\chapter{Expressions\label{expressions}}
\index{expression}

This chapter explains the meaning of the elements of expressions in
Python.

\strong{Syntax Notes:} In this and the following chapters, extended
BNF\index{BNF} notation will be used to describe syntax, not lexical
analysis.  When (one alternative of) a syntax rule has the form

\begin{productionlist}[*]
  \production{name}{\token{othername}}
\end{productionlist}

and no semantics are given, the semantics of this form of \code{name}
are the same as for \code{othername}.
\index{syntax}

\section{Arithmetic conversions\label{conversions}}
\indexii{arithmetic}{conversion}

When a description of an arithmetic operator below uses the phrase
``the numeric arguments are converted to a common type,'' the
arguments are coerced using the coercion rules listed at
~\ref{coercion-rules}.  If both arguments are standard numeric types,
the following coercions are applied:

\begin{itemize}
\item	If either argument is a complex number, the other is converted
	to complex;
\item	otherwise, if either argument is a floating point number,
	the other is converted to floating point;
\item	otherwise, if either argument is a long integer,
	the other is converted to long integer;
\item	otherwise, both must be plain integers and no conversion
	is necessary.
\end{itemize}

Some additional rules apply for certain operators (e.g., a string left
argument to the `\%' operator). Extensions can define their own
coercions.

\section{Atoms\label{atoms}}
\index{atom}

Atoms are the most basic elements of expressions.  The simplest atoms
are identifiers or literals.  Forms enclosed in
reverse quotes or in parentheses, brackets or braces are also
categorized syntactically as atoms.  The syntax for atoms is:

\begin{productionlist}
  \production{atom}
             {\token{identifier} | \token{literal} | \token{enclosure}}
  \production{enclosure}
             {\token{parenth_form} | \token{list_display}}
  \productioncont{| \token{generator_expression} | \token{dict_display}}
  \productioncont{| \token{string_conversion}}
\end{productionlist}

\subsection{Identifiers (Names)\label{atom-identifiers}}
\index{name}
\index{identifier}

An identifier occurring as an atom is a name.  See
section~\ref{naming} for documentation of naming and binding.

When the name is bound to an object, evaluation of the atom yields
that object.  When a name is not bound, an attempt to evaluate it
raises a \exception{NameError} exception.
\exindex{NameError}

\strong{Private name mangling:}
\indexii{name}{mangling}%
\indexii{private}{names}%
When an identifier that textually occurs in a class definition begins
with two or more underscore characters and does not end in two or more
underscores, it is considered a \dfn{private name} of that class.
Private names are transformed to a longer form before code is
generated for them.  The transformation inserts the class name in
front of the name, with leading underscores removed, and a single
underscore inserted in front of the class name.  For example, the
identifier \code{__spam} occurring in a class named \code{Ham} will be
transformed to \code{_Ham__spam}.  This transformation is independent
of the syntactical context in which the identifier is used.  If the
transformed name is extremely long (longer than 255 characters),
implementation defined truncation may happen.  If the class name
consists only of underscores, no transformation is done.

\subsection{Literals\label{atom-literals}}
\index{literal}

Python supports string literals and various numeric literals:

\begin{productionlist}
  \production{literal}
             {\token{stringliteral} | \token{integer} | \token{longinteger}}
  \productioncont{| \token{floatnumber} | \token{imagnumber}}
\end{productionlist}

Evaluation of a literal yields an object of the given type (string,
integer, long integer, floating point number, complex number) with the
given value.  The value may be approximated in the case of floating
point and imaginary (complex) literals.  See section \ref{literals}
for details.

All literals correspond to immutable data types, and hence the
object's identity is less important than its value.  Multiple
evaluations of literals with the same value (either the same
occurrence in the program text or a different occurrence) may obtain
the same object or a different object with the same value.
\indexiii{immutable}{data}{type}
\indexii{immutable}{object}

\subsection{Parenthesized forms\label{parenthesized}}
\index{parenthesized form}

A parenthesized form is an optional expression list enclosed in
parentheses:

\begin{productionlist}
  \production{parenth_form}
             {"(" [\token{expression_list}] ")"}
\end{productionlist}

A parenthesized expression list yields whatever that expression list
yields: if the list contains at least one comma, it yields a tuple;
otherwise, it yields the single expression that makes up the
expression list.

An empty pair of parentheses yields an empty tuple object.  Since
tuples are immutable, the rules for literals apply (i.e., two
occurrences of the empty tuple may or may not yield the same object).
\indexii{empty}{tuple}

Note that tuples are not formed by the parentheses, but rather by use
of the comma operator.  The exception is the empty tuple, for which
parentheses \emph{are} required --- allowing unparenthesized ``nothing''
in expressions would cause ambiguities and allow common typos to
pass uncaught.
\index{comma}
\indexii{tuple}{display}

\subsection{List displays\label{lists}}
\indexii{list}{display}
\indexii{list}{comprehensions}

A list display is a possibly empty series of expressions enclosed in
square brackets:

\begin{productionlist}
  \production{test}
             {\token{or_test} | \token{lambda_form}}
  \production{testlist}
             {\token{test} ( "," \token{test} )* [ "," ]}
  \production{list_display}
             {"[" [\token{listmaker}] "]"}
  \production{listmaker}
             {\token{expression} ( \token{list_for}
              | ( "," \token{expression} )* [","] )}
  \production{list_iter}
             {\token{list_for} | \token{list_if}}
  \production{list_for}
             {"for" \token{expression_list} "in" \token{testlist}
              [\token{list_iter}]}
  \production{list_if}
             {"if" \token{test} [\token{list_iter}]}
\end{productionlist}

A list display yields a new list object.  Its contents are specified
by providing either a list of expressions or a list comprehension.
\indexii{list}{comprehensions}
When a comma-separated list of expressions is supplied, its elements are
evaluated from left to right and placed into the list object in that
order.  When a list comprehension is supplied, it consists of a
single expression followed by at least one \keyword{for} clause and zero or
more \keyword{for} or \keyword{if} clauses.  In this
case, the elements of the new list are those that would be produced
by considering each of the \keyword{for} or \keyword{if} clauses a block,
nesting from
left to right, and evaluating the expression to produce a list element
each time the innermost block is reached\footnote{In Python 2.3, a
list comprehension "leaks" the control variables of each
\samp{for} it contains into the containing scope.  However, this
behavior is deprecated, and relying on it will not work once this
bug is fixed in a future release}.
\obindex{list}
\indexii{empty}{list}

\subsection{Generator expressions\label{genexpr}}
\indexii{generator}{expression}

A generator expression is a compact generator notation in parentheses:

\begin{productionlist}
  \production{generator_expression}
             {"(" \token{test} \token{genexpr_for} ")"}
  \production{genexpr_for}
             {"for" \token{expression_list} "in" \token{test}
              [\token{genexpr_iter}]}
  \production{genexpr_iter}
             {\token{genexpr_for} | \token{genexpr_if}}
  \production{genexpr_if}
             {"if" \token{test} [\token{genexpr_iter}]}
\end{productionlist}

A generator expression yields a new generator object.
\obindex{generator}
\obindex{generator expression}
It consists of a single expression followed by at least one
\keyword{for} clause and zero or more \keyword{for} or \keyword{if}
clauses.  The iterating values of the new generator are those that
would be produced by considering each of the \keyword{for} or
\keyword{if} clauses a block, nesting from left to right, and
evaluating the expression to yield a value that is reached the
innermost block for each iteration.

Variables used in the generator expression are evaluated lazily
when the \method{next()} method is called for generator object
(in the same fashion as normal generators). However, the leftmost
\keyword{for} clause is immediately evaluated so that error produced
by it can be seen before any other possible error in the code that
handles the generator expression.
Subsequent \keyword{for} clauses cannot be evaluated immediately since
they may depend on the previous \keyword{for} loop.
For example: \samp{(x*y for x in range(10) for y in bar(x))}.

The parentheses can be omitted on calls with only one argument.
See section \ref{calls} for the detail.

\subsection{Dictionary displays\label{dict}}
\indexii{dictionary}{display}

A dictionary display is a possibly empty series of key/datum pairs
enclosed in curly braces:
\index{key}
\index{datum}
\index{key/datum pair}

\begin{productionlist}
  \production{dict_display}
             {"\{" [\token{key_datum_list}] "\}"}
  \production{key_datum_list}
             {\token{key_datum} ("," \token{key_datum})* [","]}
  \production{key_datum}
             {\token{expression} ":" \token{expression}}
\end{productionlist}

A dictionary display yields a new dictionary object.
\obindex{dictionary}

The key/datum pairs are evaluated from left to right to define the
entries of the dictionary: each key object is used as a key into the
dictionary to store the corresponding datum.

