Typed Arithmetic Expressions

Transcription

1 Typed Arithmetic Expressions CS 550 Programming Languages Jeremy Johnson TAPL Chapters 3 and 5 1

2 Types and Safety Evaluation rules provide operational semantics for programming languages. The rules provide state transitions for an abstract machine that evaluates terms in the language. Evaluating a term can result in a value or get stuck in an erroneous state. We would like to be able to tell, without actually evaluating the term, whether or not it will get stuck. Type rules are introduced to associate types with terms. A term t is well-typed if there is some type T such that t has type T. The safety property says that well-typed terms do not lead to erroneous states, i.e. do not go wrong Safety is progress plus preservation. A well-typed term is either a value or it transitions to another well-typed term. 2

3 Outline Arithmetic Expressions Syntax and evaluation rules Untyped arithmetic expressions Typed arithmetic expressions Safety well typed terms do not get stuck 3

4 Boolean Expressions Syntax t ::= true false if t then t else t v ::= true false Evaluation if true then t 2 else t3 t 2 (E-IfTrue) if false then t 2 else t 3 t 3 (E-IfFalse) t 1 t 1 (E-If) if t 1 then t 2 else t 3 if t 1 then t 2 else t 3

5 Derivation s = if true then false else false t = if s then true else true u = if false then true else true E-IfTrue s false E-If t u E-If if t then false else false if u then false else false

6 Determinacy Theorem [Determinacy of one-step evaluation]. If t t and t t, then t = t. Proof. By induction on a derivation of t t.

7 Determinacy Proof. By induction on a derivation of t t. Three cases: E-IfTrue, E-IfFalse and E-If. The first two are base cases. 1) E-IfTrue. t = if true then t 2 else t 3 and t = t 2 E-IfFalse and E-If are not applicable, so t = t = t 2

8 Determinacy Proof. By induction on a derivation of t t. Three cases: E-IfTrue, E-IfFalse and E-If. 2) E-IfFalse. t = if false then t 2 else t 3 and t = t 3 E-IfTrue and E-If are not applicable, so t = t = t 3

9 Determinacy Proof. By induction on a derivation of t t. Three cases: E-IfTrue, E-IfFalse and E-If. 3) E-If. t = if t 1 then t 2 else t 3 and t = if t 1 then t 2 else t 3 and t 1 t 1 E-IfTrue and E-IfFalse are not applicable since t 1 is not a normal form.

10 Determinacy Proof. By induction on a derivation of t t. 3) E-If. t = if t 1 then t 2 else t 3 and t = if t 1 then t 2 else t 3 with t 1 t 1 E-IfTrue and E-IfFalse are not applicable since t 1 is not a normal form, so t = if t 1 then t 2 else t 3 with t 1 t 1. By induction t 1 = t 1 and t = t.

11 Normal Forms Definition. A term t is in normal form if no evaluation rule applies to it. Theorem. Every value is in a normal form and every term t that is in normal form is a value. Proof. Immediate from rules. Show contrapositive by induction on t.

12 Normal Forms Theorem. Every value is in a normal form and every term t that is in normal form is a value. Show contrapositive by induction on t. If t is not a value then it is not a normal form t = if t 1 then t 2 else t 3 1. t 1 = true E-IfTrue applicable 2. t 1 = false E-IfFalse applicable 3. else by induction t 1 t 1 and E-If is applicable

13 Multi-step Evaluation Definition. Multistep evaluation * is the reflexive, transitive closure of one step evaluation.

14 Uniqueness of Normal Forms Theorem [Uniqueness of normal forms]. If t * u and t * v, where u and v are normal forms, then u = v. Proof. Corollary of determinacy.

15 Termination of Evaluation Theorem [Termination of evaluation]. For every term t, there is a normal form t such that t * t. Proof. Each evaluation step decreases the size and since size is natural number and the natural numbers are well founded the size must reach 1.

