Database System Concepts
7th Edition
ISBN: 9780078022159
Author: Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher: McGraw-Hill Education
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Input : a set S of generators of a nontransitive group G acting on f~;
Output : whether G is semiregular;
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- 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…arrow_forwardPumping Lemmaarrow_forwardFor each of the following languages LS {a, b}*, show that the elements of the infinite set {a"|n 2 0} are pairwise L-distinguishable. (a) L={a"ba²"|n 2 0}arrow_forward
- Let LPrime = {I the number of strings accepted by M is prime}. Classify this language as either (i) decidable, (ii) recognizable but not decidable, or (iii) not recognizable. Prove your answer. You may give a high-level description of any TM programs used in your proof. ( You should not use Rice's theorem.)arrow_forwardQuestion 34 The correct statements are: Decidable languages are closed under union. Decidable languages are closed under intersection. Decidable languages are closed under complement. Semi-decidable languages are closed under complement.arrow_forwardState whether the following claim is true or false and prove your answer. Claim: For any (non-empty) alphabet Σ, there is no language L ⊆ Σ∗ which is both regular and inherently ambiguous.arrow_forward
- Specify and implement a rational number type. Give the rep invariant and abstraction function and implement repOk and tostringarrow_forwardUse 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}arrow_forwardDesign a Pushdown Automaton to accept the language anbn for n≥2. Submit the following: Analysis of the problem Algorithmic principles Solution PDAarrow_forward
- 5. Find a closed form representation for the function defined recursively by f(0) = 5 and f(n+1)=2f(n)+4". Prove that your representation is correct using an inductive argument.arrow_forwardAlgorithm to Element as Product of Strong GeneratorsInput : a group G acting on f~ = { 1,2 ..... n };an element g of G;a base and strong generating set for G;Schreier vectors v (i), 1 < i < k, for the stabiliser chain;Output : a symbolic product, word, expressing g in terms of the strong generators;arrow_forwardFor each of the following regular expressions over Σ = {a, b}, state 5 strings that are in the language of the regular expression, and 5 strings that are not. 1. (a∗b)∗|a∗2. (Σ∗aaΣ∗)|(Σ∗bbΣ∗)|(ab)∗|(ba)∗|a|barrow_forward
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