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Diffstat (limited to 'report')
| -rw-r--r-- | report/example_derivation2.graffle | 536 | ||||
| -rw-r--r-- | report/example_derivation2.pdf | bin | 0 -> 19177 bytes | |||
| -rw-r--r-- | report/introduction.tex | 53 | ||||
| -rwxr-xr-x | report/report.tex | 1 |
4 files changed, 568 insertions, 22 deletions
diff --git a/report/example_derivation2.graffle b/report/example_derivation2.graffle new file mode 100644 index 00000000..9a32db8b --- /dev/null +++ b/report/example_derivation2.graffle @@ -0,0 +1,536 @@ +<?xml version="1.0" encoding="UTF-8"?> +<!DOCTYPE plist PUBLIC "-//Apple//DTD PLIST 1.0//EN" "http://www.apple.com/DTDs/PropertyList-1.0.dtd"> +<plist version="1.0"> +<dict> + <key>ActiveLayerIndex</key> + <integer>0</integer> + <key>AutoAdjust</key> + <true/> + <key>CanvasColor</key> + <dict> + <key>w</key> + <string>1</string> + </dict> + <key>CanvasOrigin</key> + <string>{0, 0}</string> + <key>CanvasScale</key> + <real>1</real> + <key>ColumnAlign</key> + <integer>1</integer> + <key>ColumnSpacing</key> + <real>36</real> + <key>CreationDate</key> + <string>2008-07-01 20:57:54 +0100</string> + <key>Creator</key> + <string>Phil Blunsom</string> + <key>DisplayScale</key> + <string>1 cm = 1 cm</string> + <key>GraphDocumentVersion</key> + <integer>5</integer> + <key>GraphicsList</key> + <array> + <dict> + <key>Bounds</key> + <string>{{234, 191}, {146, 23}}</string> + <key>Class</key> + <string>ShapedGraphic</string> + <key>FitText</key> + <string>YES</string> + <key>Flow</key> + <string>Resize</string> + <key>ID</key> + <integer>142</integer> + <key>Shape</key> + <string>Rectangle</string> + <key>Style</key> + <dict> + <key>fill</key> + <dict> + <key>Draws</key> + <string>NO</string> + </dict> + <key>shadow</key> + <dict> + <key>Draws</key> + <string>NO</string> + </dict> + <key>stroke</key> + <dict> + <key>Draws</key> + <string>NO</string> + </dict> + </dict> + <key>Text</key> + <dict> + <key>Text</key> + <string>{\rtf1\ansi\ansicpg1252\cocoartf949\cocoasubrtf330 +{\fonttbl\f0\fnil\fcharset128 HiraMinProN-W3;\f1\fnil\fcharset128 HiraKakuProN-W3;\f2\fnil\fcharset134 STXihei; +} +{\colortbl;\red255\green255\blue255;} +\pard\tx560\tx1120\tx1680\tx2240\tx2800\tx3360\tx3920\tx4480\tx5040\tx5600\tx6160\tx6720\qc\pardirnatural + +\f0\b\fs28 \cf0 S +\b0 \'81\'cb \'81\'71 +\f1\b\fs24 X +\b0 \sub 1 +\f2 \nosupersub \'b5\'c4 +\f1 \sub +\f2 \nosupersub \'a1\'a3 +\f1 , +\b X +\b0 \sub 1 +\f2 \nosupersub +\f1\b . +\f0\b0\fs28 \'81\'72}</string> + </dict> + <key>Wrap</key> + <string>NO</string> + </dict> + <dict> + <key>Bounds</key> + <string>{{234, 220}, {308, 23}}</string> + <key>Class</key> + <string>ShapedGraphic</string> + <key>FitText</key> + <string>YES</string> + <key>Flow</key> + <string>Resize</string> + <key>ID</key> + <integer>141</integer> + <key>Shape</key> + <string>Rectangle</string> + <key>Style</key> + <dict> + <key>fill</key> + <dict> + <key>Draws</key> + <string>NO</string> + </dict> + <key>shadow</key> + <dict> + <key>Draws</key> + <string>NO</string> + </dict> + <key>stroke</key> + <dict> + <key>Draws</key> + <string>NO</string> + </dict> + </dict> + <key>Text</key> + <dict> + <key>Text</key> + <string>{\rtf1\ansi\ansicpg1252\cocoartf949\cocoasubrtf330 +{\fonttbl\f0\fnil\fcharset128 HiraMinProN-W3;\f1\fnil\fcharset128 