10-reducibility

10-reducibility - Reducibility We say that a problem Q...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
Reducibility We say that a problem Q reduces to problem P if we can use P to solve Q . In the context of decidability we have the following “templates”: If A reduces to B and B is decidable, then so is A . (1) and If A reduces to B and A is undecidable, then so is B . (2) The template (1) allows us to set up proofs by contradiction to prove undecidability. –p
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Reducibility Theorem: The language HALT TM = {h M,w i| M is a TM and M halts on input w } is undecidable. Proof: Proof by contradiction. We assume that HALT TM is decidable. We show that A TM is reducible to HALT TM by constructing a machine based on HALT TM that will decide A TM . Let Q be a TM that decides HALT TM . The we can construct a decider S that decides A TM as follows, S = "On input h M,w i , where M is a TM and w a string: 1. Run Q on h M,w i . 2. If Q rejects, reject . 3. If Q accepts, simulate M on w until it halts. 4. If M has accepted, accept ;i f M has rejected, reject ." We have shown that A TM is undecidable, therefore this is a contradiction and our assumption that HALT TM is decidable must be incorrect. 2
Background image of page 2
L ( M ) Theorem: The language E TM = {h M i| M is a TM and L ( M )= ∅} is undecidable. Proof: By contradiction. Assume E TM is decidable and Q is the decider. We show that A TM reduces to E TM by constructed the following decider S for A TM , S = "On input h M,w i , where M isaTMand w a string: 1. Build the machine M 1 as follows, M 1 = "On input x : 1. If x 6 = w , reject . 2. If x = w ,run M on input w and accept if M does." 2. Run Q in h M 1 i . 3. If
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 4
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 10/03/2011 for the course CSC 544 taught by Professor Staff during the Spring '11 term at Rhode Island.

Page1 / 13

10-reducibility - Reducibility We say that a problem Q...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online