Let be a (non-empty) alphabet and let w € Σ* be a string. We say that x = Σ* is aprefix of the string w if there exists a string u = Σ* such that w = xu.Consider the following languageL= {w = {a,b}*|for every prefix x ofwn(x) > na(x)}Prove that L is a context-free language. Your proof should rely on mathematical induction.

Proof: Language L is Context-FreeLanguage Definition:The language L is defined as:L = { w ∈ {a, b}*…

Answered: Let be a (non-empty) alphabet and let w…

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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Consider the language L = {a'bck | i + j,j # k, i + k}; it is not a context- free language. Show that it nevertheless satisfies the conditions of the context-free pumping lemma, that i s, show that there exists an m so that for all strings w in L of length at least m, we can write w = uvxyz with |vxy| 1, such that Vi > 0 : uv'xy'z € L.
Consider the language L of strings over the alphabet Σ = {a, b} which contain an equal number of a’s and b’s, and the set of b’s is consecutive in the string, e.g., ab, abba, aabb, abbbaa, aaabbbbbaa, etc. Note that ε ∈ L for completeness (0 number of a’s is equal to 0 number of b’s, which are also consecutive). Show that this language is context-free by giving a context-free grammar. Please make sure to test the answers onhttps://web.stanford.edu/class/archive/cs/cs103/cs103.1156/tools/cfg/
Consider the language L of strings over the alphabet Σ = {a, b} which contain an equal number of a’s and b’s, and the set of b’s is consecutive in the string, e.g., ab, abba, aabb, abbbaa, aaabbbbbaa, etc. Note that ε ∈ L for completeness (0 number of a’s is equal to 0 number of b’s, which are also consecutive). Show that this language is context-free by giving a context-free grammar.
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