Undecidability

Undecidability - Harvard CS 121 and CSCI E-207 Lecture 15:...

Info iconThis preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon

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

View Full DocumentRight Arrow Icon

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

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

Unformatted text preview: Harvard CS 121 and CSCI E-207 Lecture 15: Undecidability Harry Lewis November 3, 2009 Reading: Sipser 4.2, 5.1. Harvard CS 121 & CSCI E-207 November 3, 2009 Motivation Goal : to find an explicit undecidable language By the ChurchTuring thesis, such a language has a membership problem that cannot be solved by any kind of algorithm We know such languages exist, by a counting argument. Every decidable language is decided by a TM There are only countably many TMs There are uncountably many languages Most languages are not decidable (or even Turing-recognizable) 1 Harvard CS 121 & CSCI E-207 November 3, 2009 Is every Turing-recognizable set decidable? This would be true if there were an algorithm to solve The Acceptance Problem: Given a TM M and an input w , does M accept input w ? Formally, A TM = {h M,w i : M accepts w } . Proposition: If A TM is decidable, then every Turing-recognizable language is decidable. A TM is the hardest Turing-recognizable language. 2 Harvard CS 121 & CSCI E-207 November 3, 2009 A simplifying detail: every string represents some TM Let be the alphabet over which TMs are represented (that is, h M i * for any TM M ) Let w * if w = h M i for some TM M then w represents M Otherwise w represents some fixed TM M (say the simplest possible TM)....
View Full Document

This document was uploaded on 12/24/2011.

Page1 / 15

Undecidability - Harvard CS 121 and CSCI E-207 Lecture 15:...

This preview shows document pages 1 - 5. Sign up to view the full document.

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