t9 - T of a relation R , given the relation R as a matrix....

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CS3230 Tutorial 9 1. Compute the longest common subsequence of (a) SLWOV NNDK and ALWGQV NBBK . (b) AGCGATAGC and ACAGATGAG 2. Show that if X and Y are two sequences starting with a , then the longest common subsequence of X and Y starts with an a . 3. Give a counterexample to the following claims: (a) If X = X [1] ...X [ m ] and Y = Y [1] ...Y [ n ], and X [ m ] 6 = Y [ n ], then the longest common subsequence of X and Y must end in either X [ m ] or Y [ n ]. (b) If X = X [1] ...X [ m ] and Y = Y [1] ...Y [ n ], and X [ m ] 6 = Y [ n ], then the longest common subsequence of X and Y must not end with either X [ m ] or Y [ n ]. 4. Consider the following problem: A relation on a set A is a subset of A × A . A relation T is called transitive if the following holds for all a,b,c A : if ( a,b ) T and ( b,c ) T , then ( a,c ) T . A relation T is called a transitive closure of a relation R on a set A if it is the smallest rela- tion (on A ) which is a superset of R and is transitive. In other words, ( a,b ) is in T iff there exist b 1 ,b 2 ,...,b k such that, a = b 1 ,b = b k , and ( b 1 ,b 2 ) , ( b 2 ,b 3 ) , ( b 3 ,b 4 ) ,..., ( b k - 1 ,b k ) are all in R (here k may be equal to 2). Give a dynamic programming algorithm to compute transitive closure
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Unformatted text preview: T of a relation R , given the relation R as a matrix. 5. (a) A Chomsky Normal Form grammar G is of the form G = (Σ ,V,S,δ ), where Σ is the alphabet set, V is a set of non-termimals (where V ∩ Σ = ∅ ), S ∈ V is a starting symbol and δ is a set of productions of the form: A → BC or A → a , where A,B,C ∈ V and a ∈ Σ is a terminal. In the following α,β,γ,w are strings in (Σ ∪ V ) * . 1 (b) We say that αAβ ⇒ G αwβ , where A → w is a production in G (that is member of δ ). (c) We say that α ⇒ * G β if one of the following holds: (i) α = β , (ii) α ⇒ G β or (iii) for some γ , α ⇒ G γ and γ ⇒ * G β . (d) We say that L ( G ) = { w : w ∈ Σ * and S ⇒ * G w } . Given a Chomsky Normal Form grammar G , give a dynamic programming algorithm to determine if a string w ∈ Σ * is a member of L ( G ). 2...
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t9 - T of a relation R , given the relation R as a matrix....

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