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

Type Systems

CSE 340 – Principles of Programming LanguagesFall 2015

Adam DoupéArizona State Universityhttp://adamdoupe.com

2Adam 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


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 TypesType cm : integer;Type RGBA : array [0..4] of int;Type png : array [0..256] of RGBA;

• Anonymous Typesarray [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

1. Same built-in types2. Pointers to structurally equivalent typesType cm : integer;Type inch : integer;cm x;inch y;x = y? // Allowed!



int* a;float* b;

• a = b?– Not structurally equivalent, because int and

float are not structurally equivalent



3. 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



struct A { a: int, b: float }struct B { b: int, a: float }

A foo;B bar;• foo = bar?



struct A { a: int, b: float }struct B { b: float, a: int }

A foo;B bar;• a = b?



4. 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



5. 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 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


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 T3T1 true true trueT2 true true trueT3 true true true









T1 T2 T3T1 true true falseT2 true true trueT3 true true true



T1 T2 T3T1 true true falseT2 true true trueT3 false true true



T1 T2 T3T1 true true falseT2 true true trueT3 false true true



T1 T2 T3T1 true true falseT2 true true falseT3 false true true



T1 T2 T3T1 true true falseT2 true true falseT3 false false true



T1 T2 T3T1 true true falseT2 true true falseT3 false false true



T1 T2 T3T1 true false falseT2 true true falseT3 false false true



T1 T2 T3T1 true false falseT2 false true falseT3 false false true



T1 T2 T3T1 true false falseT2 false true falseT3 false false true

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

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