6.5: Induction in Computer Science

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Page ID: 49340

Eric Lehman, F. Thomson Leighton, & Alberty R. Meyer
Google and Massachusetts Institute of Technology via MIT OpenCourseWare

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Induction is a powerful and widely applicable proof technique, which is why we’ve devoted two entire chapters to it. Strong induction and its special case of ordinary induction are applicable to any kind of thing with nonnegative integer sizes—which is an awful lot of things, including all step-by-step computational processes.

Structural induction then goes beyond number counting, and offers a simple, natural approach to proving things about recursive data types and recursive computation.

In many cases, a nonnegative integer size can be defined for a recursively defined datum, such as the length of a string, or the number of operations in an Aexp. It is then possible to prove properties of data by ordinary induction on their size. But this approach often produces more cumbersome proofs than structural induction.

In fact, structural induction is theoretically more powerful than ordinary induction. However, it’s only more powerful when it comes to reasoning about infinite data types—like infinite trees, for example—so this greater power doesn’t matter in practice. What does matter is that for recursively defined data types, structural induction is a simple and natural approach. This makes it a technique every computer scientist should embrace.