"So, entropy is uncertainty, coding is compression, channels have noise, and..."
Riku started explaining but got confused midway.
"Um, so Huffman codes are... no, that's not right. Reed-Solomon is..."
Aoi smiled wryly. "Riku, your explanation is too redundant."
"Redundant? I thought I was explaining properly."
"Redundancy is an important concept in information theory," Yuki laughed.
"What do you mean?"
Aoi wrote sentences on the whiteboard.
"Tomorrow will be sunny." "Tomorrow will probably be sunny." "Tomorrow, weather, sunny."
"They all convey the same information. But the information content differs."
"Oh, the third one is shortest," Riku noticed.
"Short, but hard to understand, right?" Yuki pointed out.
"Yes. Natural language has intentional redundancy. To aid understanding, to prevent misunderstanding."
Aoi continued. "English has about 75 percent redundancy. Meaning, 75 percent of characters are predictable."
"That much!" Riku was surprised.
"So even with parts missing, we can understand. 'T_e w_ath_r is n_ce t_d_y'"
Yuki read. "The weather is nice today! Readable even without vowels."
"That's the power of redundancy. In communication, it becomes resistance to noise."
Riku became serious. "So redundancy is a good thing?"
"It depends. In data compression, redundancy is the enemy. In communication, it's an ally."
"Isn't that contradictory?"
"That's the trade-off. Efficiency and robustness often conflict," Aoi drew a diagram.
"Compression: remove redundancy → reduce data → but weak to noise Error correction: add redundancy → increase data → but strong against noise"
Yuki wrote in the notebook. "We can't have both simultaneously..."
"But Shannon said compression and error correction can be separated."
"Separated?"
"First, compress with source coding. Next, add error-correcting codes with channel coding. It can be divided into two stages."
Riku thought. "Remove waste first, then add necessary waste?"
"Perfect understanding," Aoi was impressed. "The data's inherent redundancy and redundancy for communication are different things."
"Which kind is Riku's redundant explanation?" Yuki teased.
"Uh..." Riku was at a loss for words.
Aoi laughed. "Riku's redundancy is actually increasing confusion. Harmful redundancy."
"That's harsh!"
"But in the learning process, redundant explanations help too. Repetition deepens understanding."
Yuki said seriously, "True, when I can't understand once, rephrasing helps."
"Yes. Redundancy has quality. Good redundancy and bad redundancy."
Aoi showed another example.
"'Today is December 25, Christmas.' 'Today is December 25.'
The former is redundant. If it's December 25, Christmas is obvious."
"But it might not be obvious in some cultures," Riku pointed out.
"Sharp. Redundancy depends on the receiver's knowledge."
"The amount of information needed changes based on what the other person knows..." Yuki murmured.
"That's conditional entropy. H(X|Y). The information needed to convey X when Y is known."
Riku said happily, "So when I repeat the same thing, it's because I think the other person doesn't understand?"
"Maybe an expression of kindness," Aoi acknowledged. "But it's inefficient."
"A trade-off between efficiency and kindness."
"Human communication can also be analyzed with information theory."
Yuki looked out the window. "Conversation is complex."
"That's what makes it interesting. Finding optimal redundancy is good communication."
Riku resolved. "Next time I'll explain more concisely!"
"I'm looking forward to it," Aoi smiled.
"But what if it's too concise and doesn't get through?" Yuki worried.
"Then you can add redundancy."
The three laughed. Redundancy, too, was an art of balance.