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HW2.lyx
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HW2.lyx
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#LyX 2.0 created this file. For more info see http://www.lyx.org/
\lyxformat 413
\begin_document
\begin_header
\textclass article
\begin_preamble
\usepackage{stmaryrd}
\end_preamble
\use_default_options true
\begin_modules
theorems-ams
\end_modules
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\index Index
\shortcut idx
\color #008000
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\html_be_strict false
\end_header
\begin_body
\begin_layout Title
Program Analysis and Synthesis
\begin_inset Newline newline
\end_inset
HW 2
\end_layout
\begin_layout Author
Igor Zarivach 306831835
\begin_inset Newline newline
\end_inset
Jalil Moraney 302872833
\end_layout
\begin_layout Section
Domain
\end_layout
\begin_layout Standard
The abstract domain is:
\begin_inset Formula $\left(L,\le_{i},\vee_{i},\wedge_{i},\bot_{i},\left[-\infty,\infty\right]\right)$m
\end_inset
, where we define the following as well:
\end_layout
\begin_layout Itemize
\noindent
\begin_inset Formula $Z_{\infty}=Z\cup\left\{ -\infty,\infty\right\} $
\end_inset
\end_layout
\begin_layout Itemize
\noindent
\begin_inset Formula $L=\left\{ \left[x,y\right]\,|\, x,y\in Z_{\infty},y\ge_{\infty}x\right\} \cup\bot_{i}$
\end_inset
.
\end_layout
\begin_layout Itemize
The relation
\begin_inset Formula $\le_{\infty}$
\end_inset
for
\begin_inset Formula $Z_{\infty}$
\end_inset
:
\begin_inset Formula $x\le_{\infty}y\iff\left(x,y\in Z,\, x\le y\right)\vee\left(x=-\infty\right)\vee\left(y=\infty\right)$
\end_inset
.
\end_layout
\begin_layout Itemize
\begin_inset Formula $\forall z\in Z_{\infty}\backslash\left\{ -\infty\right\} :\,\infty+z=\infty,\infty-z=\infty,z+\infty=\infty,z-\infty=\infty.$
\end_inset
\end_layout
\begin_layout Itemize
\begin_inset Formula $\forall z\in Z_{\infty}\backslash\left\{ \infty\right\} :\,\left(-\infty\right)+z=-\infty,\left(-\infty\right)-z=-\infty,z+\left(-\infty\right)=-\infty,z-\left(-\infty\right)=\infty.$
\end_inset
\end_layout
\begin_layout Itemize
\begin_inset Formula $\forall z\in Z_{\infty}:\, z*0=0*z=0$
\end_inset
.
\end_layout
\begin_layout Itemize
\begin_inset Formula $\forall z\in Z_{\infty}\backslash\left\{ -\infty\right\} \cup\left\{ z\in Z\,|\, z<0\right\} :\,\infty*z=\infty,z*\infty=\infty.$
\end_inset
\end_layout
\begin_layout Itemize
\begin_inset Formula $\forall z\in Z_{\infty}\backslash\left\{ \infty\right\} \cup\left\{ z\in Z\,|\, z>0\right\} :\,\infty*z=-\infty,z*\infty=-\infty.$
\end_inset
\end_layout
\begin_layout Itemize
\begin_inset Formula $\forall z\in Z_{\infty}\backslash\left\{ -\infty\right\} \cup\left\{ z\in Z\,|\, z<0\right\} :\,\left(-\infty\right)*z=-\infty,z*\left(-\infty\right)=-\infty.$
\end_inset
\end_layout
\begin_layout Itemize
\begin_inset Formula $\forall z\in Z_{\infty}\backslash\left\{ \infty\right\} \cup\left\{ z\in Z\,|\, z>0\right\} :\,\left(-\infty\right)*z=-\infty,z*\left(-\infty\right)=\infty.$
\end_inset
\end_layout
\begin_layout Itemize
\begin_inset Formula $\forall z\in\left\{ -\infty,\infty\right\} $
\end_inset
,
\begin_inset Formula $\forall x\in\left\{ z\in Z\,|\, z>0\right\} :$
\end_inset
\begin_inset space ~
\end_inset
\begin_inset Formula $\frac{z}{x}=z$
\end_inset
.
