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A: Hello student Greetings Hope you are doing great. Thank You!!!
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A: Let's see the solution in the next steps
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- The binomial coefficient C(N,k) can be defined recursively as follows: C(N,0) = 1, C(N,N) = 1, and for 0 < k < N, C(N,k) = C(N-1,k) + C(N - 1,k - 1). Write a function and give an analysis of the running time to compute the binomial coefficients as follows: A. The function is written recursively.Show Let f(.) be a computable, strictly monotonic function, that is, f(n+ 1) > f(n) for all n. Show B = {f(n) | n ∈ N} is recursive.Determine whether the proposed definition isa valid recursive definition of a function f from the setof nonnegative integers to the set of integers. If f is welldefined, find a formula for f(n) when n is a nonnegativeinteger and prove that your formula is valid. f(0) = 1, f(n) = −f(n − 1) for n ≥ 1
- The binomial coefficient C(N,k) can be defined recursively as follows: C(N,0) = 1, C(N,N) = 1, and for 0 < k < N, C(N,k) = C(N-1,k) + C(N - 1,k - 1). Write a function and give an analysis of the running time to compute the binomial coefficients as follows: A. The function is written using dynamic programming.Refine the is_palindrome function to work with arbitrary strings, by ignoring non-letter characters and the distinction between upper- and lowercase letters. For example, if the input string is "Madam, I’m Adam!" then you’d first strip off the last character because it isn’t a letter, and recursively check whether the shorter string "Madam, I’m Adam" is a palindrome. c++Determine whether the proposed definition isa valid recursive definition of a function f from the setof nonnegative integers to the set of integers. If f is welldefined, find a formula for f(n) when n is a nonnegativeinteger and prove that your formula is valid. Do NOT use proofs. f(0) = 1, f(n) = −f(n − 1) for n ≥ 1
- Give a recursive definition for the set of all strings of a’s and b’s that begins with an a and ends in a b. Say, S = { ab, aab, abb, aaab, aabb, abbb, abab..} Let S be the set of all strings of a’s and b’s that begins with a and ends in a b. The recursive definition is as follows – Base:... Recursion: If u ∈ S, then... Restriction: There are no elements of S other than those obtained from the base and recursion of S.For the problems given below, determine whether it is more efficient to use a divide and conquer strategy or a dynamic programming strategy, explain your reason. Give the recursion formula for each. (3*4=12) Find the average of a given set of unsorted numbers. Find whether there is a subset of integers in a given set that adds up to 8 Given a set of numbers, can they set be partitioned into two groups such that the sum of each group is equal; i.e 1,5,11,5 can be partitioned to 1,5,5 and 11Computer Science Let a M be a finite set of messages, and let S(M) denote the set of all permutations of M (all bijective functions f : M → M). We’ll assume that if given a description of σ ∈ S(M), both σ and σ −1 are efficiently computable. Suppose P ⊂ S(M) is such that ∀x, y ∈ M, ∃σ ∈ P such that σ(x) = y. (a) Show that |P| ≥ |M|. (This is easy, but makes sure you’ve parsed the definition.) (b) Show that if |P| = |M|, then the following encryption scheme is perfectly secure, provided you only use it once: 1 Key generation: select a random σ ∈ P; Encryption: m 7→ σ(m) Decryption: c 7→ σ −1 (c) (c) Show that the above is false if |M| < |P| < 2|M|. (d) Observe that for any finite group G and any g ∈ G, the map x 7→ gx is a permutation of G. By viewing G itself as a set of permutations of G in this way, show that the above property is satisfied (with M = P = G). (e) The traditional xor one time pad is a special case of the above. What is the finite group in this case?
- Recursively define the set P of strings (composed of lowercase letters) that have an even number of each vowel (i.e. a word with 2 a's and 4 e's is valid but not 1 a and 5 e's). Use set A to represent all alphabet characters, set V to represent all vowels.Let ∑={a,b} and T be the set of words in ∑* that have no consecutive a’s (i.e. there cannot be 2 or more a’s in a row). (a) Give a recursive definition for the set T. (b) Use your recursive definition to show that the string “abbaba” is in T. (c) Is your recursive definition uniquely determined? Explain why or why not.Please written by computer source Let x be a binary string. The minimal description of x, written d(x), is the shortest string ⟨M, w⟩ where TM M on input w halts with x on its tape. If several such strings exist, select the lexicographically first among them. The descriptive complexity, or Kolmogorov complexity, of x, written K(x), is K(x) = |d(x)|. Show that the function K(x) is not a computable function. HINTS: If K is a computable function, there is some TM which computes it. That TM can be used to find strings of large complexity. Try to design a program which outputs “complex” strings but which contradicts their supposed complexity, and even contradicting the supposed complexity of a single string suffices.
