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\title{Pattern-matching and substitution in electronic circuits}
\author{Théophile Bastian, under supervision of Carl-Johan Seger
        and Mary Sheeran\\
    \small{Chalmers University, Göteborg, Sweden}}
\date{February~--~June 2017}

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\begin{document}
\maketitle

\begin{abstract}
    \todo{abstract}
\end{abstract}

\tableofcontents

\pagebreak

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\section{Introduction}

In the previous years, verification and proved software has gathered an
increasing interest in the computer science community, as people realised how
hard bugs are to track down. But hardware bugs are even more tedious to find
and fix, and can easily lead to disastrous consequences, as those cannot be
patched on existing hardware. For instance, the well-known Pentium
``\textsc{fdiv}'' bug~\cite{pratt1995fdiv} that affected a large number of
Pentium processors lead to wrong results for some floating point divisions.
Intel had to offer to replace all the defective CPUs, leading to an announced
loss of 475 million dollars~\cite{nicely_fdiv}. Even recently, the Skylake and
Kaby Lake hyperthreading bug had to be patched using microcode, loosing
performance and reliability.

To avoid such disasters, the industry nowadays uses a wide range of techniques
to catch bugs as early as possible --- which, hopefully, is before the product's
release date. Among those are \todo{list + cite}, but also proved hardware. On
circuits as complex as processors, usually, only sub-components are proved
correct in a specified context --- that should, but is not proved to, be
respected by the other parts of the circuit. Yet, this trade-off between proved
correctness and engineer's work time already gives a pretty good confidence in
the circuit.

In this context, Carl Seger was one of the main developers of fl at Intel, a
functional ml-inspired programming language integrating many features useful to
get insights of a circuit, testing it and proving it. It mostly features
symbolic trajectory evaluation based model checking and theorem proving, and is
intended to be an all-purpose toolbox for the hardware prover.

Among other features, it includes a ``search and replace'' feature, which can
search every occurrence of a given gates pattern in a circuit, and replace it
by some other gates pattern, proved observationally equivalent beforehand.
Time has proved this method very efficient to design circuits: this way, one
can start from an inefficient, yet simple circuit, prove it, and then refine it
into an equivalent, yet efficient one, through proved transformations. It is
also possible to go the other way, and start with an optimized circuit, hard to
understand, and make it easier to understand to work more efficiently.

\paragraph{My internship} lies amid a project shared between Carl-Johan Seger
and Mary Sheeran, aiming to develop tools for proved design of FPGA circuits.
One of the keystones of this project is an open-sourced and publicly available
version of fl, used for the proving part, and is still at the moment under
development.

My part of the work resided on this ``search and replace'' tool. More
specifically, I focused on writing a C++ library, \emph{isomatch}, which is
interfaced with the C core of fl and provides it with low-level and very fast
functions for this task.

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\section{Context \& AST}

\todo{Rename this section}

\begin{figure}[!h]
    \begin{align*}
        \textbf{Integer constant } n, m, \ldots \qquad& \\
        \\
        \textbf{Wire } in0, out0, ctl0, \ldots \qquad& \\
        \\
        \textbf{Vector } \evec{v}{n} & & \textit{($n$ elements of type $v$)} \\
        \\
        \textbf{Circuit } c, d, \ldots ::=~
            &\text{delay} (in0, out0) &\textit{(delay 1 clock tick)} \\
            \vert~&\text{tristate} (in0, out0, ctl0)
                &\textit{(three-state gate)} \\
            \vert~&\text{comb} (\evec{in0}{n}, \evec{out0}{m}, \evec{e}{m})
                &\textit{(combinatorial gate)} \\
            \vert~&\text{assert} (\evec{in0}{n}, \evec{e}{m)}
                &\textit{(assertion gate)} \\
            \vert~&\text{group} (\evec{c}{n})
                &\textit{(circuit hierarchical group)} \\
        \\
        \textbf{Binary operator } \otimes ::=~
            &\wedge & \textit{(and)} \\
            \vert~&\vee & \textit{(or)} \\
            \vert~&\oplus & \textit{(xor)} \\
            \vert~&+ & \textit{(add)} \\
            \vert~&- & \textit{(sub)} \\
            \vert~&\times & \textit{(times)} \\
            \vert~&\div & \textit{(div)} \\
            \vert~&\% & \textit{(mod)} \\
            \vert~&\lsl & \textit{(logical shift left)} \\
            \vert~&\lsr & \textit{(logical shift right)} \\
            \vert~&\asr & \textit{(arithmetic shift right)} \\
        \\
        \textbf{Unary and constant operator } \otimes_0 ::=~
            &\clsl & \textit{(logical shift left of constant)} \\
            \vert~&\clsr & \textit{(logical shift right of constant)} \\
            \vert~&\casr & \textit{(arithmetic shift right of constant)} \\
        \\
        \textbf{Unary operator } \ominus ::=~
            &\lnot & \textit{(logical negation)} \\
        \\
        \textbf{Expression } e, f, \ldots ::=~
            & x & \textit{(variable)} \\
            \vert~& n & \textit{(integer constant)} \\
            \vert~& e \otimes f & \textit{(binary operator)} \\
            \vert~& e \otimes_0 n & \textit{(unary operator with constant)} \\
            \vert~& \ominus e & \textit{(unary operator)} \\
            \vert~& e_{\vert~n \ldots m} & \textit{(slicing: take a subword)}\\
            \vert~& e~\vert~f & \textit{(merging: concatenate two words)} \\
    \end{align*}
    \caption{AST of circuits used}\label{fig:ast}
\end{figure}

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\section{General approach}

\todo{}

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\section{Signatures}
\todo{}

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\section{Group equality}
\todo{}

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\section{Pattern-match}
\todo{}

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\section{Performance}
\todo{}

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