lecture22

# lecture22 - COMPSCI 102 Introduction to Discrete...

This preview shows pages 1–11. Sign up to view the full content.

COMPSCI 102 Introduction to Discrete Mathematics

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

View Full Document
Cantor’s Legacy: Infinity And Diagonalization Lecture 22 (November 22, 2010)
The Theoretical Computer: no bound on amount of memory no bound on amount of time Ideal Computer is defined as a computer with infinite RAM You can run a Java program and never have any overflow, or out of memory errors

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

View Full Document
An Ideal Computer It can be programmed to print out: 2: 2.0000000000000000000000… 1/3: 0.33333333333333333333… φ : 1.6180339887498948482045… e: 2.7182818284559045235336… π : 3.14159265358979323846264…
Printing Out An Infinite Sequence A program P prints out the infinite sequence s 0 , s 1 , s 2 , …, s k , … if when P is executed on an ideal computer, it outputs a sequence of symbols such that - The k th symbol that it outputs is s k - For every k , P eventually outputs the k th symbol. I.e., the delay between symbol k and symbol k+1 is not infinite

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

View Full Document
Computable Real Numbers A real number R is computable if there is a (finite) program that prints out the decimal representation of R from left to right. Thus, each digit of R will eventually be output. Are all real numbers computable?
Describable Numbers A real number R is describable if it can be denoted unambiguously by a finite piece of English text 2: “Two.” π : “The area of a circle of radius one.” Are all real numbers describable?

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

View Full Document
Is every computable real number , also a describable real number ? And what about the other way? Computable r : some program outputs r Describable r : some sentence denotes r
Computable Describable Theorem: Every computable real is also describable Proof: Let R be a computable real that is output by a program P. The following is an unambiguous description of R: “The real number output by the following program:” P

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

View Full Document
Are all reals computable?
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 55

lecture22 - COMPSCI 102 Introduction to Discrete...

This preview shows document pages 1 - 11. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online