\end_layout
\begin_layout Itemize
\begin_inset Formula $\forall z\in\left\{ -\infty,\infty\right\} ,\forall x\in\left\{ z\in Z\,|\, z<0\right\} :\,\frac{z}{x}=-z.$
\end_inset
\end_layout
\begin_layout Itemize
\begin_inset Formula $\forall x\in Z,\forall z\in\left\{ -\infty,\infty\right\} :\,\frac{x}{z}=0.$
\end_inset
\end_layout
\begin_layout Itemize
We also note that the only thing we can say about
\begin_inset Formula $\frac{\infty}{\infty}$
\end_inset
and
\begin_inset Formula $\frac{-\infty}{-\infty}$
\end_inset
is that they are positive.
i.e., greater or equal to 1.
Thus to simple the definition of the transformer we define the
\begin_inset Formula $\frac{\infty}{\infty}=\frac{-\infty}{-\infty}=1$
\end_inset
.
\end_layout
\begin_layout Itemize
We also note that the only thing we can say about
\begin_inset Formula $\frac{\infty}{-\infty}$
\end_inset
and
\begin_inset Formula $\frac{-\infty}{\infty}$
\end_inset
is that they are negative.
i.e., less or equal to -1.
Thus to simple the definition of the transformer we define the
\begin_inset Formula $\frac{-\infty}{\infty}=\frac{\infty}{-\infty}=-1$
\end_inset
.
\end_layout
\begin_layout Itemize
For a set
\begin_inset Formula $S\subseteq Z_{\infty}$
\end_inset
,
\begin_inset Formula $min_{\infty}\left(S\right)$
\end_inset
is the minimal number in
\begin_inset Formula $S$
\end_inset
according to
\begin_inset Formula $\le_{\infty}$
\end_inset
.
\end_layout
\begin_layout Itemize
For a set
\begin_inset Formula $S\subseteq Z_{\infty}$
\end_inset
,
\begin_inset Formula $max_{\infty}\left(S\right)$
\end_inset
is the maximal number in
\begin_inset Formula $S$
\end_inset
according to
\begin_inset Formula $\le_{\infty}$
\end_inset
.
\end_layout
\begin_layout Itemize
\begin_inset Formula $\left[a,b\right]\le_{i}\left[c,d\right]\Longleftrightarrow\left(c\le_{\infty}a\right)\wedge\left(b\le_{\infty}d\right)$
\end_inset
.
\end_layout
\begin_layout Itemize
\begin_inset Formula $\left[a,b\right]\vee_{i}\left[c,d\right]=\left[min_{\infty}\left(\left\{ a,c\right\} \right),max_{\infty}\left(\left\{ b,d\right\} \right)\right]$
\end_inset
.
\end_layout
\begin_layout Itemize
\begin_inset Formula $\left[a,b\right]\wedge_{i}\left[c,d\right]=\left[meet\left(max_{\infty}\left(\left\{ a,c\right\} \right),min_{\infty}\left(\left\{ b,d\right\} \right)\right)\right]$
\end_inset
where
\begin_inset Formula $meet\left(a,b\right)$
\end_inset
returns
\begin_inset Formula $\left[a,b\right]$
\end_inset
if
\begin_inset Formula $a\le_{\infty}b$
\end_inset
and
\begin_inset Formula $\bot_{i}$
\end_inset
otherwise.
\end_layout
\begin_layout Subsection
Interval abstraction
\end_layout
\begin_layout Standard
\begin_inset Formula $\alpha_{i}$
\end_inset
: (Label
\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\strikeout off
\uuline off
\uwave off
\noun off
\color none
\begin_inset Formula $\rightarrow$
\end_inset
\family default
\series default
\shape default
\size default
\emph default
\bar default
\strikeout default
\uuline default
\uwave default
\noun default
\color inherit
(Var
\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\strikeout off
\uuline off
\uwave off
\noun off
\color none
\begin_inset Formula $\rightarrow$
\end_inset
\family default
\series default
\shape default
\size default
\emph default
\bar default
\strikeout default
\uuline default
\uwave default
\noun default
\color inherit
\begin_inset Formula $\wp$
\end_inset
(Z)))
\begin_inset Formula $\longrightarrow$
\end_inset
(Label
\family roman
\series medium
\shape up
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\emph off
\bar no
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\noun off
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\begin_inset Formula $\rightarrow$
\end_inset
\family default
\series default
\shape default
\size default
\emph default
\bar default
\strikeout default
\uuline default
\uwave default
\noun default
\color inherit
(Var
\family roman
\series medium
\shape up
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\emph off
\bar no
\strikeout off
\uuline off
\uwave off
\noun off
\color none
\begin_inset Formula $\rightarrow$
\end_inset
\family default
\series default
\shape default
\size default
\emph default
\bar default
\strikeout default
\uuline default
\uwave default
\noun default
\color inherit
\begin_inset Formula $L$
\end_inset
))
\end_layout
\begin_layout Standard
\noindent
\begin_inset Formula $\alpha_{i}(C)(Label)x=\begin{cases}
[min(C(Label)x),max(C(Label)x)] & C(Label)x\ne\textrm{Ø}\\
\bot_{i} & C(Label)x=\textrm{Ø}
\end{cases}$
\end_inset
\end_layout
\begin_layout Standard
\noindent
\begin_inset Formula $\alpha_{i}$
\end_inset
maps for each program Label and local variable and a set of integers
\begin_inset Formula $A$
\end_inset
to interval that contains every
\begin_inset Formula $a\in A$
\end_inset
.
