\chapter{Concept and Design}
\label{cha:conceptdesign}
% introduction to what needs to be done. also clarify terms "Host" and "Device" here
To be able to determine whether evaluating mathematical expressions on the GPU is better suited than on the CPU, a prototype needs to be implemented. More specifically, a prototype for interpreting these expressions on the GPU, as well as a prototype that transpiles expressions into code that can be executed by the GPU. The goal of this chapter, is to describe how these two prototypes can be implemented conceptually. First the requirements for the prototypes as well as the data they operate on are explained. This is followed by the design of the interpreter and the transpiler. The CPU interpreter will not be described, as it already exists.

% TODO: maybe describe CPU interpreter too? We will see

\section[Requirements]{Requirements and Data}
The main goal of both prototypes or evaluators is to provide a speed-up compared to the CPU interpreter already in use. However, it is also important to determine which evaluator provides the most speed-up. This also means that if one of the evaluators is faster, it is intended to replace the CPU interpreter. Therefore, they must have similar capabilities, and therefore meet the following requirements:

\begin{itemize}
	\item Multiple expressions as input.
	\item All input expressions have the same number of variables ($x_n$), but can have a different number of parameters ($p_n$).
	\item The variables are parametrised using a matrix of the form $k \times N$, where $k$ is the number of variables in the expressions and $N$ is the number of different parametrisations for the variables. This matrix is the same for all expressions. 
	\item The parameters are parametrised using a vector of vectors. Each vector $v_i$ corresponds to an expression $e_i$.
	\item The following operations must be supported: $x + y$, $x - y$, $x * y$, $x / y$, $x ^ y$, $|x|$, $\log(x)$, $e^x$ and $\sqrt{x}$. Note that $x$ and $y$ can either stand for a value, a variable, or another operation.
	\item The results of the evaluations are returned in a matrix of the form $k \times N$. In this case, $k$ is equal to the $N$ of the variable matrix and $N$ is equal to the number of input expressions.
\end{itemize}

\begin{figure}
	\centering
	\includegraphics[width=.9\textwidth]{input_output_explanation.png}
	\caption{This diagram shows how the input and output looks like and how they interact with each other.}
	\label{fig:input_output_explanation}
\end{figure}


With this, the required capabilities are outlined. However, the input and output data need to further be explained for a better understanding. The first input contains the expressions that need to be evaluated. These can have any length and can contain constant values, variables and parameters and all of these are linked together with the supported operations. In the example shown in Figure \ref{fig:input_output_explanation}, there are six expressions $e_1$ through $e_6$. Next is the variable matrix. One entry in this matrix, corresponds to one variable in every expression. The row indicates which variable it holds the value for. For example the values in row three, are used to parametrise the variable $x_3$. Each column holds a different set of variables. Each expression must be evaluated using every variable set. In the provided example, there are three variable sets, each holding the values for four variables $x_1$ through $x_4$. After all expressions are evaluated using all variable sets the results of these evaluations must be stored in the results matrix. Each entry in this matrix holds the resulting value of the evaluation of one expression parametrised with one variable set. The row indicates the variable set while the column indicates the expression.

This is the minimal functionality needed to evaluate expressions with variables generated by a symbolic regression algorithm. In the case of parameter optimisation, it is useful to have a different type of variable, called parameter. For parameter optimisation it is important that for the given variable sets, the best fitting parameters need to be found. To achieve this, the evaluator is called multiple times with different parameters, but the same variables. The results are then evaluated for their fitness by the caller. In this case, the parameters do not change within one call. Parameters could therefore be treated as constant values of the expressions, and no separate input for them would be needed. However, providing the possibility to have the parameters as an input, makes the process of parameter optimisation easier. Unlike variables, not all expressions need to have the same number of parameters. Therefore, they are structured as a vector of vectors and not a matrix. The example in Figure \ref{fig:input_output_explanation} shows how the parameters are structured. For example one expression has zero parameters, while another has six parameters $p_1$ through $p_6$. It needs to be mentioned that just like the number of variables, the number of parameters per expression is not limited. It is also possible to completely omit the parameters if they are not needed. Because these evaluators will primarily be used in parameter optimisation use-cases, allowing parameters as an input is required.

