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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Define a NPDA for the language L= {uv € {a, b}*||u|l = |v| and u # v*}. Another way to interpret L is that it is the set of all cven length strings that are not a palindrome.
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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A palindrome is a string that reads the same backward as forward, i.e., w = wR. A palindromic language is one in which all the strings are palindromes, i.e., some subset of P = {w | w ∈ Σ∗ and w = wR} Consider the language L ={<B> | B is a DFA with input alphabet Σ and L(B) contains a palindrome}. Show that L is decidable. Explain step by step.
Consider the language L = {a i b j c k | i 6= j, j 6= k, i 6= 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| ≤ m, |vy| ≥ 1, such that ∀i ≥ 0 : uvixyi z ∈ L.
Let language L over alphabet Σ = { a, b } be defined as L = { a'b* }, where i is a multiple of some given n and k is a multiple of some given m, all non- negative integers. In other words: L contains all strings consisting of: o A number of a's that is a multiple of n o followed by a number of b's that is a multiple of m. Lis Using a constructive method, show that for any given values of n and m,. regular.
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False For each language L over a given alphabet E, let 1(L, E) denote the language over E obtained from L by generating all permutations of its strings - that is, r(L, E) -{we E*: there is a string v in L such that w is a permutation of v ). Specify which one of the following statements is true. Choose None of the other statements is true. The language r((0b1)*, (0, b,1}) is decidable. The language n((Ob1)*, (0, b,1}) is regular. There is no regular language L over an alphabet E such that the language n(L, E) fails to be context-free. Consider alnhaber E-Lo and context-free grammar G over E with precisely one
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