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A short proof that O2 is an MCFL

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Figure

Figure 1: Derivation in G. The numbers indicatethe rules that were used.
Figure 2: Two strings xtogether represent a closed path, consisting of a pathfrom = ababab and y = ba to- (0, 0) to (−1, −1) and a path from (−1, −1) (0, 0).
Figure 3: With xted) path = a b a b b and y = a a b, thebeginning of path A[0] and the beginning of (dot- B[0] have an 180◦ angle in P[0], whichimplies x and y start with complementing symbols(here a and a; the other possibility is b and b)
Figure 6: Excursion from the right at ℓ[k].
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