% \subsection{Non-Goals}
% Probably a good idea. Probably move this to "introduction"
\section{Architecture}

Based on the requirements above, the architecture of both prototypes can be designed. While the requirements only specify the input and output, the components and workflow also need to be specified. This section aims at giving an architectural overview of both prototypes, alongside their design decisions. 

\begin{figure}
	\centering
	\includegraphics[width=.9\textwidth]{kernel_architecture.png}
	\caption{The interpreter has one kernel that is dispatched multiple times, while the transpiler, has multiple kernels that are dispatched once. This helps to eliminate thread divergence.}
	\label{fig:kernel_architecture}
\end{figure}

A design decision that has been made for both prototypes is to split the evaluation of each expression into a separate kernel or kernel dispatch as seen in Figure \ref{fig:kernel_architecture}. As explained in Section \ref{sec:thread_hierarchy}, it is desirable to reduce the occurrence of thread divergence as much as possible. Although the SIMT programming model tries to mitigate the negative effects of thread divergence, it is still a good idea to avoid it when possible. For this use-case, thread divergence can easily be avoided by not evaluating all expressions in a single kernel or kernel dispatch. GPUs are able to have multiple resident grids, with modern GPUs being able to accommodate 128 grids concurrently \parencite{nvidia_cuda_2025}. One grid corresponds to one kernel dispatch, and therefore allows up-to 128 kernels to be run concurrently. Therefore, dispatching a kernel for each expression, has the possibility to improve the performance. In the case of the interpreter, having only one kernel that can be dispatched for each expression, also simplifies the kernel itself. This is because the kernel can focus on evaluating one expression and does not require additional code to handle multiple expressions at once. Similarly, the transpiler can also be simplified, as it can generate many smaller kernels than one big kernel. Additionally, the smaller kernels do not need any branching, because the generated code only needs to perform the operations as they occur in the expression itself.

\subsection{Pre-Processing}
The first step in both prototypes is the pre-processing step. It is needed, as it simplifies working with the expressions in the later steps. One of the responsibilities of the pre-processor is to verify that only allowed operators and symbols are present in the given expressions. This is comparable to the work a scanner like Flex\footnote{\url{https://github.com/westes/flex}} performs. Additionally, this step also converts the expression into an intermediate representation. In essence, the pre-processing step can be compared to the front-end of a compiler as described in Section \ref{sec:compilers}. The conversion into the intermediate representation transforms the expressions from infix-notation into postfix-notation. This further allows the later parts to more easily evaluate the expressions. One of the major benefits of this notation is the implicit operator precedence. It allows the evaluators to evaluate the expressions token by token from left to right, without needing to worry about the correct order of operations. One token represents either an operator, a constant value, a variable or a parameter. Apart from the intermediate representation containing the expression in postfix-notation, it also contains the information about the types of the tokens themselves. This is all that is needed for the interpretation and transpilation steps. A simple expression like $x + 2$ would look like depicted in figure \ref{fig:pre-processing_results} after the pre-processing step.

\begin{figure}
	\centering
	\includegraphics[width=.9\textwidth]{pre-processing_result.png}
	\caption{This diagram shows how an expression will be transformed in the pre-processing step.}
	\label{fig:pre-processing_results}
\end{figure}

It would have also been possible to perform the pre-processing step on the GPU. However, pre-processing only one expression can not easily be split into multiple threads, which means one GPU thread would need to process one expression. As described in Section \ref{sec:gpgpu} a single GPU thread is slower than a single CPU thread and as a result means the processing will also be slower. Furthermore, it wouldn't make sense to process all expressions in a single kernel. This would lead to a lot of thread divergence, which essentially means processing one expression after the other. The SIMT programming model might help with parallelising at least some parts of the processing work. However, the generated expressions can differ a lot from each other and restricting them to be similar and therefore SIMT friendly, would likely reduce the overall quality of the symbolic regression algorithm. Therefore, it does not make sense to perform the processing step on the GPU. This is a typical example of code that is better run on the CPU, also because the parallelisation possibilities of one thread per expression can be applied to the CPU as well. Concepts like caching processed expressions, or caching parts of the processed expressions can also be employed on the CPU. This would not be possible on the GPU, because a GPU can not save state between two kernel dispatches.

