CSE321 Programming Languages Inductive Proofs POSTECH March 20

CSE-321 Programming Languages Inductive Proofs 박성우 POSTECH March 20, 2006

Inductive Definitions of Syntactic Categories • Natural numbers • Regular binary trees 2

Inductive Definitions of Judgments • Judgment • Inference rules 3

Even and Odd Numbers • Judgments • Inference rules 4

Derivable Rule and Admissible Rule • Derivable rule • Admissible rule 5

But. . . • What is the point of specifying a system and doing nothing else? – E. g. , why do we define the two judgments n even and n odd at all? • What if the definition is wrong? – E. g. , what if we mistakenly introduced the rule: • So we need "inductive proofs. " 6

Outline • Inductive proofs – Structural induction – Rule induction 7

Structural Induction • Prove a property of a syntactic category by analyzing the structure of its definition • I want to prove P(n) for every natural number n. – Examples of P(n) • n has a successor. • n is zero or has a predecessor n'. • n is a product of prime numbers. • n is even (which cannot be proven). 8

Structural Induction - Secret Revealed 9

Structural Induction on Trees 10

Example 11

Here is the first theorem we prove in this course!

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Outline • Inductive proofs – Structural induction V – Rule induction 14

Rule Induction • Similar to structural induction but applied to derivation trees • You will be using rule induction millions of times in the study of programming language theory – because programming languages themselves are defined inductively. 15

Rule Induction • A judgment J with two inference rules: 16

Then what? 17

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Are proofs by rule induction always this boring? Are proofs by rule induction always about something obvious?

mparen and lparen • From • We obtain 21

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Sometimes we need a lemma if a direct proof attempt fails.

• But it is not of the form "If J holds, then P(J) holds. " • Trick: prove instead • That is, 24

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Assignments • Assignment 2 – The due date is this Friday, but finish it quick! – Because. . . • Assignment 3 – Programming part • Our first attempt to implement something non-trivial – Huffman Code – Written part • About rule inductions • You are required to typeset your answers. • We accept only PS and PDF files. • Learn to use Latex. 27