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First content on group equality

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Théophile Bastian 2017-08-20 18:00:45 +02:00
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report/report.tex
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@ -350,6 +350,11 @@ The ``IO adjacency'' term is an additional term in the signatures of order
above $0$, indicating what input and output pins of the circuit group
containing the current gate are adjacent to it.
The default order of signature used in all computations, unless more is useful,
is 2, after a few benchmarks.
\todo{explain range of $n$}
\paragraph{Efficiency.} Every circuit memoizes all it can concerning its
signature: the inner signature, the IO adjacency, the signatures of order $n$
already computed, etc.
@ -373,7 +378,33 @@ would be to try to multithread this computation.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Group equality}
\todo{}
Given two circuit group gates, the task of group equality is to determine
whether the two groups are structurally equivalent, as discussed above.
Group equality itself is handled as a simple backtracking algorithm, trying to
establish a match (an isomorphism, that is, a permutation of the gates of one
of the groups) between the two groups given.
The gates of the two groups are matched by equal signatures, equal number of
inputs and outputs, based on the signature of default order (that is, 2). A few
checks are made, \eg{} every matching group must have the same size on both
sides (if not, then, necessary, the two groups won't match). Then, the worst
case of number of permutations to check is evaluated.
If this number is too high, the signature order will be incremented, and the
matching groups re-created accordingly, until a satisfyingly low number of
permutations is reached (or the diameter of the circuit is reached, meaning
that increasing the order of signature won't have any additional impact). This
order increase ``on-demand'' proved itself very efficient, effectively lowering
the number of permutations examined to no more than $4$ in studied cases.
Once a permutation is judged worth to be examined, the group equality is run
recursively on all its matched gates. If this step succeeds, the graph
structure is then checked. If both steps succeeds, the permutation is correct
and an isomorphism has been found; if not, we move on to the next permutation.
\todo{Anything more to tell here?}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Pattern-match}
@ -381,6 +412,11 @@ would be to try to multithread this computation.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Performance}
\todo{}
\subsection{Corner cases}
\todo{}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%