\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, for a better understanding, the input and output data need to be explained further. The first input contains the expressions that need to be evaluated. These can be of any length and can contain constant values, variables and parameters, all of which are linked together with the supported operations. In the simplified example shown in Figure \ref{fig:input_output_explanation}, there are six expressions $e_1$ to $e_6$. 

Next is the variable matrix. An 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 parameterise the variable $x_3$. Each column holds a different set of variables. Each expression must be evaluated using each set of variables. In the provided example, there are three variable sets, each containing the values for four variables $x_1$ to $x_4$. 

After all expressions have been evaluated using all variable sets, the results of these evaluations must be stored in the result matrix. Each entry in this matrix holds the result of the evaluation of one expression parameterised with one variable set. The row indicates the variable set and the column indicates the expression.

The prototypes developed in this thesis, are part of a GP algorithm for symbolic regression. This means that the expressions that are evaluated, represent parts of the search space of all expressions being made up of any combination of allowed operators, the set of input variables, a set of parameters and constants. This means that the size of the search space grows exponentially. Exploring this search space by simply generating expressions, evaluating them once and then generating the next set of expressions leaves much of the search space unexplored. To combat this, parameters are introduced. These allow the algorithm to perform some kind of local search. To enable this, the prototypes must support not only variables, but also parameters.

The parameters themselves are unique to each expression, meaning they have a one-to-one mapping to an expression. Furthermore, as can be seen in Figure \ref{fig:input_output_explanation}, each expression can have a different number of parameters, or even no parameters at all. However, with no parameters, it wouldn't be possible to perform parameter optimisation. This is in contrast to variables, where each expression must have the same number of variables. Because parameters are unique to each expression and can vary in size, they are not structured as a matrix, but as a vector of vectors.

An important thing to consider, is the volume and volatility of the data itself. The example used above has been drastically simplified. It is expected, that there are hundreds of expressions evaluate per GP generation. Each of these expressions may contain between ten and 50 tokens. A token is equivalent to either a variable, a parameter, a constant value or an operator.

Usually, the number of variables per expression is around ten. However, the number of variable sets can increase drastically. It can be considered, that $1\,000$ variable sets is the lower limit. On the other hand, $100\,000$ can be considered as the upper limit. Considering that one variable takes up 4 bytes of space and 10 variables are needed per expression, at least $4 * 10 * 1\,000 = 40\,000$ bytes and at most $4 * 10 * 100\,000 = 400\,000$ bytes need to be transferred to the GPU for the variables.

These variables do not change during the runtime of the symbolic regression algorithm. As a result the data only needs to be sent to the GPU once. This means that the impact of this data transfer is minimal. On the other hand, the data for the parameters is much more volatile. As explained above, they are used for parameter optimisation and therefore vary from evaluation to evaluation and need to be sent to the GPU very frequently. However, the amount of data that needs to be sent is also much smaller. TODO: ONCE I GET THE DATA SEE HOW MANY BYTES PARAMETERS TAKE ON AVERAGE

\section{Architecture}
\label{sec:architecture}
Based on the requirements and data structure 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, further increases GPU utilisation. 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 rather 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. This also reduces the overhead on the GPU. One drawback of generating a kernel for each expression, is the generation itself. Especially for smaller variable sets, it is possible, that the time it takes to transpile an expression is greater than the time it takes to evaluate it. However, for larger variable sets this should not be a concern.

%
% TODO: Probably include a diagram that shows how the evaluators are integrated in the symbolic regression algorithm (assuming its a GP variant), to show the bigger picture
%

\subsection{Pre-Processing}
\label{sec: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. Secondly, this step also converts the expression into an intermediate representation. In essence, the pre-processing step can be compared to the frontend of a compiler as described in Section \ref{sec:compilers}. If new operators are required, the pre-processor must be extended as well. Otherwise, expressions containing these operators would be treated as invalid and never reach the evaluator.

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 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. 

