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  • Print publication year: 2009
  • Online publication date: June 2012

Hints and selected exercises

Summary

Chapter 0

0.2 Answers are: (a) n (b) n2 (c) 2n (d) log n (e) n (f) n log n (g) nlog 3 (h) n2.

Chapter 1

1.1 Follow the grade-school algorithms.

1.5 Use the proof of Claim 1.6.

1.6 show that the universal TM U obtained by the proof of Theorem 1.9 can be tweaked to be oblivious.

1.12. b. By possibly changing from S to its complement, we may assume that the empty function φ (that is not defined on any input) is in S and that there is some function f that is defined on some input x that is not in S. Use this to show that an algorithm to compute fS can compute the function HALTx, which outputs 1 on input α iff Mα halts on input x. Then reduce computing HALT to computing HALTx thereby deriving Rice's Theorem from Theorem 1.11.

Chapter 2

2.2 CONNECTED and 2COL are shown to be in P in Exercise 1.14 (though 2COL is called BIPARTITE there). 3COL is shown to be NP-complete in Exercise 2.21, and hence it is unlikely that it is in P.

2.3 First show that for every rational matrix A, the determinant of A can always be represented using a number of bits that is polynomial in the representation of A. Then use Cramer's rule for expressing the solution of linear equations in terms of determinants.