上海交通大学试卷 ( A 卷 ) ( 2015 至 2016 学年第 2 学期 )

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1 上海交通大学试卷 ( A 卷 ) ( 2015 至 2016 学年第 2 学期 ) 班级号 (class) 学号 (id) 姓名 (name) 课程名称 程序语言理论 Theory of Programming Languages 成绩 (score) 1. (14 points) (a) Please give the value of a0, a1 after running the following Ocaml program: let f0 x = x + x + x let x0 = ref 3 let a0 = f0 (x0:=!x0+1;!x0) let f1 x = x() + x() + x() let x1 = ref 3 let a1 = f1 (fun () -> (x1:=!x1+1;!x1)) (b) Please argue the differences between call-by-value and call-by-name strategies. (c) Can you give one piece of Ocaml code that diverges(does not terminate) by using call-by-value strategy, but terminates by using call-by-name strategy? (d) Assume we have the following evaluation rules for untyped λ-calculus. Please select the rules for call-by-value strategy; and select those for call-by-name strategy. β-red1 (λx.t1 ) t 2 [x t 2 ] t 1 β-red2 (λx.t) v [x v] t App1 t 1 t 1 t 1 t 2 t 1 t 2 App2 t 2 t 2 t 1 t 2 t 1 t 2 App3 t 2 t 2 v 1 t 2 v 1 t 2 1

2 我承诺, 我将严格遵守考试纪律 承诺人 : 题号得分批阅人 ( 流水阅卷教师签名处 ) 总分 2. (12 points) 2. (12 ) Given This problem the subtyping considers rules this in Ruby Fig. class: 1 on page 9. class A (a) How many attr_accessor different supertypes :x does {a : Top, b : Top} have? (b) Can youdef find m1 an infinite descending chain in the subtype relation? i.e., an infinite sequence of types = 4 0, S 1,... such that for any i, S i+1 <: S i. (c) What about end an infinite ascending chain? (d) If we add def a new m2 typing rule m1 > 4 please figure end out the relation between Nat <: Float def m3 = 4 1 = {a : Nat Float, b : {}} > 4 T end 2 = {a : Float Nat, b : {c : Nat}} If T 1 andef T 2 has m4 subtyping relation, give the derivation tree; If not, explain the reason briefly. self.x = > 4 end end 1) (4 ) Is it possible to define a class B such that evaluating B.new.m2 causes the method m2 defined in class A (not an override of m2) to return true? If so, define class B as such, else explain why it is not possible. 2) (4 ) Is it possible to define a class B such that evaluating B.new.m3 causes the method m3 defined in class A (not an override of m3) to return true? If so, define class B as such, else explain why it is not possible. 3) (4 ) Is it possible to define a class B such that evaluating B.new.m4 causes the method 2 m4 defined in class A (not an override of m4) to return true? If so, define class B as such, else explain why it is not possible.

3 A Name:, Id: 3. (14 points) Recursive type is very powerful, it can make infinite data. This problem consider a typed λ-calculus with variant types, pairs, recursive types, fix, Unit, and Nat. Please refer to Fig. 2 on page 9 for syntax of variants. (a) Define a type NatList for a list of natural numbers where each list is either a nil of unit or cons of a pair of type Nat NatList. (b) Give a value of type NatList that has two elements 0 and succ0, and 0 is the head of this list. (c) Define a function hd of type NatList Nat that returns the head of a list. Return 0 if the given list is nil. (d) Define a function sum of type NatList Nat that returns the summation of all the numbers in a list. You can use (plus m n) directly for m + n. 3

4 4. (10 points) Suppose a generating function F on the universe {a, b, c, d} is defined by the following inference rules: c d b c b d c a A set S is F -closed if F (S) S; it is F -consistent if S F (S); it is a fixed point if S = F (S). (a) List all the F -closed sets. (b) List all the F -consistent sets. (c) What is the least fixed point of F? What is the greatest fixed point of F? 4

