CSE-321 Programming Languages

Inductive Definitions

POSTECH

March 15, 2006

박성우

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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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• 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 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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• 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

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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