1. Applied Computer Science II Chapter 5: Reducability Prof. Dr. Luc De Raedt Institut für Informatik Albert-Ludwigs Universität Freiburg Germany

2. Overview Examine several other undecidable problems Reducibility –Basic method to relate two problems to one another in the light of “(un)solvability” –Reducibility is used for various types of “unsolvability”, cf. complexity Mapping reducibility The Post Correspondence Problem PCP

3. Reducability

5. Undecidable problems from language theory

6. Computable functions Cf. Loop-programs Examples : –f( )=m+n –f( ) = M’ where M’ accepts the same language as TM M except that it does not move its head against the left “wall”; if M is not a TM then return epsilon

13. Reductions via computation histories Deterministic versus non- deterministic machines From now on we focus on deterministic machines

14. Linear bound automaton Only limited memory available

15. LBA LBAs are quite powerful, e.g.

16. So, LBAs are fundamentally different than TMs !

24. Theorem 5.24

25. Conclusions Examine several other undecidable problems Reducibility –Basic method to relate two problems to one another in the light of “(un)solvability” –Reducibility is used for various types of “unsolvability”, cf. complexity Mapping reducibility The Post Correspondence Problem PCP