Redundant Inequalities:prove statement Let e1, e2 be two symbolic expressions defined in V ∪P and q ∈ V ∪P. c1, c2 are two inequalities of I such that c1 is defined by e1 REL q and c2 by e2 REL q. REL is an inequality relationship in {≤, <}. Then c2 can be eliminated from I, iff low(e1 − e2) ≥ 0.

2. What should the pre-condition P be in each of the following correctness statements for the statement to be an instance of Hoare's assignment axiom scheme? (a) P { x (b) P { y = 1; } x 5 (c) P { i = i (d) P { i = i k; k; } For All (i=0; i 0 Exists (k=0%; k i

Assignment 6: Problem Set on TMs. 1. Given the following definition of a Turing machine M = (Q, E, I, 6, 91, accept, Greject): • Q={91, 92, 98, Jaccept, greject} « Σ = {0, 1, #} • T = EU {z, u} 0-0, R 1-1,R ## .R 0-2,R 9 0-2, L 9: 97 0-0, L 1-1, L #-#.R 1-2, R 9s 3. Prove that every regular language is decidable. ##.R 9 1-3, L 0-0, R 1-1, R 0-0, L 1-1,4 z-z, R Simulate the Turing machine on input (a) 010#1 and (b) 1#1. Use the concept of configurations to illustrate the transitions. z-z.R Garape Note: To simplify the figure, the reject state and transitions going to the reject state are not shown. Those transitions occur implicitly whenever a state lacks an outgoing transition for a particular symbol. R 2. Let A be the set of all strings over the alphabet (a, b) where the number of a's is twice as much as the number of b's. Construct the state diagram of a Turing machine that recognizes A.

Question 38 The correct statements are: If If Σ □ If If Ø, then Σ* has a subset that is not regular. Ø, then Σ* has a subset that is not context free. Ø, then Σ* has a subset that is not in D. Ø, then Σ* has a subset that is not in SD.

Which property guarantees f:X--->Y is surjective? Of is injective O the preimage of Y is X O f(x)=Y O X=Y

Let S = {x | x = 3n, n ∈ Z+}. Which of the following expression(s) is/are true?a. 3 ∈ S b. 3 + 6 ∈ S c. ∅ ⊂ S d. S ⊆ S e. 1 ∈ S

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Use the Pumping Lemma to show that the the following language over {a,b,c}* is not CF  {wct | t ∈ {a,b}* ∧ w is a substring of t}

Problem 5: Disjoint sets. Consider a list of cities c1, c2, . . . , cn. Assume we have a relationR such that, for any i, j, R(ci, cj ) is 1 if cities ci and cj are in the same province, and 0otherwise.(i) Using a disjoint set data structure, write an algorithm that puts each city in a setsuch that ci and cj are in the same set if and only if they are in the same province.Justify corectness of your algorithm.(ii) If a disjoint set data structure is implemented by trees and in union the name of thenew set is always the name of the larger set, what is the worst-case running time ofyour algorithm.

I need help solving 1 and 2