\subsection{Interpreter}

\begin{figure}
	\centering
	\includegraphics[width=.9\textwidth]{component_diagram_interpreter.png}
	\caption{This diagram depicts the coarse-grained workflow of the interpreter. It shows how the parts interact with each other and with the system it will operate in.}
	\label{fig:component_diagram_interpreter}
\end{figure}

The interpreter consists of two parts. The CPU side is the part of the program, that interacts with both the GPU and the caller. An overview on the components and the workflow of the interpreter can be seen in Figure \ref{fig:component_diagram_interpreter}. Once the interpreter receives the expressions, they are pre-processed. This ensures the expressions are valid, and that they are transformed into the intermediate representation needed for evaluating them. The results of this pre-processing are then sent to the GPU, which performs the actual interpretation of the expressions. Alongside the expressions, the data for the variables and parameters also needs to be sent to the GPU. Once all the data resides on the GPU, the interpreter kernel can be dispatched. It needs to be noted, that for each of the expressions, a separate kernel will be dispatched. As already described, this decision has been made to reduce thread divergence and therefore increase performance. In fact, dispatching the same kernel multiple times with different expressions, means, there will not occur any thread divergence as explained later. Once the GPU has finished evaluating all expressions with all variable sets, the result will be stored in a matrix on the GPU. The CPU then retrieves the results and returns them to the caller in the format specified by the requirements.

Evaluating the expressions is relatively straight forward. Due to the expressions being in postfix-notation, the actual interpreter must only iterate over all tokens once and perform the appropriate tasks. If the interpreter encounters a binary operator, it must simply read the previous two values and perform the operation specified by the operator. For unary operators, only the previous value must be read. As already mentioned, expressions in postfix-notation implicitly contain the operator precedence, therefore no look-ahead or other strategies need to be used to ensure correct evaluation. The Algorithm \ref{alg:eval_interpreter} shows how the interpreter works. Note that this is a simplified version, that only works with additions, multiplications, constant values and variables.

\begin{algorithm}
	\caption{Interpreting an equation in postfix-notation}\label{alg:eval_interpreter}
	\begin{algorithmic}[1]
		\Procedure{Evaluate}{\textit{expr}: PostfixExpression}
			\State $\textit{stack} \gets []$

			\While{HasTokenLeft(\textit{expr})}
				\State $\textit{token} \gets \text{GetNextToken}(\textit{expr})$ 
				\If{$\textit{token.Type} = \text{Constant}$} 
					\State Push($\textit{stack}$, $\textit{token.Value}$)
				\ElsIf{$\textit{token.Type} = \text{Variable}$}
					\State Push($\textit{stack}$, GetVariable($\textit{token.Value}$))
				\ElsIf{$\textit{token.Type} = \text{Operator}$} 
					\If{$\textit{token.Value} = \text{Addition}$}
						\State $\textit{right} \gets \text{Pop}(\textit{stack})$
						\State $\textit{left} \gets \text{Pop}(\textit{stack})$
						\State Push($\textit{stack}$, $\textit{left} + \textit{right}$)
					\ElsIf{$\textit{token.Value} = \text{Multiplication}$}
						\State $\textit{right} \gets \text{Pop}(\textit{stack})$
						\State $\textit{left} \gets \text{Pop}(\textit{stack})$
						\State Push($\textit{stack}$, $\textit{left} * \textit{right}$)
					\EndIf
				\EndIf
			\EndWhile

			\Return $\text{Pop}(\textit{stack})$
		\EndProcedure
	\end{algorithmic}
\end{algorithm}

If a new operator is needed, it must simply be added as another else-if block inside the operator branch. New token types like variables or parameters, can also be added by adding a new outer else-if block that checks for these token types. However, the pre-processing step also needs to be extended with these new operators and token types. Otherwise, the expression will never reach the evaluation step as they would be seen as invalid. It is also possible to add unary operators like $\log()$. In this case only one value would be read from the stack, the operation would be applied, and the result would be written back to the stack.

