MIT6_042JS10_lec19

MIT6_042JS10_lec19 - Mathematics for Computer Science...

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1 Albert R Meyer, March 17, 2010 lec 7W.1 Recursive Definitions & Structural Induction Mathematics for Computer Science MIT 6.042J/18.062J Albert R Meyer, March 17, 2010 lec 7W.2 Recursive DeFnitions Define something in terms of a simpler version of the same thing: Base case(s) that don’t depend on anything else. Constructor case(s) that depend on simpler cases. Albert R Meyer, March 17, 2010 lec 7W.9 Matched Paren Strings, M set of strings, M ! { ] , [ } * ! Base: ! " M , (the empty string) ! Constructor: If s , t " M , then [ s ] t " M Albert R Meyer, March 17, 2010 strings in M [] s = ! t = ! [[]] s = [] t = ! [][] s = ! t = [] [[]][] s = [] t = [] [[[]]] s = [[]] t = ! 7W.10 Matched Paren Strings M ! ! ! Albert R Meyer, March 17, 2010 not in M strings starting with ] are not in M because ! ! does not start with ] ! [ s ] t does not start with ] and everything in M arises in one of these two ways 7W.11 Albert R Meyer, March 17, 2010 lec 7W.12 Matched Paren Strings, M set of strings, M ! { ] , [ } * ! Base: ! " M , ! Constructor: If s , t " M , then [ s ] t " M ! That’s all Extremal Clause (Implicit part of definition)
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2 Albert R Meyer, March 17, 2010 Structural Induction 7W.13 To prove P( x
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This note was uploaded on 05/27/2011 for the course CS 6.042J taught by Professor Prof.albertr.meyer during the Spring '11 term at MIT.

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MIT6_042JS10_lec19 - Mathematics for Computer Science...

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