Restrictions on the types of the key values are listed earlier in
section \ref{types}.  (To summarize, the key type should be hashable,
which excludes all mutable objects.)  Clashes between duplicate keys
are not detected; the last datum (textually rightmost in the display)
stored for a given key value prevails.
\indexii{immutable}{object}

\subsection{String conversions\label{string-conversions}}
\indexii{string}{conversion}
\indexii{reverse}{quotes}
\indexii{backward}{quotes}
\index{back-quotes}

A string conversion is an expression list enclosed in reverse (a.k.a.
backward) quotes:

\begin{productionlist}
  \production{string_conversion}
             {"`" \token{expression_list} "`"}
\end{productionlist}

A string conversion evaluates the contained expression list and
converts the resulting object into a string according to rules
specific to its type.

If the object is a string, a number, \code{None}, or a tuple, list or
dictionary containing only objects whose type is one of these, the
resulting string is a valid Python expression which can be passed to
the built-in function \function{eval()} to yield an expression with the
same value (or an approximation, if floating point numbers are
involved).

(In particular, converting a string adds quotes around it and converts
``funny'' characters to escape sequences that are safe to print.)

Recursive objects (for example, lists or dictionaries that contain a
reference to themselves, directly or indirectly) use \samp{...} to
indicate a recursive reference, and the result cannot be passed to
\function{eval()} to get an equal value (\exception{SyntaxError} will
be raised instead).
\obindex{recursive}

The built-in function \function{repr()} performs exactly the same
conversion in its argument as enclosing it in parentheses and reverse
quotes does.  The built-in function \function{str()} performs a
similar but more user-friendly conversion.
\bifuncindex{repr}
\bifuncindex{str}

\section{Primaries\label{primaries}}
\index{primary}

Primaries represent the most tightly bound operations of the language.
Their syntax is:

\begin{productionlist}
  \production{primary}
             {\token{atom} | \token{attributeref}
              | \token{subscription} | \token{slicing} | \token{call}}
\end{productionlist}

\subsection{Attribute references\label{attribute-references}}
\indexii{attribute}{reference}

An attribute reference is a primary followed by a period and a name:

\begin{productionlist}
  \production{attributeref}
             {\token{primary} "." \token{identifier}}
\end{productionlist}

The primary must evaluate to an object of a type that supports
attribute references, e.g., a module, list, or an instance.  This
object is then asked to produce the attribute whose name is the
identifier.  If this attribute is not available, the exception
\exception{AttributeError}\exindex{AttributeError} is raised.
Otherwise, the type and value of the object produced is determined by
the object.  Multiple evaluations of the same attribute reference may
yield different objects.
\obindex{module}
\obindex{list}

\subsection{Subscriptions\label{subscriptions}}
\index{subscription}

A subscription selects an item of a sequence (string, tuple or list)
or mapping (dictionary) object:
\obindex{sequence}
\obindex{mapping}
\obindex{string}
\obindex{tuple}
\obindex{list}
\obindex{dictionary}
\indexii{sequence}{item}

\begin{productionlist}
  \production{subscription}
             {\token{primary} "[" \token{expression_list} "]"}
\end{productionlist}

The primary must evaluate to an object of a sequence or mapping type.

If the primary is a mapping, the expression list must evaluate to an
object whose value is one of the keys of the mapping, and the
subscription selects the value in the mapping that corresponds to that
key.  (The expression list is a tuple except if it has exactly one
item.)

If the primary is a sequence, the expression (list) must evaluate to a
plain integer.  If this value is negative, the length of the sequence
is added to it (so that, e.g., \code{x[-1]} selects the last item of
\code{x}.)  The resulting value must be a nonnegative integer less
than the number of items in the sequence, and the subscription selects
the item whose index is that value (counting from zero).

A string's items are characters.  A character is not a separate data
type but a string of exactly one character.
\index{character}
\indexii{string}{item}

\subsection{Slicings\label{slicings}}
\index{slicing}
\index{slice}

A slicing selects a range of items in a sequence object (e.g., a
string, tuple or list).  Slicings may be used as expressions or as
targets in assignment or \keyword{del} statements.  The syntax for a
slicing:
\obindex{sequence}
\obindex{string}
\obindex{tuple}
\obindex{list}

\begin{productionlist}
  \production{slicing}
             {\token{simple_slicing} | \token{extended_slicing}}
  \production{simple_slicing}
             {\token{primary} "[" \token{short_slice} "]"}
  \production{extended_slicing}
             {\token{primary} "[" \token{slice_list} "]" }
  \production{slice_list}
             {\token{slice_item} ("," \token{slice_item})* [","]}
  \production{slice_item}
             {\token{expression} | \token{proper_slice} | \token{ellipsis}}
  \production{proper_slice}
             {\token{short_slice} | \token{long_slice}}
  \production{short_slice}
             {[\token{lower_bound}] ":" [\token{upper_bound}]}
  \production{long_slice}
             {\token{short_slice} ":" [\token{stride}]}
  \production{lower_bound}
             {\token{expression}}
  \production{upper_bound}
             {\token{expression}}
  \production{stride}
             {\token{expression}}
  \production{ellipsis}
             {"..."}
\end{productionlist}

There is ambiguity in the formal syntax here: anything that looks like
an expression list also looks like a slice list, so any subscription
can be interpreted as a slicing.  Rather than further complicating the
syntax, this is disambiguated by defining that in this case the
interpretation as a subscription takes priority over the
interpretation as a slicing (this is the case if the slice list
contains no proper slice nor ellipses).  Similarly, when the slice
list has exactly one short slice and no trailing comma, the
interpretation as a simple slicing takes priority over that as an
extended slicing.\indexii{extended}{slicing}

The semantics for a simple slicing are as follows.  The primary must
evaluate to a sequence object.  The lower and upper bound expressions,
if present, must evaluate to plain integers; defaults are zero and the
\code{sys.maxint}, respectively.  If either bound is negative, the
sequence's length is added to it.  The slicing now selects all items
with index \var{k} such that
\code{\var{i} <= \var{k} < \var{j}} where \var{i}
and \var{j} are the specified lower and upper bounds.  This may be an
empty sequence.  It is not an error if \var{i} or \var{j} lie outside the
range of valid indexes (such items don't exist so they aren't
selected).

The semantics for an extended slicing are as follows.  The primary
must evaluate to a mapping object, and it is indexed with a key that
is constructed from the slice list, as follows.  If the slice list
contains at least one comma, the key is a tuple containing the
conversion of the slice items; otherwise, the conversion of the lone
slice item is the key.  The conversion of a slice item that is an
expression is that expression.  The conversion of an ellipsis slice
item is the built-in \code{Ellipsis} object.  The conversion of a
proper slice is a slice object (see section \ref{types}) whose
\member{start}, \member{stop} and \member{step} attributes are the
values of the expressions given as lower bound, upper bound and
stride, respectively, substituting \code{None} for missing
expressions.
\withsubitem{(slice object attribute)}{\ttindex{start}
  \ttindex{stop}\ttindex{step}}

\subsection{Calls\label{calls}}
\index{call}

A call calls a callable object (e.g., a function) with a possibly empty
series of arguments:
\obindex{callable}

\begin{productionlist}
  \production{call}
             {\token{primary} "(" [\token{argument_list} [","]] ")"}
             {\token{primary} "(" [\token{argument_list} [","] |
	      \token{test} \token{genexpr_for} ] ")"}
  \production{argument_list}
             {\token{positional_arguments} ["," \token{keyword_arguments}]}
  \productioncont{                     ["," "*" \token{expression}]}
  \productioncont{                     ["," "**" \token{expression}]}
  \productioncont{| \token{keyword_arguments} ["," "*" \token{expression}]}
  \productioncont{                    ["," "**" \token{expression}]}
  \productioncont{| "*" \token{expression} ["," "**" \token{expression}]}
  \productioncont{| "**" \token{expression}}
  \production{positional_arguments}
             {\token{expression} ("," \token{expression})*}
  \production{keyword_arguments}
             {\token{keyword_item} ("," \token{keyword_item})*}
  \production{keyword_item}
             {\token{identifier} "=" \token{expression}}
\end{productionlist}

A trailing comma may be present after the positional and keyword
arguments but does not affect the semantics.

The primary must evaluate to a callable object (user-defined
functions, built-in functions, methods of built-in objects, class
objects, methods of class instances, and certain class instances
themselves are callable; extensions may define additional callable
object types).  All argument expressions are evaluated before the call
is attempted.  Please refer to section \ref{function} for the syntax
of formal parameter lists.

If keyword arguments are present, they are first converted to
positional arguments, as follows.  First, a list of unfilled slots is
created for the formal parameters.  If there are N positional
arguments, they are placed in the first N slots.  Next, for each
keyword argument, the identifier is used to determine the
corresponding slot (if the identifier is the same as the first formal
parameter name, the first slot is used, and so on).  If the slot is
already filled, a \exception{TypeError} exception is raised.
Otherwise, the value of the argument is placed in the slot, filling it
(even if the expression is \code{None}, it fills the slot).  When all
arguments have been processed, the slots that are still unfilled are
filled with the corresponding default value from the function
definition.  (Default values are calculated, once, when the function
is defined; thus, a mutable object such as a list or dictionary used
as default value will be shared by all calls that don't specify an
argument value for the corresponding slot; this should usually be
avoided.)  If there are any unfilled slots for which no default value
is specified, a \exception{TypeError} exception is raised.  Otherwise,
the list of filled slots is used as the argument list for the call.

