Specify initial ring buffer contents when rolling

This allows us to remove the restriction `S_min >= W`, since the
checksum values produced before `W` bytes have been consumed are now
precisely determined.

Author: Cole Miller

spec.md

@@ -82,7 +82,8 @@ $\operatorname{SPLIT}_C \in V_8 \rightarrow V_v$
 - $W \in U_{32}$, the window size
 - $T \in U_{32}$, the threshold
 
-The configuration must satisfy $S_{\text{max}} \ge S_{\text{min}} \ge W > 0$.
+The configuration must satisfy $S_{\text{max}} \ge S_{\text{min}} > 0$ and $W >
+0$.
 
 ## Definitions
 
@@ -230,7 +231,7 @@ package `go4.org/rollsum`.
 #### Rolling
 
 `rrs` is a family of _rolling_ hashes. We can compute hashes in a
-rolling fashion by taking advantage of the fact that:
+rolling fashion by taking advantage of the fact that, for $l \geq k \geq 0$:
 
 $a(k + 1, l + 1) = (a(k, l) - (X_k + c) + (X_{l+1} + c)) \mod M$
 
@@ -246,6 +247,16 @@ So, a typical implementation will work like this:
     $a(k + 1, l + 1)$ and $b(k + 1, l + 1)$. Then use those values to
     compute $s(k + 1, l + 1)$ and also store them for future use.
 
+In all cases the ring buffer should initially contain all zero bytes.
+
+In some cases it's helpful to use the function $S(l) = s(l - W + 1, l)$, where
+$W$ is the window size (see ["Splitting"](#splitting) above). This function can
+be defined for all $l \geq 0$ by applying the previous formulas for $a(k, l)$
+and $b(k, l)$ and putting $X_i = 0$ for all $i < 0$. The resulting values for
+$S$ will be the same as when $a$ and $b$ are computed using the rolling
+implementation just described, provided that the ring buffer is initially zeroed
+as specified.
+
 #### Choice of M
 
 Choosing $M = 2^{16}$ has the advantages of simplicity and efficiency,