5 A Name:, Id: 5. (12 points) This problem uses System F extended with addition. Please refer to Fig. 3 on page 10 for definitions of System F. (a) Give the appropriate System F typing rule(s) and evaluation rule(s) for addition expressions of the form t 1 + t 2. (This should be easy.) (b) Consider a typing context Γ = x : X.X X: i. What does T need to be for the program fragment x [T] (λy : Nat.y + 7) 11 to typecheck? ii. Given your choice for T, what type does x [T] (λy : Nat.y + 7) have? Give a typing derivation for Γ x [T] (λy : Nat.y + 7). (c) Given your choice for T. If v is an arbitrary value of type X.X X, then what might v [T] (λy : Nat.y + 7) 11 evaluate to? (d) If v is an arbitrary value of type (Nat Nat) Nat Nat then what might v (λy : Nat.y + 7) 11 evaluate to? 5

6 6. (12 points) Assume we have a typed programming language formally defined by a small-step operational semantics and a typing judgment. Assume the Preservation and Progress Theorems hold for this language. Preservation: If t : T and t t, then t : T. Progress: If t : T for some T. Then either t is a value or else there is some t with t t. Consider each question below separately and explain your answers briefly. (a) Suppose we change the operational semantics by adding a new inference rule. i. Is it possible that Preservation does not hold any more? ii. Is it possible that Progress does not hold any more? (b) Suppose we change the type system by adding a new typing rule. i. Is it possible that Preservation does not hold any more? ii. Is it possible that Progress does not hold any more? 6

7 A Name:, Id: 7. (10 points) This problem considers to translate a λ-term to the corresponding de Bruijn term. Here are some definitions for your reference. A naming context Γ = x n, x n 1,, x 1, x 0 assigns to each x i the de Bruijn index i. The d-place shift of a term t about cutoff c, written d c (t) is defined as d c (k) = { k if k < c k + d if k c d c (λ.t 1 ) = λ. d c+1 (t 1 ) d c (t 1 t 2 ) = d c (t 1 ) d c (t 2 ) The substitution of a term s for variable number j in a term t, written [j s]t, is defined as follows: { s if k = j [j s]k = k otherwise [j s](λ.t 1 ) = λ.[j s]t 1 [j s](t 1 t 2 ) = [j s]t 1 [j s]t 2 The β-reduction rule on de Bruijn terms is: (λ.t 1 ) t 2 1 [0 1 t 2 ]t 1 (a) Let Γ = wa, give the nameless representation of (λb.w (λa.b a)) (λb.b a). (b) Show the computation (β-reduction) of this de Bruijn term step by step. 7

8 8. (16 points) This problem consider a simple programming language on booleans. Syntax: t::= true false if t then t else t diverg v::= true false Evaluation: E-IfTrue if true then t2 else t 3 t 2 E-IfFalse if false then t2 else t 3 t 3 E-If t 1 t 1 if t 1 then t 2 else t 3 if t 1 then t 2 else t 3 E-Diverg diverg diverg Below is one possible encoding from this small language to untyped call-by-value λ- calculus. true 1 false 1 if t_1 then t_2 else t_3 1 diverg 1 def = λx.λy.x (1) def = λx.λy.y (2) def = t 1 1 t 2 1 t 3 1 (3) def = (λx.xx)(λx.xx) (4) An encoding is termination preserving if for any term t, t terminates if and only if t terminates. (a) Give a counter example to show that the above encoding 1 is NOT termination preserving; (b) Give a termination preserving encoding 2 from this small language to untyped call-by-value λ-calculus, and prove it. (c) If we want to keep the encoding 1, please modify the evaluation rules for this small language to make 1 termination preserving, and prove it. 8

