Programming Languages and Compilers CS 421 William Mansky

Why Data Types? n Data types play a key role in: Data abstraction in

Terminology n Type: A type t defines a set of possible data values E.

Types as Specifications n n n Types describe properties Different type systems describe different

Sound Type System n n n If an expression is assigned type t, and

Strongly Typed Language n When no application of an operator to arguments can lead

Type Checking n When is op(arg 1, …, argn) allowed? n Type checking assures

Static Type Checking n n Performed after parsing, before code generation Type of every

Static Type Checking n Typically places restrictions on languages Garbage collection n References instead

Type Declarations n Type declarations: explicit assignment of types to variables (signatures to functions)

Type Inference n Type inference: A program analysis to assign a type to an

Format of Type Judgments n n A type judgment has the form Ⱶ exp

Example Valid Type Judgments n n n [ ] Ⱶ true or false :

Format of Typing Rules Assumptions 1 Ⱶ exp 1 : 1. . . n

Axioms - Constants Ⱶ n : int (assuming n is an integer constant) Ⱶ

Axioms - Variables Notation: Let (x) = if x : and there is no

Simple Rules - Arithmetic Primitive operators ( { +, -, *, …}): Ⱶ e

Simple Rules - Booleans Connectives Ⱶ e 1 : bool Ⱶ e 2 :

Simple Example n n n Let = [ x: int ; y: bool ]

Type Variables in Rules n n n If_then_else rule: Ⱶ e 1 : bool

Function Application n n Application rule: Ⱶ e 1 : 1 2 Ⱶ e

Application Examples n n Ⱶ print_int : int unit Ⱶ 5 : int Ⱶ

Fun Rules describe types, but also how the environment may change n Can only

Fun Examples [y : int ] + Ⱶ y + 3 : int Ⱶ

(Monomorphic) Let and Let Rec n n let rule: Ⱶ e 1 : 1

Example n Which rule do we apply? ? Ⱶ (let rec one = 1

Example Let rec rule: 2 [one : int list] Ⱶ 1 (let x =

Proof of 1 n Which rule? [one : int list] Ⱶ (1 : :

Proof of 1 n 3 Application 4 [one : int list] Ⱶ ((: :

Proof of 3 Constants Rule [one : int list] Ⱶ (: : ) :

Proof of 4 n Rule for variables [one : int list] Ⱶ one :

Proof of 2 5 [x : int; one : int list] Ⱶ n Constant

Proof of 5 ? [x : int; one : int list] Ⱶ fun y

Proof of 5 ? [y : int; x : int; one : int list]

Proof of 5 6 7 [y: int; x: int; one : int list] Ⱶ

Proof of 6 Constant Variable […] Ⱶ (: : ) : int list […;

Proof of 7 Pf of 6 [y/x] Variable [y : int; …] Ⱶ ((:

Curry-Howard Isomorphism n n Type Systems are logics; logics are type systems Types are

Curry - Howard Isomorphism n Modus Ponens A B • Application Ⱶ e 1

Remaining Problems n n n The above system can’t handle polymorphism as in OCAML

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Programming Languages and Compilers (CS 421) William Mansky http: //courses. engr. illinois. edu/cs 421/ Based in part on slides by Mattox Beckman, as updated by Vikram Adve, Gul Agha, Elsa Gunter, and Dennis Griffith 9/11/2021 1

Why Data Types? n Data types play a key role in: Data abstraction in the design of programs n Type checking in the analysis of programs n Compile-time code generation in the n translation and execution of programs 9/11/2021 2

Terminology n Type: A type t defines a set of possible data values E. g. short in C is {x| 215 - 1 x -215} n A value in this set is said to have type t n n Type system: rules of a language assigning types to expressions 9/11/2021 3

Types as Specifications n n n Types describe properties Different type systems describe different properties, e. g. : n Data is read-write versus read-only n Operation has authority to access data n Data came from “right” source n Operation might or could not raise an exception Common type systems focus on types describing same data layout and access methods 9/11/2021 4