If there are more positional arguments than there are formal parameter
slots, a \exception{TypeError} exception is raised, unless a formal
parameter using the syntax \samp{*identifier} is present; in this
case, that formal parameter receives a tuple containing the excess
positional arguments (or an empty tuple if there were no excess
positional arguments).

If any keyword argument does not correspond to a formal parameter
name, a \exception{TypeError} exception is raised, unless a formal
parameter using the syntax \samp{**identifier} is present; in this
case, that formal parameter receives a dictionary containing the
excess keyword arguments (using the keywords as keys and the argument
values as corresponding values), or a (new) empty dictionary if there
were no excess keyword arguments.

If the syntax \samp{*expression} appears in the function call,
\samp{expression} must evaluate to a sequence.  Elements from this
sequence are treated as if they were additional positional arguments;
if there are postional arguments \var{x1},...,\var{xN} , and
\samp{expression} evaluates to a sequence \var{y1},...,\var{yM}, this
is equivalent to a call with M+N positional arguments
\var{x1},...,\var{xN},\var{y1},...,\var{yM}.

A consequence of this is that although the \samp{*expression} syntax
appears \emph{after} any keyword arguments, it is processed
\emph{before} the keyword arguments (and the
\samp{**expression} argument, if any -- see below).  So:

\begin{verbatim}
>>> def f(a, b):
...  print a, b
...
>>> f(b=1, *(2,))
2 1
>>> f(a=1, *(2,))
Traceback (most recent call last):
  File "<stdin>", line 1, in ?
TypeError: f() got multiple values for keyword argument 'a'
>>> f(1, *(2,))
1 2
\end{verbatim}

It is unusual for both keyword arguments and the
\samp{*expression} syntax to be used in the same call, so in practice
this confusion does not arise.

If the syntax \samp{**expression} appears in the function call,
\samp{expression} must evaluate to a (subclass of) dictionary, the
contents of which are treated as additional keyword arguments.  In the
case of a keyword appearing in both \samp{expression} and as an
explicit keyword argument, a \exception{TypeError} exception is
raised.

Formal parameters using the syntax \samp{*identifier} or
\samp{**identifier} cannot be used as positional argument slots or
as keyword argument names.  Formal parameters using the syntax
\samp{(sublist)} cannot be used as keyword argument names; the
outermost sublist corresponds to a single unnamed argument slot, and
the argument value is assigned to the sublist using the usual tuple
assignment rules after all other parameter processing is done.

A call always returns some value, possibly \code{None}, unless it
raises an exception.  How this value is computed depends on the type
of the callable object.

If it is---

\begin{description}

\item[a user-defined function:] The code block for the function is
executed, passing it the argument list.  The first thing the code
block will do is bind the formal parameters to the arguments; this is
described in section \ref{function}.  When the code block executes a
\keyword{return} statement, this specifies the return value of the
function call.
\indexii{function}{call}
\indexiii{user-defined}{function}{call}
\obindex{user-defined function}
\obindex{function}

\item[a built-in function or method:] The result is up to the
interpreter; see the \citetitle[../lib/built-in-funcs.html]{Python
Library Reference} for the descriptions of built-in functions and
methods.
\indexii{function}{call}
\indexii{built-in function}{call}
\indexii{method}{call}
\indexii{built-in method}{call}
\obindex{built-in method}
\obindex{built-in function}
\obindex{method}
\obindex{function}

\item[a class object:] A new instance of that class is returned.
\obindex{class}
\indexii{class object}{call}

\item[a class instance method:] The corresponding user-defined
function is called, with an argument list that is one longer than the
argument list of the call: the instance becomes the first argument.
\obindex{class instance}
\obindex{instance}
\indexii{class instance}{call}

\item[a class instance:] The class must define a \method{__call__()}
method; the effect is then the same as if that method was called.
\indexii{instance}{call}
\withsubitem{(object method)}{\ttindex{__call__()}}

\end{description}

\section{The power operator\label{power}}

The power operator binds more tightly than unary operators on its
left; it binds less tightly than unary operators on its right.  The
syntax is:

\begin{productionlist}
  \production{power}
             {\token{primary} ["**" \token{u_expr}]}
\end{productionlist}

Thus, in an unparenthesized sequence of power and unary operators, the
operators are evaluated from right to left (this does not constrain
the evaluation order for the operands).

The power operator has the same semantics as the built-in
\function{pow()} function, when called with two arguments: it yields
its left argument raised to the power of its right argument.  The
numeric arguments are first converted to a common type.  The result
type is that of the arguments after coercion.

With mixed operand types, the coercion rules for binary arithmetic
operators apply. For int and long int operands, the result has the
same type as the operands (after coercion) unless the second argument
is negative; in that case, all arguments are converted to float and a
float result is delivered. For example, \code{10**2} returns \code{100},
but \code{10**-2} returns \code{0.01}. (This last feature was added in
Python 2.2. In Python 2.1 and before, if both arguments were of integer
types and the second argument was negative, an exception was raised).

Raising \code{0.0} to a negative power results in a
\exception{ZeroDivisionError}.  Raising a negative number to a
fractional power results in a \exception{ValueError}.

\section{Unary arithmetic operations \label{unary}}
\indexiii{unary}{arithmetic}{operation}
\indexiii{unary}{bit-wise}{operation}

All unary arithmetic (and bit-wise) operations have the same priority:

\begin{productionlist}
  \production{u_expr}
             {\token{power} | "-" \token{u_expr}
              | "+" \token{u_expr} | "{\~}" \token{u_expr}}
\end{productionlist}

The unary \code{-} (minus) operator yields the negation of its
numeric argument.
\index{negation}
\index{minus}

The unary \code{+} (plus) operator yields its numeric argument
unchanged.
\index{plus}

The unary \code{\~} (invert) operator yields the bit-wise inversion
of its plain or long integer argument.  The bit-wise inversion of
\code{x} is defined as \code{-(x+1)}.  It only applies to integral
numbers.
\index{inversion}

In all three cases, if the argument does not have the proper type,
a \exception{TypeError} exception is raised.
\exindex{TypeError}

\section{Binary arithmetic operations\label{binary}}
\indexiii{binary}{arithmetic}{operation}

The binary arithmetic operations have the conventional priority
levels.  Note that some of these operations also apply to certain
non-numeric types.  Apart from the power operator, there are only two
levels, one for multiplicative operators and one for additive
operators:

\begin{productionlist}
  \production{m_expr}
             {\token{u_expr} | \token{m_expr} "*" \token{u_expr}
              | \token{m_expr} "//" \token{u_expr}
              | \token{m_expr} "/" \token{u_expr}}
  \productioncont{| \token{m_expr} "\%" \token{u_expr}}
  \production{a_expr}
             {\token{m_expr} | \token{a_expr} "+" \token{m_expr}
              | \token{a_expr} "-" \token{m_expr}}
\end{productionlist}

The \code{*} (multiplication) operator yields the product of its
arguments.  The arguments must either both be numbers, or one argument
must be an integer (plain or long) and the other must be a sequence.
In the former case, the numbers are converted to a common type and
then multiplied together.  In the latter case, sequence repetition is
performed; a negative repetition factor yields an empty sequence.
\index{multiplication}

The \code{/} (division) and \code{//} (floor division) operators yield
the quotient of their arguments.  The numeric arguments are first
converted to a common type.  Plain or long integer division yields an
integer of the same type; the result is that of mathematical division
with the `floor' function applied to the result.  Division by zero
raises the
\exception{ZeroDivisionError} exception.
\exindex{ZeroDivisionError}
\index{division}

The \code{\%} (modulo) operator yields the remainder from the
division of the first argument by the second.  The numeric arguments
are first converted to a common type.  A zero right argument raises
the \exception{ZeroDivisionError} exception.  The arguments may be floating
point numbers, e.g., \code{3.14\%0.7} equals \code{0.34} (since
\code{3.14} equals \code{4*0.7 + 0.34}.)  The modulo operator always
yields a result with the same sign as its second operand (or zero);
the absolute value of the result is strictly smaller than the absolute
value of the second operand\footnote{
    While \code{abs(x\%y) < abs(y)} is true mathematically, for
    floats it may not be true numerically due to roundoff.  For
    example, and assuming a platform on which a Python float is an
    IEEE 754 double-precision number, in order that \code{-1e-100 \% 1e100}
    have the same sign as \code{1e100}, the computed result is
    \code{-1e-100 + 1e100}, which is numerically exactly equal
    to \code{1e100}.  Function \function{fmod()} in the \module{math}
    module returns a result whose sign matches the sign of the
    first argument instead, and so returns \code{-1e-100} in this case.
    Which approach is more appropriate depends on the application.
}.
\index{modulo}

The integer division and modulo operators are connected by the
following identity: \code{x == (x/y)*y + (x\%y)}.  Integer division and
modulo are also connected with the built-in function \function{divmod()}:
\code{divmod(x, y) == (x/y, x\%y)}.  These identities don't hold for
floating point numbers; there similar identities hold
approximately where \code{x/y} is replaced by \code{floor(x/y)} or
\code{floor(x/y) - 1}\footnote{
    If x is very close to an exact integer multiple of y, it's
    possible for \code{floor(x/y)} to be one larger than
    \code{(x-x\%y)/y} due to rounding.  In such cases, Python returns
    the latter result, in order to preserve that \code{divmod(x,y)[0]
    * y + x \%{} y} be very close to \code{x}.
}.

