CSE-321 Programming Languages Review POSTECH June 14, 2011 박성우

2 Topics Covered SML programming Inductive proofs -Calculus Operational semantics De Bruijn indexes Simply-typed -calculus Abstract machine Subtyping, Object-orientation (Assignment 7) Recursive type Polymorphism Type reconstruction, W-algorithm Compilation (Assignment 8, Part 2)

3 321 강의의 범위 CMU Principles of Programming Languages 수준 – – e.html

4 Assignments A1: 26, 95.27(SML programming 1) A2: 26, 92.4(SML programming 2) A3: 26, 81.54(SML programming 3) A4: 25, 89.6(Implementing -calculus) A5: 25, 91.4(Typechecking) A6: 22, 70.18(De Bruijn indexes) A7: 18, 79.17(Featherweight Java) A8: ?, ?(Type reconstruction and translation)

vs vs vs Assignment 4 (Implementing -calculus) –average 2006: 88.0 –average 2007: –average 2009: –average 2010: –average 2011: 89.6 Assignment 6 (De Bruijn indexes) –average 2007: –average 2009: –average 2010: –average 2011: Assignment 7(Featherweight Java) –average 2006: –average 2007: –average 2009: –average 2010: –average 2011: 79.17

6 절대 평가 Exam 20% * 2 Quiz 10% Assignment 50% –A1, A2, A3: 2.5% –A4: 5% –A5: 5% –A6: 10% –A7: 7.5% –A8 (Part 1): 15% A8 part 2: Extra 25%

7 학점

LICS 2011 (Logic in Computer Science)

9 Paper-and-Pencil Proof

10 Proof in Coq Strings of matched parentheses Inductive T : Set := | eps : T | cons : E -> T -> T. Inductive mparen : T -> Prop :=... Inductive lparen : T -> Prop :=... Theorem mparen2lparen : forall s:T, mparen s -> lparen s. Proof. intros s H. induction H. apply Leps. rewrite <- concat_eps. auto using Leps, Lseq. auto using lparenConcat. Qed.

11 Pythagoras' theorem Proof Theorem even_is_even_times_even: forall (n : nat), even (n * n) -> even n. Theorem main_thm: forall (n p : nat), n * n = double (p * p) -> p = 0. Definition irrational (x : R) : Prop := forall (p : Z) (q : nat), q <> 0 -> x <> (p / q)%R. Theorem irrational_sqrt_2: irrational (sqrt 2%nat).

12 Simply-typed -Calculus Proof of determinism of evaluations Theorem eval_deterministic : forall t t' t'', eval t t' -> eval t t'' -> t' = t''. Proof of type safety in simply-typed lambda-calculus

13 Four Color Problem "Every map can be colored with no more than 4 colors." Proposed in 1852 Computer-based solution published in 1976 Formalization of the proof in 1994 –theorem prover Coq 54

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