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CS 476 – Programming Language Design

CS 476 – Programming Language Design. William Mansky. Language #1: Expressions. Simple arithmetic and boolean operations Every term computes to a value , either int or bool Arithmetic operators: plus, minus, times Boolean operators: and, or, not, comparison, if-then-else

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CS 476 – Programming Language Design

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  1. CS 476 – Programming Language Design William Mansky

  2. Language #1: Expressions • Simple arithmetic and boolean operations • Every term computes to a value, either int or bool • Arithmetic operators: plus, minus, times • Boolean operators: and, or, not, comparison, if-then-else • 3 + 5 * 9 should compute to 48 • if 1 = 0 or 1 = 1 then 2 else 4 should compute to 2

  3. Expressions: Syntax • Arithmetic operators: plus, minus, times • Boolean operators: and, or, not, comparison, if-then-else

  4. Expressions: Syntax type exp = Int of int | Add of exp * exp | … | Bool of bool | And of exp * exp | … | Not of exp | Eq of exp * exp | If of exp * exp * exp E ::= <#> | E + E | E – E | E * E | <bool> | E and E | E or E | not E | E = E | if E then E else E

  5. Expressions: Interpreter • Every term computes to a value, either int or bool type value = IntV of int | BoolV of bool let rec eval (e : exp) : value = (* let rec eval e = *) match e with | Int i -> | Add (e1, e2) -> | …

  6. Expressions: Interpreter • Every term computes to a value, either int or bool type value = IntV of int | BoolV of bool let rec eval (e : exp) : value = (* let rec eval e = *) match e with | Int i -> IntVi | Add (e1, e2) -> | …

  7. Expressions: Interpreter • Every term computes to a value, either int or bool type value = IntV of int | BoolV of bool let rec eval (e : exp) : value = (* let rec eval e = *) match e with | Int i -> IntVi | Add (e1, e2) -> eval e1 + eval e2 | … Type error!

  8. Expressions: Interpreter let rec eval (e : exp) : value = match e with | Int i -> IntVi | Add (e1, e2) -> (match eval e1, eval e2 with | IntV i1, IntV i2 -> IntV (i1 + i2) | _, _ -> ?) • Should our language have runtime errors?

  9. Expressions: Interpreter with Errors let rec eval (e : exp) : value = match e with | Int i -> IntVi | Add (e1, e2) -> (match eval e1, eval e2 with | IntV i1, IntV i2 -> IntV (i1 + i2) | _, _ -> None) type ‘a option = Some of ‘a | None

  10. Expressions: Interpreter with Errors let rec eval (e : exp) : value option = match e with | Int i -> IntVi | Add (e1, e2) -> (match eval e1, eval e2 with | IntV i1, IntV i2 -> IntV (i1 + i2) | _, _ -> None) type ‘a option = Some of ‘a | None

  11. Expressions: Interpreter with Errors let rec eval (e : exp) : value option = match e with | Int i -> Some (IntVi) | Add (e1, e2) -> (match eval e1, eval e2 with | Some (IntV i1), Some (IntV i2) -> Some (IntV (i1 + i2)) | _, _ -> None) type ‘a option = Some of ‘a | None

  12. Expressions: Interpreter with Errors let rec eval (e : exp) : value option = match e with | … | Bool b -> Some (BoolV b) | And (e1, e2) -> (match eval e1, eval e2 with | Some (BoolV b1), Some (BoolV b2) -> Some (BoolV (b1 && b2)) | _, _ -> None)

  13. Expressions: Interpreter with Errors let rec eval (e : exp) : value option = match e with | … | Eq (e1, e2) -> (match eval e1, eval e2 with | Some (IntV i1), Some (IntV i2) -> Some (BoolV (i1 = i2)) | …) • What kinds of results should we be able to compare?

  14. Expressions: Interpreting Comparison let rec eval (e : exp) : value option = match e with | … | Eq (e1, e2) -> (match eval e1, eval e2 with | Some v1, Some v2 -> Some (BoolV (v1 = v2)) | …)

  15. Expressions: Interpreting Comparison let rec eval (e : exp) : value option = match e with | … | Eq (e1, e2) -> Some (BoolV (eval e1 = eval e2)) • Should we be able to compare ints and bools? Should two erroneous expressions be equal? Depends on what kind of language we want!

  16. Expressions: Interpreter with Errors let rec eval (e : exp) : value option = match e with | … | If (e, e1, e2) -> (match eval e with | Some (BoolV b) -> if b then eval e1 else eval e2 | _ -> None)

  17. Expressions: Int-only Interpreter let rec eval (e : exp) : int = match e with | Int i -> i | Add (e1, e2) -> eval e1 + eval e2 | Bool b -> if b then 1 else 0 | … | If (e, e1, e2) -> if eval e <> 0 then eval e1 else eval e2 • Simpler interpreter, but behavior may surprise programmers!

  18. Structure of a language • Syntax • Concrete: what do programs look like? • Abstract: what are the pieces of a program? • Semantics • Static: which programs make sense? • Dynamic: what do programs do when we run them? • Pragmatics • Implementation: how can we actually make the semantics happen? • IDE, tool support, etc.

  19. Static Semantics: Types Type systems do several things: • Help programmers tell different kinds of program expressions apart • Stop bad programs from running/catch errors early • Tell us what kind of value to expect from a piece of code

  20. Static Semantics: Types • A type system is a relation between terms and types • We write for “term has type ” • We define type systems using inference rules: • Often sound: implies that evaluates to a value of type • Often conservative: not all programs that evaluate are well-typed

  21. Expressions: Types • Types: int, bool • Rules:

  22. Expressions: Types • Types: int, bool • Rules:

  23. Expressions: Types • Types: int, bool • Rules:

  24. Proof Trees

  25. Proof Trees

  26. Proof Trees

  27. Proof Trees

  28. Proof Trees

  29. Proof Trees

  30. Proof Trees

  31. Proof Trees

  32. Proof Trees

  33. Expressions: Type Checking • Types: int, bool type typ = TInt | TBool • These are “object-level” (expression) types, not “meta-level” (OCaml) types!

  34. Expressions: Type Checking type typ = TInt | TBool • These are “object-level” (expression) types, not “meta-level” (OCaml) types! let rec typecheck (e : exp) (t : typ) : bool = match e with | Int i -> | Bool b -> | Add (e1, e2) -> …

  35. Expressions: Type Checking type typ = TInt | TBool • These are “object-level” (expression) types, not “meta-level” (OCaml) types! let rec typecheck (e : exp) (t : typ) : bool = match e with | Int i -> t = TInt | Bool b -> | Add (e1, e2) -> …

  36. Expressions: Type Checking type typ = TInt | TBool • These are “object-level” (expression) types, not “meta-level” (OCaml) types! let rec typecheck (e : exp) (t : typ) : bool = match e with | Int i -> t = TInt | Bool b -> t = TBool | Add (e1, e2) -> …

  37. Expressions: Type Checking type typ = TInt | TBool • These are “object-level” (expression) types, not “meta-level” (OCaml) types! let rec typecheck (e : exp) (t : typ) : bool = match e with | Int i -> t = TInt | Bool b -> t = TBool | Add (e1, e2) -> …

  38. Expressions: Type Checking type typ = TInt | TBool • These are “object-level” (expression) types, not “meta-level” (OCaml) types! let rec typecheck (e : exp) (t : typ) : bool = match e with | Int i -> t = TInt | Bool b -> t = TBool | Add (e1, e2) -> typecheck e1 TInt && typecheck e2 Tint && t = TInt …

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