0}\ (\yhkeyword{case(} e \yhkeyword{) } []\ (\yhconstant{Some}\ s) \yhkeywordsp{endcase})\ \sigma_{1}}\\
%
\inferrule[Blocking Reg]{\yhfunction{name}\ \textit{lhs} = \yhconstant{OK}\ n \\ \textit{erun}\ \Gamma_{\rm r}\ \Gamma_{\rm a}\ \textit{rhs}\ v_{\rm rhs}}{\textit{srun}\ (\Gamma_{\rm r},\Gamma_{\rm a},\Delta_{\rm r},\Delta_{\rm a})\ (\textit{lhs} = \textit{rhs})\ (\Gamma_{\rm r} [n \mapsto v_{\rm rhs}], \Gamma_{\rm a}, \Delta_{\rm r}, \Delta_{\rm a})}\\
%
\inferrule[Nonblocking Reg]{\yhfunction{name}\ \textit{lhs} = \yhconstant{OK}\ n \\ \textit{erun}\ \Gamma_{\rm r}\ \Gamma_{\rm a}\ \textit{rhs}\ v_{\rm rhs}}{\textit{srun}\ (\Gamma_{\rm r}, \Gamma_{\rm a}, \Delta_{\rm r}, \Delta_{\rm a})\ (\textit{lhs} \Leftarrow \textit{rhs})\ (\Gamma_{\rm r}, \Gamma_{\rm a}, \Delta_{\rm r} [n \mapsto v_{\rm rhs}], \Delta_{\rm a})}
%
% \inferrule[Blocking Array]{\yhkeyword{name}\ \textit{lhs} = \yhkeyword{OK}\ n \\ \textit{erun}\ \Gamma_{r}\ \Gamma_{a}\ \textit{rhs}\ v_{\textit{rhs}}}{\textit{srun}\ (\Gamma_{r},\Gamma_{a},\Delta_{r},\Delta_{a})\ (\textit{lhs} = \textit{rhs})\ (\Gamma_{r} // \{n \rightarrow v_{\textit{rhs}}\}, \Gamma_{a}, \Delta_{r}, \Delta_{a})}\\
%
% \inferrule[Nonblocking Array]{\yhkeyword{name}\ \textit{lhs} = \yhkeyword{OK}\ n \\ \textit{erun}\ \Gamma\ \textit{rhs}\ v_{\textit{rhs}}}{\textit{srun}\ (\Gamma_{r}, \Gamma_{a}, \Delta_{r}, \Delta_{a})\ (\textit{lhs} \Leftarrow \textit{rhs})\ (\Gamma_{r}, \Gamma_{a}, \Delta_{r} // \{n \rightarrow v_{\textit{rhs}}\}, \Delta_{a})}
\end{gather*}
Taking the \textsc{CondTrue} rule as an example, this rule will only apply if the Boolean result of running the expression results in a \texttt{true} value. It then also states that the statement in the true branch of the conditional statement \textit{stt} runs from state $\sigma_{0}$ to state $\sigma_{1}$. If both of these conditions hold, we then get that the conditional statement will also run from state $\sigma_{0}$ to state $\sigma_{1}$. The \textsc{Blocking} and \textsc{Nonblocking} rules are a bit more interesting, as these modify the blocking and nonblocking association maps respectively.
One main difference between these semantics and the Verilog semantics by \citet{loow19_verif_compil_verif_proces} is that there is no function for external nondeterministic effects, such as memories and inputs and outputs. These are instead handled explicitly in the semantics by using two dimensional unpacked arrays to model memories and assuming that inputs to modules cannot change. Another difference with these semantics is that partial updates to arrays are fully supported, due to the fact that there are two different queues for arrays and variables. Originally, if there was a blocking assignment to an array, and then a nonblocking assignment to a different region in the array, then the blocking assignment would disappear at the end of the clock cycle. This is because the complete array would be overwritten with the updated array in the nonblocking association maps. However, in our semantics, only the values that were changed in the array are actually recorded in the nonblocking assignment queue, meaning once the blocking and nonblocking array association maps are merged, only the actual indices that changed with nonblocking assignment are updated in the blocking assignment map.
