First content on group equality

report/report.tex

@@ -350,9 +350,61 @@ 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.
+
+This memoization, alongside with the exclusive use of elementary operations,
+makes the computation of a signature very fast. The computation is linear in
+the number of gates in a circuit, times the order computed; the computation is
+lazy.
+
+To keep those memoized values up to date whenever the structure of the circuit
+is changed (since this is meant to be integrated in a programming language, fl,
+meaning the structure of the circuit will possibly be created, checked for
+signature, altered, then checked again), each circuit keeps track of a
+``timestamp'' of last modification, which is incremented whenever the circuit
+or its children are modified. A memoized data is always stored alongside with a
+timestamp of computation, which invalidates a previous result when needed.
+
+One possible path of investigation for future work, if the computation turns
+out to be still too slow in real-world cases --- which looks unlikely ---,
+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}
@@ -360,6 +412,11 @@ containing the current gate are adjacent to it.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Performance}
+
+\todo{}
+
+\subsection{Corner cases}
+
 \todo{}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%