CSE-321 Programming Languages
Inductive Proofs
박성우
POSTECH
March 20, 2006
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?
mparen and lparen
• From
• We obtain
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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,
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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.
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