patternMinor
Infinite prefix-closed context-free languages contain an infinite regular subset
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infinitefreelanguagescontainregularclosedcontextprefixsubset
Problem
The Problem:
Say that a language is prefix-closed if all prefixes of every string
in the language are also in the language. Let C be an infinite,
prefix-closed, context-free language. Show that C contains an infinite
regular subset.
Can we show this by using Myhill-Nerode Theorem?
Say that a language is prefix-closed if all prefixes of every string
in the language are also in the language. Let C be an infinite,
prefix-closed, context-free language. Show that C contains an infinite
regular subset.
Can we show this by using Myhill-Nerode Theorem?
Solution
Given a normal form grammer $G$ for an infinite prefix-closed $L$, examine the (almost) regular grammer $G'$ obtained by transforming rules of the form $A\rightarrow BC$ into $A\rightarrow B$. I leave it to you to show that $L(G')$ satisfies your requirements.
Context
StackExchange Computer Science Q#139333, answer score: 2
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