HiraKakuProN-W3;} +{\colortbl;\red255\green255\blue255;} +\pard\tx560\tx1120\tx1680\tx2240\tx2800\tx3360\tx3920\tx4480\tx5040\tx5600\tx6160\tx6720\qc\pardirnatural + +\f0\b\fs28 \cf0 X +\f1\b0\fs24 \sub 1 +\f0\fs28 \nosupersub \'81\'cb \'81\'71 +\f1\b\fs24 X +\b0 \sub 2 \nosupersub \'90\'a5 \'8d\'f0\'93\'56 \'90\'5b\'96\'e9 +\b X +\b0 \sub 3 \nosupersub , +\b X +\b0 \sub 2 +\b \nosupersub X +\b0 \sub 3 \nosupersub late last night +\f0\fs28 \'81\'72}</string> + </dict> + <key>Wrap</key> + <string>NO</string> + </dict> + <dict> + <key>Bounds</key> + <string>{{234, 336}, {150, 23}}</string> + <key>Class</key> + <string>ShapedGraphic</string> + <key>FitText</key> + <string>YES</string> + <key>Flow</key> + <string>Resize</string> + <key>ID</key> + <integer>140</integer> + <key>Shape</key> + <string>Rectangle</string> + <key>Style</key> + <dict> + <key>fill</key> + <dict> + <key>Draws</key> + <string>NO</string> + </dict> + <key>shadow</key> + <dict> + <key>Draws</key> + <string>NO</string> + </dict> + <key>stroke</key> + <dict> + <key>Draws</key> + <string>NO</string> + </dict> + </dict> + <key>Text</key> + <dict> + <key>Text</key> + <string>{\rtf1\ansi\ansicpg1252\cocoartf949\cocoasubrtf330 +{\fonttbl\f0\fnil\fcharset128 HiraMinProN-W3;\f1\fnil\fcharset128 HiraKakuProN-W3;} +{\colortbl;\red255\green255\blue255;} +\pard\tx560\tx1120\tx1680\tx2240\tx2800\tx3360\tx3920\tx4480\tx5040\tx5600\tx6160\tx6720\qc\pardirnatural + +\f0\b\fs28 \cf0 X +\f1\b0\fs24 \sub 5 +\f0\fs28 \nosupersub \'81\'cb \'81\'71 +\f1\fs24 \'96\'6b\'8b\'9e , Beijing +\f0\fs28 \'81\'72}</string> + </dict> + <key>Wrap</key> + <string>NO</string> + </dict> + <dict> + <key>Bounds</key> + <string>{{234, 307}, {165, 23}}</string> + <key>Class</key> + <string>ShapedGraphic</string> + <key>FitText</key> + <string>YES</string> + <key>Flow</key> + <string>Resize</string> + <key>ID</key> + <integer>138</integer> + <key>Shape</key> + <string>Rectangle</string> + <key>Style</key> + <dict> + <key>fill</key> + <dict> + <key>Draws</key> + <string>NO</string> + </dict> + <key>shadow</key> + <dict> + <key>Draws</key> + <string>NO</string> + </dict> + <key>stroke</key> + <dict> + <key>Draws</key> + <string>NO</string> + </dict> + </dict> + <key>Text</key> + <dict> + <key>Text</key> + <string>{\rtf1\ansi\ansicpg1252\cocoartf949\cocoasubrtf330 +{\fonttbl\f0\fnil\fcharset128 HiraMinProN-W3;\f1\fnil\fcharset128 HiraKakuProN-W3;\f2\fnil\fcharset134 STXihei; +} +{\colortbl;\red255\green255\blue255;} +\pard\tx560\tx1120\tx1680\tx2240\tx2800\tx3360\tx3920\tx4480\tx5040\tx5600\tx6160\tx6720\qc\pardirnatural + +\f0\b\fs28 \cf0 X +\f1\b0\fs24 \sub 4 +\f0\fs28 \nosupersub \'81\'cb \'81\'71 +\f2\fs24 \'c9\'cf\'ba\'a3 +\f1 , Shanghai +\f0\fs28 \'81\'72}</string> + </dict> + <key>Wrap</key> + <string>NO</string> + </dict> + <dict> + <key>Bounds</key> + <string>{{234, 278}, {287, 23}}</string> + <key>Class</key> + <string>ShapedGraphic</string> + <key>FitText</key> + <string>YES</string> + <key>Flow</key> + <string>Resize</string> + <key>ID</key> + <integer>137</integer> + <key>Shape</key> + <string>Rectangle</string> + <key>Style</key> + <dict> + <key>fill</key> + <dict> + <key>Draws</key> + <string>NO</string> + </dict> + <key>shadow</key> + <dict> + <key>Draws</key> + <string>NO</string> + </dict> + <key>stroke</key> + <dict> + <key>Draws</key> + <string>NO</string> + </dict> + </dict> + <key>Text</key> + <dict> + <key>Text</key> + <string>{\rtf1\ansi\ansicpg1252\cocoartf949\cocoasubrtf330 +{\fonttbl\f0\fnil\fcharset128 HiraMinProN-W3;\f1\fnil\fcharset128 HiraKakuProN-W3;\f2\fnil\fcharset134 STXihei; +} +{\colortbl;\red255\green255\blue255;} +\pard\tx560\tx1120\tx1680\tx2240\tx2800\tx3360\tx3920\tx4480\tx5040\tx5600\tx6160\tx6720\qc\pardirnatural + +\f0\b\fs28 \cf0 X +\f1\b0\fs24 \sub 3 +\f0\fs28 \nosupersub \'81\'cb \'81\'71 +\f1\fs24 \'98\'b8 +\b X +\b0 \sub 4 \nosupersub \'92\'ef +\f2 \'b4\'ef +\f1\b X +\b0 \sub 5 \nosupersub , arrived in +\b X +\b0 \sub 5\nosupersub from +\b X +\b0 \sub 4 +\f0\fs28 \nosupersub \'81\'72}</string> + </dict> + <key>Wrap</key> + <string>NO</string> + </dict> + <dict> + <key>Bounds</key> + <string>{{234, 249}, {148, 23}}</string> + <key>Class</key> + <string>ShapedGraphic</string> + <key>FitText</key> + <string>YES</string> + <key>Flow</key> + <string>Resize</string> + <key>ID</key> + <integer>101</integer> + <key>Shape</key> + <string>Rectangle</string> + <key>Style</key> + <dict> + <key>fill</key> + <dict> + <key>Draws</key> + <string>NO</string> + </dict> + <key>shadow</key> + <dict> + <key>Draws</key> + <string>NO</string> + </dict> + <key>stroke</key> + <dict> + <key>Draws</key> + <string>NO</string> + </dict> + </dict> + <key>Text</key> + <dict> + <key>Text</key> + <string>{\rtf1\ansi\ansicpg1252\cocoartf949\cocoasubrtf330 +{\fonttbl\f0\fnil\fcharset128 HiraMinProN-W3;\f1\fnil\fcharset128 HiraKakuProN-W3;} +{\colortbl;\red255\green255\blue255;} +\pard\tx560\tx1120\tx1680\tx2240\tx2800\tx3360\tx3920\tx4480\tx5040\tx5600\tx6160\tx6720\qc\pardirnatural + +\f0\b\fs28 \cf0 X +\f1\b0\fs24 \sub 2 +\f0\fs28 \nosupersub \'81\'cb \'81\'71 +\f1\fs24 \'95\'7a\'98\'4e , Brown +\f0\fs28 \'81\'72}</string> + </dict> + <key>Wrap</key> + <string>NO</string> + </dict> + </array> + <key>GridInfo</key> + <dict/> + <key>GuidesLocked</key> + <string>NO</string> + <key>GuidesVisible</key> + <string>YES</string> + <key>HPages</key> + <integer>1</integer> + <key>ImageCounter</key> + <integer>1</integer> + <key>IsPalette</key> + <string>NO</string> + <key>KeepToScale</key> + <false/> + <key>Layers</key> + <array> + <dict> + <key>Lock</key> + <string>NO</string> + <key>Name</key> + <string>Layer 1</string> + <key>Print</key> + <string>YES</string> + <key>View</key> + <string>YES</string> + </dict> + </array> + <key>LayoutInfo</key> + <dict/> + <key>LinksVisible</key> + <string>NO</string> + <key>MagnetsVisible</key> + <string>NO</string> + <key>MasterSheet</key> + <string>Master 1</string> + <key>MasterSheets</key> + <array> + <dict> + <key>ActiveLayerIndex</key> + <integer>0</integer> + <key>AutoAdjust</key> + <true/> + <key>CanvasColor</key> + <dict> + <key>w</key> + <string>1</string> + </dict> + <key>CanvasOrigin</key> + <string>{0, 0}</string> + <key>CanvasScale</key> + <real>1</real> + <key>ColumnAlign</key> + <integer>1</integer> + <key>ColumnSpacing</key> + <real>36</real> + <key>DisplayScale</key> + <string>1 cm = 1 cm</string> + <key>GraphicsList</key> + <array/> + <key>GridInfo</key> + <dict/> + <key>HPages</key> + <integer>1</integer> + <key>IsPalette</key> + <string>NO</string> + <key>KeepToScale</key> + <false/> + <key>Layers</key> + <array> + <dict> + <key>Lock</key> + <string>NO</string> + <key>Name</key> + <string>Layer 