Suppose L = {w/w # 0^n 1^n for any ne N} is a language over the alphabet {0, 1}. Prove that L is context free by giving a PDA that accepts it. Examples: • E&L, 0000 € L, . 0011 L, • 111100 € L, • 01111 € L, • 1001€ L

Suppose L = {w/w # 0^n 1^n for any ne N} is a language over the alphabet {0, 1}. Prove that L is context free by giving a PDA that accepts it. Examples: • E&L, 0000 € L, . 0011 L, • 111100 € L, • 01111 € L, • 1001€ L

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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Suppose L ⊆ Σ. Define end(L) = {w ∈ Σ∗ | ∃w′ ∈ Σ∗.(w′.w ∈ L)}. For example, if L = {1, 123} then end(L) = {1, 123, 23, 3, ε}. Show that if L is regular, then end(L) is regular. Hint: use ε-transitions.
Suppose a Genetic Algorithm uses chromosomes of the form x=abcdef with a fixed length of six genes.Each gene can be any digit between 0 and 9 . Let the fitness of individual x be calculated as : f(x) =(a+b)-(c+d)-(e+f) And let the initial population consist of four individuals x1, ... ,x4 with the following chromosomes : x1 = 3 5 3 2 6 5, x2 = 9 8 0 1 2 2, x3 = 1 2 2 1 2 3, x4 = 7 9 2 3 1 1 1. 1.Evaluate the fitness of each individual, showing all your workings, and arrange them in order with the fittest first and the least fit last. 2. calculate the average fitness. 3. Perform the following crossover operations:a- Cross the fittest two individuals using one point crossover at the middle point.b- Cross the second and third fittest individuals using a two point crossover (points b and e). Suppose the new population consists of the four offspring individuals received by the crossover operations in the above question. Evaluate the fitness of the new population, showing all your…
Given strings x = “Deep”, y = “Learning” and an empty string ε, the following are some operations undertaken on the strings: i) Lengths of x, y, ε are |x| = |“Deep”| = 4, |y| = |“Learning”| = 8 and |ε| = 0. ii) The concatenation operation applied over x, y and ε as follows yields the resulting strings and results: x + y = “DeepLearning” y + x = “LearningDeep” x + ε = “Deep” ε + y = “Learning” |x + y| = 12 | ε + y| = 8 iii) “Learn” is a prefix of string y, and “Deep” is a prefix of string x + y. iv) “earning” is a suffix of string y, and “Learning” is a suffix of string x + y. v) “earn” is a substring of string y. vi) x = y is false; x = x+ ε is true. vii) “Lenin” is a subsequence of string y. implement a program using given data
Given strings x = “Deep”, y = “Learning” and an empty string ε, the following are some operations undertaken on the strings: i) Lengths of x, y, ε are |x| = |“Deep”| = 4, |y| = |“Learning”| = 8 and |ε| = 0. ii) The concatenation operation applied over x, y and ε as follows yields the resulting strings and results: x + y = “DeepLearning” y + x = “LearningDeep” x + ε = “Deep” ε + y = “Learning” |x + y| = 12 | ε + y| = 8 iii) “Learn” is a prefix of string y, and “Deep” is a prefix of string x + y. iv) “earning” is a suffix of string y, and “Learning” is a suffix of string x + y. v) “earn” is a substring of string y. vi) x = y is false; x = x+ ε is true. vii) “Lenin” is a subsequence of string y. make a program using given data:
Let s = σ1. . . σk be a binary string of length k > 0. We say that a binary string w = w1 . . . wn contains s as a subsequence if there are k indices 1 ≤ i1 < i2, . . . < ik ≤ n such that wik = sr for every 1 ≤ r ≤ k. For example, if s = 11 then 10001, 1010 and 110 contain s as a subsequence whereas 000 and 1000 do not. Prove that the language of all binary strings containing a fixed binary string s of length k as a subsequence is a regular language.
