1. 10 December 2002BU CAS CS520 1 Principles of Programming Languages (Final Review) Hongwei Xi Comp. Sci. Dept. Boston University

2. 10 December 2002BU CAS CS520BU CAS CS520 2 Untyped Lambda-Calculus Terms t ::= x |  x.t | t 1 (t 2 ) Values v ::= x | x.t Evaluation rules: t 1  t 1 ’ t 2  t 2 ’ ----------------- ------------------ t 1 (t 2 )  t 1 ’(t 2 ) v 1 (t 2 )  v 1 (t 2 ’) -------------------------- ( x.t)(v)  [x -> v] t

3. 10 December 2002BU CAS CS520BU CAS CS520 3 Nameless Representation Terms t ::= One | Shift (t) | Lam (t) | App(t 1, t 2 ) For instance, the term x. y.y(x) is represented as: Lam(Lam(App(One, Shift(One))))

4. 10 December 2002BU CAS CS520BU CAS CS520 4 Simply Typed Lambda-Calculus We use B for base types. Types T ::= B | T 1   T 2 Terms t ::= x | x:T.t | t 1 (t 2 ) Contexts  ::=  | , x:T Static semantics: typing rules Dynamic semantics: evaluation rules

5. 10 December 2002BU CAS CS520BU CAS CS520 5 Some important properties of STLC Canonical forms Substitution lemma: if , x:S |- t 1 :T and  |- t 2 :S, then  |- [x  t 2 ] t 1 :T Subject reduction: if  |- t:T and t  t’, then  |- t’:T Progress if  |- t:T, then either t is a value or t  t’ for some term t’

6. 10 December 2002BU CAS CS520BU CAS CS520 6 Simple Extensions to STLC Tuples: terms t ::= … | {t 1, t 2 } | t.1 | t.2 types T ::= T 1  T 2 Local binding: terms t ::= … | let x = t 1 in t 2 Sums: terms t ::= … | inl(t) | inr(t) | (case t of inl(x) => t 1 | inr(x) => t 2 ) types T ::= T 1 + T 2 Recursion: terms t ::= … | fix (t) … …

7. 10 December 2002BU CAS CS520BU CAS CS520 7 Normalization of STLC Reducibility predicates R T for types T – T is a base type: R T (t) if t terminates. – T = T1 * T2: R T (t) if t terminates and R T 1 (v 1 ) and R T 2 (v 2 ) whenever t  * {v 1, v 2 }. – T = T1  T2: R T (t) if t terminates and R T 2 ([x  v 1 ] t 0 ]) whenever t  * x:T 1.t 0 and R T 1 (v 1 ).

8. 10 December 2002BU CAS CS520BU CAS CS520 8 References Terms t ::= … | ref (t) | !t | t 1 := t 2 | l Values v ::= … | l Types T ::= … | Ref (T) Stores  ::= [] |  [l  v] Store typings  ::= [] | [l 

9. 10 December 2002BU CAS CS520BU CAS CS520 9 Exceptions Terms t ::= raise (t) | try t 1 with t 2 Answers ans ::= v | raise v Evaluation rules: ------------------------------- ----------------------------- (raise v) (t)  raise v v(raise v’)  raise v’ … … The typing rule for ‘raise’:  |- t: Exn -------------------  |- raise t: T

10. 10 December 2002BU CAS CS520BU CAS CS520 10 Recursive Types Types t ::= … | X |  X. T Typing rules: fold/unfold

11. 10 December 2002BU CAS CS520BU CAS CS520 11 Universal Types System F / The 2 nd order polymorphically typed lambda-calculus ( 2 ) Terms t ::= …|  X.t | t[T] Types T ::= … | X |  X.T Value restriction: only values can occur under 

12. 10 December 2002BU CAS CS520BU CAS CS520 12 Existential Types Terms t ::= {*T, t} | open t 1 as {*X,x} in t 2 Types T ::= … |  X.T Encoding existential types via universal types

13. 10 December 2002BU CAS CS520BU CAS CS520 13 Type Reconstruction

14. 10 December 2002BU CAS CS520BU CAS CS520 14 Subtyping Coercion: a subtyping proof of T1 <= T2 can be translated into a function of type T1  T2. Coherence: there may be different subtyping proofs of T1 <= T2 for some T1 and T2.

15. 10 December 2002BU CAS CS520BU CAS CS520 15 Evaluation Contexts Evaluation contexts: E ::= [] | E(t) | v(E) | if E then t1 else t2 | {E, t} | {v, E} | E.1 | E.2 | let x = E in t | … Redexes and their reductions: – ( x.t)(v)  [x->v] t – {v 1, v 2 }.1  v 1 – {v 1, v 2 }.2  v 2 – … …

16. 10 December 2002BU CAS CS520BU CAS CS520 16 Callcc and Throw Terms t ::= … | *E | callcc (t) | throw (t 1, t 2 ) Values v ::= … | *E Evaluation contexts E ::= … | callcc(E) | throw (E, t) E[callcc( k.t)]  E[[k  *E]t] E[throw(*E’, t)] -> E’[t]

17. 10 December 2002BU CAS CS520BU CAS CS520 17 Imperative Objects Subtyping Inheritance Open recursion

18. 10 December 2002BU CAS CS520BU CAS CS520 18 The End Questions?