\end_layout
\begin_layout Paragraph
low and high
\begin_inset Newline newline
\end_inset
\end_layout
\begin_layout Standard
Define for local
\begin_inset Formula $x$
\end_inset
and abstract state
\begin_inset Formula $\sigma$
\end_inset
\begin_inset Newline newline
\end_inset
\end_layout
\begin_layout Standard
\noindent
\begin_inset Formula $\sigma(x).low$
\end_inset
:
\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\strikeout off
\uuline off
\uwave off
\noun off
\color none
\begin_inset Formula $\sigma(x)$
\end_inset
=[a,b]
\begin_inset Formula $\Rightarrow$
\end_inset
\family default
\series default
\shape default
\size default
\emph default
\bar default
\strikeout default
\uuline default
\uwave default
\noun default
\color inherit
\begin_inset Formula $\sigma(x).low$
\end_inset
= a
\end_layout
\begin_layout Standard
\noindent
\begin_inset Formula $\sigma(x).high$
\end_inset
:
\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\strikeout off
\uuline off
\uwave off
\noun off
\color none
\begin_inset Formula $\sigma(x)$
\end_inset
=[a,b]
\begin_inset Formula $\Rightarrow$
\end_inset
\family default
\series default
\shape default
\size default
\emph default
\bar default
\strikeout default
\uuline default
\uwave default
\noun default
\color inherit
\begin_inset Formula $\sigma(x).high$
\end_inset
= b
\end_layout
\begin_layout Subsection
Logical Transformations - If
\end_layout
\begin_layout Subsubsection
[x>y]
\end_layout
\begin_layout Paragraph*
True
\begin_inset Newline newline
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $[$
\end_inset
if (e1
\begin_inset Formula $>$
\end_inset
e2)
\begin_inset Formula $]_{true}$
\end_inset
(
\begin_inset Formula $\sigma$
\end_inset
) =
\begin_inset Formula $\begin{cases}
below & \text{\text{e1 and e2 are constants}}\\
below & \text{e1 is local, e2 is constant}\\
below & \text{e1 is constant, e2 is local}\\
below & \text{e1,e2 are locals}
\end{cases}$
\end_inset
\begin_inset Newline newline
\end_inset
\end_layout
\begin_layout Standard
x is local, a is constant
\end_layout
\begin_layout Standard
\noindent
\begin_inset Formula $[$
\end_inset
if (x
\begin_inset Formula $>$
\end_inset
a)
\begin_inset Formula $]_{true}$
\end_inset
(
\begin_inset Formula $\sigma$
\end_inset
) =
\begin_inset Formula $\begin{cases}
\sigma\wedge_{i}\{x\rightarrow[a+1,\infty] & \text{\text{\ensuremath{\sigma(x).high}> a}}\\
\bot & else
\end{cases}$
\end_inset
\begin_inset Newline newline
\end_inset
\end_layout
\begin_layout Standard
x is local, a is constant
\end_layout
\begin_layout Standard
\noindent
\begin_inset Formula $[$
\end_inset
if (a
\begin_inset Formula $>$
\end_inset
x)
\begin_inset Formula $]_{true}$
\end_inset
(
\begin_inset Formula $\sigma$
\end_inset
) =
\begin_inset Formula $\begin{cases}
\sigma\wedge_{i}\{x\rightarrow[-\infty,a-1] & \text{\text{a > \ensuremath{\sigma(x).low}}}\\
\bot & else
\end{cases}$
\end_inset
\end_layout
\begin_layout Standard
\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\strikeout off
\uuline off
\uwave off
\noun off
\color none
a,b are constants
\end_layout
\begin_layout Standard
\noindent
\begin_inset Formula $[$
\end_inset
if (a
\begin_inset Formula $>$
\end_inset
b)
\begin_inset Formula $]_{true}$
\end_inset
(
\begin_inset Formula $\sigma$
\end_inset
) =
\begin_inset Formula $\begin{cases}
\sigma & \text{\text{a > b}}\\