In addition to performing the modulo operation on numbers, the \code{\%}
operator is also overloaded by string and unicode objects to perform
string formatting (also known as interpolation). The syntax for string
formatting is described in the
\citetitle[../lib/typesseq-strings.html]{Python Library Reference},
section ``Sequence Types''.

\deprecated{2.3}{The floor division operator, the modulo operator,
and the \function{divmod()} function are no longer defined for complex
numbers.  Instead, convert to a floating point number using the
\function{abs()} function if appropriate.}

The \code{+} (addition) operator yields the sum of its arguments.
The arguments must either both be numbers or both sequences of the
same type.  In the former case, the numbers are converted to a common
type and then added together.  In the latter case, the sequences are
concatenated.
\index{addition}

The \code{-} (subtraction) operator yields the difference of its
arguments.  The numeric arguments are first converted to a common
type.
\index{subtraction}

\section{Shifting operations\label{shifting}}
\indexii{shifting}{operation}

The shifting operations have lower priority than the arithmetic
operations:

\begin{productionlist}
  % The empty groups below prevent conversion to guillemets.
  \production{shift_expr}
             {\token{a_expr}
              | \token{shift_expr} ( "<{}<" | ">{}>" ) \token{a_expr}}
\end{productionlist}

These operators accept plain or long integers as arguments.  The
arguments are converted to a common type.  They shift the first
argument to the left or right by the number of bits given by the
second argument.

A right shift by \var{n} bits is defined as division by
\code{pow(2,\var{n})}.  A left shift by \var{n} bits is defined as
multiplication with \code{pow(2,\var{n})}; for plain integers there is
no overflow check so in that case the operation drops bits and flips
the sign if the result is not less than \code{pow(2,31)} in absolute
value.  Negative shift counts raise a \exception{ValueError}
exception.
\exindex{ValueError}

\section{Binary bit-wise operations\label{bitwise}}
\indexiii{binary}{bit-wise}{operation}

Each of the three bitwise operations has a different priority level:

\begin{productionlist}
  \production{and_expr}
             {\token{shift_expr} | \token{and_expr} "\&" \token{shift_expr}}
  \production{xor_expr}
             {\token{and_expr} | \token{xor_expr} "\textasciicircum" \token{and_expr}}
  \production{or_expr}
             {\token{xor_expr} | \token{or_expr} "|" \token{xor_expr}}
\end{productionlist}

The \code{\&} operator yields the bitwise AND of its arguments, which
must be plain or long integers.  The arguments are converted to a
common type.
\indexii{bit-wise}{and}

The \code{\^} operator yields the bitwise XOR (exclusive OR) of its
arguments, which must be plain or long integers.  The arguments are
converted to a common type.
\indexii{bit-wise}{xor}
\indexii{exclusive}{or}

The \code{|} operator yields the bitwise (inclusive) OR of its
arguments, which must be plain or long integers.  The arguments are
converted to a common type.
\indexii{bit-wise}{or}
\indexii{inclusive}{or}

\section{Comparisons\label{comparisons}}
\index{comparison}

Unlike C, all comparison operations in Python have the same priority,
which is lower than that of any arithmetic, shifting or bitwise
operation.  Also unlike C, expressions like \code{a < b < c} have the
interpretation that is conventional in mathematics:
\indexii{C}{language}

\begin{productionlist}
  \production{comparison}
             {\token{or_expr} ( \token{comp_operator} \token{or_expr} )*}
  \production{comp_operator}
             {"<" | ">" | "==" | ">=" | "<=" | "<>" | "!="}
  \productioncont{| "is" ["not"] | ["not"] "in"}
\end{productionlist}

Comparisons yield boolean values: \code{True} or \code{False}.

Comparisons can be chained arbitrarily, e.g., \code{x < y <= z} is
equivalent to \code{x < y and y <= z}, except that \code{y} is
evaluated only once (but in both cases \code{z} is not evaluated at all
when \code{x < y} is found to be false).
\indexii{chaining}{comparisons}

Formally, if \var{a}, \var{b}, \var{c}, \ldots, \var{y}, \var{z} are
expressions and \var{opa}, \var{opb}, \ldots, \var{opy} are comparison
operators, then \var{a opa b opb c} \ldots \var{y opy z} is equivalent
to \var{a opa b} \keyword{and} \var{b opb c} \keyword{and} \ldots
\var{y opy z}, except that each expression is evaluated at most once.

Note that \var{a opa b opb c} doesn't imply any kind of comparison
between \var{a} and \var{c}, so that, e.g., \code{x < y > z} is
perfectly legal (though perhaps not pretty).

The forms \code{<>} and \code{!=} are equivalent; for consistency with
C, \code{!=} is preferred; where \code{!=} is mentioned below
\code{<>} is also accepted.  The \code{<>} spelling is considered
obsolescent.

The operators \code{<}, \code{>}, \code{==}, \code{>=}, \code{<=}, and
\code{!=} compare
the values of two objects.  The objects need not have the same type.
If both are numbers, they are converted to a common type.  Otherwise,
objects of different types \emph{always} compare unequal, and are
ordered consistently but arbitrarily.  You can control comparison
behavior of objects of non-builtin types by defining a \code{__cmp__}
method or rich comparison methods like \code{__gt__}, described in
section~\ref{specialnames}.

(This unusual definition of comparison was used to simplify the
definition of operations like sorting and the \keyword{in} and
\keyword{not in} operators.  In the future, the comparison rules for
objects of different types are likely to change.)

Comparison of objects of the same type depends on the type:

\begin{itemize}

\item
Numbers are compared arithmetically.

\item
Strings are compared lexicographically using the numeric equivalents
(the result of the built-in function \function{ord()}) of their
characters.  Unicode and 8-bit strings are fully interoperable in this
behavior.

\item
Tuples and lists are compared lexicographically using comparison of
corresponding elements.  This means that to compare equal, each
element must compare equal and the two sequences must be of the same
type and have the same length.

If not equal, the sequences are ordered the same as their first
differing elements.  For example, \code{cmp([1,2,x], [1,2,y])} returns
the same as \code{cmp(x,y)}.  If the corresponding element does not
exist, the shorter sequence is ordered first (for example,
\code{[1,2] < [1,2,3]}).

\item
Mappings (dictionaries) compare equal if and only if their sorted
(key, value) lists compare equal.\footnote{The implementation computes
   this efficiently, without constructing lists or sorting.}
Outcomes other than equality are resolved consistently, but are not
otherwise defined.\footnote{Earlier versions of Python used
  lexicographic comparison of the sorted (key, value) lists, but this
  was very expensive for the common case of comparing for equality.  An
  even earlier version of Python compared dictionaries by identity only,
  but this caused surprises because people expected to be able to test
  a dictionary for emptiness by comparing it to \code{\{\}}.}

\item
Most other objects of builtin types compare unequal unless they are
the same object;
the choice whether one object is considered smaller or larger than
another one is made arbitrarily but consistently within one
execution of a program.

\end{itemize}

The operators \keyword{in} and \keyword{not in} test for set
membership.  \code{\var{x} in \var{s}} evaluates to true if \var{x}
is a member of the set \var{s}, and false otherwise.  \code{\var{x}
not in \var{s}} returns the negation of \code{\var{x} in \var{s}}.
The set membership test has traditionally been bound to sequences; an
object is a member of a set if the set is a sequence and contains an
element equal to that object.  However, it is possible for an object
to support membership tests without being a sequence.  In particular,
dictionaries support membership testing as a nicer way of spelling
\code{\var{key} in \var{dict}}; other mapping types may follow suit.

For the list and tuple types, \code{\var{x} in \var{y}} is true if and
only if there exists an index \var{i} such that
\code{\var{x} == \var{y}[\var{i}]} is true.

For the Unicode and string types, \code{\var{x} in \var{y}} is true if
and only if \var{x} is a substring of \var{y}.  An equivalent test is
\code{y.find(x) != -1}.  Note, \var{x} and \var{y} need not be the
same type; consequently, \code{u'ab' in 'abc'} will return \code{True}.
Empty strings are always considered to be a substring of any other string,
so \code{"" in "abc"} will return \code{True}.
\versionchanged[Previously, \var{x} was required to be a string of
length \code{1}]{2.3}

For user-defined classes which define the \method{__contains__()} method,
\code{\var{x} in \var{y}} is true if and only if
\code{\var{y}.__contains__(\var{x})} is true.

