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# Lec6 - CMPSC 461 Programming Language Concepts Lecture 6...

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CMPSC 461: Programming Language Concepts Lecture 6 Instructor: Prof. Swarat Chaudhuri

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Bottom-up inductive definition of data Define a set inductively. That is, give rules to build it. As usual, the base case and inductive case. Let us take the set S of all even natural numbers. First try: the set that satisfies: 0 is in S if n is in S , then (n+2) is in S Doesn’t work. Many such sets. The least such set. That is, the set that is the subset of all sets satisfying these conditions.
Growing sets… S 1 S 2 S 3 S 0 = empty Finally, some set S k such that S k = S k+1 . This set is S. We are defining S as a limit of a sequence of sets , f(Ø), f(f(Ø)),… > where f(S) is obtained by adding to S all numbers (n +2), where n is in S.

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f is a (mathematical) function from sets to sets. f : Sets → Sets We want a solution to the equation f(S) = S But there are many possible solutions to this equation. We want the
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Lec6 - CMPSC 461 Programming Language Concepts Lecture 6...

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