1</string> + <key>Print</key> + <string>YES</string> + <key>View</key> + <string>YES</string> + </dict> + </array> + <key>LayoutInfo</key> + <dict/> + <key>Orientation</key> + <integer>2</integer> + <key>OutlineStyle</key> + <string>Basic</string> + <key>RowAlign</key> + <integer>1</integer> + <key>RowSpacing</key> + <real>36</real> + <key>SheetTitle</key> + <string>Master 1</string> + <key>UniqueID</key> + <integer>1</integer> + <key>VPages</key> + <integer>1</integer> + </dict> + </array> + <key>ModificationDate</key> + <string>2008-07-01 21:05:14 +0100</string> + <key>Modifier</key> + <string>Phil Blunsom</string> + <key>NotesVisible</key> + <string>NO</string> + <key>Orientation</key> + <integer>2</integer> + <key>OriginVisible</key> + <string>NO</string> + <key>OutlineStyle</key> + <string>Basic</string> + <key>PageBreaks</key> + <string>YES</string> + <key>PrintInfo</key> + <dict> + <key>NSBottomMargin</key> + <array> + <string>coded</string> + <string>BAtzdHJlYW10eXBlZIHoA4QBQISEhAhOU051bWJlcgCEhAdOU1ZhbHVlAISECE5TT2JqZWN0AIWEASqEhAFklwCG</string> + </array> + <key>NSLeftMargin</key> + <array> + <string>coded</string> + <string>BAtzdHJlYW10eXBlZIHoA4QBQISEhAhOU051bWJlcgCEhAdOU1ZhbHVlAISECE5TT2JqZWN0AIWEASqEhAFklwCG</string> + </array> + <key>NSOrientation</key> + <array> + <string>int</string> + <string>1</string> + </array> + <key>NSPaperSize</key> + <array> + <string>size</string> + <string>{842, 595}</string> + </array> + <key>NSRightMargin</key> + <array> + <string>coded</string> + <string>BAtzdHJlYW10eXBlZIHoA4QBQISEhAhOU051bWJlcgCEhAdOU1ZhbHVlAISECE5TT2JqZWN0AIWEASqEhAFklwCG</string> + </array> + <key>NSTopMargin</key> + <array> + <string>coded</string> + <string>BAtzdHJlYW10eXBlZIHoA4QBQISEhAhOU051bWJlcgCEhAdOU1ZhbHVlAISECE5TT2JqZWN0AIWEASqEhAFklwCG</string> + </array> + </dict> + <key>ReadOnly</key> + <string>NO</string> + <key>RowAlign</key> + <integer>1</integer> + <key>RowSpacing</key> + <real>36</real> + <key>SheetTitle</key> + <string>Canvas 1</string> + <key>SmartAlignmentGuidesActive</key> + <string>YES</string> + <key>SmartDistanceGuidesActive</key> + <string>YES</string> + <key>UniqueID</key> + <integer>1</integer> + <key>UseEntirePage</key> + <true/> + <key>VPages</key> + <integer>1</integer> + <key>WindowInfo</key> + <dict> + <key>CurrentSheet</key> + <string>0</string> + <key>DrawerOpen</key> + <false/> + <key>DrawerTab</key> + <string>Outline</string> + <key>DrawerWidth</key> + <real>209</real> + <key>FitInWindow</key> + <false/> + <key>Frame</key> + <string>{{68, 11}, {846, 735}}</string> + <key>ShowRuler</key> + <false/> + <key>ShowStatusBar</key> + <true/> + <key>VisibleRegion</key> + <string>{{-6, -26}, {831, 621}}</string> + <key>Zoom</key> + <string>1</string> + </dict> +</dict> +</plist> diff --git a/report/example_derivation2.pdf b/report/example_derivation2.pdf Binary files differnew file mode 100644 index 00000000..b143245c --- /dev/null +++ b/report/example_derivation2.pdf diff --git a/report/introduction.tex b/report/introduction.tex index 3b673c8e..12cc2705 100644 --- a/report/introduction.tex +++ b/report/introduction.tex @@ -81,7 +81,7 @@ We structured the workshop into three parallel but interdependent streams: \begin{figure} \centering - \subfigure{\includegraphics[scale=0.5]{intro_slides/JeNeVeuxPasTravailler-hiero-labelled.pdf}} + \includegraphics[scale=0.5]{intro_slides/JeNeVeuxPasTravailler-hiero-labelled.pdf} \caption{Example derivation using the Hiero grammar extraction heuristics where non-terminals have been clustered into unsupervised syntactic categories denoted by $X?