Type Systems CSE 340 – Principles of Programming Languages Fall 2015 Adam Doupé Arizona State University

Adam Doupé, Principles of Programming Languages 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 2

Adam Doupé, Principles of Programming Languages 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 3

Adam Doupé, Principles of Programming Languages Type Constructors Pointer to T, where T is a type struct { a 1 : T 1 ; a 2 : T 2 ; …, a k : T k ; } –Where a i is a field name and T i is a previously defined type array range of T –Where range can be single or multi dimensional function of T 1, T 2,..., T k returns T –Type is a function, the types of the parameters are T 1... T k and the return type is T 4

Adam Doupé, Principles of Programming Languages 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; 5

Adam Doupé, Principles of Programming Languages Type Compatibility Which assignments are allowed by the type system? –a = b;? –int a; float b; –float a; int b; 6

Adam Doupé, Principles of Programming Languages 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 7

Adam Doupé, Principles of Programming Languages 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? 8

Adam Doupé, Principles of Programming Languages 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 9

Adam Doupé, Principles of Programming Languages Name Equivalence a: array [0..4] of int; b: array [0..4] of int; a = b? –Not allowed under name equivalence 10

Adam Doupé, Principles of Programming Languages Name Equivalence a, b: array [0..4] of int; a = b? –Not allowed because array [0..4] of int is not named 11

Adam Doupé, Principles of Programming Languages 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 12

Adam Doupé, Principles of Programming Languages 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 13

Adam Doupé, Principles of Programming Languages Structural Equivalence 1.Same built-in types 2.Pointers to structurally equivalent types Type cm : integer; Type inch : integer; cm x; inch y; x = y? // Allowed! 14

Adam Doupé, Principles of Programming Languages Structural Equivalence int* a; float* b; a = b? –Not structurally equivalent, because int and float are not structurally equivalent 15

Adam Doupé, Principles of Programming Languages Structural Equivalence 3.Determining struct structural equivalence –Two structures –st1 { x 1 : W 1, x 2 : W 2, …, x k : W k } –st2 { y 1 : Q 1, y 2 : Q 2,..., y k : Q k } –st1 and st2 are structurally equivalent iff W 1 structurally equivalent to Q 1 W 2 structurally equivalent to Q 2... W k structurally equivalent to Q k 16

Adam Doupé, Principles of Programming Languages Structural Equivalence struct A { a: int, b: float } struct B { b: int, a: float } A foo; B bar; foo = bar? 17

Adam Doupé, Principles of Programming Languages Structural Equivalence struct A { a: int, b: float } struct B { b: float, a: int } A foo; B bar; a = b? 18

Adam Doupé, Principles of Programming Languages Structural Equivalence 4.Determining array structural equivalence –Two Arrays –T1 = array range1 of t 1 –T2 = array range2 of t 2 –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 t 1 and t 2 are structurally equivalent 19

Adam Doupé, Principles of Programming Languages Structural Equivalence 5.Determining function structural equivalence –Two functions –T1 = function of (t 1, t 2, t 3, …, t k ) returns t –T2 = function of (v 1, v 2, v 3,..., v k ) returns v –T1 and T2 are structurally equivalent iff: For all i from 1 to k, t i is structurally equivalent to v i t is structurally equivalent to v 20

Adam Doupé, Principles of Programming Languages 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 or 2 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 if T1 is structurally equivalent to T2 21

Adam Doupé, Principles of Programming Languages 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 22

Adam Doupé, Principles of Programming Languages 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 T i and T j are not structurally equivalent, then set the entry i, j in the table to false Note that T i and T j are the i th and j th types in the program 23

Adam Doupé, Principles of Programming Languages T1 = struct { a: int; p: pointer to T2; } T2 = struct { c: int; q: pointer to T3; } T3 = struct { a : float; p: pointer to T1; } 24 T1T2T3 T1true T2true T3true

Adam Doupé, Principles of Programming Languages T1 = struct { a: int, p: pointer to T2 } T2 = struct { c: int, q: pointer to T3 } T3 = struct { a: float, p: pointer to T1 } 25 T1T2T3 T1true T2true T3true

Adam Doupé, Principles of Programming Languages T1 = struct { a: int, p: pointer to T2 } T2 = struct { c: int, q: pointer to T3 } T3 = struct { a: float, p: pointer to T1 } 26 T1T2T3 T1true T2true T3true

Adam Doupé, Principles of Programming Languages T1 = struct { a: int, p: pointer to T2 } T2 = struct { c: int, q: pointer to T3 } T3 = struct { a: float, p: pointer to T1 } 27 T1T2T3 T1true T2true T3true

Adam Doupé, Principles of Programming Languages T1 = struct { a: int, p: pointer to T2 } T2 = struct { c: int, q: pointer to T3 } T3 = struct { a: float, p: pointer to T1 } 28 T1T2T3 T1true false T2true T3true

Adam Doupé, Principles of Programming Languages T1 = struct { a: int, p: pointer to T2 } T2 = struct { c: int, q: pointer to T3 } T3 = struct { a: float, p: pointer to T1 } 29 T1T2T3 T1true false T2true T3falsetrue

Adam Doupé, Principles of Programming Languages T1 = struct { a: int, p: pointer to T2 } T2 = struct { c: int, q: pointer to T3 } T3 = struct { a: float, p: pointer to T1 } 30 T1T2T3 T1true false T2true T3falsetrue

Adam Doupé, Principles of Programming Languages T1 = struct { a: int, p: pointer to T2 } T2 = struct { c: int, q: pointer to T3 } T3 = struct { a: float, p: pointer to T1 } 31 T1T2T3 T1true false T2true false T3falsetrue

Adam Doupé, Principles of Programming Languages T1 = struct { a: int, p: pointer to T2 } T2 = struct { c: int, q: pointer to T3 } T3 = struct { a: float, p: pointer to T1 } 32 T1T2T3 T1true false T2true false T3false true

Adam Doupé, Principles of Programming Languages T1 = struct { a: int, p: pointer to T2 } T2 = struct { c: int, q: pointer to T3 } T3 = struct { a: float, p: pointer to T1 } 33 T1T2T3 T1true false T2true false T3false true

Adam Doupé, Principles of Programming Languages T1 = struct { a: int, p: pointer to T2 } T2 = struct { c: int, q: pointer to T3 } T3 = struct { a: float, p: pointer to T1 } 34 T1T2T3 T1truefalse T2true false T3false true

Adam Doupé, Principles of Programming Languages T1 = struct { a: int, p: pointer to T2 } T2 = struct { c: int, q: pointer to T3 } T3 = struct { a: float, p: pointer to T1 } 35 T1T2T3 T1truefalse T2falsetruefalse T3false true

Adam Doupé, Principles of Programming Languages T1 = struct { a: int, p: pointer to T2 } T2 = struct { c: int, q: pointer to T3 } T3 = struct { a: float, p: pointer to T1 } 36 T1T2T3 T1truefalse T2falsetruefalse T3false true