An Inherently Ambiguous Context-Free Language
Here we break the Easy Theory record on the longest "content" video! We prove that a particular context-free language is inherently ambiguous (meaning that every grammar for it is ambiguous, which is deriving some string in at least two ways). The proof is straightforward although technical. We subdivide the proof into 10 sections to assist with moving through the proof. Note that this is a first on Youtube (and other than the paper from which it originates, the only complete one explained on the internet). The original paper is "A Direct Proof of the Inherent Ambiguity of a Simple Context-Free Language" by Maurer, link here: https://dl.acm.org/doi/abs/10.1145/32....
(Note: there is a "simpler" proof we will eventually show on the channel using a tool called "Ogden's Lemma", but this is a direct and self-contained proof.)
Timestamps:
0:00 - Intro
0:28 - Part 1: What is the language?
4:23 - Part 2: Proof L is context-free
10:13 - Part 3: What is a "reduced" grammar?
17:00 - Part 4: What is an "almost looping" grammar?
20:45 - Part 5: Proof of unambiguous almost looping CFG
32:16 - Part 6: The "Big" Lemma Statement
41:07 - Part 7: Every variable is at least one of the three types
56:40 - Part 8: Every variable is exactly one of the three types
1:02:55 - Part 9: Proving the last two cases
1:09:18 - Part 10: Proving the final theorem
For more information about CFGs and CNF, check out the following playlist:
Context-Free Grammars: • Context-Free Grammars (CFG) and Conte...
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I am a professor of Computer Science, and am passionate about CS theory. I have taught many courses at several different universities, including several sections of undergraduate and graduate theory-level classes.