Fill out the table for executing the polynomial-time dynamic programming algo- rithm for deciding whether the string 1001 is in the context-free language generated by the | following CFG. Fill the table completely do not stop the algorithm early. (Note: This CFG is not quite in Chomsky Normal Form since A appears on the right-hand side of a rule, but the same algorithm still works.) A + BC | CC B → BA| 0 С + АВ | BB | 1

a. Correctness of dynamic programming algorithm: Usually, a dynamic programming algorithm can be seen as a recursion and proof by induction is one of the easiest way to show its correctness. The structure of a proof by strong induction for one variable, say n, contains three parts. First, we define the Proposition P(n) that we want to prove for the variable n. Next, we show that the proposition holds for Base case(s), such as n = 0, 1, . . . etc. Finally, in the Inductive step, we assume that P(n) holds for any value of n strictly smaller than n' , then we prove that P(n') also holds. Use the proof by strong induction properly to show that the algorithm of the Knapsack problem above is correct. b. Bounded Knapsack Problem: Let us consider a similar problem, in which each item i has ci > 0 copies (ci is an integer). Thus, xi is no longer a binary value, but a non-negative integer at most equal to ci , 0 ≤ xi ≤ ci . Modify the dynamic programming algorithm seen at class for this...

Can you please help me solve problem 2?

Write the pseudocode for an algorithm that determines if a function is injective, assuming that you can iterate through all the elements in the domain and the co-domain in a finite number of steps.

Write a recursive implementation of Euclid's Algorith for finding the greatest common divisor(GCD) of two intergers. Descriptions of this algorithm are available in algebra books and on theweb. (Note: A nonrecursive version of the GCD problem was given in the programming exercisesfor Chapter 7.) Write a test program that calls your GCD procedure five times, using thefollowing pairs of integers: (5,20),(24,18),(11,7),(432,226),(26,13). After each procedure call,display the GCD.

code a python function (sat) that solves the satisfiability of the clause set by running through all truth assignments. In case the clause set is satisfiable it should output a satisfying assignment. A full (truth) assignment should be represented by a list of literals.

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.

write a c++ recursive function to solve the following: str contains a single pair of parenthesis, return a new string // made of only the parenthesis and whatever those parensthesis contain. You can use substr in this problem. // // Pseudocode Example:// findParen("abc(ghj)789") => "(ghj)" // findParen("(x)7") => "(x)" // findParen("4agh(y)") => "(y)" // string findParen(string str) { }

binary search c++ recursive

Consider the recursive function provided above. After the initial call to the function ham has been made with argument 26 then how many subsequent recursive calls will be made before a value a final result can be returned? This question is internally recognized as variant 235. Q11) 1 2 3 4 5 6 7 8 none of the above

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.

The following recursive definition of a set S over {a, b, c, d}. Basis: "", a, b, c E S. Recursive Step: If ax E S, then bax ES and cax E S. If bx € S, then cbx € S. If cx = S, then bax ES and bcx = S. If dx E S, then ddx E S. Closure: SE S only if it is a, b or c or it can be obtained from a, b or c using finitely many operations of the Recursive Step. a) What recursive step can be removed from the above definition and why? b) Derive from the definition of S that bcbc S. c) Explain why baa & S.

Write a recursive function, numMoves(), that takes a number representing the number of disks in the Hanoi problem, and returns the number of moves involved. I recommend you start writing this function by modifying the hanoi.py program given. Test cases: numMoves(3) should return 7 and numMoves(4) should return 15. Do not use any global variables. hanoi.py