$.} \label{fig:intro_labelled_hiero} \end{figure} @@ -124,31 +124,24 @@ Chapter \ref{chap:training} describes this work. The remainder of this introductory chapter provides a formal definition of SCFGs and describes the language pairs that we experimented with. \section{Synchronous context free grammar} \label{sec:scfg} +{\em This section will be moved to the start of Chapter 2} + +%\subsubsection*{Synchronous context free grammar} \label{sec:scfg} +\begin{figure}[t] +\begin{center} +\includegraphics[width=0.6\columnwidth]{example_derivation2.pdf} +\end{center} +\caption[Derivation]{An example SCFG derivation from a Chinese source sentence which yields the English sentence: {\em ``Brown arrived in Shanghai from Beijing late last night.''}. The non-terminal alignment $\mathbf{a}$ is specified by the variable subscripts.} +\label{fig:intro_example_derivation} +\end{figure} + +The translation models discussed explored in this workshop are based on synchronous grammars. +Here we provide a short definition of the formalism we've employed: synchronous context free grammar (SCFG). A synchronous context free grammar (SCFG, \cite{lewis68scfg}) generalizes context-free grammars to generate strings concurrently in two (or more) languages. A string pair is generated by applying a series of paired rewrite rules of the form, $X \rightarrow \langle \mathbf{e}, \mathbf{f}, \mathbf{a} \rangle$, where $X$ is a non-terminal, $\mathbf{e}$ and $\mathbf{f}$ are strings of terminals and non-terminals and $\mathbf{a}$ specifies a one-to-one alignment between non-terminals in $\mathbf{e}$ and $\mathbf{f}$. In the context of SMT, by assigning the source and target languages to the respective sides of a probabilistic SCFG it is possible to describe translation as the process of parsing the source sentence, which induces a parallel tree structure and translation in the target language \cite{chiang07hierarchical}. Terminal are rewritten as pairs of strings of terminal symbols in the source and target languages. Additionally, one side of a terminal expansion may be the special symbol $\epsilon$, which indicates a null alignment which permits arbitrary insertions and deletions. -Figure \ref{fig:scfg} shows an example derivation for Japanese to English translation using an SCFG. - -\begin{figure}Grammar fragment: -\begin{eqnarray*} -\label{rule:discont}X & \rightarrow & \langle \nt{X}{1}\ \nt{X}{2}\ \nt{X}{3},\ \nt{X}{1}\ \nt{X}{3}\ \nt{X}{2} \rangle \\ -X & \rightarrow & \langle \textrm{\emph{John-ga}},\ \textrm{\emph{John}} \rangle \\ -X & \rightarrow & \langle \textrm{\emph{ringo-o}},\ \textrm{\emph{an apple}} \rangle \\ -X & \rightarrow & \langle \textrm{\emph{tabeta}},\ \textrm{\emph{ate}} \rangle -\end{eqnarray*} -Sample derivation: -\begin{eqnarray*} -\label{derivationt} -& &\langle \nt{S}{1},\nt{S}{1} \rangle \Rightarrow \langle \nt{X}{2},\ \nt{X}{2} \rangle \\ - & \Rightarrow& \langle \nt{X}{3}\ \nt{X}{4}\ \nt{X}{5},\ \nt{X}{3}\ \nt{X}{5}\ \nt{X}{4} \rangle \\ - & \Rightarrow &\langle \textrm{\emph{John-ga}}\ \nt{X}{4}\ \nt{X}{5},\ \textrm{\emph{John}}\ \nt{X}{5}\ \nt{X}{4} \rangle \\ - & \Rightarrow &\langle \textrm{\emph{John-ga}}\ \textrm{\emph{ringo-o}}\ \nt{X}{5},\ \textrm{\emph{John}}\ \nt{X}{5}\ \textrm{\emph{an apple}} \rangle \\ - & \Rightarrow &\langle \textrm{\emph{John-ga ringo-o tabeta}},\ \textrm{\emph{John ate an apple}} \rangle -\end{eqnarray*} -\caption{A fragment of an SCFG with a ternary non-terminal expansion and three terminal rules.