lecture22

lecture22 - COMPSCI 102 Introduction to Discrete...

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COMPSCI 102 Introduction to Discrete Mathematics
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Cantor’s Legacy: Infinity And Diagonalization Lecture 22 (November 22, 2010)
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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
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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…
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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
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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?
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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?
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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
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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
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Are all reals computable?
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lecture22 - COMPSCI 102 Introduction to Discrete...

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