16 Arithmetic Expressions Syntax t ::= 0 succ t pred t iszero t nv ::= 0 succ nv Evaluation t 1 t 1 (E-Succ) succ t 1 succ t 1 pred 0 0 (E-PredZero) pred (succ nv 1 ) nv 1 (E-PredSucc) t 1 t 1 (E-Pred) pred t 1 pred t 1 iszero 0 true (E-IsZeroZero) iszero (succ nv 1 ) false (E-IsZeroSucc) t 1 t 1 (E-IsZero) iszero t 1 iszero t 1

17 Derivation pred (succ (pred 0)) * 0 pred (succ (pred 0)) pred (succ 0) 0 E-PredZero pred 0 0 E-Succ succ (pred 0) succ 0 E-Pred pred (succ (pred 0)) pred (succ 0)

18 Stuck Terms Definition. A stuck term is a term that is in normal form but is not a value. E.G. pred false if 0 then true else false

19 Typing Relation Definition. A typing relation, t : T, is defined by a set of inference rules assigning types to terms. A term is well-typed if there is some T such that t : T. For arithmetic expressions there are two types: Bool and Nat Insisting that evaluation rules are only applied to proper types prevents things from going wrong (getting stuck).

20 Typing Relation Syntax T ::= Bool Typing Rules true : Bool false : Bool (T-True) (T-False) t 1 : Bool, t 2 : T, t 3 : T if t 1 then t 2 else t 3 : T (T-If)

21 Typing Relation Syntax T ::= Nat *The typing relation is conservative. I.E. some terms that do not get stuck are not well-typed. Typing Rules 0 : Nat (T-Zero) t 1 : Nat (T-Succ) succ t 1 : Nat t 1 : Nat (T-Pred) pred t 1 : Nat if (iszero 0) then 0 else false Want type checking to be easy t 1 : Nat iszero t 1 : Bool (T-IsZero)

22 Inversion Lemma 1. If true : R, then R = Bool. 2. If false : R, then R = Bool. 3. If if t 1 then t 2 else t 3 : R, then t 1 : Bool and t 2 : R and t 3 : R. 4. If 0 : R, then R = Nat. 5. If succ t 1 : R, then R = Nat and t 1 : Nat 6. If pred t 1 : R, then R = Nat and t 1 : Nat 7. If iszero t 1 : R, then R = Bool and t 1 : Nat

23 Uniqueness of Types Theorem. Each term t has at most one type. Proof. By induction on t using the inversion lemma. The inversion lemma provides a recursive algorithm for computing types.

24 Safety = Progress + Preservation Progress. A well-typed term is not stuck (either it is a value or it can take a step according to the evaluation rules). Preservation. If a well-typed term takes a step of evaluation, then the resulting term is well-typed.

25 Progress Theorem. If t : T, then t is a value or there is a t such that t t. Proof. By induction on the derivation of t : T.

26 Progress Theorem. If t : T, then t is a value or there is a t such that t t. Proof. By induction on the derivation of t : T. Base Cases: T-True, T-False, T-Zero, then t is a value.

27 Progress Theorem. If t : T, then t is a value or there is a t such that t t. Proof. By induction on the derivation of t : T. Case: T-If t = if t 1 then t 2 else t 3, t 1 : Bool, t 2, t 3 : T By induction either t 1 has a value E-IfTrue or E-IfFalse is applicable. Else t 1 t 1 and E-If is applicable

28 Progress Theorem. If t : T, then t is a value or there is a t such that t t. Proof. By induction on the derivation of t : T. Case: T-Succ t = succ t 1 and t 1 : Nat By induction either t 1 has a value t is a value else t 1 t 1 and E-Succ is applicable

29 Progress Theorem. If t : T, then t is a value or there is a t such that t t. Proof. By induction on the derivation of t : T. Case: T-Pred t = pred t 1 and t 1 : Nat By induction either t 1 has a value E-PredZero or E-PredSucc is applicable else t 1 t 1 and E- Pred is applicable

30 Progress Theorem. If t : T, then t is a value or there is a t such that t t. Proof. By induction on the derivation of t : T. Case: T-IsZero t = iszero t 1 and t 1 : Nat By induction either t 1 is a value either E- IsZeroZero or E-IsZeroSucc is applicable else t 1 t 1 and E-IsZero is applicable

31 Preservation Theorem. If t : T and t t, then t : T. Proof. By induction on t : T.

32 Preservation Theorem. If t : T and t t, then t : T. Proof. By induction on t : T. Case T-True, T-False, and T-Zero. t = true and T = Bool. t = false and T = Bool. t = 0 and T = Nat In all of these cases, t is a value and there is no t t and theorem is vacuously true.

