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Problem 1 . Consider the following inductively defined set of good strings of a s and b s: Seed: The empty string epsi is

Problem 1. Consider the following inductively defined set of good strings of as and bs:
Seed: The empty string \epsi is good.
Rule 1: If s is good, then asb is good.
Rule 2: If s is good, then bsa is good.
Rule 3: If s1,s2 are good, then s1s2 is good.
(a) Prove that abaababbba is good.
(b) Give a characterization of the good strings: i.e., a property P such that a string s is good if
and only if s has property P.
(Your characterization should be non-trivial; e.g., P cannot be all good strings.)
(c) Prove that every good string satisfies your P.(Soundness.)
(d) Prove that every string that satisfies your P is good. (Completeness.)
(e) Prove that abababbba is not good.

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