chore: Some polishing on the skippable document

barretenberg/cpp/pil/vm2/docs/skippable.md

@@ -2,7 +2,7 @@
 
 ## Introduction
 
-For each sub-trace defined in a .pil file, one can optionally add so-called "skippable" condition which allows to improve performance on prover side whenever the "skippable" condition is satisfied. It basically skips some accumulation computation in sumcheck protocol to all sub-relations pertaining to the sub-trace. More on how to define a valid "skippable" condition in the next section. We emphasize that the "skippable" mechanism does not change the behavior of the verifier and therefore does not present any security risk about soundness, i.e., it does not help a malicious prover to prove a wrong statement even if the "skippable" condition is too relaxed. What can however happen is that the verification fails when it should not (perfect completeness is not guarenteed anymore).
+For each sub-trace defined in a .pil file, one can optionally add so-called "skippable" condition which allows to improve performance on prover side whenever the "skippable" condition is satisfied. It basically skips some accumulation computation in sumcheck protocol to all sub-relations pertaining to the sub-trace. More on how to define a valid "skippable" condition in the next section. We emphasize that the "skippable" mechanism does not change the behavior of the verifier and therefore does not present any security risk about soundness, i.e., it does not help a malicious prover to prove a wrong statement even if the "skippable" condition is too relaxed. What can however happen is that the verification fails when it should not (perfect completeness is not guarenteed anymore if we wrongly skip).
 
 ## Explanations
 
@@ -13,7 +13,13 @@ A valid "skippable" condition is a condition defined on some columns which guara
 ## Strong Skippable Condition
 
 As accumulated column values in sumcheck will be in general unknown, a way to ensure that a skippable condition guarantees that all sub-relations will be satisfied is to choose a condition which algebraically implies that all sub-relations will be zero. Typically, if
-each sub-relation is of the form $sel * (\ldots) = 0$ then setting $sel == 0$ as "skippable" condition does the job. Similarly, for a list of boolean selectors $sel_1,\ldots sel_n$ and sub-relations of the form $sel_i * (\ldots) = 0$, a "skippable" condition can be picked as
+each sub-relation is of the form $sel \cdot (\ldots) = 0$ then setting $sel == 0$ as "skippable" condition does the job. Similarly, for a list of boolean selectors $sel_1,\ldots sel_n$ and sub-relations of the form
+$$sel_1 \cdot (\ldots) = 0$$
+$$sel_2 \cdot (\ldots) = 0$$
+$$ \ldots $$
+$$sel_n \cdot (\ldots) = 0$$
+
+a "skippable" condition can be picked as
 $$ sel_1 + sel_2 + \cdots sel_n == 0$$
 We name such a condition "strong" and show in next section that this can be relaxed.
 
@@ -25,7 +31,7 @@ $$ ColMerged{_i} = (1 - \alpha) \cdot Col_i + \alpha \cdot Col_{i+1} $$
 
 for every even i ($Col_i$ denotes the ith row element of $Col$).
 Then, each "merged value" is evaluated in your sub-relation.
-Note that $ColMerged_i$ is more or less random except when $Col_i$ and $Col*{i+1}$ are zeros.
+Note that $ColMerged_i$ is more or less random except when $Col_i$ and $Col_{i+1}$ are zeros.
 Assume that for a given sub-relation all $ColMerged_i$ are "random" except for the term satisfying the skippable condition. Then it will evaluate to zero and can be effectively skipped. (Assuming the strong definition of skippable where the skippable condition can nullify a sub-relation no matter what are the other values.)
 Now, one can leverage on the fact that we are using our witness generator to use the skippable in a more generous manner by leveraging columns that are guaranteed to be zero whenever the skippable condition is true and that this particular column being zero nullifies a sub-relation (in other words the column entry is multiplicative factor of the sub-relation).
 Let us take an example of a subtrace with two subrelations over columns $a$, $b$, $c$: