Type Systems CSE 340 – Principles of Programming Languages Fall 2016 Adam Doupé Arizona State University http://adamdoupe.com

Type Systems Informally, a type in a programming language specifies a set of values and operations that can be applied on those values A type can be associated with a variables or a constant Values are not necessarily numeric values, for example we can specify function types A type system consists of Basic types Type constructors Type inference Type compatibility

Type Declaration Programming language will typically include Basic types Included in the programming language and available to any program written in that language Type constructors Way for a programmer to define new types

Type Constructors Pointer to T, where T is a type struct { a1: T1; a2 : T2; …, ak: Tk; } Where ai is a field name and Ti is a previously defined type array range of T Where range can be single or multi dimensional function of T1, T2, ..., Tk returns T Type is a function, the types of the parameters are T1 ... Tk and the return type is T

Using Type Constructors Declaring Types Type cm : integer; Type RGBA : array [0..4] of int; Type png : array [0..256] of RGBA; Anonymous Types array [0..4] of int x; struct { int a; char b; } y;

Type Compatibility Which assignments are allowed by the type system? a = b;? int a; float b; float a; int b;

Type Inference Types of expressions or other constructs as a function of subexpression types a + b a int; b float Returns a float in C Error in ML a * b a string; b int Error in most languages Returns a string in Python

Type Compatibility Principally about type equivalence How to determine if two types are equal? Type cm : integer; Type inch : integer; cm x; inch y; x = y?

Name Equivalence Types must have the exact same name to be equivalent Type cm : integer; Type inch : integer; cm x; inch y; x = y? // ERROR

Name Equivalence a: array [0..4] of int; b: array [0..4] of int; a = b? Not allowed under name equivalence

Name Equivalence a, b: array [0..4] of int; a = b? Not allowed because array [0..4] of int is not named

Name Equivalence Type A: array [0..4] of int; a: A; b: A; a = b? Allowed, because both a and b have the same name

Internal Name Equivalence If the program interpreter gives the same internal name to two different variables, then they share the same type a, b: array [0..4] of int; c: array [0..4] of int; a = b? Yes, because interpreter/compiler gives the same internal name to a and b a = c? No, because interpreter/compiler gives different internal name to c than to a and b

Structural Equivalence Same built-in types Pointers to structurally equivalent types Type cm : integer; Type inch : integer; cm x; inch y; x = y? // Allowed!

Structural Equivalence int* a; float* b; a = b? Not structurally equivalent, because int and float are not structurally equivalent

Structural Equivalence Determining struct structural equivalence Two structures st1 { x1: W1, x2: W2, …, xk: Wk } st2 { y1: Q1, y2: Q2, ..., yk: Qk } st1 and st2 are structurally equivalent iff W1 structurally equivalent to Q1 W2 structurally equivalent to Q2 ... Wk structurally equivalent to Qk

Structural Equivalence struct A { a: int, b: float } struct B { b: int, a: float } A foo; B bar; foo = bar?

Structural Equivalence struct A { a: int, b: float } struct B { b: float, a: int } A foo; B bar; foo = bar?

Structural Equivalence Determining array structural equivalence Two Arrays T1 = array range1 of t1 T2 = array range2 of t2 T1 and T2 are structurally equivalent iff: range1 and range2 have (1) the same number of dimensions and (2) the same number of entries in each dimension t1 and t2 are structurally equivalent

Structural Equivalence Determining function structural equivalence Two functions T1 = function of (t1, t2, t3, …, tk) returns t T2 = function of (v1, v2, v3, ..., vk) returns v T1 and T2 are structurally equivalent iff: For all i from 1 to k, ti is structurally equivalent to vi t is structurally equivalent to v

Determining Structural Equivalence The goal is to determine, for every pair of types in the program, if they are structurally equivalent Seems fairly simple, just keep applying the previous 5 rules until the base case 1 is reached How to handle the following case: T1 = struct { a: int; p: pointer to T2; } T2 = struct { a: int; p: pointer to T1; } Applying the rules states that T1 is structurally equivalent to T2 iff pointer to T1 is structurally equivalent to pointer to T2, which is true iff T1 is structurally equivalent to T2 …

Structural Equivalence Algorithm The way to break the stalemate is to assume that T1 and T2 are structurally equivalent because no rule contradicts them being structurally equivalent Our goal is to create an n X n table, where n is the number of types in the program, and each entry in the table is true if the types are structurally equivalent and false otherwise

Structural Equivalence Algorithm To support cyclical definitions, we first initialize all entries in the table to true We assume that types are structurally equivalent unless we have proof otherwise Algorithm is fairly simple Set the n X n table to have each entry as true While table has not changed Check each entry i, j in the table, and if Ti and Tj are not structurally equivalent, then set the entry i, j in the table to false Note that Ti and Tj are the ith and jth types in the program

T1 = struct { a: int, p: pointer to T2 } T2 = struct { c: int, q: pointer to T3 } T3 = struct { a: float, p: pointer to T1 } T1 T2 T3 true

T1 = struct { a: int, p: pointer to T2 } T2 = struct { c: int, q: pointer to T3 } T3 = struct { a: float, p: pointer to T1 } T1 T2 T3 true

T1 = struct { a: int, p: pointer to T2 } T2 = struct { c: int, q: pointer to T3 } T3 = struct { a: float, p: pointer to T1 } T1 T2 T3 true

T1 = struct { a: int, p: pointer to T2 } T2 = struct { c: int, q: pointer to T3 } T3 = struct { a: float, p: pointer to T1 } T1 T2 T3 true false

T1 = struct { a: int, p: pointer to T2 } T2 = struct { c: int, q: pointer to T3 } T3 = struct { a: float, p: pointer to T1 } T1 T2 T3 true false

T1 = struct { a: int, p: pointer to T2 } T2 = struct { c: int, q: pointer to T3 } T3 = struct { a: float, p: pointer to T1 } T1 T2 T3 true false

T1 = struct { a: int, p: pointer to T2 } T2 = struct { c: int, q: pointer to T3 } T3 = struct { a: float, p: pointer to T1 } T1 T2 T3 true false

T1 = struct { a: int, p: pointer to T2 } T2 = struct { c: int, q: pointer to T3 } T3 = struct { a: float, p: pointer to T1 } T1 T2 T3 true false

T1 = struct { a: int, p: pointer to T2 } T2 = struct { c: int, q: pointer to T3 } T3 = struct { a: float, p: pointer to T1 } T1 T2 T3 true false

T1 = struct { a: int, p: pointer to T2 } T2 = struct { c: int, q: pointer to T3 } T3 = struct { a: float, p: pointer to T1 } T1 T2 T3 true false

T1 = struct { a: int, p: pointer to T2 } T2 = struct { c: int, q: pointer to T3 } T3 = struct { a: float, p: pointer to T1 } T1 T2 T3 true false

T1 = struct { a: int, p: pointer to T2 } T2 = struct { c: int, q: pointer to T3 } T3 = struct { a: float, p: pointer to T1 } T1 T2 T3 true false