Sound Type System n n n If an expression is assigned type t, and it evaluates to a value v, then v is in the set of values defined by t SML, OCaml, Scheme and Ada have sound type systems Most implementations of C and C++ do not 9/11/2021 5

Strongly Typed Language n When no application of an operator to arguments can lead to a run-time type error, language is strongly typed n n E. g. : 1 + 2. 3; ; Depends on definition of “type error” n Is an array bounds error a type error? 9/11/2021 6

Type Checking n When is op(arg 1, …, argn) allowed? n Type checking assures that operations are applied to the right number of arguments of the right types n n May include implicit coercion Used to resolve overloaded operations and catch type errors 9/11/2021 7

Static Type Checking n n Performed after parsing, before code generation Type of every variable and signature of every operator must be known at compile time Catches many programming errors at earliest point Can’t check types that depend on dynamically computed values n E. g. array bounds 9/11/2021 8

Static Type Checking n Typically places restrictions on languages Garbage collection n References instead of pointers n All variables initialized when created n Variable only used at one type n 9/11/2021 9

Type Declarations n Type declarations: explicit assignment of types to variables (signatures to functions) in the code of a program Must be checked in a strongly typed language n May not be necessary for strong typing or even static n 9/11/2021 10

Type Inference n Type inference: A program analysis to assign a type to an expression from the program context of the expression Fully static type inference first introduced by Robin Milner in ML n Haskell, OCaml, SML use type inference n 9/11/2021 11

Format of Type Judgments n n A type judgment has the form Ⱶ exp : is a typing environment n n n Supplies the types of variables and functions is a list of the form [ x : , . . . ] exp is a program expression is a type to be assigned to exp Ⱶ pronounced “turnstile” or “entails” 9/11/2021 12

Example Valid Type Judgments n n n [ ] Ⱶ true or false : bool [ x : int ] Ⱶ x + 3 : int [ p : int -> string ] Ⱶ p 5 : string 9/11/2021 13

Format of Typing Rules Assumptions 1 Ⱶ exp 1 : 1. . . n Ⱶ expn : n Ⱶ exp : Conclusion n Idea: Type of expression determined by type of components n Rule without assumptions is called an axiom n may be omitted when empty 9/11/2021 14

Format of Typing Rules Assumptions 1 Ⱶ exp 1 : 1. . . n Ⱶ expn : n Ⱶ exp : Conclusion n , exp, are parameterized environments, expressions and types - i. e. may contain meta-variables 9/11/2021 15

Axioms - Constants Ⱶ n : int (assuming n is an integer constant) Ⱶ true : bool n n Ⱶ false : bool These rules hold under any typing environment n is a meta-variable 9/11/2021 16

Axioms - Variables Notation: Let (x) = if x : and there is no x : to the left of x : in Variable axiom: Ⱶx: 9/11/2021 if (x) = 17

Simple Rules - Arithmetic Primitive operators ( { +, -, *, …}): Ⱶ e 1 : int Ⱶ e 2 : int Ⱶ e 1 e 2 : int Relations ( { < , > , =, <=, >= }): ˜ Ⱶ e 1 : int Ⱶ e 2 : int Ⱶ e 1 e 2 : bool ˜ 9/11/2021 18

Simple Rules - Booleans Connectives Ⱶ e 1 : bool Ⱶ e 2 : bool Ⱶ e 1 && e 2 : bool Ⱶ e 1 : bool Ⱶ e 2 : bool Ⱶ e 1 || e 2 : bool 9/11/2021 19

Simple Example n n n Let = [ x: int ; y: bool ] Show Ⱶ y || (x + 3 > 6) : bool Start building the proof tree from the bottom up ? Ⱶ y || (x + 3 > 6) : bool 9/11/2021 20

Simple Example n n n Let = [ x: int ; y: bool ] Show Ⱶ y || (x + 3 > 6) : bool Which rule has this as a conclusion? ? Ⱶ y || (x + 3 > 6) : bool 9/11/2021 21