The Algorithm \ref{alg:eval_interpreter} in this case resembles the kernel. This kernel will be dispatched for every expression that needs to be evaluated, to eliminate thread divergence. Thread divergence can only happen on data dependent branches. In this case, the while loop and every if and else-if statement contains a data dependent branch. Depending on the expression passed to the kernel, the while loop may run longer than for another expression. Similarly, not all expressions have the same constants, operators and variables in the same order and would therefore lead to each thread, taking different paths. However, one expression, always has the same constants, operators and variables in the same locations, meaning all threads will take the same paths. This also means that despite the interpreter containing many data dependent branches, these branches only depend on the expression itself. Because of this, all threads will take the same paths and therefore will never diverge from one another if they execute the same expression.

\subsection{Transpiler}

\begin{figure}
	\centering
	\includegraphics[width=.9\textwidth]{component_diagram_transpiler.png}
	\caption{This diagram depicts the coarse-grained workflow of the transpiler. It shows how the parts interact with each other and with the system it will operate in.}
	\label{fig:component_diagram_transpiler}
\end{figure}

Similar to the interpreter, the transpiler also consists of a part that runs on the CPU and a part that runs on the GPU. When looking at the components and workflow of the transpiler, as shown in Figure \ref{fig:component_diagram_transpiler}, it is almost identical to the interpreter. However, the key difference between the two, is the additional code generation, or transpilation step. Apart from that, the transpiler also needs the same pre-processing step and also the GPU to evaluate the expressions. However, the GPU evaluator generated by the transpiler works differently to the GPU evaluator for the interpreter. The difference between these evaluators will be explained later.

Before the expressions can be transpiled into PTX code, they need to be pre-processed. As already described, this step ensures the validity of the expressions and transforms them into the intermediate representation described above. As with the interpreter, this also simplifies the code generation step at the cost of some performance because the validity has to be ensured, and the intermediate representation needs to be generated. However, in this case the benefit of having a simple code generation step was more important than performance. By transforming the expressions into postfix-notation, the code generation follows a similar pattern to the interpretation already described. Algorithm \ref{alg:transpile} shows how the transpiler takes an expression, transpiles it and then returns the finished code. It can be seen that the while loop is the same as the while loop of the interpreter. The main difference is in the operator branches. Because now code needs to be generated, the branches themselves call their designated code generation function, such as $\textit{GetAddition}$. However, this function can not only return the code that performs the addition for example. When executed, this addition also returns a value which will be needed as an input by other operators. Therefore, not only the code fragment must be returned, but also the reference to the result. This reference can then be put on the stack for later use the same as the interpreter stores the value for later use. The code fragment must also be added to the already generated code so that it can be returned to the caller. As with the interpreter, there is a final value on the stack when the loop has finished. Once the code is executed, this value is the reference to the result of the expression. This value then needs to be stored in the results matrix, so that it can be retrieved by the CPU after all expressions have been executed on the GPU. Therefore, one last code fragment must be generated to handle the storage of this value in the results matrix. This fragment must then be added to the code already generated, and the transpilation process is completed.

\begin{algorithm}
	\caption{Transpiling an equation in postfix-notation}\label{alg:transpile}
	\begin{algorithmic}[1]
		\Procedure{Transpile}{\textit{expr}: PostfixExpression}: String
			\State $\textit{stack} \gets []$
			\State $\textit{code} \gets$ ""