In these semantics, module instantiations are not supported, as they can be modelled by inlining the logic that modules would have produced. This therefore means that function calls are implemented by inlining all the functions as well. Consequently, recursive function calls are not supported, however, these are not supported by other high-level synthesis tools either, as allocating memory dynamically is not possible with fixed size RAM blocks.
To integrate our semantics with CompCert, we need to define the same states that CompCert uses to define their semantics, which are the \texttt{State}, \texttt{Callstate} and \texttt{Returnstate}. The \texttt{Callstate} and \texttt{Returnstate} are needed because the initial state describes a call to main, and the final state results in the return value from the main function. Even though the Verilog semantics do not support function calls, semantics for the initial function call and final return state need to be implemented to be properly shown equivalent to the Verilog semantics. Firstly, the \texttt{State} is contains all the information necessary to identify the current execution state of the function or module. In Verilog, we therefore keep track of the current module $m$ we are working on, the current value of the program counter $v$, which translates to the current value of the state register $s_{t}$, the current contents of the stack frame \textit{sf} and finally the current states of the association maps for variables $\Gamma_{r}$ and arrays $\Gamma_{a}$. We can therefore define the state as \texttt{State} \textit{sf} $m$ $v$ $\Gamma_{r}$ $\Gamma_{a}$. The \texttt{Callstate} only needs to be supported for the initial call to the function and the \texttt{Returnstate} only needs to be supported for the return of the main function. As function calls are not supported in the semantics, there cannot be any other possible calling state or return state in the program.
We then define the semantics of running the module for one clock cycle in the following way:
\begin{figure*}
\centering
\begin{gather*}
\inferrule[Module]{\Gamma_{r} ! s_{t} = \texttt{Some } v \\ (m_{i}, \Gamma_{r}^{0}, \Gamma_{a}^{0}, \epsilon, \epsilon\ l)\ \longrightarrow_{\vec{m}} (m_{i}, \Gamma_{r}^{1}, \Gamma_{a}^{1}, \Delta_{r}^{1}, \Delta_{a}^{1}) \\ (\Gamma_{r}^{1} // \Delta_{r}^{1}) ! s_{t} = \texttt{Some } v'}{\texttt{State } \textit{sf }\ m\ v\ \Gamma_{r}^{0}\ \Gamma_{a}^{0} \longrightarrow \texttt{State } \textit{sf }\ m\ v'\ (\Gamma_{r}^{1} // \Delta_{r}^{1})\ (\Gamma_{a}^{1} // \Delta_{a}^{1})}\\
%
\inferrule[Finish]{\Gamma_{r}!\textit{fin} = \texttt{Some } 1 \\ \Gamma_{r}!\textit{ret} = \texttt{Some } r}{\texttt{State } \textit{sf }\ m\ s_{t}\ \Gamma_{r}\ \Gamma_{a} \longrightarrow \texttt{Returnstate } \textit{sf }\ r}\\
%
\inferrule[Call]{ }{\texttt{Callstate } \textit{sf }\ m\ \vec{r} \longrightarrow \texttt{State } \textit{sf }\ m\ n\ (\textit{init\_params }\ \vec{r}\ a // \{s_{t} \rightarrow n\})}\\
%
\inferrule[Return]{ }{\texttt{Returnstate } (\texttt{Stackframe } r\ m\ \textit{pc }\ \Gamma_{r}\ \Gamma_{a} :: \textit{sf}) \longrightarrow \texttt{State } \textit{sf }\ m\ \textit{pc }\ (\Gamma_{r} // \{ \textit{st} \rightarrow \textit{pc}, r \rightarrow i \})\ \epsilon}
\end{gather*}
\caption{Inferrence rules for modules}%
\label{fig:inferrence_module}
\end{figure*}
\YH{TODO:\@ Need to fix the last rule, as it is actually only used for a case that shouldn't ever be hit.}
The \textsc{Module} rule is the main rule for the execution of one clock cycle of the module. Given that the value of the $s_{t}$ register is the value of the program counter at the current instruction and that the value of the $s_{t}$ register in the resulting association map is equal to the next program counter value, we can then say that if all the module items in the body go from one state to another, that the whole module will step from that state to the other.
%%\input{verilog_notes}
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