1 / 27

Mapping Reducibility

Mapping Reducibility. Lecture 34 Section 5.3 Mon, Nov 12, 2007. Reducibility. We have been “reducing” one problem to another. Then we claimed that if the reduced problem was decidable, then so was the original problem. Reducibility.

arden-willis
Download Presentation

Mapping Reducibility

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Mapping Reducibility Lecture 34 Section 5.3 Mon, Nov 12, 2007

  2. Reducibility • We have been “reducing” one problem to another. • Then we claimed that if the reduced problem was decidable, then so was the original problem.

  3. Reducibility • But how do we know that a Turing machine can carry out the reduction? • If a Turing machine cannot perform the reduction, then the original problem may not really be decidable.

  4. Computable Functions • A function f : ** is a computable function if there exists a Turing machine M such that on every input w, M halts with f(w), and only f(w), on its tape.

  5. Examples of Computable Functions • The function INCR(n) = n + 1. • The function DECR(n) = n – 1 if n > 0 and 0 if n = 0. • The function ADD(m, n) = m + n. • The function SUB(m, n) = m – n if mn and 0 if m < n. • The function MULT(m, n) = mn.

  6. Mapping Reducibility • A language A is mapping reducible (or just reducible) to a language B if there exists a computable function f : ** such that for every w in *, wAf(w) B.

  7. Mapping Reducibility • f is called a reduction of A to B. • Use the notation AmB to mean that A is mapping reducible to B. • This notation is a bit counter-intuitive since the symbol “points” the other way.

  8. Example of a Reduction • Let  = {0, 1}. • Let EVEN be the language of binary representations of even integers. • Let ODD be the language of binary representations of odd integers.

  9. Example of a Reduction • The function incr() is a reduction of EVEN to ODD. • Thus, EVENmODD. • Why is the function decr() not a reduction of ODD to EVEN? • What Turing machine would give us a reduction of ODD to EVEN?

  10. Decidability and Reducibility • Theorem: If B is decidable and AmB, then A is decidable. • Proof: • Let DB be a decider for B. • Let f be a reduction of A to B. • Then build a decider DA for A as follows.

  11. Decidability and Reducibility DA acc acc w f(w) f DB rej rej A m B

  12. Mw acc acc x = w M, w U x COMP rej rej xw The Emptiness Problem • We showed that ATMmETMc by building the following Turing machine.

  13. DA acc rej M, w Mw CONVERT DE acc rej The Emptiness Problem

  14. The Emptiness Problem • Notice that we needed to reverse the “accept” and “reject” outputs. • That is, M, w ATM Mw  ETM. • In other words, we reduced ATM to ETMc, not ETM.

  15. Mapping Reducible • It is ok to build this reversal into the design of a decider. • However, we cannot assume that wAf(w) B. when convenient. • In designing a recognizer, the difference may matter.

  16. Reducibility, Decidability, and Recognizability • Theorem: If B is decidable and AmB, then A is decidable. • Theorem: If B is recognizable and AmB, then A is recognizable. • Theorem: If A is unrecognizable and AmB, then B is unrecognizable.

  17. Reducibility and Unrecognizability • What about the following? • Theorem: If B is recognizable and AmBc, then Ac is recognizable. • Conjecture: If B is unrecognizable and AmBc, then A is recognizable.

  18. The Unrecognizability of EQTM and EQTMc • Theorem: EQTM and EQTMc are both unrecognizable. • Proof: • Suppose that EQTM is recognizable. • Let REQ be a recognizer of EQTM . • We will build a recognizer for ATMc. • Given M and w, build the following two machines M and MM,w.

  19. M x rej MM, w x M, w acc acc U The Unrecognizability of EQTM and EQTMc

  20. The Unrecognizability of EQTM and EQTMc • What is the language of M? • What is the language of MM, w?

  21. M M, w M, MM,w acc acc CON REQ The Unrecognizability of EQTM and EQTMc • Now build this Turing machine.

  22. The Unrecognizability of EQTM and EQTMc • If M rejects w, then M and MM,w are equivalent. • If M and MM,w are equivalent, then REQ will accept M, MM,w. • If REQ accepts M, MM,w, then M accepts M, w, and conversely. • Thus, Mrecognizes ATMc, a contradiction.

  23. M* x acc The Unrecognizability of EQTM and EQTMc • To show that EQTMc is also unrecognizable, replace M with the machine and repeat the argument.

  24. Irreducibility of ATM to ETM • Theorem: ATM is not mapping reducible to ETM. • Lemma: If AmB, then AcmBc.

  25. Irreducibility of ATM to ETM • Proof: • The statement wAf(w) B is equivalent to the statement wAf(w) B which is equivalent to wAcf(w) Bc.

  26. Irreducibility of ATM to ETM • Theorem: ATM is not mapping reducible to ETM. • Proof: • Suppose it is. • Then ATMcmETMc. • But we know that ETMc is recognizable and that ATMc is not recognizable. (How?)

  27. Irreducibility of ATM to ETM • This is a contradiction. (Why?) • Therefore, ATM is not mapping reducible to ETM.

More Related