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Q1: Show that ATM is not mapping reducible to ETM. 10 marks
Spring 2016 – Mid

Q2: Let LALL = {M is a TM with input alphabet ∑ and L(M)∑*}, prove that LALLis not Turing recognizable. 10 marks
Spring 2016 – Mid

Q3: In the Silly Post Correspondence Problem (SPCP), in each pair the top string has the same length as the bottom string. Show that the SPCP is decidable, 10 marks
Spring 2016 – Mid

Q4: Lets recall that a language A is Truing recognizable if there is a TM M such that L(M)=A. 5 marks
Spring 2016 – Mid

Q5: Let X be the set{1,2,3,4,5} and Y be the set (6,7,8,9,10}. We describe the functions f:(X-->Y) and g:(X-->Y) in the following table.
 N 1 2 3 4 5 F(n) 6 7 6 7 6
 N 1 2 3 4 5 G(n) 10 9 8 7 6
(a) Is f one-to-one?          (b) Is f onto?       (c) Is f a correspondence?
(d). Is g one-to-one?        (e). Is g onto?     (f) Is g a correspondence?
Spring 2016 – Mid

Q6: Let B be the set of all infinite sequences over {0,1}. Show that B is uncountable, using a proof by diagonalization. 10 marks
Spring 2016 – Mid
Q7: Show that the set of possible statements in TH(N,+,X) is turing recognizable. 10 marks
Spring 2016 – Mid
Q8: if ∀Y∃X[R1(x,x,y)], if we assign plus to (a,c,c) whenever a+b=c. if R is (universe) real number then whether the sentence is TRUE, justify you answer. 10 marks
Spring 2016 – Mid
Q9: Show that the set of all off all odd integers has one-to-one correspondence with the set of all even integers. 5 marks
Spring 2016 – Mid
Q10: Show that all positive numbers has one-to-one correspondence to real number. 5 marks
Spring 2016 – Mid
Q11: if A belong to a language that contains infinity pairs, prove that its uncountable. 5 marks
Spring 2016 – Mid
Q12: Consider following instance of PCP. Is it possible to find a match? If YES then give the dominos arrangements, If NO then prove. 5 marks
1/0, 101/1, 1/001
Spring 2016 – Mid
Q13: Show that 234 and 399 are relatively prime or not. 5 marks
Spring 2016 – Mid
Q14: Show that some TRUE statements in TH(N,+,X) are not proveable. 10 marks
Spring 2016 – Mid
Q15: Is PCP decidable over unary alphabet? 5 marks
Spring 2016 – Mid
Q16: Show that A≤TB and B≤TC then A≤TC. 10 marks
Spring 2016 – Mid
Q17: Prove that Turing recogniable languages are closed under concatenation. 10 marks
Fall 2015 – Mid
 Q18: Show that directed Hamiltonian cycle is NP-Complete. 10 marksSpring 2016 – Final Q19: Prove cycle- length problem is NL-Complete. 10 marksSpring 2016 – Final Q20: Let CNFK = { | ? is a satisfiable CNF-Formula where each variable appears in at most k places}. 15 marks1. Show that CNF2 ∈ P.2. Show that CNF3 ∈ NP-complete.Spring 2016 – Final Q21: A directed graph is strongly connected if every two nodes are connected by a directed path in each direction. Let STRONGLY-CONNECTED = { |G| is a strongly connected graph}. Show that STRONGLY-CONNECTED is NL-complete. 10 marksSpring 2016 – Final Q22: Show that ALBA is P-Space Complete. 10 marksSpring 2016 – Final Q23: Let ADD={(x,y,z) | x,y,z > 0 are binary integers and x+y=z}. Show that ADD ∈ LSPACE. 10 marksSpring 2016 – Final Q24: Let t(n) be a function where t(n) ≥ n , Show that t(n) time k tape TM has an equivalent O(t2(n)) time single tape TM. 10 marksSpring 2016 – Final Q25: Show that 3-Color problem is in NP Complete. 10 marksSpring 2016 – Final Q26: Show that EDFA is in NL-Complete. 10 marksSpring 2016 – Final Q27: Graphs G and H are called isomorphic if the nodes of G may be reordered so that it is identical to H. Let ISO = { |G| and |H| are isomorphic graphs}. Show that ISO is in NP. 10 marksSpring 2016 – Final Q28: Let SET-SPLITTING = { |S| is a finite set and C = {C1, ... ,Ck} is a collection of subsets of S, for some k > 0, such that elements of S can be colored red or blue so that no Ci has all its elements colored with the same. Show that SET-SPLITTING is in NP. 10 marksSpring 2016 – Final Q29: Find Max-clique in DP. 10 marksSpring 2016 – Final Q30: Let CONNECTED={|G| is connected undirected graph}. Show that it is in P. 10 marksSpring 2016 – Final Q31: Let HALF-CLIQUE = { |G| is an undirected graph having a complete subgraph with at least m/2 nodes, where m is the number of nodes in G}.  Show that Half Clique is in NP-complete. 10 marksSpring 2016 – Final Q32: Show that, if P = NP, then every language A is in P, except A = ∅ and A = ∑*, is in NP-complete. 10 marksSpring 2016 – Final Q33: Let PAL-ADD = { | x, y > 0 are binary integers, where x + y is an integer whose binary representation is palindrome}. Show that PAL-ADD ϵ LSPACE. 10 marksSpring 2016 – Final

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