} -\label{fig:scfg} -\end{figure} +Figure \ref{fig:intro_example_derivation} is an example derivation for Chinese to English translation using an SCFG of the form that I propose to learn using non-parametric Bayesian models. The generative story is as follows. In the beginning was the grammar, in which we allow two types of rules: {\emph non-terminal} and {\emph terminal} expansions. @@ -159,4 +152,20 @@ Rewrite each frontier non-terminal, $c$, using a rule chosen from our grammar ex Repeat until there are no remaining frontier non-terminals. The sentences in both languages can then be read off the leaves, using the rules' alignments to find the right ordering. +\begin{figure}[t] + \centering + \subfigure{\includegraphics[scale=0.7]{intro_slides/PhraseExtraction1.pdf}} + \subfigure{\includegraphics[scale=0.7]{intro_slides/HieroExtraction2.pdf}} +\caption{Extracting translation rules from aligned sentences. All the phrases obtained using the standard phrase extraction heuristics are depicted in the left figure, these are: $\langle$ Je, I $\rangle$, $\langle$ veux, want to $\rangle$, $\langle$ travailler, work $\rangle$, $\langle$ ne veux pas, do not want to $\rangle$, $\langle$ ne veux pas travailler, do not want to work $\rangle$, $\langle$ Je ne veux pas, I do not want to $\rangle$, $\langle$ Je ne veux pas travailler, I do not want to work $\rangle$. On the right is shown how a discontiguous SCFG rule is created by generalising a phrase embedded in another phrase, the extracted rule is: X $\rightarrow$ $\langle$ ne X$_1$ pas, do not X$_1$ $\rangle$.} +\label{fig:intro_rule_extraction} +\end{figure} + +The process for extracting SCFG rules is based on that used to extract translation phrases in phrase based translation systems. +The phrase based approach \cite{koehn03} uses heuristics to extract phrase translation pairs from a word-aligned corpus. +The phrase extraction heuristic is illustrated in Figure \ref{fig:intro_rule_extraction}. +This heuristic extracts all phrases whose words are either not aligned, or aligned with only other words in the same phrase. +The phrase translation probabilities are then calculated using a maximum likelihood estimation. +The Hiero \cite{chiang07hierarchical} SCFG extraction heuristic starts from a grammar consisting of the set of contiguous phrases, wherever a phrase is wholly embedded within another a new rule is add with the embedded phrase replace by the non-terminal X. +This process continues until all possible rules have been extracted, subject to the constraints that every rule must contain a terminal on the source side, a rule may only contain two non-terminals on its right side and that those non-terminals may not be adjacent. +The left example in Figure \ref{fig:intro_rule_extraction} depicts this rule generalisation process. diff --git a/report/report.tex b/report/report.tex index 9e8cc185..2b8e045f 100755 --- a/report/report.tex +++ b/report/report.tex @@ -29,6 +29,7 @@ \theoremstyle{plain} \newcommand{\nt}[2]{\textrm{#1}_{\framebox[5pt]{\scriptsize #2}}} +\newcommand{\ind}[1]{{\fboxsep1pt\raisebox{-.5ex}{\fbox{{\tiny #1}}}}} \begin{document} \title{\vspace{-15mm}\LARGE {\bf Final Report}\\[2mm] |