For user-defined classes which do not define \method{__contains__()} and
do define \method{__getitem__()}, \code{\var{x} in \var{y}} is true if
and only if there is a non-negative integer index \var{i} such that
\code{\var{x} == \var{y}[\var{i}]}, and all lower integer indices
do not raise \exception{IndexError} exception. (If any other exception
is raised, it is as if \keyword{in} raised that exception).

The operator \keyword{not in} is defined to have the inverse true value
of \keyword{in}.
\opindex{in}
\opindex{not in}
\indexii{membership}{test}
\obindex{sequence}

The operators \keyword{is} and \keyword{is not} test for object identity:
\code{\var{x} is \var{y}} is true if and only if \var{x} and \var{y}
are the same object.  \code{\var{x} is not \var{y}} yields the inverse
truth value.
\opindex{is}
\opindex{is not}
\indexii{identity}{test}

\section{Boolean operations\label{Booleans}}
\indexii{Boolean}{operation}

Boolean operations have the lowest priority of all Python operations:

\begin{productionlist}
  \production{expression}
             {\token{or_test} [\token{if} \token{or_test} \token{else}
              \token{test}] | \token{lambda_form}}
  \production{or_test}
             {\token{and_test} | \token{or_test} "or" \token{and_test}}
  \production{and_test}
             {\token{not_test} | \token{and_test} "and" \token{not_test}}
  \production{not_test}
             {\token{comparison} | "not" \token{not_test}}
\end{productionlist}

In the context of Boolean operations, and also when expressions are
used by control flow statements, the following values are interpreted
as false: \code{False}, \code{None}, numeric zero of all types, and empty
strings and containers (including strings, tuples, lists, dictionaries,
sets and frozensets).  All other values are interpreted as true.

The operator \keyword{not} yields \code{True} if its argument is false,
\code{False} otherwise.
\opindex{not}

The expression \code{\var{x} if \var{C} else \var{y}} first evaluates
\var{C} (\emph{not} \var{x}); if \var{C} is true, \var{x} is evaluated and
its value is returned; otherwise, \var{y} is evaluated and its value is
returned.  \versionadded{2.5}

The expression \code{\var{x} and \var{y}} first evaluates \var{x}; if
\var{x} is false, its value is returned; otherwise, \var{y} is
evaluated and the resulting value is returned.
\opindex{and}

The expression \code{\var{x} or \var{y}} first evaluates \var{x}; if
\var{x} is true, its value is returned; otherwise, \var{y} is
evaluated and the resulting value is returned.
\opindex{or}

(Note that neither \keyword{and} nor \keyword{or} restrict the value
and type they return to \code{False} and \code{True}, but rather return the
last evaluated argument.
This is sometimes useful, e.g., if \code{s} is a string that should be
replaced by a default value if it is empty, the expression
\code{s or 'foo'} yields the desired value.  Because \keyword{not} has to
invent a value anyway, it does not bother to return a value of the
same type as its argument, so e.g., \code{not 'foo'} yields \code{False},
not \code{''}.)

\section{Lambdas\label{lambdas}}
\indexii{lambda}{expression}
\indexii{lambda}{form}
\indexii{anonymous}{function}

\begin{productionlist}
  \production{lambda_form}
             {"lambda" [\token{parameter_list}]: \token{expression}}
\end{productionlist}

Lambda forms (lambda expressions) have the same syntactic position as
expressions.  They are a shorthand to create anonymous functions; the
expression \code{lambda \var{arguments}: \var{expression}}
yields a function object.  The unnamed object behaves like a function
object defined with

\begin{verbatim}
def name(arguments):
    return expression
\end{verbatim}

See section \ref{function} for the syntax of parameter lists.  Note
that functions created with lambda forms cannot contain statements.
\label{lambda}

\section{Expression lists\label{exprlists}}
\indexii{expression}{list}

\begin{productionlist}
  \production{expression_list}
             {\token{expression} ( "," \token{expression} )* [","]}
\end{productionlist}

An expression list containing at least one comma yields a
tuple.  The length of the tuple is the number of expressions in the
list.  The expressions are evaluated from left to right.
\obindex{tuple}

The trailing comma is required only to create a single tuple (a.k.a. a
\emph{singleton}); it is optional in all other cases.  A single
expression without a trailing comma doesn't create a
tuple, but rather yields the value of that expression.
(To create an empty tuple, use an empty pair of parentheses:
\code{()}.)
\indexii{trailing}{comma}

\section{Evaluation order\label{evalorder}}
\indexii{evaluation}{order}

Python evaluates expressions from left to right. Notice that while
evaluating an assignment, the right-hand side is evaluated before
the left-hand side.

In the following lines, expressions will be evaluated in the
arithmetic order of their suffixes:

\begin{verbatim}
expr1, expr2, expr3, expr4
(expr1, expr2, expr3, expr4)
{expr1: expr2, expr3: expr4}
expr1 + expr2 * (expr3 - expr4)
func(expr1, expr2, *expr3, **expr4)
expr3, expr4 = expr1, expr2
\end{verbatim}

\section{Summary\label{summary}}

The following table summarizes the operator
precedences\indexii{operator}{precedence} in Python, from lowest
precedence (least binding) to highest precedence (most binding).
Operators in the same box have the same precedence.  Unless the syntax
is explicitly given, operators are binary.  Operators in the same box
group left to right (except for comparisons, including tests, which all
have the same precedence and chain from left to right --- see section
\ref{comparisons} -- and exponentiation, which groups from right to left).

\begin{tableii}{c|l}{textrm}{Operator}{Description}
    \lineii{\keyword{lambda}}			{Lambda expression}
  \hline
    \lineii{\keyword{or}}			{Boolean OR}
  \hline
    \lineii{\keyword{and}}			{Boolean AND}
  \hline
    \lineii{\keyword{not} \var{x}}		{Boolean NOT}
  \hline
    \lineii{\keyword{in}, \keyword{not} \keyword{in}}{Membership tests}
    \lineii{\keyword{is}, \keyword{is not}}{Identity tests}
    \lineii{\code{<}, \code{<=}, \code{>}, \code{>=},
            \code{<>}, \code{!=}, \code{==}}
	   {Comparisons}
  \hline
    \lineii{\code{|}}				{Bitwise OR}
  \hline
    \lineii{\code{\^}}				{Bitwise XOR}
  \hline
    \lineii{\code{\&}}				{Bitwise AND}
  \hline
    \lineii{\code{<<}, \code{>>}}		{Shifts}
  \hline
    \lineii{\code{+}, \code{-}}{Addition and subtraction}
  \hline
    \lineii{\code{*}, \code{/}, \code{\%}}
           {Multiplication, division, remainder}
  \hline
    \lineii{\code{+\var{x}}, \code{-\var{x}}}	{Positive, negative}
    \lineii{\code{\~\var{x}}}			{Bitwise not}
  \hline
    \lineii{\code{**}}				{Exponentiation}
  \hline
    \lineii{\code{\var{x}.\var{attribute}}}	{Attribute reference}
    \lineii{\code{\var{x}[\var{index}]}}	{Subscription}
    \lineii{\code{\var{x}[\var{index}:\var{index}]}}	{Slicing}
    \lineii{\code{\var{f}(\var{arguments}...)}}	{Function call}
  \hline
    \lineii{\code{(\var{expressions}\ldots)}}	{Binding or tuple display}
    \lineii{\code{[\var{expressions}\ldots]}}	{List display}
    \lineii{\code{\{\var{key}:\var{datum}\ldots\}}}{Dictionary display}
    \lineii{\code{`\var{expressions}\ldots`}}	{String conversion}
\end{tableii}


\section*{Abbreviations}
We are going to use standard notation $\qz, \zz, \nz = \zz_{\geq0}, \nz_0 = \zz_{>0}$ for the set of rational numbers, integer numbers, non-negative integers and natural numbers, respectively. Additionally, we define
\begin{longtable}{rcl}
$\trm{Set}$ &:& category of sets\\
$\trm{Top}$ &:& category of topological spaces\\
$\trm{Grp}$ &:& category of groups\\
$\trm{Abel}$ &:& category of abelian groups\\
$\trm{Rng}$ &:& category of rings\\
$\trm{CRng}$ &:& category of commutative rings\\
$\trm{Mod}_R$ &:& category of $R$-modules\\
$\trm{Alg}_R$ &:& category of $R$-algebras\\
$\trm{CAlg}_R$ &:& category of commutative $R$-algebras\\
$\trm{Sch}$ &:& category of schemes\\
$\trm{GrpSch}$ &:& category of group schemes.\\
\end{longtable}
The algebraic closure of a field $C$ is denoted by $\ov{C}$. The completion of a topological space $(X, \tau)$ is denoted by $(\hat{X}, \hat{\tau})$.


