<p>In computer science theory some textbooks introduce fractional/quotient regular languages. In particular, given L1 and L2 languages on the same alphabet, the right quotient of L1 with L2 is defined as</p>
<p>L1/L2 = {x : xy ∈ L1 for some y ∈ L2}.</p>
<p>The left quotient is defined in a similar way. I was wondering if there are some real-life cases in which quotient languages can be useful. Any ideas?</p>

CC BY-SA 4.0
2021-09-29T16:22:57.417Z
Do you mean " L1 and L2  are _regular_ languages on the same alphabet". I haven't seen the concept before. Which textbook(s). Hopefully plural. I'd expect the book author(s) to give some reasonable examples.
Buffy
CC BY-SA 4.0
2021-09-29T18:49:43.713Z
@Buffy, the definition makes sense for all languages, not just regular ones. And proving that such constructions give regular languages if $L1$ and $L2$ are regular is a popular exercise.
vonbrand
<cs-theory>
What are the real-life use cases for for fractional/quotient regular languages?
# What are the real-life use cases for for fractional/quotient regular languages?

(Asked by: username_0 on 2021-09-29)

In computer science theory some textbooks introduce fractional/quotient regular languages. In particular, given L1 and L2 languages on the same alphabet, the right quotient of L1 with L2 is defined as

L1/L2 = {x : xy ∈ L1 for some y ∈ L2}.

The left quotient is defined in a similar way. I was wondering if there are some real-life cases in which quotient languages can be useful. Any ideas?

**Tags:** cs-theory

> **Comments:**
> **username_1**: Do you mean " L1 and L2  are _regular_ languages on the same alphabet". I haven't seen the concept before. Which textbook(s). Hopefully plural. I'd expect the book author(s) to give some reasonable examples.  
> **username_2**: @username_1, the definition makes sense for all languages, not just regular ones. And proving that such constructions give regular languages if $L1$ and $L2$ are regular is a popular exercise.  

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