\bot & else
\end{cases}$
\end_inset
\begin_inset Newline newline
\end_inset
\end_layout
\begin_layout Standard
\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\strikeout off
\uuline off
\uwave off
\noun off
\color none
x,y are locals
\end_layout
\begin_layout Standard
\noindent
\begin_inset Formula $[$
\end_inset
if (x
\begin_inset Formula $>$
\end_inset
y)
\begin_inset Formula $]_{true}$
\end_inset
(
\begin_inset Formula $\sigma$
\end_inset
) =
\begin_inset Formula $\begin{cases}
\bot & \text{x and y are the same local}\\
\bot & \sigma(x).high\le\ \sigma(y).low\\
\sigma\wedge_{i}\{x\rightarrow[\sigma(y).low+1,\infty],y\rightarrow[-\infty,\sigma(x).high-1]\} & else
\end{cases}$
\end_inset
\end_layout
\begin_layout Paragraph*
\noindent
Explanation:
\end_layout
\begin_layout Standard
\noindent
Integer
\begin_inset Formula $n$
\end_inset
\begin_inset Formula $\in$
\end_inset
\begin_inset Formula $[$
\end_inset
if (x
\begin_inset Formula $>$
\end_inset
y)
\begin_inset Formula $]_{true}$
\end_inset
(
\begin_inset Formula $\sigma$
\end_inset
)
\begin_inset Formula $(x)$
\end_inset
iff
\begin_inset Formula $n\in\sigma(x)$
\end_inset
and
\begin_inset Formula $\exists m\in\sigma(y)$
\end_inset
,
\begin_inset Formula $n>m$
\end_inset
.
\begin_inset Newline newline
\end_inset
We have
\begin_inset Formula $\sigma(y).low\le m\le\sigma(y).low$
\end_inset
, so
\begin_inset Formula $n\ge\sigma(y).low+1$
\end_inset
, which implies that
\begin_inset Formula $n\in[\sigma(y).low+1,\infty]$
\end_inset
.
\end_layout
\begin_layout Standard
\noindent
So
\begin_inset Formula $n$
\end_inset
is in the interval
\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\strikeout off
\uuline off
\uwave off
\noun off
\color none
\begin_inset Formula $\sigma(x)\wedge_{i}[\sigma(y).low+1,\infty].$
\end_inset
\family default
\series default
\shape default
\size default
\emph default
\bar default
\strikeout default
\uuline default
\uwave default
\noun default
\color inherit
The same logic works for
\begin_inset Formula $y$
\end_inset
.
\end_layout
\begin_layout Standard
\noindent
\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\strikeout off
\uuline off
\uwave off
\noun off
\color none
We can also check if
\begin_inset Formula $\text{\sigma(x).high \le\sigma(y).low}$
\end_inset
\family default
\series default
\shape default
\size default
\emph default
\bar default
\strikeout default
\uuline default
\uwave default
\noun default
\color inherit
\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\strikeout off
\uuline off
\uwave off
\noun off
\color none
by
\begin_inset Formula
\[
\text{\sigma(x).high \le\sigma(y).low \ensuremath{\Longleftrightarrow\sigma(x)\wedge_{i}[\sigma(y).low+1,\infty]=\bot_{i}}}
\]
\end_inset
So the rule to implement is
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
[\text{if(x>y)}]_{true}(\sigma)=\begin{cases}
\bot & \text{x and y are the same local}\\
\bot & \sigma(x)\wedge_{i}[\sigma(y).low+1,\infty]=\bot_{i}\\
\sigma\wedge_{i}\{x\rightarrow[\sigma(y).low+1,\infty], & else\\
y\rightarrow[-\infty,\sigma(x).high-1]\}
\end{cases}
\]
\end_inset
\end_layout
\begin_layout Paragraph*
False
\begin_inset Newline newline
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $[$
\end_inset
if (e1
\begin_inset Formula $>$
\end_inset
e2)
\begin_inset Formula $]_{false}$
\end_inset