cws/cw05.tex
author Christian Urban <christian.urban@kcl.ac.uk>
Mon, 29 Aug 2022 08:26:59 +0200
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updated
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
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% !TEX program = xelatex
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\documentclass{article}
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\usepackage{../style}
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\usepackage{../graphicss}
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\usepackage{../langs}
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\definecolor{navyblue}{rgb}{0.0, 0.0, 0.5}
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\definecolor{pansypurple}{rgb}{0.47, 0.09, 0.29}
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\begin{document}
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\color{pansypurple}
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\section*{RESIT / REPLACEMENT}
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{\bf
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The resit / replacement task is essentially CW5 (listed below) with
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the exception that the lexer and parser is already provided. The
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parser will generate an AST (see file \texttt{fun\_llvm.sc}). Your task
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is to generate an AST for the K-intermediate language and supply
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sufficient type annotations such that you can generate valid code for
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the LLVM-IR. The submission deadline is 9th August at 16:00. At the
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deadline, please send me an email containing a zip-file with your
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files.
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Feel free to reuse the files I have uploaded on KEATS (especially
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the files generating simple LLVM-IR code). Of help might also be the
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videos of Week~10.\bigskip
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\noindent
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Good Luck!}
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\color{black}
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\section*{Coursework 5}
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\noindent This coursework is worth 25\% and is due on \cwFIVE{} at
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18:00. You are asked to implement a compiler targeting the LLVM-IR.
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Be careful that this CW needs some material about the LLVM-IR
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that has not been shown in the lectures and your own experiments
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might be required. You can find information about the LLVM-IR at
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\begin{itemize}
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\item \url{https://bit.ly/3rheZYr}
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\item \url{https://llvm.org/docs/LangRef.html}  
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\end{itemize}  
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\noindent
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You can do the implementation of your compiler in any programming
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language you like, but you need to submit the source code with which
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you generated the LLVM-IR files, otherwise a mark of 0\% will be
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awarded. You are asked to submit the code of your compiler, but also
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the generated \texttt{.ll} files. No PDF is needed for this
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coursework.  You should use the lexer and parser from the previous
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courseworks, but you need to make some modifications to them for the
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`typed' version of the Fun-language. I will award up to 5\% if a lexer
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and a parser are correctly implemented. At the end, please package
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everything(!) in a zip-file that creates a directory with the name
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\begin{center}
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\texttt{YournameYourFamilyname}
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\end{center}
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\noindent
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on my end. You will be marked according to the input files
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\begin{itemize}
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\item\href{https://talisker.nms.kcl.ac.uk/cgi-bin/repos.cgi/afl-material/raw-file/tip/cwtests/cw05/sqr.fun}{sqr.fun}  
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\item\href{https://talisker.nms.kcl.ac.uk/cgi-bin/repos.cgi/afl-material/raw-file/tip/cwtests/cw05/fact.fun}{fact.fun}
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\item\href{https://talisker.nms.kcl.ac.uk/cgi-bin/repos.cgi/afl-material/raw-file/tip/cwtests/cw05/mand.fun}{mand.fun}
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\item\href{https://talisker.nms.kcl.ac.uk/cgi-bin/repos.cgi/afl-material/raw-file/tip/cwtests/cw05/mand2.fun}{mand2.fun}
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\item\href{https://talisker.nms.kcl.ac.uk/cgi-bin/repos.cgi/afl-material/raw-file/tip/cwtests/cw05/hanoi.fun}{hanoi.fun}    
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\end{itemize}  
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\noindent
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which are uploaded to KEATS.
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\subsection*{Disclaimer\alert}
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It should be understood that the work you submit represents your own
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effort. You have not copied from anyone else. An exception is the
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Scala code I showed during the lectures or uploaded to KEATS, which
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you can both use. You can also use your own code from the CW~1 --
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CW~4.
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\subsection*{Task}
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The goal is to lex and parse 5 Fun-programs, including the
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Mandelbrot program shown in Figure~\ref{mand}, and generate
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corresponding code for the LLVM-IR. Unfortunately the calculations for
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the Mandelbrot Set require floating point arithmetic and therefore we
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cannot be as simple-minded about types as we have been so far
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(remember the LLVM-IR is a fully-typed language and needs to know the
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exact types of each expression). The idea is to deal appropriately
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with three types, namely \texttt{Int}, \texttt{Double} and
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\texttt{Void} (they are represented in the LLVM-IR as \texttt{i32},
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\texttt{double} and \texttt{void}). You need to extend the lexer and
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parser accordingly in order to deal with type annotations. The
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Fun-language includes global constants, such as
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\begin{lstlisting}[numbers=none]
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  val Ymin: Double = -1.3;
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  val Maxiters: Int = 1000;
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\end{lstlisting}
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\noindent
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where you can assume that they are `normal' identifiers, just
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starting with a capital letter---all other identifiers should have
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lower-case letters. Function definitions can take arguments of
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type \texttt{Int} or \texttt{Double}, and need to specify a return
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type, which can be \texttt{Void}, for example
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\begin{lstlisting}[numbers=none]
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  def foo(n: Int, x: Double) : Double = ...
