CSE321 Programming Languages Inductive Definitions

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CSE-321 Programming LanguagesInductive
Definitions
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• POSTECH
• March 15, 2006

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Why Inductive Definitions?

• Definition of SML
• finite in size

• SML programs
• infinite in number

• We need a mechanism by which a finite description
  of SML produces infinitely many SML programs.
• We need inductive definitions.

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Outline

• Inductive definitions of syntactic categories
• Inductive definitions of judgments
• Derivable rules and admissible rules
• Inductive proofs

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Natural Numbers
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Natural Numbers - Examples
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Regular Binary Trees
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Mutual Induction
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Strings of Parentheses
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Outline

• Inductive definitions of syntactic categories V
• Inductive definitions of judgments
• Derivable rules and admissible rules
• Inductive proofs

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Judgments

• An object of knowledge that may or may not be
  provable.
• Examples
• "1 - 1 is equal to 0."
• "1 1 is equal to 0."
• "It is raining."
• "succ succ zero belongs to the syntactic category
  nat."
• ...

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Inference Rules
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Examples of Inference Rules
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Examples of Axioms
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Remember

• Judgments make sense only if there are
  inference rules for proving or refuting them.
• Example
• Without arithmetic rules, what is the meaning of
• "1 - 1 is equal to 0"?

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Natural Numbers using Judgments

• Judgment
• Inference rules

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Metavariables

• is called a metavariable.
• It is just a placeholder for a sequence of zero
  and succ.
• We are not talking about " nat."
• We are talking about "succ succ zero nat."

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Derivation Tree
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Even and Odd Numbers
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Regular Binary Trees
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Full Regular Binary Trees
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A Full Regular Binary Tree
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Proof
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Outline

• Inductive definitions of syntactic categories V
• Inductive definitions of judgments V
• Derivable rules and admissible rules
• Inductive proofs

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Question

• We know arithmetic rules.
• But do we use arithmetic rules to calculate 4
  9?
• Why not just use 4 9 36 from the
  multiplication table?

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From a Derivation Tree

• Inference rules
• A derivation tree
• May I use the following rule?

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Derivable Rule

• There is a derivation tree from the premises to
  the conclusion.
• May be used as if it were an original inference
  rule.

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Is this rule derivable?

• No!
• because the premise is always smaller than the
  conclusion in the inference rules

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But does this rule make sense?

• Yes!
• because the only way to prove the premise is by
  first proving the conclusion

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Admissible Rules

• There is a proof that the premise implies the
  conclusion.
• May be used as if it were an original inference
  rule
• if the system does not change.

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Derivable Rules vs. Admissible Rules

• Which is stronger?
• A derivable rule is an admissible rule?
• Yes
• No
• An admissible rule is a derivable rule?
• Yes
• No

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Derivable Rules vs. Admissible Rules

• A derivable rule remains valid no matter what.
• An admissible rule may be invalidated when a
  new inference rule is introduced.
• Now is the rule below still admissible?

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Homework 1

• Good News
• Everyone did a great job!
• Solution and statistics will be uploaded today.
• Bad News
• Everyone did such a wonderful job that
  Assignment 2 will be out tonight.
• Assignment 2 will be due next Monday.
• Good or Bad News
• Assignment 2 will be as much fun.
• Many problems require a bit/lot of thinking
• tail recursion
• structures and signatures
• functional objects