1. Consider the following dynamic programming implementation of the Knapsack problem: #include int find_max(int a, int b) { if(a > b) return a; return b; } int knapsack(int W, int *wt, int *val,int n) { int ans[n + 1][W + 1]; int itm,w; for(itm = 0; itm <= n; itm++) ans[itm][0] = 0; for(w = 0;w <= W; w++) ans[0][w] = 0; for(itm = 1; itm <= n; itm++) { for(w = 1; w <= W; w++) { if(wt[itm - 1] <= w) ans[itm][w] = ______________; else ans[itm][w] = ans[itm - 1][w]; } } return ans[n][W]; } int main() { int w[] = {10,20,30}, v[] = {60, 100, 120}, W = 50; int ans = knapsack(W, w, v, 3); printf("%d",ans); return 0; } Which of the following lines completes the above code? A. find_max(ans[itm – 1][w – wt[itm – 1]] + val[itm – 1], ans[itm – 1][w]) B. find_max(ans[itm – 1][w –…
point p is on line l if and only if point l∗is on line p∗.1. Using the property above, prove the following properties:(a) If n points p1, ..., pnare on a common line l, then p∗1, p∗2, ..., p∗nintersect at a common point l∗.(b) If n lines l1, ..., lnintersect at a common point p, then l∗1, l∗2, ..., l∗nare on a common line p∗.2. If we have a line segment s connecting two points p1 and p2, describe a region s∗corresponding to thedual of s in terms of p∗1 and p∗2. (No rigorous proof is needed.)3. If a line l intersects the line segment s, prove that l∗is in the region s∗.
Given a set of n positive integers, C = {c1,c2, ..., cn} and a positive integer K, is there a subset of C whose elements sum to K? A dynamic program for solving this problem uses a 2-dimensional Boolean table T, with n rows and k + 1 columns. T[i,j] 1≤ i ≤ n, 0 ≤ j ≤ K, is TRUE if and only if there is a subset of C = {c1,c2, ..., ci} whose elements sum to j. Which of the following is valid for 2 ≤ i ≤ n, ci ≤ j ≤ K? a) T[i, j] = ( T[i − 1, j] or T[i, j − ci]) b) T[i, j] = ( T[i − 1, j] and T[i, j − ci ]) c) T[i, j] = ( T[i − 1, j] or T[i − 1, j − ci ]) d) T[i, j] = ( T[i − 1, j] and T[i − 1, j − cj ]) In the above problem, which entry of the table T, if TRUE, implies that there is a subset whose elements sum to K? a) T[1, K + 1] b) T[n, K] c) T[n, 0] d) T[n, K + 1]
Given a set of n positive integers, C = {c1,c2, ..., cn} and a positive integer K, is there a subset of C whose elements sum to K? A dynamic program for solving this problem uses a 2-dimensional Boolean table T, with n rows and k + 1 columns. T[i,j] 1≤ i ≤ n, 0 ≤ j ≤ K, is TRUE if and only if there is a subset of C = {c1,c2, ..., ci} whose elements sum to j. Which of the following is valid for 2 ≤ i ≤ n, ci ≤ j ≤ K? a) ?[?, ?] = ( ?[? − 1, ?] ?? ?[?, ? − ?? ]) b) ?[?, ?] = ( ?[? − 1, ?] ??? ?[?, ? − ?? ]) c) ?[?, ?] = ( ?[? − 1, ?] ?? ?[? − 1, ? − ?? ]) d) ?[?, ?] = ( ?[? − 1, ?] ??? ?[? − 1, ? − ?? ]) In the above problem, which entry of the table T, if TRUE, implies that there is a subset whose elements sum to K? a) ?[1, ? + 1] b) ?[?, ?] c) ?[?, 0] d) ?[?, ? + 1]
Draw out a Turing machine that computes a mapping reduction from L1 ≤ m L2 where L1 ＝ {w:w ∈ {a,b}* and |W| is even} and where L2 ＝ {anbn:n ≥ 0}.
For each language below, indicate whether the language is: 1) in P, 2) in NP but probably not in P, or 3) probably not in NP. Give a brief informal justification for your answer 1. L = { (M, w) | M is a Turing machine, w is a string and M should accept the input w}. 2. L = {[w,x] | w is list of cities with distance between them, x is a visiting order and x must be the shortest path that travels between all the cities visiting each one only once}. 3. L = {[M, w] | M is a context free language in Chomsky normal form, w is a string and w is accepted by the language M}.
Problem Alice was given a number line containing every positive integer, x, where 1<=x<=n. She was also given m line segments, each having a left endpoint, l (1<=l<=n), a right endpoint, r (1<=r<=n), and an integer weight, w. Her task is to cover all numbers on the given number line using a subset of the line segments, such that the maximum weight among all the line segments in the chosen subset is minimal. In other words, let S=(l1,r1,w1),(l2,r2,w2),....,(lk,rk,wk), represent a set of k line segments chosen by Alice, max(w1,w2,...,wk) should be minimized. All numbers 1,2,....n should be covered by at least one of k the chosen line segments. It is okay for the chosen line segments to overlap. You program should output the minimized maximum weight in the chosen subset that covers all the numbers on the number line, or -1 if it is not possible to cover the number line. Input format The first line of the input contains an integer, n - denoting the range of numbers…