CSE-321 Programming Languages Inductive Definitions

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CSE-321 Programming LanguagesInductive
Definitions
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• POSTECH
• March 14, 2007

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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
• Judgments and inference rules
• 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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Strings of Matched Parentheses
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Strings of Matched Parentheses
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Strings of Parentheses
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Outline

• Inductive definitions of syntactic categories
• Judgments and inference rules
• 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."
• "S S O belongs to the syntactic category nat."
• ...

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Question

• "1 - 1 is equal to 0" is a judgment.
• What is the meaning of "1 - 1 is equal to
  0" without arithmetic rules?
• Judgments make sense only if there are
  inference rules for proving or refuting them.

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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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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 O and
  S.
• We are not talking about " nat."
• We are talking about "S S O 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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Outline

• Inductive definitions of syntactic categories
• Judgments and inference rules
• 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 was an original inference
  rule.

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Is this rule derivable?

• No!
• no derivation tree like

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But does this rule make sense?

• Yes!

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Admissible Rules

• There is a proof that the premise implies the
  conclusion.
• May be used as if it was 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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Announcement

• Assignment 2 is out.
• Due at 1am, March 21 (next Wednesday)
• tail-recursive functions, structures, ...
• Course notes
• Sections 2.1 - 2.3 cover today's lecture.
• Sections 2.4 and 2.5 might be expanded a bit this
  weekend.