CSE321 Programming Languages Inductive Proofs

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CSE-321 Programming LanguagesInductive Proofs
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• POSTECH
• March 20, 2006

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Inductive Definitions of Syntactic Categories

• Natural numbers
• Regular binary trees

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Inductive Definitions of Judgments

• Judgment
• Inference rules

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Even and Odd Numbers

• Judgments
• Inference rules

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Derivable Rule and Admissible Rule

• Derivable rule
• Admissible rule

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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."

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Outline

• Inductive proofs
• Structural induction
• Rule induction

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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).

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Structural Induction - Secret Revealed
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Structural Induction on Trees
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Example
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Here is the first theorem we prove in this
course!
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Outline

• Inductive proofs
• Structural induction V
• Rule induction

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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.

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Rule Induction

• A judgment J with two inference rules

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Then what?
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Are proofs by rule induction always this boring?
Are proofs by rule induction always about
something obvious?
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mparen and lparen

• From
• We obtain

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Sometimes we need a lemmaif a direct proof
attempt fails.
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• But it is not of the form "If J holds, then
  P(J) holds."
• Trick prove instead
• That is,

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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.