\documentclass[11pt,a4paper,fleqn]{article}
\usepackage{geometry}
\usepackage{tabularx}
\usepackage{amsmath}
\usepackage{amssymb}
\geometry{
   a4paper,
   left=20mm,
   right=20mm,
   top=20mm,
   bottom=20mm,
}

\setlength\parskip{11pt}
\setlength\parindent{0pt}
\title{Assignment 1}
\author{Hady Januar Willi}
\date{March 2019}
\begin{document}
   % \maketitle
   \begin{center}
      \LARGE Assignment 1
   \end{center}
   \section{Student Information}

   \begin{tabular}{l l}
      Name           & : Hady Januar Willi      \\
      Student ID     & : A0174396R              \\
      School Email   & : E0210429@u.nus.edu     \\
      Email          & : hadywilli@outlook.com  \\
   \end{tabular}

   \section{Initialized Variables Analysis Design}

   This analysis is a may analysis where the variable listed may or may not be initialized in that block from the previous block.
   There is no kill in this analysis as all the kill function always results in empty set.

   \subsection{Entry and Exit}

   \begin{equation}
      IV_{entry}(\ell)=
      \begin{cases}
         \varnothing & if \ell = init(S_*) \\
         \cup \{ IV_{exit}(\ell')|(\ell',\ell) \in flow(S_*) \} & otherwise \\
      \end{cases}
   \end{equation}
   \begin{equation}
      IV_{exit}(\ell)=(IV_{entry}(\ell)) \cup gen_{IV}(B^{\ell}) where B^{\ell} \in blocks(S_*)
   \end{equation}

   \subsection{Generate}

   \begin{equation}
      gen_{IV}=([z:=a]^{\ell})=\{z\}
   \end{equation}
   \begin{equation}
      gen_{IV}=([skip]^{\ell})=\varnothing
   \end{equation}
   \begin{equation}
      gen_{IV}=([b]^{\ell})=IV(b)
   \end{equation}

   \subsection{Kill}

   \begin{equation}
      kill_{IV}=([z:=a]^{\ell})=\varnothing
   \end{equation}
   \begin{equation}
      kill_{IV}=([skip]^{\ell})=\varnothing
   \end{equation}
   \begin{equation}
      kill_{IV}=([b]^{\ell})=\varnothing
   \end{equation}

   \subsection{Template}

   \begin{tabularx}{\linewidth}{| l | X |}
      \hline
      \textbf{Pattern} & \textbf{Instance} \\
      \hline\hline
      \( L \)  & \( P(Var_*) \) \\
      \hline
      \( \sqsubseteq \)  & \( \subseteq \) \\
      \hline
      \( \sqcup \)  & \( \cup \) \\
      \hline
      \( \perp \)  & \( \varnothing \) \\
      \hline
      \( l \)  & \( \varnothing \) \\
      \hline
      \( E \)  & \( init(S_*) \) \\
      \hline
      \( F \)  & \( flow(S_*) \) \\
      \hline
      \( \tilde{F} \)  & \( \{f : L \rightarrow L | \exists l_g : f(l) = l \cup l_g \} \) \\
      \hline
      \( f_{\ell} \)  & \( f_{\ell}(l) = l \cup gen(B^{\ell}) \ where \ B^{\ell} \in blocks (S_*) \) \\
      \hline
   \end{tabularx}

   \section{Analysis Implementation}

   The analysis designed above is implemented using LLVM version 7 in a clean Ubuntu 18.04 virtual machine.
   The full LLVM installation script is included together with the submission package.

   The implementation for both task 2 and task 3 uses simple depth first search by recursive function.
   The notion of node is LLVM basic block and path is LLVM basic block parent - child relation in LLVM flow.
   For each store instruction in the basic block, the operand is added into the current node set.
   Graph traversal continues if there is a successor, and if either one of the two conditions below is met:

   \begin{enumerate}
      \item The current node is not traversed yet
      \item The initialized variable set size increased for the current node
   \end{enumerate}

   \section{Submission Folder Structure}

   \begin{tabularx}{\linewidth}{l l l X}
      $>$ zip  &                    & -- & Folder containing all files                   \\
               & $>$ bin            & -- & Folder containing compiled assignment binary  \\
               & $>$ graphs         & -- & Folder containing dot and pdf files generated from LLVM intermediate representation  \\
               & $>$ installation   & -- & Folder containing LLVM 7 installation script \\
               & $>$ llvms          & -- & Folder containing LLVM intermediate representation  \\
               & $>$ source         & -- & Folder containing assignment source code  \\
               & $>$ target         & -- & Folder containing target C code  \\
   \end{tabularx}

   \section {Running the Implementation}

   After extracting the file, the following commands can be run in sequence:

   \begin{verbatim}
      ./compile
      ./llvm
      ./task2
      ./task3
   \end{verbatim}

   After compilation, to run against any other LLVM intermediate representation file, the following command can be used:

   \begin{verbatim}
      ./bin/assignment1 other-file.ll
   \end{verbatim}

   To generate the graphs, the following command can be run:

   \begin{verbatim}
      ./graph
   \end{verbatim}

   If all else fails, the compiled binary is also included in the assignment, and can be used with the following command which hopefully runs with or without LLVM libraries:

   \begin{verbatim}
      ./bin/assignment1/test.ll
   \end{verbatim}

   \section{Notes}

   \begin{enumerate}
      \item \%1, which is the return variable is always set in all blocks
      \item LLVM 7 outputs variables as \%N instead of variable name from source code
      \item Task 2 command executes against example 1 in assignment handout
      \item Task 3 command executes against example 3 in assignment handout
   \end{enumerate}

\end{document}


\chapter{Conclusions}
\label{ch:conclusion}

\section{Summary}
This thesis has presented learning based solutions for

\begin{itemize}
    \item transparent object matting from a single image (\Cref{ch:tomnet}),
    \item calibrated photometric stereo for non-Lambertian surfaces (\Cref{ch:psfcn}), and
    \item lighting calibration for uncalibrated photometric stereo (\Cref{ch:lcnet}).
\end{itemize}

A brief summary of the algorithms and techniques introduced is given below.

The problem of transparent object matting from a single image was addressed in \Cref{ch:tomnet}. We have introduced a simple and efficient model for transparent object matting, and proposed a CNN architecture, named TOM-Net, that takes a single image as input and predicts environment matte as an object mask, an attenuation mask, and a refractive flow field in a fast feed-forward pass. We created a large-scale synthetic dataset and a real dataset as a benchmark for learning transparent object matting. We have also shown that TOM-Net can perform better by incorporating a trimap or a background image in the input. Promising results have been achieved on both synthetic and real data, which clearly demonstrate the feasibility and effectiveness of the proposed approach.

The problem of calibrated photometric stereo for non-Lambertian surfaces under directional lightings was addressed in \Cref{ch:psfcn}.
We have proposed a flexible deep fully convolutional network, named PS-FCN, that accepts an arbitrary number of images and their associated light directions as input and regresses an accurate normal map.
Our PS-FCN does not require a pre-defined set of light directions during training and testing, and can handle multiple images and light directions in an order-agnostic manner.
A data normalization strategy was introduced to better handle surfaces with SVBRDFs.
In order to train PS-FCN, two synthetic datasets with various realistic shapes and materials have been created. Results on diverse real datasets have clearly shown that our method outperforms previous calibrated photometric stereo methods.

The problem of lighting estimation for uncalibrated photometric stereo was addressed in \Cref{ch:lcnet}.
We have first introduced a lighting calibration network, named LCNet, to estimate directional lightings for uncalibrated photometric stereo.
To understand what have been learned by LCNet for lighting estimation, we analyse the features learned by the network, and find that attached shadows, shadings, and specular highlights are key elements for lighting estimation.
Based on our findings, we then introduced the guided calibration network, named GCNet, that explicitly leverages inter-image information of object shape and intra-image information of shading to estimate more reliable lightings.
Experiments on both synthetic and real datasets showed that GCNet achieves significantly better results than LCNet, and demonstrated that our method can be integrated with existing calibrated photometric stereo methods to handle uncalibrated setups.

\section{Future Work}
Although the methods proposed in this thesis are novel and achieve promising results on their specific tasks, there are rooms for improvement.

\begin{itemize}
\item Colored transparent object matting under natural illumination

    First, our transparent object matting method assumes objects to be colorless and is not applicable to colored transparent object. Second, we assume a single planar background as the only light source (following most of the previous works), and does not consider the more sophisticated refractive properties of a transparent object under natural illumination (\eg, specular highlight, Fresnel effect, and acoustic shadow).
It would be very useful to develop a method for colored transparent object matting under natural illumination in the future.

\item Joint estimation of surface normals, reflectances, and lightings from photometric stereo images

    We have proposed methods for estimating surface normals and directional lightings from multiple input images of an object. However, we did not explicitly estimate the surface reflectance of the object. By knowing the surface reflectance of an object, we can perform some interesting applications like object appearance editing. It would be helpful to consider the problem of joint estimation of surface normals, reflectances, and lightings from photometric stereo images.