Simple Example n n n Let = [ x: int ; y: bool ] Show Ⱶ y || (x + 3 > 6) : bool Booleans: || Ⱶ y : bool Ⱶ x + 3 > 6 : bool Ⱶ y || (x + 3 > 6) : bool 9/11/2021 22

Simple Example n n n Let = [ x: int ; y: bool ] Show Ⱶ y || (x + 3 > 6) : bool Pick an assumption to prove ? Ⱶ y : bool Ⱶ x + 3 > 6 : bool Ⱶ y || (x + 3 > 6) : bool 9/11/2021 23

Simple Example n n n Let = [ x: int ; y: bool ] Show Ⱶ y || (x + 3 > 6) : bool Which rule has this as a conclusion? ? Ⱶ y : bool Ⱶ x + 3 > 6 : bool Ⱶ y || (x + 3 > 6) : bool 9/11/2021 24

Simple Example n n n Let = [ x: int ; y: bool ] Show Ⱶ y || (x + 3 > 6) : bool Axiom for variables Ⱶ y : bool Ⱶ x + 3 > 6 : bool Ⱶ y || (x + 3 > 6) : bool 9/11/2021 25

Simple Example n n n Let = [ x: int ; y: bool ] Show Ⱶ y || (x + 3 > 6) : bool Pick an assumption to prove ? Ⱶ y : bool Ⱶ x + 3 > 6 : bool Ⱶ y || (x + 3 > 6) : bool 9/11/2021 26

Simple Example n n n Let = [ x: int ; y: bool ] Show Ⱶ y || (x + 3 > 6) : bool Which rule has this as a conclusion? ? Ⱶ y : bool Ⱶ x + 3 > 6 : bool Ⱶ y || (x + 3 > 6) : bool 9/11/2021 27

Simple Example n n n Let = [ x: int ; y: bool ] Show Ⱶ y || (x + 3 > 6) : bool Arithmetic relations Ⱶ x + 3 : int Ⱶ 6 : int Ⱶ y : bool Ⱶ x + 3 > 6 : bool Ⱶ y || (x + 3 > 6) : bool 9/11/2021 28

Simple Example n n n Let = [ x: int ; y: bool ] Show Ⱶ y || (x + 3 > 6) : bool Pick an assumption to prove ? Ⱶ x + 3 : int Ⱶ 6 : int Ⱶ y : bool Ⱶ x + 3 > 6 : bool Ⱶ y || (x + 3 > 6) : bool 9/11/2021 29

Simple Example n n n Let = [ x: int ; y: bool ] Show Ⱶ y || (x + 3 > 6) : bool Which rule has this as a conclusion? ? Ⱶ x + 3 : int Ⱶ 6 : int Ⱶ y : bool Ⱶ x + 3 > 6 : bool Ⱶ y || (x + 3 > 6) : bool 9/11/2021 30

Simple Example n n n Let = [ x: int ; y: bool ] Show Ⱶ y || (x + 3 > 6) : bool Axiom for constants Ⱶ x + 3 : int Ⱶ 6 : int Ⱶ y : bool Ⱶ x + 3 > 6 : bool Ⱶ y || (x + 3 > 6) : bool 9/11/2021 31

Simple Example n n n Let = [ x: int ; y: bool ] Show Ⱶ y || (x + 3 > 6) : bool Pick an assumption to prove ? Ⱶ x + 3 : int Ⱶ 6 : int Ⱶ y : bool Ⱶ x + 3 > 6 : bool Ⱶ y || (x + 3 > 6) : bool 9/11/2021 32

Simple Example n n n Let = [ x: int ; y: bool ] Show Ⱶ y || (x + 3 > 6) : bool Which rule has this as a conclusion? ? Ⱶ x + 3 : int Ⱶ 6 : int Ⱶ y : bool Ⱶ x + 3 > 6 : bool Ⱶ y || (x + 3 > 6) : bool 9/11/2021 33

