review3sol - Computer Science 340 Reasoning about...

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Computer Science 340 Reasoning about Computation Review Session 3 Monday, January 14, 2008 Problem 1 An algebraic number is a complex number that is a root of a polynomial with integer coefficients. Is the set of algebraic numbers countable ? Solution: The set of polynomials with integer coefficients is countable. A polynomial of degree n can have at most n roots. Hence the set of algebraic numbers is countable. Problem 2 A useless state in a Turing Machine is one that is never entered on any input string. Consider the problem of testing whether a Turing machine has any useless states. Formulate this problem as a language and show that it is undecidable. Solution: U = { M | M is a TM and M has a useless state } . We give a reduction from E TM to U . Recall that E TM = { M | M is a TM and L ( M ) = ∅} . Given M we construct the description of a new Turing Machine M which behaves as follows on input w : M simulates M on input w . In addition, M has a special state s . M enters the state s iff the simulation of M on w accepts. We can ensure that the only possible useless state in M is s . (Note that this does not depend on whether or not M has useless states). Thus M has a useless state iff L ( M ) = . This is a reduction from E TM to U . Since E TM is undecidable, U is undecidable.
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  • Fall '07
  • CharikarandChazelle
  • Rational number, Countable set, NP-complete, Boolean satisfiability problem, Algebraic number, MAX CUT

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