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  def id(n: Int) : Int = ...
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  def bar() : Void = ...
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\end{lstlisting}
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\noindent
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The idea is to record all typing information that is given
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in the Fun-program, but then delay any further typing inference to
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after the CPS-translation. That means the parser should
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generate ASTs given by the Scala dataypes:
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\begin{lstlisting}[numbers=none,language=Scala]
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abstract class Exp 
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abstract class BExp  
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abstract class Decl 
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case class Def(name: String, args: List[(String, String)],
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               ty: String, body: Exp) extends Decl
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case class Main(e: Exp) extends Decl
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case class Const(name: String, v: Int) extends Decl
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case class FConst(name: String, x: Double) extends Decl
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case class Call(name: String, args: List[Exp]) extends Exp
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case class If(a: BExp, e1: Exp, e2: Exp) extends Exp
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case class Var(s: String) extends Exp
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case class Num(i: Int) extends Exp     // integer numbers
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case class FNum(i: Double) extends Exp  // floating numbers
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case class ChConst(c: Int) extends Exp // char constants
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case class Aop(o: String, a1: Exp, a2: Exp) extends Exp
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case class Sequence(e1: Exp, e2: Exp) extends Exp
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case class Bop(o: String, a1: Exp, a2: Exp) extends BExp
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   145
\end{lstlisting}
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   146
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   147
\noindent
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   148
This datatype distinguishes whether the global constant is an integer
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   149
constant or floating constant. Also a function definition needs to
9d5e4fa0c64d updated
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   150
record the return type of the function, namely the argument
9d5e4fa0c64d updated
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\texttt{ty} in \texttt{Def}, and the arguments consist of an pairs of
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   152
identifier names and types (\texttt{Int} or \texttt{Double}). The hard
9d5e4fa0c64d updated
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   153
part of the CW is to design the K-intermediate language and infer all
9d5e4fa0c64d updated
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   154
necessary types in order to generate LLVM-IR code. You can check
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
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   155
your LLVM-IR code by running it with the interpreter \texttt{lli}.
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   156
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Christian Urban <christian.urban@kcl.ac.uk>
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   157
\begin{figure}[t]
857
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   158
\lstinputlisting[language=Scala]{../cwtests/cw05/mand.fun}
820
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   159
\caption{The Mandelbrot program in the `typed' Fun-language.\label{mand}}
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   160
\end{figure}
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parents: 752
diff changeset
   161
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Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   162
\begin{figure}[t]
857
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   163
\includegraphics[scale=0.35]{../solution/cw5/out.png}
865
bd27a5768c60 updated
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parents: 858
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\caption{Ascii output of the Mandelbrot program.\label{mand2}}
820
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   165
\end{figure}
9d5e4fa0c64d updated
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   166
853
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   167
Also note that the second version of the Mandelbrot program and also
858
67e4eae871a9 updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 857
diff changeset
   168
the Tower of Hanoi program use character constants, like \texttt{'a'},
853
851d8c00f033 updated
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   169
\texttt{'1'}, \texttt{'$\backslash$n'} and so on. When they are tokenised,
851d8c00f033 updated
Christian Urban <christian.urban@kcl.ac.uk>
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   170
such characters should be interpreted as the corresponding ASCII code (an
851d8c00f033 updated
Christian Urban <christian.urban@kcl.ac.uk>
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   171
integer), such that we can use them in calculations like \texttt{'a' + 10}
851d8c00f033 updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 836
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   172
where the result should be 107. As usual, the character \texttt{'$\backslash$n'} is the
851d8c00f033 updated
Christian Urban <christian.urban@kcl.ac.uk>
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   173
ASCII code 10.
851d8c00f033 updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 836
diff changeset
   174
851d8c00f033 updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 836
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   175
820
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   176
\subsection*{LLVM-IR}
9d5e4fa0c64d updated
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   177
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Christian Urban <christian.urban@kcl.ac.uk>
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   178
There are some subtleties in the LLVM-IR you need to be aware of:
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   179
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Christian Urban <christian.urban@kcl.ac.uk>
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   180
\begin{itemize}
9d5e4fa0c64d updated
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   181
\item \textbf{Global constants}: While global constants such as
9d5e4fa0c64d updated
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diff changeset
   182
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Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   183
\begin{lstlisting}[numbers=none]  
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Christian Urban <christian.urban@kcl.ac.uk>
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   184
val Max : Int = 10;
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   185
\end{lstlisting}
200
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff changeset
   186
820
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   187
\noindent
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   188
can be easily defined in the LLVM-IR as follows
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   189
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   190
\begin{lstlisting}[numbers=none]  
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   191
@Max = global i32 10
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   192
\end{lstlisting}
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   193
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   194
\noindent
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   195
they cannot easily be referenced. If you want to use
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   196
this constant then you need to generate code such as
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   197
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   198
\begin{lstlisting}[numbers=none]  
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   199
%tmp_22 = load i32, i32* @Max
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   200
\end{lstlisting}
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   201
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   202
\noindent
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   203
first, which treats \texttt{@Max} as an Integer-pointer (type
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   204
\texttt{i32*}) that needs to be loaded into a local variable,
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   205
here \texttt{\%tmp\_22}.