\item Photometric stereo under natural illumination

    Most of the existing methods for photometric stereo (also in our methods) assume a directional lighting model, and this requires objects to be placed indoor with controllable illuminations. It would be interesting to consider a more general lighting model (\eg, natural illumination), as it allows accurate surface normal estimation of an object outside the laboratory environment.
\end{itemize}


\par
\section{Exercising the objects' driver programs}
\par
Each object has a {\tt src/} subdirectory that contains source code,
and a {\tt doc} subdirectory that contains \LaTeX\ files for
documentation.
Most objects also have a {\tt drivers/} subdirectory that contain
driver programs that exercise and verify the objects' behavior.
\par
Once the global {\tt spooles.a} library is built, one can compile,
link and load the driver programs.
This can be done inside the {\tt drivers/} subdirectory of each
individual object, or all the executables for the driver programs
can be created by typing {\tt make all\_drivers} in the top level
directory.
\par
Let us look at one particular example.
The {\tt Chv} object is used during the sparse factorization to
store a ``front'', a submatrix of the sparse matrix that is worked
on at one time.
The most expensive part of the factorization is the dense
matrix-matrix operations that are done to a front.
The {\tt Chv/drivers/test\_update.c} program exercises this
functionality.
To create the {\tt test\_update} executable, type
{\tt make test\_update} while in the {\tt Chv/drivers/}{ directory.
To execute this program, there is a {\tt do\_update} shell script.
Here we can test the accuracy and performance of this particular
computation.
\par
The shell script has a number of parameters ---
real or complex entries, symmetric, Hermitian or nonsymmetric
matrices, sparse or dense fronts, sizes of the updating and
updated fronts, etc.
The message level parameter {\tt msglvl} governs the amount of
output sent to the message file.
When {\tt msglvl} is {\tt 1}, only timing information is written.
When {\tt msglvl} is {\tt 2} or greater, the program dumps output
in a form that is readable by Matlab, (and, with some tweaking,
by the GNU Octave program that is very similar and free.)
This is our usual practice to validate the behavior of small parts
of the computations before we link them into larger pieces.
\par
While the source code is very well documented, and the driver
programs are fairly well documented, the shell scripts that
exercise the driver programs are rarely documented.
Consult the source of each driver program to understand their
accompanying shell script.
\par
If you want to get a feel for the speed of the {\bf SPOOLES}
library on your system, see the {\tt test\_update} program in
the {\tt Chv/drivers/} directory
and the {\tt test\_solveupd*} programs in
the {\tt SubMtx/drivers/} directory.
These are the programs that exercise the BLAS3 computations used
during the factors and solves.
Remember to set {\tt msglvl} to {\tt 1} to get just timings.
To test out the numeric factorizations and solves, see the
{\tt testGrid} program.
in the {\tt FrontMtx/drivers} directory.
There are similar programs in the {\tt MT/drivers} and {\tt
MPI/drivers} directories.
Many shell scripts, particularly those that deal with the ordering
and factorization objects, attempt to read in matrix and graph
files you will not find on your systems.\footnote{
What you are seeing is a snapshot of my development environment.
You should be able to customize your environment.
}
There is a small {\tt Matrices} directory that contains test
matrices, which can be used in some of the shell scripts,
once the files are un-zip'ed.
\par


\subsection{Evaluation}
\label{sec:evaluation}

\begin{figure}
	\vspace*{-\figskipabove px}
	\centering
	\hspace*{-4px}
	\includegraphics[width=1.025\linewidth]{fig_architectures}
	\vspace*{-12px}
	\caption{{\bf Network Architectures.} We use different resolutions for ShapeNet and KITTI as well as ModelNet and \Kinect (bottom and top, respectively). In both cases, architectures for higher resolutions employ one additional stage in the en- and decoder (in {\color{gray}gray}). Each convolutional layer is followed by $\text{ReLU}$ activations and batch normalization \citep{Ioffe2015ICML}; the window sizes for max pooling and nearest-neighbor upsampling can be derived from the context; the number of channels are given in parentheses.}
	\label{fig:architectures}
	\vspace*{-\figskipbelow px}
\end{figure}

For occupancy grids, we use Hamming distance (\Abs) and intersection-over-union (\IoU) between the (thresholded) predictions and the ground truth; note that lower \Abs is better, while lower \IoU is worse. For SDFs, we consider a mesh-to-mesh distance on ShapeNet and a mesh-to-point distance on KITTI. We follow \citep{Jensen2014CVPR} and consider accuracy (\Acc) and completeness (\Compl). To measure \Acc, we uniformly sample roughly $10\text{k}$  points on the reconstructed mesh and average their distance to the target mesh. Analogously, \Compl is the distance from the target mesh (or the ground truth points on KITTI) to the reconstructed mesh. Note that for both \Acc and \Compl, lower is better. On ShapeNet and ModelNet, we report both \Acc and \Compl in voxels, \ie, in multiples of the voxel edge length (\ie, in [vx], as we do not know the absolute scale of the models); on KITTI, we report \Compl in meters (\ie, in [m]).


\section{Integration by parts}


\documentclass[aspectratio=169]{beamer}

\usetheme{sii}

\title{SSL/TLS: Encryption}
\author{Michael Purcell}
\date{01 June 2021}
\begin{document}

\begin{frame}[Triangle=siiorange]
\titlepage
\end{frame}

\section{Cipher Suites}
\begin{frame}[Triangle=siiorange]
	\tocpage
\end{frame}

\begin{frame}[triangle=siiblue]
	\frametitle{Cipher Suites}
	A cipher suite is a set of cryptographic algorithms that can be used to secure a SSL/TLS session. An SSL/TLS cipher suite consists of:
	\begin{itemize}
		\item A key exchange algorithm
		\item An authentication algorithm
		\item A bulk encryption algorithm
		\item A message authentication code (MAC) algorithm
	\end{itemize}

	\vfill

	The primary purpose of the SSL/TLS Handshake is to establish which cipher suite will be used during the subsequent session.
\end{frame}

\begin{frame}[triangle=siiblue]
	\frametitle{Example Cipher Suites}
	Modern cipher suites that are believed to be secure:
	\begin{itemize}
		\item \texttt{TLS\_ECDHE\_ECDSA\_WITH\_AES\_256\_GCM\_SHA384}
		\item \texttt{TLS\_ECDHE\_RSA\_WITH\_AES\_128\_CBC\_SHA256}
		\item \texttt{TLS\_DHE\_RSA\_WITH\_CHACHA20\_POLY1305\_SHA256}
	\end{itemize}

	\vfill

	Legacy cipher suites that may be insecure:
	\begin{itemize}
		\item \texttt{TLS\_RSA\_WITH\_RC4\_128\_MD5}
		\begin{itemize}
			\item Both \texttt{RC4} and \texttt{MD5} are known to be insecure and should not be used in modern applications.
		\end{itemize}
		\item \texttt{TLS\_DH\_DSS\_WITH\_3DES\_EDE\_CBC\_SHA}
		\begin{itemize}
			\item The small block size of \texttt{3DES} (64 bits) makes it unsuitable to handle high-volume traffic as as demonstrated by the "Sweet32" attack.
		\end{itemize}
	\end{itemize}
\end{frame}

\section{The Pseudorandom Function}
\begin{frame}[Triangle=siiorange]
	\tocpage
\end{frame}

\begin{frame}[triangle=siiblue]
	\frametitle{HMAC}
	Given a hash function \texttt{hash}, we define
	\begin{equation} \nonumber
		\texttt{HMAC\_hash}(\texttt{k}, \texttt{m}) = \texttt{hash}\left((\texttt{k'} \oplus \texttt{opad}) \ \Vert \  \texttt{hash}\left((\texttt{k'} \oplus \texttt{ipad}) \ \Vert \ \texttt{m} \right)\right)
	\end{equation}
	where $\texttt{opad} = \texttt{0x5c5c\ldots 5c}$, $\texttt{ipad} = \texttt{0x3636\ldots 36}$, and \texttt{k'} is a block-sized key derived from \texttt{k}.  That is
	\begin{align*}
		\texttt{k'} &=
		\begin{cases}
			\texttt{hash}(\texttt{k}) & \text{if \texttt{k} is larger than the block size}; \\
			\texttt{k} & \text{otherwise}.
		\end{cases} \\
	\end{align*}
\end{frame}

\begin{frame}[triangle=siiblue]
	\frametitle{The Pseudorandom Function}
	We have
	\begin{equation} \nonumber
		\texttt{PRF}(\texttt{secret}, \texttt{label}, \texttt{seed}) = \texttt{P\_hash}(\texttt{secret}, \texttt{label} \ \Vert \  \texttt{seed})
	\end{equation}
	where
	\begin{equation} \nonumber
	\begin{split}
		\texttt{P\_hash}(\texttt{secret}, \texttt{seed}) = \ &\texttt{HMAC\_hash}(\texttt{secret}, \texttt{A}(1) \ \Vert \  \texttt{seed})\\
		&\ \Vert \ \texttt{HMAC\_hash}(\texttt{secret}, \texttt{A}(2) \ \Vert \  \texttt{seed}) \\
		&\ \Vert \ \texttt{HMAC\_hash}(\texttt{secret}, \texttt{A}(3) \ \Vert \  \texttt{seed}) \\
		&\qquad \vdots
	\end{split}
	\end{equation}
\end{frame}