Simple Example n n n Let = [ x: int ; y: bool ] Show Ⱶ y || (x + 3 > 6) : bool Arithmetic operations Ⱶ x : int Ⱶ 3 : int Ⱶ x + 3 : int Ⱶ 6 : int Ⱶ y : bool Ⱶ x + 3 > 6 : bool Ⱶ y || (x + 3 > 6) : bool 9/11/2021 34

Simple Example n n n Let = [ x: int ; y: bool ] Show Ⱶ y || (x + 3 > 6) : bool Pick an assumption to prove ? Ⱶ x : int Ⱶ 3 : int Ⱶ x + 3 : int Ⱶ 6 : int Ⱶ y : bool Ⱶ x + 3 > 6 : bool Ⱶ y || (x + 3 > 6) : bool 9/11/2021 35

Simple Example n n n Let = [ x: int ; y: bool ] Show Ⱶ y || (x + 3 > 6) : bool Pick an assumption to prove ? Ⱶ x : int Ⱶ 3 : int Ⱶ x + 3 : int Ⱶ 6 : int Ⱶ y : bool Ⱶ x + 3 > 6 : bool Ⱶ y || (x + 3 > 6) : bool 9/11/2021 36

Simple Example n n n Let = [ x: int ; y: bool ] Show Ⱶ y || (x + 3 > 6) : bool Axiom for constants Ⱶ x : int Ⱶ 3 : int Ⱶ x + 3 : int Ⱶ 6 : int Ⱶ y : bool Ⱶ x + 3 > 6 : bool Ⱶ y || (x + 3 > 6) : bool 9/11/2021 37

Simple Example n n n Let = [ x: int ; y: bool ] Show Ⱶ y || (x + 3 > 6) : bool Pick an assumption to prove ? Ⱶ x : int Ⱶ 3 : int Ⱶ x + 3 : int Ⱶ 6 : int Ⱶ y : bool Ⱶ x + 3 > 6 : bool Ⱶ y || (x + 3 > 6) : bool 9/11/2021 38

Simple Example n n n Let = [ x: int ; y: bool ] Show Ⱶ y || (x + 3 > 6) : bool Axiom for variables Ⱶ x : int Ⱶ 3 : int Ⱶ x + 3 : int Ⱶ 6 : int Ⱶ y : bool Ⱶ x + 3 > 6 : bool Ⱶ y || (x + 3 > 6) : bool 9/11/2021 39

Simple Example n n n Let = [ x: int ; y: bool ] Show Ⱶ y || (x + 3 > 6) : bool No more assumptions! QED! Ⱶ x : int Ⱶ 3 : int Ⱶ x + 3 : int Ⱶ 6 : int Ⱶ y : bool Ⱶ x + 3 > 6 : bool Ⱶ y || (x + 3 > 6) : bool 9/11/2021 40

Type Variables in Rules n n n If_then_else rule: Ⱶ e 1 : bool Ⱶ e 2 : Ⱶ e 3 : Ⱶ (if e 1 then e 2 else e 3) : is a type variable (meta-variable) Can take any type at all All instances in a rule application must get same type Then branch, else branch and if_then_else must all have same type 9/11/2021 41

Function Application n n Application rule: Ⱶ e 1 : 1 2 Ⱶ e 2 : 1 Ⱶ ( e 1 e 2 ) : 2 If you have a function expression e 1 of type 1 2 applied to an argument of type 1, the resulting expression has type 2 9/11/2021 42

Application Examples n n Ⱶ print_int : int unit Ⱶ 5 : int Ⱶ (print_int 5) : unit e 1 = print_int, e 2 = 5, 1 = int, 2 = unit Ⱶ map print_int : int list unit list Ⱶ [3; 7] : int list Ⱶ (map print_int [3; 7]) : unit list n n e 1 = map print_int, e 2 = [3; 7], 1 = int list, 2 = unit list 9/11/2021 43

Fun Rules describe types, but also how the environment may change n Can only do what rule allows! n fun rule: [ x : 1 ] + Ⱶ e : 2 Ⱶ fun x -> e : 1 2 n 9/11/2021 44