33 Preservation Theorem. If t : T and t t, then t : T. Proof. By induction on t : T. Case T-If t = if t 1 then t 2 else t 3, t 1 : Bool, t 2, t 3 : T E-IfTrue E-IfFalse t 1 = true t = t 2 : T t 1 = false t = t 3 : T E-If t 1 t 1 and by induction t 1 : Bool t = if t 1 then t 2 else t 3 : T (by T-If)

34 Preservation Theorem. If t : T and t t, then t : T. Proof. By induction on t : T. Case T-Succ t = succ t 1 T = Nat t 1 : Nat E-Succ t 1 t 1 and by induction t 1 : Nat t = succ t 1 : Nat (by T-Succ)

35 Preservation Theorem. If t : T and t t, then t : T. Proof. By induction on t : T. Case T-Pred t = pred t 1 T = Nat t 1 : Nat E-PredZero t 1 = 0 and t = 0 : Nat (by T-Zero) E-PredSucc t 1 = succ nv 1 and t = nv 1 : Nat (by T- Zero or T-Succ) E-Pred t 1 t 1 and by induction t 1 : Nat t = pred t 1 : Nat (by T-Pred)

36 Preservation Theorem. If t : T and t t, then t : T. Proof. By induction on t : T. Case T-IsZero t = iszero t 1 T = Bool t 1 : Nat E-ZeroZero t 1 = 0 and t = true : Bool (by T-True) E-IsZeroSucc t 1 = succ nv 1 and t = false : Bool (by T-False) E-IsZero t 1 t 1 and by induction t 1 : Nat t = iszero t 1 : Bool (by T-IsZero)

37 Untyped Lambda Calculus Syntax Evaluation t ::= terms: x variable t 1 t 1 (E-App1) λx.t abstraction t 1 t 2 t 1 t 2 t t application t 2 t 2 (E-App2) v ::= values v 1 t 2 v 1 t 2 λx.t abstraction value (λx.t 12 )v 2 [x v 2 ] t 12 (E-AppAbs) Call by value

38 Substitution Definition. [x s] x = s [x s] y = y if y x [x s] (λy.t 1 ) = λy. [x s] t 1 if y x & y FV(s) [x s] (t 1 t 2 ) = [x s] t 1 [x s] t 2

39 Function Types Extend types to include functions λx:true [returns Bool] λx. λy.y [returns function] λx.if x then 0 else 1 [returns Nat, expects Bool] T T Definition. T ::= T T Bool Bool Bool, (Bool Bool) (Bool Bool) T 1 T 2 T 3 T 1 (T 2 T 3 )

40 Typing Relation Type rules for functions x : T 1 t 2 : T 2 λx:t 1.t 2 : T 1 T 2 Need three term relation, Γ t : T, that includes typing context Γ (set of assumptions on free variables) Γ, x : T 1 t 2 : T 2 Γ λx:t 1.t 2 : T 1 T 2

41 Simply Typed Lambda Calculus Syntax Evaluation t ::= terms: x variable t 1 t 1 (E-App1) λx:t.t abstraction t 1 t 2 t 1 t 2 t t application t 2 t 2 (E-App2) v ::= values v 1 t 2 v 1 t 2 λx:t.t abstraction value (λx:t 11.t 12 )v 2 [x v 2 ] t 12 (E-AppAbs)

42 Lambda Calculus Typing Syntax T ::= types: T T function types Γ ::= contexts empty context Γ, x:t term variable binding Type Rules x : T (T-Var) Γ x : T Γ, x : T 1 t 2 : T 2 (T-Abs) Γ λx:t 1.t 2 : T 1 T 2 Γ t 1 : T 1 T 2 Γ t 2 : T 1 (T-App) Γ t 1 t 2 : T 2

43 Substitution Lemma Theorem. If Γ, x : S t : T and Γ s : S, then Γ [x s]t : T.

44 Safety Theorem [Progress]. Suppose t is a closed (no free variables), well-typed term (i.e. t : T). Then either t is a value or else there is some t with t t. Theorem [Preservation]. If Γ t : T and t t, then Γ t : T.

45 Erasure Definition. The erasure of a simply typed term t is defined by erase(x) = x erase(λx:t 1.t 2 ) = λx.erase(t 2 ) erase(t 1 t 2 ) = erase(t 1 ) erase(t 2 ) Theorem. If t t, then erase(t) erase(t ). If erase(t) m, then there is a simply typed term t such that t t and erase(t ) = m.

46 Curry-Howard Correspondence Logic 1. Propositions 2. P Q 3. P Q 4. Proof of P 5. Prop P is provable Programming Languages 1. Types 2. Type P Q 3. Type P Q 4. Term t of type P 5. Type P is inhabited by some term A term of λ- is a proof of a logical proposition corresponding to its type.