\section{Key Calculation}
\begin{frame}[Triangle=siiorange]
	\tocpage
\end{frame}

\begin{frame}[triangle=siiblue]
	\frametitle{Compute the Master Secret}
	Let $\texttt{cs\_random}$ be the concatenation of the random bytes supplied by the client and server during the TLS Handshake. That is
	\begin{equation} \nonumber
	\texttt{cs\_random} = \texttt{client\_random} \ \Vert \ \texttt{server\_random}.
	\end{equation}
	Then we compute the value of \texttt{master\_secret} via
	\begin{align*}
		\texttt{master\_secret} = \texttt{PRF}(&\texttt{pre\_master\_secret},\\
		&\texttt{"master secret"},\\
		&\texttt{cs\_random})[0\ldots 47].
	\end{align*}

	\vfill

	Notice that \texttt{master\_secret} is always 48 bytes long.
\end{frame}

\begin{frame}[triangle=siiblue]
	\frametitle{Key Calculation}
	Let $\texttt{sc\_random}$ be the concatenation of the random bytes supplied by the server and client during the TLS Handshake. Notice that the order is reversed form that used in the computation of \texttt{master\_secret}. That is
	\begin{equation} \nonumber
	\texttt{sc\_random} = \texttt{server\_random} \ \Vert \ \texttt{client\_random}.
	\end{equation}
	Then we compute the value of \texttt{key\_block} via
	\begin{equation} \nonumber
	\begin{split}
		\texttt{key\_block} = \texttt{PRF}(&\texttt{master\_secret}, \\
		&\texttt{"key expansion"},\\
		&\texttt{sc\_random})[0\ldots \texttt{n-1}]
	\end{split}
	\end{equation}
	where \texttt{n} is the required number of bytes of key material.
\end{frame}

\section{Record Payload Protection}
\begin{frame}[Triangle=siiorange]
	\tocpage
\end{frame}

\subsection{Legacy Cipher Suites}
\begin{frame}[triangle=siiblue]
	\frametitle{MAC-then-encrypt}
	All of the cipher suites defined in the original description of TLSv1.2 use \texttt{HMAC\_hash} as their MAC algorithm.

	\vfill

	The MAC is computed on the unencrypted record data.

	\vfill

	The bulk encryption algorithm is then used to encrypt both the record data and MAC.

	\vfill

	This is commonly called the MAC-then-Encrypt (MtE) construction.
\end{frame}

\begin{frame}[triangle=siiblue]
	\frametitle{Computing The MAC}
	The MAC is computed on a payload that consists of a sequence number, parameters describing any TLS compression that has been applied, and a TLS record fragment. That is
	\begin{equation}\nonumber
	\begin{split}
		\texttt{payload} = \texttt{seq\_num} &\ \Vert \ \texttt{compressed\_type} \\
		&\ \Vert \ \texttt{compressed\_version} \\
		&\ \Vert \ \texttt{compressed\_length} \\
		&\ \Vert \ \texttt{record\_fragment}.
	\end{split}
	\end{equation}

	\vfill

	For cipher suites that use the \texttt{HMAC\_hash} as their MAC algorithm, the MAC is given by:
	\begin{equation}\nonumber
		\texttt{MAC} = \texttt{HMAC\_hash}\left(\texttt{MAC\_write\_key}, \texttt{payload}\right).
	\end{equation}
\end{frame}

\begin{frame}[triangle=siiblue]
	\frametitle{Stream Ciphers}
	A \emph{stream cipher} generates a key stream which is XORed with the plaintext to produce the ciphertext.

	\vfill

	Prior to 2016, RC4 was the only stream cipher used in SSL/TLS cipher suites.

	\vfill

	In 2016, cipher suites which use ChaCha20 were introduced by RFC 7905.

	\vfill

	Use of RC4 was prohibited in 2015 by RFC 7465
\end{frame}

\begin{frame}[triangle=siiblue]
	\frametitle{CBC Block Ciphers}
	A block cipher applies a key-dependent permutation to a \emph{block} of plaintext to produce a block of ciphertext.

	\vfill

	AES is used in many SSL/TLS cipher suites.

	\vfill

	A \emph{block cipher mode of operation} is an algorithm that allows a block cipher to be used to encrypt a plaintext that consists of more than one block.

	\vfill

	The original specification for TLSv1.2 included algorithms for using block ciphers in CBC (cipher block chaining) mode to encrypt TLS records.
\end{frame}

%\begin{frame}
%	\frametitle{CBC Details}
%	Encryption:
%
%	\vfill
%
%	Decryption:
%\end{frame}

\begin{frame}[triangle=siiblue]
	\frametitle{Padding}
	Block ciphers can only be applied to plaintexts whose length is an integer multiple of the block size.

	\vfill

	If a TLS record is not an appropriate length then \emph{padding} must be added.

	\vfill

	The padding may be any length up to 255 bytes.

	\vfill

	Each byte of padding is filled with the padding length value.
\end{frame}

\begin{frame}[triangle=siiblue]
	\frametitle{Initialization Vector}
	Using a block cipher in CBC mode requires an initialization vector (IV).

	\vfill

	The IV is not secret and is sent in-the-clear immediately prior to the encrypted contents of its associated TLS record.

	\vfill

	Despite being sent in-the-clear, the IV for each record must be unpredictable prior to transmission.

	\vfill

	Early versions of TLS did not use explicit IVs. Instead, they used the last ciphertext block of the previous record. This allows an attacker compromise the security of the connection as demonstrated by  the so-called "BEAST" attack.

\end{frame}
\subsection{AEAD Cipher Suites}
\begin{frame}[triangle=siiblue]
	\frametitle{Authenticated Encryption}
	Authenticated encryption (AE) combines the functionalities of bulk encryption and message authentication into a single algorithm.

	\vfill

	One simple way to create an authenticated encryption algorithm is via the encrypt-then-MAC (EtM) construction.

	\vfill

	With EtM, the plaintext is first encrypted and then a MAC is computed (using a different key) for the resulting ciphertext.

\end{frame}

\begin{frame}[triangle=siiblue]
	\frametitle{Additional Data}
	Authenticated encryption with associated data (AEAD) is an extension of authenticated encryption that allows a sender to include some data that must be authenticated but does not need to be (or cannot be) encrypted.

	\vfill

	One common example of associated data is packet routing information.  If the destination address and/or ports are encrypted, then the packet cannot be delivered.  It can be useful, however, to authenticated the routing information to prevent an attacker from manipulating network traffic flows.
\end{frame}

\begin{frame}[triangle=siiblue]
	\frametitle{AEAD Algorithms}
	AEAD algorithms used in SSL/TLS cipher suites include:
	\begin{itemize}
		\item AES-GCM (AES - Galois/Counter Mode)
		\item AES-CCM (AES - Counter with CBC-MAC)
		\item ChaCha20-Poly1305
	\end{itemize}

	\vfill

	The plaintext for a MAC computing these AEAD algorithm consists of a TLS record fragment.

	\vfill

	The associated data consists of a sequence number and information about any TLS compression that has been applied.
\end{frame}

\section{References}
\begin{frame}[Triangle=siiorange]
	\tocpage
\end{frame}

\begin{frame}[triangle=siiblue]
	\frametitle{RFCs}
	\begin{itemize}
		\item RFC 5246 - ``The Transport Layer Security (TLS) Protocol Version 1.2''

		\vfill

		\item RFC 2104 - ``HMAC: Keyed-Hashing for Message Authentication''

		\vfill

		\item RFC 7465 - ``Prohibiting RC4 Cipher Suites''

		\vfill

		\item RFC 7905 - ``ChaCha20-Poly1305 Cipher Suites for Transport Layer Security (TLS)''

		\vfill

		\item RFC 5116 - ``An Interface and Algorithms for Authenticated Encryption''
	\end{itemize}
\end{frame}

\begin{frame}[triangle=siiblue]
	\frametitle{Other Resources}
	\begin{itemize}
		\item The paper ``On the Practical (In-)Security of 64-bit Block Ciphers: Collision Attacks on HTTP over TLS and OpenVPN'' by Karthikeyan Bhargavan and Ga\"etan Leurent

		\vfill

		\item The blog ``Command Line Fanatic'' by Jeremy Davies
		\begin{itemize}
			\item ``An Illustrated Guide to the BEAST Attack''
		\end{itemize}

		\vfill

		\item The blog ``hashedout'' by The SSL Store
		\begin{itemize}
			\item ``Cipher Suites: Ciphers, Algorithms and Negotiating Security Settings'' by Patrick Nohe
		\end{itemize}

		\vfill

		\item The blog ``Moxie Marlinspike'' by Moxie Marlinspike
		\begin{itemize}
			\item ``The Cryptographic Doom Principle''
		\end{itemize}
	\end{itemize}

\end{frame}
\end{document}