1. CS Master – Introduction to the Theory of Computation Jan Maluszynski - HT 20077.1 Lecture 7 Undecidability cont. Jan Maluszynski, IDA, 2007 http://www.ida.liu.se/~janma

2. CS Master – Introduction to the Theory of Computation Jan Maluszynski - HT 20077.2 Outline (Sipser 5.1,5.3, 6.2) 1.Reducibility 2.Examples of undecidable problems 3.Mapping reducibility 4.Decidability of Logical Theories

3. CS Master – Introduction to the Theory of Computation Jan Maluszynski - HT 20077.3 Reduction techniques For proving B is undecidable Assume B is decidable Find an undecidable problem A that can be decided using the decision procedure for B A reducible to B Example: HALT TM = { | M is a TM and M halts on w} We show: A TM = { | M is a TM and M accepts w} is reducible to HALT TM

4. CS Master – Introduction to the Theory of Computation Jan Maluszynski - HT 20077.4 Reduction techniques example Assume TM R decides HALT TM = On input run R on it: If R rejects – reject If R accepts ie. M halts on w: simulate M on w –If M accepts w – accept –If M rejects w - reject Thus with R we could decide A TM = { | M is a TM and M accepts w} But we know A TM is undecidable Hence R cannot exist and HALT TM is undecidable

5. CS Master – Introduction to the Theory of Computation Jan Maluszynski - HT 20077.5 E TM (emptiness) is undecidable E TM = { | M is a TM and L(M) is empty} Assume R decides E TM. Reduce A TM to E TM. Given M and w construct M1 that on any input x : If x = w simulates M on w Otherwise rejects x. Run R on : M1 is rejected iff M accepts w. Thus with R we could decide A TM = { | M is a TM and M accepts w} But we know A TM is undecidable Hence R cannot exist and E TM is undecidable

6. CS Master – Introduction to the Theory of Computation Jan Maluszynski - HT 20077.6 Rice’s theorem Let P be any nontrivial property of the language of a Turing machine M. Nontrivial: it contains some but not all TM descriptions. Theorem: { | L(M) satisfies P} is undecidable e.g given M it is undecidable if L(M) is: -empty, -regular, -Context-free ….

7. CS Master – Introduction to the Theory of Computation Jan Maluszynski - HT 20077.7 Mapping reducibility A function f:  *   * is computable if some TM M on every input w halts with f(w) on its tape Language A is mapping reducible to B written A  m B if there is a computable function f s.that for every w w  A  f(w)  B f is called reduction of A to B.

8. CS Master – Introduction to the Theory of Computation Jan Maluszynski - HT 20077.8 Mapping reducibility cont. Theorem: If A  m B and A is undecidable then B is undecidable Example: f( ) = where M1 rejects all inputs reduces emptiness problem to TM equivalence

9. CS Master – Introduction to the Theory of Computation Jan Maluszynski - HT 20077.9 Mapping reducibility cont. Theorem: If A  m B and A is not Turing-recognizable then B is not Turing-recognizable Remark: A  m B iff complement(A)  m complement(B) To prove B not Turing-recognizable show: A  m complement(B) for some undecidable A

10. CS Master – Introduction to the Theory of Computation Jan Maluszynski - HT 20077.10 TM equivalence is not T-recognizable Reduce A TM to (negation) EQ TM For given construct TMs: (M1 rejects all inputs) M3 accepts all inputs M2: On any input simulates M on w accepts if M accepts w 1. equivalent iff M does not accept w 2. equivalent iff M accepts w Hence: by (1) EQ TM not Turing-recognizable by (2) complement of EQ TM not Turing-recognizable

11. CS Master – Introduction to the Theory of Computation Jan Maluszynski - HT 20077.11 More examples of undecidable languages Hilbert’s tenth problem: Does a given polynomial with integer coefficients has an integral root? Stated early 1900 proved undecidable 1970 N The language Th( N,+,  ) of true closed arithmetic formulae interpreted on natural numbers e.g.  x  y  z[x=y+z]