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   206
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   207
\item \textbf{Void-Functions}: While integer and double functions
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   208
  can easily be called and their results can be allocated to a
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   209
  temporary variable:
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   210
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   211
  \begin{lstlisting}[numbers=none]  
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
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   212
   %tmp_23 = call i32 @sqr (i32 %n)
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   213
  \end{lstlisting}
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   214
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   215
  void-functions cannot be allocated to a variable. They need to be
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   216
  called just as
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   217
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   218
  \begin{lstlisting}[numbers=none]  
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   219
  call void @print_int (i32 %tmp_23)
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   220
\end{lstlisting}
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   221
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   222
\item \textbf{Floating-Point Operations}: While integer operations
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   223
  are specified in the LLVM-IR as
201
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents: 200
diff changeset
   224
820
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   225
  \begin{lstlisting}[numbers=none,language=Scala]
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   226
  def compile_op(op: String) = op match {
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   227
    case "+" => "add i32 "
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   228
    case "*" => "mul i32 "
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   229
    case "-" => "sub i32 "
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   230
    case "==" => "icmp eq i32 "
853
851d8c00f033 updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 836
diff changeset
   231
    case "!=" => "icmp ne i32 "
820
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   232
    case "<=" => "icmp sle i32 " // signed less or equal
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   233
    case "<"  => "icmp slt i32 " // signed less than
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   234
  }\end{lstlisting}
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   235
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   236
  the corresponding operations on doubles are
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   237
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   238
  \begin{lstlisting}[numbers=none,language=Scala]
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   239
  def compile_dop(op: String) = op match {
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   240
    case "+" => "fadd double "
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   241
    case "*" => "fmul double "
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   242
    case "-" => "fsub double "
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   243
    case "==" => "fcmp oeq double "
853
851d8c00f033 updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 836
diff changeset
   244
    case "!=" => "fcmp one double "
820
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   245
    case "<=" => "fcmp ole double "   
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   246
    case "<"  => "fcmp olt double "   
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   247
  }\end{lstlisting}
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   248
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   249
\item \textbf{Typing}: In order to leave the CPS-translations
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   250
  as is, it makes sense to defer the full type-inference to the
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   251
  K-intermediate-language. For this it is good to define
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   252
  the \texttt{KVar} constructor as
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   253
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   254
\begin{lstlisting}[numbers=none,language=Scala]  
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   255
case class KVar(s: String, ty: Ty = "UNDEF") extends KVal\end{lstlisting}
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   256
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   257
  where first a default type, for example \texttt{UNDEF}, is
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   258
  given. Then you need to define two typing functions
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   259
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   260
  \begin{lstlisting}[numbers=none,language=Scala]  
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   261
    def typ_val(v: KVal, ts: TyEnv) = ???
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   262
    def typ_exp(a: KExp, ts: TyEnv) = ???
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   263
  \end{lstlisting}
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   264
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   265
  Both functions require a typing-environment that updates
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   266
  the information about what type each variable, operation
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   267
  and so on receives. Once the types are inferred, the
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   268
  LLVM-IR code can be generated. Since we are dealing only
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   269
  with simple first-order functions, nothing on the scale
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   270
  as the `Hindley-Milner' typing-algorithm is needed. I suggest
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   271
  to just look at what data is avaliable and generate all
836
c8c30949e06f updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 821
diff changeset
   272
  missing information by ``simple means''\ldots rather than
c8c30949e06f updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 821
diff changeset
   273
  looking at the literature which solves the problem
c8c30949e06f updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 821
diff changeset
   274
  with much heavier machinery.
820
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   275
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   276
\item \textbf{Build-In Functions}: The `prelude' comes
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   277
  with several build-in functions: \texttt{new\_line()},
853
851d8c00f033 updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 836
diff changeset
   278
  \texttt{skip}, \texttt{print\_int(n)}, \texttt{print\_space()},
851d8c00f033 updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 836
diff changeset
   279
  \texttt{print\_star()} and \texttt{print\_char(n)}. You can find the `prelude' for
821
c3f9e0fe08cb updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 820
diff changeset
   280
  example in the file \texttt{sqr.ll}.
820
9d5e4fa0c64d updated
Christian Urban <christian.urban@kcl.ac.uk>
parents: 752
diff changeset
   281
\end{itemize}  
205
0b59588d28d2 updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents: 204
diff changeset
   282
200
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff changeset
   283
\end{document}
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff changeset
   284
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff changeset
   285
%%% Local Variables: 
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff changeset
   286
%%% mode: latex
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff changeset
   287
%%% TeX-master: t
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff changeset
   288
%%% End: