Inductive Definitions

Kangwon National University

임현승

Programming Languages

These slides were originally created by Prof. Sungwoo Park at POSTECH.

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Why Inductive Definitions?• Definition of Ocaml

– finite in size• Ocaml programs

– infinite in number

• We need a mechanism by which a finite description of OCaml produces infinitely many OCaml programs.

– We need inductive definitions.

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Outline• Inductive Definitions of Syntactic Categories• Judgments and Inference Rules• Derivable Rules and Admissible Rules

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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

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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

Full Regular Binary Trees• Also known as perfect 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

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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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• Inference rules

• A derivation tree

• May I use the following rule?

From a Derivation Tree

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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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• 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?

Derivable Rules vs. Admissible Rules