9 A References 16.1 Algorithmic Subtyping 211 Figure 1: Subtype relation with records {} <: Extends 15-1 and 15-3 S <: S S<:U U<:T S<:T S <: Top (S-Refl) (S-Trans) (S-Top) T 1 <:S 1 S 2 <:T 2 S 1 S 2 <:T 1 T 2 {l i } {k j j 1..m } k j =l i impliess j <:T i {k j :S j j 1..m }<:{l i :T i } (S-Arrow) (S-Rcd) 136 Figure 16-1: Subtype relation with records (compact version) 11 Simple Extensions Figure 2: Variants <> Extendsλ (9-1) derivations for records involving combinations of depth, width, and permutation subtyping. Before we can drop S-Trans, we must first add a rule that New syntactic forms bundles depth, width, and permutation subtyping t 0 t into 0 one: t ::=... terms: (E-Case) j 1..m case t {l i } {k j } k j =l i impliess 0 of <l i =x <l=t> as T tagging j <:T i > t i i case t (S-Rcd) j 1..m 0 case tof <l i =x i > t i {k j :S case j }<:{l i :T i } of<l i=x i > t i t i t i T ::= Lemma: If S <: T is derivable types: from the subtyping rules including S-RcdDepth, <l i =t i > as T <l i =t (E-Variant) i > as T <l i :T i S-Rcd-Width, > type and of S-Rcd-Perm variants (but not S-Rcd), then it can also be derived using S-Rcd (and not S-RcdDepth, New typing S-Rcd-Width, rules or S-Rcd-Perm), Γand t:t vice New evaluation rules versa. t t Γ t j :T j case (<l j =v j > ast) Proof: of <l Straightforward i =x i > t i induction Γ <l on derivations. j =t j >as <l i :T i >:<l i :T i > [x j v j ]t j (T-Variant) Lemma justifies eliminating rules S-RcdDepth, S-Rcd-Width, and (E-CaseVariant) S-Rcd-Perm in favor of S-Rcd. Figure 16-1 Γ t summarizes 0 :<l i :T i the> resulting system. Next, we show that, in the systemfor of eachi Figure 16-1, Γ,x i the :T i reflexivity t i :T and transitivity rules are inessential. Γ case t 0 of <l i =x i > t i (T-Case) :T Lemma: Figure 11-11: Variants 1. S<:S can be derived for every type S without using S-Refl. 2. If S<:T is derivable, then it can be derived without using S-Trans Variants Proof: Exercise [Recommended, ]. Binary sums generalize to labeled variants just as products generalize to labeled records. [ ]: IfInstead we addoft the 1 type +T 2, we Bool, write<l how do 1 :Tthese 1,l 2 :T properties 2 >, wherel change? 1 andl 2 are Exercise field labels. Instead of inl tas T 1 +T 2, we write <l 1 =t>as <l 1 :T 1,l 2 :T 2 >. This Andbrings insteadusof tolabeling the definition the branches of the algorithmic of the casesubtype with inl relation. and inr, we use the same labels as the corresponding sum type. With these generalizations, Definition: The algorithmic subtyping relation is the least relation on types thegetaddr example from the previous section becomes: closed under the rules in Figure Addr = <physical:physicaladdr, virtual:virtualaddr>; a = <physical=pa> as Addr; a : Addr 9 getname = λa:addr. case a of <physical=x> x.firstlast <virtual=y> y.name;

10 23.3 System F 343 Figure 3: System F Based onλ (9-1) Syntax t ::= x λx:t.t t t λx.t t [T] terms: variable abstraction application type abstraction type application Evaluation t t t 1 t 1 t 1 t 2 t 1 t 2 t 2 t 2 v 1 t 2 v 1 t 2 (E-App1) (E-App2) (λx:t 11.t 12 ) v 2 [x v 2 ]t 12 (E-AppAbs) v ::= λx:t.t λx.t values: abstraction value type abstraction value t 1 t 1 t 1 [T 2 ] t 1 [T 2] (E-TApp) T ::= X T T X.T types: type variable type of functions universal type (λx.t 12 )[T 2 ] [X T 2 ]t 12 (E-TappTabs) Typing x:t Γ Γ x:t Γ t:t (T-Var) Γ ::= Γ,x:T Γ,X contexts: empty context term variable binding type variable binding Γ,x:T 1 t 2 :T 2 Γ λx:t 1.t 2 :T 1 T 2 Γ t 1 :T 11 T 12 Γ t 2 :T 11 Γ t 1 t 2 :T 12 (T-Abs) (T-App) Γ,X t 2 :T 2 Γ λx.t 2 : X.T 2 (T-TAbs) Γ t 1 : X.T 12 Γ t 1 [T 2 ]:[X T 2 ]T 12 (T-TApp) Figure 23-1: Polymorphic lambda-calculus (System F) is to keep track of scopes and make sure that the same type variable is not added twice to a context. In later chapters, we will annotate type variables with information of various kinds, such as bounds (Chapter 26) and kinds (Chapter 29). Figure 23-1 shows the complete definition of the polymorphic lambdacalculus, with differences fromλ highlighted. As usual, this summary de- 10 fines just the pure calculus, omitting other type constructors such as records,

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