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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{abstract}
    \todo{abstract}
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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 replace \todo{how many?} 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 \todo{the?} main developer of FL
\todo{(?, acronym?)} at Intel, a functional programming language integrating
many features useful to get insights of a circuit, testing it and proving it.
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.\\
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.

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\section{Context \& AST}

\todo{Rename this section}\\
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\section{General approach}

Among many others, one idea that proved itself efficient at Intel for circuit
verification was to prove the correctness of a ``simple'' circuit, that is, a
circuit that is not too optimized, on which the various features and parts can
be easily seen and properties can be expressed cleanly. This circuit could then
be refined afterwards, only by means of proved transformations mostly
resulting from a somewhat large database of usual transformations, along with
their proofs of correctness.

This process of replacement was mostly done through a ``search and replace''
tool. The tool would basically search every (non-overlapping) occurrence of a
given ``pattern'', remove these occurrences from the circuit, and plug the
replacement version of the pattern in its place. A pattern, in this context,
consists in a piece of circuit, along with in- and outbound wires to be
reconnected in the right place afterwards.

\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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