This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: COT 5310 Homework 5 Key Fall 2007 1. Let INF = { f : dom( f ) is infinite } and NE = { f : there is a y such that f ( y ) converges } . Show that NE ≤ m INF. Present the mapping and then explain why it works as desired. To do this define a total recursive function g such that index f is in NE iff g ( f ) is in INF. Be sure to address both cases ( f in and f not in). We need a mapping g such that f ∈ NE ⇔ g ( f ) ∈ INF. So we don’t write g ( f )( x ) we’ll use the notation g f = g ( f ). Define g f to be g f ( ( x, t ) ) = μ z [STP( x, f, t )] . So g f ( ( x, t ) ) returns 0 if STP( x, f, t ) otherwise it diverges. If f ∈ NE there exists a y such that f ( y ) converges, meaning there exists a ( y, t ) such that STP( y, f, t ) is true. Then since STP( y, f, t ) ⇒ STP( y, f, t + k ) for all k ≥ 0, g f ∈ INF. If f / ∈ NE then there does not exist a y such that f ( y ) converges, meaning that dom( f ) = ∅ . This means that for all ( y, t ) , STP( y, f, t ) is false. Since the domain of) is false....
View
Full
Document
This note was uploaded on 07/14/2011 for the course COT 5310 taught by Professor Staff during the Spring '08 term at University of Central Florida.
 Spring '08
 Staff

Click to edit the document details