Applied Computer Science II Chapter 5 Reducability Prof

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

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

Reducability

Undecidable problems from language theory

Computable functions • Cf. Loop-programs • Examples : – f(<m, n>)=m+n – f(<M>) = 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

Reductions via computation histories • Deterministic versus nondeterministic machines • From now on we focus on deterministic machines

Linear bound automaton Only limited memory available

LBA • LBAs are quite powerful, e. g.

• So, LBAs are fundamentally different than TMs !

Theorem 5. 24

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