Induction

INDUCTION As soon as an Analytical Engine exists, it will necessarily guide the future course of the science. Whenever any result is sought by its aid, the question will then arise – By what course of calculation can these results be arrived at by the machine in the shortest time? - Charles Babbage, 1864

Overview ,[object Object],[object Object],[object Object],03/12/11

Numerical Iteration ,[object Object],[object Object],03/12/11

[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],03/12/11

Recursive Fibonacci ,[object Object],[object Object],[object Object],[object Object],What’s the “problem” with this algorithm? 03/12/11

Iterative Fibonacci ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],03/12/11

Structural Induction ,[object Object],[object Object],[object Object],03/12/11

Validity of Structural Induction follows Mathematical Induction ( for the nonnegative integers) ,[object Object],[object Object],[object Object],03/12/11

Validity of Structural Induction follows Mathematical Induction ( for the nonnegative integers) ,[object Object],[object Object],[object Object],[object Object],03/12/11

Well-Formed Formulas T is a wff F is a wff p is a wff for any propositional variable p If p is a wff, then (  p) is a wff If p and q are wffs, then (p  q) is a wff If p and q are wffs, then (p  q) is a wff For example, a statement like ((  r)  (p  r)) can be proven to be a wff by arguing that (  r) and (p  r) are wffs by induction and then applying rule 5. Note: we have three recursive/construction rules to create new elements. 03/12/11 Basic Cases Recursive Step

Structural induction --- illustrative example ,[object Object],03/12/11

Structural induction --- illustrative example ,[object Object],[object Object],[object Object],[object Object],03/12/11

Double Induction ,[object Object],[object Object],03/12/11

Steps of Double Induction ,[object Object],[object Object],[object Object],[object Object],03/12/11

Example of Double Induction ,[object Object],[object Object],[object Object],[object Object],[object Object],03/12/11

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Induction

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