49+Slides--Complexity (1)

49+Slides--Complexity (1) - CS103 HO#49 Slides-Complexity...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
CS103 HO#49 Slides--Complexity 11/12/10 1 Mapping Reducibility w A f(w) B. w A f(w) B f(w) B w A f(w) B w A w A f(w) B Note that A M B A M B AB f Mapping Reducibility Theorem 5.22: If A M B and B is decidable, then A is decidable. recognizable recognizable Corollary 5.23: If A M B and A is undecidable, then B is undecidable. not recognizable not recognizable maps to A B ¬ Solvable(A) ¬ Solvable(B) A language known not to be decidable A language whose decidability we are trying to determine If B decidable Then A decidable But A not decidable So B not decidable A m B Using Mapping Reductions: The Most Common Scenario Using Mapping Reductions: A Less Common Scenario maps to A B A language known to be decidable A language whose decidability we are trying to determine A m B Solvable(B) Solvable(A) If B decidable Then A decidable E TM = { M | L(M) = Ø} (accepts nothing) NE TM ={ M | L(M) Ø} (accepts something) NE TM is Turing-recognizable Build a non-deterministic TM M' : M Guess w w Simulate M on w Accepts w Accept M M' If M accepts any string, M' will guess it, and M' accepts M . If L(M) = Ø, no guess leads to acceptance, so M' does not accept M. So L( M' ) = NE TM Big-O Notation • The exact running time of an algorithm may be a complicated function of the length of its input, so we typically just estimate it. • The question we generally ask is: how does the running time of an algorithm change as we size of the input gets very large. This is called asymptotic analysis.
Background image of page 1

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

View Full DocumentRight Arrow Icon
CS103 HO#49 Slides--Complexity 11/12/10 2 Big-O Notation Let f and g be functions f, g: N R + . We say that f(n) = O(g(n)) if there are positive integers c and n 0 such that for every integer n > n 0 f(n) c g(n).
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 4

49+Slides--Complexity (1) - CS103 HO#49 Slides-Complexity...

This preview shows document pages 1 - 3. Sign up to view the full document.

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