Let be a (non-empty) alphabet and let w € Σ* be a string. We say that x = Σ* is a prefix of the string w if there exists a string u = Σ* such that w = xu. Consider the following language L= {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.

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. I want a CFG

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.

Suppose LS {a, b}* is defined as follows: A e L; for every x and y in L, the strings axb, bxa, and xy are in L. Show that L= AE B, the language of all strings x in {a, b}* satisfying na(x) = n¿(x).

Consider the following regular expression over alphabet Σ = {a,b,c}:  Σ∗a+Σ∗bc Provide two strings that are in the language L and two strings that are not in the language L. w∈L: w∈L: w /∈L: w /∈L:

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

Let Σ be the alphabet {a,b}. Let L be the language describe by                     Σ∗ − (ε|Σ∗b)aa(ε|bΣ∗), where the hyphen denotes set difference and ε de- notes the empty string. (Hint: L consists of the strings over Σ in which no maximal substring consisting solely of a’s is of length 2.) Draw a finite automa- ton that accepts L.

Let Σ = {0, 1}. Define the language L7 = { ⟨A⟩ | A is a DFA, L(A) contains at least one string of length 7, L(A) ⊆ Σ∗ }. Show that L7 is a decidable language.