CSE-321 Programming Languages Inductive Definitions POSTECH March 15, 2006 박성우

2 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.

3 Outline Inductive definitions of syntactic categories Inductive definitions of judgments Derivable rules and admissible rules Inductive proofs

4 Natural Numbers

5 Natural Numbers - Examples

6 Regular Binary Trees

7 Mutual Induction

8 Strings of Parentheses

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

10 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." –...

11 Inference Rules

12 Examples of Inference Rules

13 Examples of Axioms

14 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"?

15 Natural Numbers using Judgments Judgment Inference rules

16 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."

17 Derivation Tree

18 Even and Odd Numbers

19 Regular Binary Trees

20 Full Regular Binary Trees

21 A Full Regular Binary Tree

22 Proof

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

24 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?

25 Inference rules A derivation tree May I use the following rule? From a Derivation Tree

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

27 Is this rule derivable? No! –because the premise is always smaller than the conclusion in the inference rules:

28 But does this rule make sense? Yes! –because the only way to prove the premise is by first proving the conclusion:

29 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.

30 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

31 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

32 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