			\While{HasTokenLeft(\textit{expr})}
				\State $\textit{token} \gets \text{GetNextToken}(\textit{expr})$ 
				\If{$\textit{token.Type} = \text{Constant}$} 
					\State Push($\textit{stack}$, $\textit{token.Value}$)
				\ElsIf{$\textit{token.Type} = \text{Variable}$} 
					\State ($\textit{codeFragment}, \textit{referenceToValue}$) $\gets$ GetVariable($\textit{token.Value}$)
					\State Push($\textit{stack}$, $\textit{referenceToValue}$)
					\State Append($\textit{code}$, $\textit{codeFragment}$)
				\ElsIf{$\textit{token.Type} = \text{Operator}$} 
					\If{$\textit{token.Value} = \text{Addition}$}
						\State $\textit{right} \gets \text{Pop}(\textit{stack})$
						\State $\textit{left} \gets \text{Pop}(\textit{stack})$
						\State $(\textit{referenceToValue}, \textit{codeFragment}) \gets \text{GetAddition}(\textit{left}, \textit{right})$
						\State Push($\textit{stack}$, $\textit{referenceToValue}$)
						\State Append($\textit{code}$, $\textit{codeFragment}$)
					\ElsIf{$\textit{token.Value} = \text{Multiplication}$}
						\State $\textit{right} \gets \text{Pop}(\textit{stack})$
						\State $\textit{left} \gets \text{Pop}(\textit{stack})$
						\State $(\textit{referenceToValue}, \textit{codeFragment}) \gets \text{GetMultiplication}(\textit{left}, \textit{right})$
						\State Push($\textit{stack}$, $\textit{referenceToValue}$)
						\State Append($\textit{code}$, $\textit{codeFragment}$)
					\EndIf
				\EndIf
			\EndWhile
						
			\State $\textit{codeFragment} \gets$ GenerateResultStoring($\text{Pop}(\textit{stack})$)
			\State Append($\textit{code}$, $\textit{codeFragment}$)

			\Return $\textit{code}$
		\EndProcedure
	\end{algorithmic}
\end{algorithm}

The code generated by the transpiler is the kernel for the transpiled expressions. This means that a new kernel must be generated for each expression that needs to be evaluated. This is in contrast to the interpreter, which has one kernel and dispatches it once for each expression. However, generating one kernel per expression results in a much simpler kernel. This allows the kernel to focus on evaluating the postfix expression from left to right. No overhead work, like branching or managing a stack is needed. However, this overhead is now offloaded to the transpilation step on the CPU as can be seen in Algorithm \ref{alg:transpile}. There is also a noticeable overhead in that a kernel has to be generated for each expression. In cases like parameter optimisation, many of the expressions will be transpiled multiple times as the transpiler is called multiple times with the same expressions.

Both the transpiler and the interpreter have their respective advantages and disadvantages. While the interpreter puts less load on the CPU, the GPU has to perform more work. Much of this work is branching or managing a stack and therefore involves many instructions that are not used to evaluate the expression itself. However, this overhead can be mitigated by the fact, that all of this overhead is performed in parallel and not sequentially. 

On the other hand, the transpiler performs more work on the CPU. The kernels are much simpler, and most of the instructions are used to evaluate the expressions themselves. Furthermore, as explained in Section \ref{sec:ptx}, any program running on the GPU, must be transpiled into PTX code before the driver can compile it into machine code. Therefore, the kernel written for the interpreter, must also be transpiled into PTX. This overhead is in addition to the branch instruction overhead. The self-written transpiler removes this intermediate step by transpiling directly to PTX. In addition, the generated code is tailored to evaluate expressions and does not need to generate generic PTX code, which can reduce transpilation time. 

Unlike the GPU, the CPU can manage state across multiple calls. Concepts such as caches can be employed by the transpiler to reduce the overhead on the CPU. In cases such as parameter optimisation, where expressions remain the same over multiple calls, the resulting PTX code can be cached. As a result the same expression doesn't need to be transpiled multiple times, drastically reducing the transpilation time and therefore improving the overall performance of the transpiler. 