Fun Examples [y : int ] + Ⱶ y + 3 : int Ⱶ fun y -> y + 3 : int [f : int bool ] + Ⱶ f 2 : : [true] : bool list Ⱶ (fun f -> f 2 : : [true]) : (int bool) bool list 9/11/2021 45

(Monomorphic) Let and Let Rec n n let rule: Ⱶ e 1 : 1 [ x : 1 ] + Ⱶ e 2 : 2 Ⱶ (let x = e 1 in e 2 ) : 2 let rec rule: [ x : 1 ] + Ⱶ e 1 : 1 [ x : 1 ] + Ⱶ e 2 : 2 Ⱶ (let rec x = e 1 in e 2 ) : 2 9/11/2021 46

Example n Which rule do we apply? ? Ⱶ (let rec one = 1 : : one in let x = 2 in fun y -> (x : : y : : one) ) : int list 9/11/2021 47

Example Let rec rule: 2 [one : int list] Ⱶ 1 (let x = 2 in [one : int list] Ⱶ fun y -> (x : : y : : one)) (1 : : one) : int list Ⱶ (let rec one = 1 : : one in let x = 2 in fun y -> (x : : y : : one)) : int list n 9/11/2021 48

Proof of 1 n Which rule? [one : int list] Ⱶ (1 : : one) : int list 9/11/2021 49

Proof of 1 n 3 Application 4 [one : int list] Ⱶ ((: : ) 1) : int list one : int list [one : int list] Ⱶ (1 : : one) : int list 9/11/2021 50

Proof of 3 Constants Rule [one : int list] Ⱶ (: : ) : int list 1 : int [one : int list] Ⱶ ((: : ) 1) : int list 9/11/2021 51

Proof of 4 n Rule for variables [one : int list] Ⱶ one : int list 9/11/2021 52

Proof of 2 5 [x : int; one : int list] Ⱶ n Constant fun y -> (x : : y : : one)) [one : int list] Ⱶ 2: int list [one : int list] Ⱶ (let x = 2 in fun y -> (x : : y : : one)) : int list 9/11/2021 53

Proof of 5 ? [x : int; one : int list] Ⱶ fun y -> (x : : y : : one)) : int list 9/11/2021 54

Proof of 5 ? [y : int; x : int; one : int list] Ⱶ (x : : y : : one) : int list [x : int; one : int list] Ⱶ fun y -> (x : : y : : one)) : int list 9/11/2021 55

Proof of 5 6 7 [y: int; x: int; one : int list] Ⱶ ((: : ) x): int list (y : : one) : int list [y : int; x: int; one : int list] Ⱶ (x : : y : : one) : int list [x : int; one : int list] Ⱶ fun y -> (x : : y : : one)) : int list 9/11/2021 56

Proof of 6 Constant Variable […] Ⱶ (: : ) : int list […; x : int; …] Ⱶ x: int [y : int; x : int; one : int list] Ⱶ ((: : ) x) : int list 9/11/2021 57

Proof of 7 Pf of 6 [y/x] Variable [y : int; …] Ⱶ ((: : ) y) […; one: int list] Ⱶ : int list one : int list [y : int; x : int; one : int list] Ⱶ (y : : one) : int list 9/11/2021 58

Curry-Howard Isomorphism n n Type Systems are logics; logics are type systems Types are propositions; propositions are types Terms (expressions) are proofs; proofs are terms Functions space arrow corresponds to implication; application corresponds to modus ponens 9/11/2021 59

Curry - Howard Isomorphism n Modus Ponens A B • Application Ⱶ e 1 : Ⱶ e 2 : Ⱶ ( e 1 e 2 ) : 9/11/2021 60

Remaining Problems n n n The above system can’t handle polymorphism as in OCAML No type variables in type language (only metavariables in the logic) Would need: n Object level type variables and some kind of type quantification n let and let rec rules to introduce polymorphism n Explicit rule to eliminate (instantiate) polymorphism 9/11/2021 61