The already mentioned concept of processing one expression per thread can also be used on the CPU, which is better designed for this type of work. Concepts such as caching processed expressions, or caching parts of the processed expressions can also be employed on the CPU to speed up pre-processing. This would not be possible on the GPU, because a GPU can not save state between two kernel dispatches. This is a typical example of code that is better run on the CPU and shows how the CPU and GPU need to work together and exploit their respective strengths to achieve the best performance. 

\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 of the components and the workflow of the interpreter is shown in Figure \ref{fig:component_diagram_interpreter}. Once the interpreter has received the expressions, they are pre-processed. This ensures that the expressions are valid, and that they are transformed into the intermediate representation needed to evaluate them. The result of this pre-processing step is then sent to the GPU, which performs the actual interpretation of the expressions. In addition to the expressions, the data for the variables and parameters must also be sent to the GPU. 

Once all the data is present on the GPU, the interpreter kernel can be dispatched. As already described, the kernel will be dispatched for each expression to reduce thread divergence. In fact, dispatching the same kernel multiple times with different expressions, means, there will not occur any thread divergence which will be explained later. 

After the GPU has finished evaluating all expressions with all variable sets, the result is 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. Because the expressions are in postfix notation, the actual interpreter just needs to iterate over all the tokens and perform the appropriate tasks. If the interpreter encounters a binary operator, it simply needs to read the previous two values and perform the operation specified by the operator. For unary operators, only the previous value needs to 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. This also means that each token is visited exactly once and no unnecessary or overhead work needs to be done. The Algorithm \ref{alg:eval_interpreter} shows how the interpreter works. Note that this is a simplified version, that only works with additions, multiplications, constants 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

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

The handling of constants and variables is very simple. Constants only need to be stored on the stack for later use. Variables also only need to be stored on the stack. However, their value must first be loaded from the variable matrix according to the token value of the variable. Since the entire variable matrix is sent to the GPU, the index of the variable set is also needed to load the variable value. However, for the sake of simplicity, it has been omitted from the algorithm.

When an operator token is encountered, the handling becomes more complex. The value of the token indicates the type of operation to be applied. For binary operators, the top two values on the stack need to be used as input to the operator. For unary operators, only the top value of the stack needs to be used as an input. Once the result has been computed, it must be stored at the top of the stack to be used as an input for the next operation. 

At the end of the algorithm, the stack contains one last entry. This entry is the value computed by the expression with the designated variable set and parameters. In order to send this value back to the CPU, it must be stored in the result matrix. The last statement performs this action. It again has been simplified to omit the index of the expression and variable set needed to store the result at the correct location.

The Algorithm \ref{alg:eval_interpreter} in this case resembles the kernel. This kernel will be dispatched for each expression that needs to be evaluated, to prevent thread divergence. Thread divergence can only occur 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 or variables in the same order, and would therefore cause each thread to take a different path. However, one expression always has the same constants, operators and variables in the same locations, meaning that all threads will take the same path. This also means that although the interpreter contains many data-dependent branches, these branches only depend on the expression itself. Because of this, all threads will follow the same path and will therefore never diverge from one another as long as they are executing 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. 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 very 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 have 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. 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 largely the same as the while loop of the interpreter. The main difference is in the operator branches, because now code needs to be generated instead of the result of computing the expression. Therefore, the branches themselves call their designated code generation function, such as $\textit{GetAddition}$. This function returns the PTX code responsible for the addition. However, this function must return more than just the code that performs the addition. 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, just 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 has been executed, this value is the reference to the result of the expression. This value then needs to be stored in the result matrix, so that it can be retrieved by the CPU after all expressions have been executed on the GPU. Therefore, a final code fragment must be generated to handle the storage of this value in the result matrix. This fragment must then be added to the code already generated, and the transpilation process is complete.

\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. There is no overhead work such as branching or managing a stack. However, this overhead is now shifted to the transpilation step on the CPU which 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 involves 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 this work is performed in parallel rather than 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 into 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 kernel dispatches. 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 across multiple calls, the resulting PTX code can be cached. As a result, the same expression doesn't need to be transpiled multiple times which drastically reducing the transpilation time. This is an important optimisation as this can improve the overall performance of the transpiler. 
