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+% Created 2023-11-04 Sat 18:02
+% Intended LaTeX compiler: pdflatex
+\documentclass[11pt]{article}
+\usepackage[utf8]{inputenc}
+\usepackage[T1]{fontenc}
+\usepackage{graphicx}
+\usepackage{longtable}
+\usepackage{wrapfig}
+\usepackage{rotating}
+\usepackage[normalem]{ulem}
+\usepackage{amsmath}
+\usepackage{amssymb}
+\usepackage{capt-of}
+\usepackage{hyperref}
+\notindent \notag  \usepackage{amsmath} \usepackage[a4paper,margin=1in,portrait]{geometry}
+\author{Elizabeth Hunt (A02364151)}
+\date{\today}
+\title{HW 07}
+\hypersetup{
+ pdfauthor={Elizabeth Hunt (A02364151)},
+ pdftitle={HW 07},
+ pdfkeywords={},
+ pdfsubject={},
+ pdfcreator={Emacs 28.2 (Org mode 9.7-pre)}, 
+ pdflang={English}}
+\begin{document}
+
+\maketitle
+\setlength\parindent{20pt}
+
+\section{Problem One}
+\label{sec:orgd2348b9}
+
+\begin{verbatim}
+1. [A1] Y  <- Y  - 1
+2.      IF Y  != 0 GOTO A
+3. [B1] IF X1 != 0 GOTO C
+4.      GOTO E
+5. [C1] X1 <- X1 - 1
+6.      Y  <- Y  + 1
+7.      Y  <- Y  + 1
+8.      Y  <- Y  + 1
+9.      GOTO B1
+\end{verbatim}
+
+\section{Problem Two}
+\label{sec:org07c7432}
+\begin{enumerate}
+\item \((1, \sigma) | \sigma = \{X_1 = 2, Y = 0, Z_1 = 0\}\)
+\item \((4, \sigma) | \sigma = \{X_1 = 2, Y = 0, Z_1 = 0\}\)
+\item \((5, \sigma) | \sigma = \{X_1 = 1, Y = 0, Z_1 = 0\}\)
+\item \((6, \sigma) | \sigma = \{X_1 = 1, Y = 1, Z_1 = 0\}\)
+\item \((7, \sigma) | \sigma = \{X_1 = 1, Y = 1, Z_1 = 1\}\)
+\item \((1, \sigma) | \sigma = \{X_1 = 1, Y = 1, Z_1 = 1\}\)
+\item \((4, \sigma) | \sigma = \{X_1 = 1, Y = 1, Z_1 = 1\}\)
+\item \((5, \sigma) | \sigma = \{X_1 = 0, Y = 1, Z_1 = 1\}\)
+\item \((6, \sigma) | \sigma = \{X_1 = 0, Y = 2, Z_1 = 1\}\)
+\item \((7, \sigma) | \sigma = \{X_1 = 0, Y = 2, Z_1 = 2\}\)
+\item \((1, \sigma) | \sigma = \{X_1 = 0, Y = 2, Z_1 = 2\}\)
+\item \((2, \sigma) | \sigma = \{X_1 = 0, Y = 2, Z_1 = 2\}\)
+\item \((3, \sigma) | \sigma = \{X_1 = 0, Y = 2, Z_1 = 3\}\)
+\item \((8, \sigma) | \sigma = \{X_1 = 0, Y = 2, Z_1 = 3\}\)
+\end{enumerate}
+
+\section{Problem Three}
+\label{sec:orgf9c18d1}
+\begin{verbatim}
+1. [A1] Y  <- Y
+2.      Y  <- Y
+3.      Y  <- Y
+4.      Y  <- Y
+5.      Y  <- Y
+6.      GOTO E
+\end{verbatim}
+
+\section{Problem Four}
+\label{sec:org49ea029}
+Let \(P\) be a program in \(L\) that computes \(g(x_1, x_2, \cdots, x_n)\); a list of instructions \([I_1, I_2, \cdots, I_k]\),
+where \(I_1\) is the first instruction and \(I_k\) the last.
+
+Then, define \(P^i | i \in N\) to be a new program such that each instruction \(I_n\) replaces \(I_{n+i}\) (when \(n=0\)
+we perform no operation), appending
+to the end of the instruction list if necessary. We then replace the sublist \([I_1, \cdots, I_i]\) with
+\([Y \leftarrow Y]^i\) in the program \(P\). As \(Y \leftarrow Y\) produces no side effects then \(P^i\) still computes \(g\).
+
+Finally, for all \(i \in N\) the length of \(P^i\) is greater than \(k\) and thus there are countably infinitely
+many \(L\) -programs to compute \(g\).
+\end{document}
\ No newline at end of file