10/31/2015IT 3271 A Type is a Set 1. A set of values 2. A low-level representation 3. A collection of operations on those values int n; Chapter 6: Types.

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A Type is a Set

1. A set of values

2. A low-level representation

3. A collection of operations on those values

int n;

Chapter 6: Types

Bertrand Russell (1872 – 1970)

(1910)

Alfred N. Whitehead (1861 – 1947)

Required for programming languages, so it is more restricted.

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Issues of types

– Type annotations (how to declare?)– Type inference (how to determine?)– Type checking (when to check?)– Type equivalence issues (what is it?)

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Primitive vs. Constructed Types Primitive type: can be used but cannot be defined.

ML: int, real, char

Constructed type: can be defined in a program

ML: type intpair = int * int

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Primitive Types

Every programming language defines its primitive types

– Some define the primitive types precisely for implementation (e.g., Java)

– Some leave it loose for implementation (e.g., C, ML)

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Integral Types in different PL’s

C:charunsigned charshort intunsigned short intintunsigned intlong intunsigned long intNo standard implementation, but longer sizes must provide at least as much range as shorter sizes.

Java:byte (1-byte signed)char (2-byte unsigned)short (2-byte signed)int (4-byte signed)long (8-byte signed)

Scheme:integerIntegers of unbounded range

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Issues of primitive types What sets of values?

– Language specification; how much left up to the implementation?

– If necessary, how can a program find out? (INT_MAX in C, Int.maxInt in ML, etc.)

What operations are supported?– Detailed definitions: rounding, exceptions, etc.

The choice of representation is a critical part in these decisions

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Constructed Types

1. enumerations2. tuples3. arrays4. strings5. lists6. unions7. subtypes8. function types

Example:

Each has a connection to a set operation in mathematics

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Making Sets by Enumeration

We can construct sets by just listing all the elements:

},,{ cbaS

C: enum coin {penny, nickel, dime, quarter};Ada: type GENDER is (MALE, FEMALE);Pascal: type primaryColors = (red, green, blue);ML: datatype day = M | Tu | W | Th | F | Sa | Su;

These define a new type (= set) They also define a collection of named constants of that type

(= elements)

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Representing Enumeration Values

A common representation is to treat the values of an enumeration as small integers

enum coin { penny = 1, nickel = 5, dime = 10, quarter = 25 };

enum escapes { BELL = '\a', BACKSPACE = '\b', TAB = '\t', NEWLINE = '\n', VTAB = '\v', RETURN = '\r' };

They are countable!!

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Operations on Enumeration Values

Equality test:

If using integer to represent elements then elements are legitimate for all integer operations:

fun isWeekend x = (x = Sa orelse x = Su);

Pascal: for C := red to blue do P(C)

C: int x = penny + nickel + dime;

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Making Sets by Tupling

The Cartesian product of two or more sets defines tuples:

YyXxyx

YXS

|,

Some language like ML supports pure tuples:

(i.e., fields without names)

fun get1 (x : real * real) = #1 x;

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Making Types by Tupling Many PLs support record types: tuples with named fields

C:

struct complex { double rp; double ip;} x;

ML:

type complex = { rp:real, ip:real};

Operations on tuples: (selection)

C: x.ipML: #ip x

fun getip (x : complex) = #ip x;

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Representing Tuple Values

A common representation is to just place the elements side-by-side in memory

But there are lots of details:– In what order?– Are there “holes” between elements (e.g. on

word boundaries) in memory?– Are these details visible to the programmer?

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What does C say about tuples (structs)?

Preface

C is a general-purpose programming language which features economy of expression, modern control flow and data structures, and a rich set of operators. C is not a “very high level” language, nor a “big” one, and is not specialized to any particular area of application. But its absence of restrictions and its generality make it more convenient and effective for many tasks than supposedly more powerful languages.

…..

The C Programming LanguageBrian W. Kernighan and Dennis M. Ritchie1978 edition

Let’s see what does C say about itself first.

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ANSI C

The members of a structure have addresses increasing in the order of their declarations. A non-field member of a structure is aligned at an addressing boundary depending on its type; therefore, there may be unnamed holes in a structure.

struct { unsigned keyword : 1;

unsigned external : 1;unsigned statics : 1;

int value; unsigned ascii : 7; char eascii;} flag1, flag2;

1 1 0 1 1 0 0 0

0 1 0 1 1 1 0 1

0 0 1 1 1 0 1 0

0 1 1 0 0 1 0 0

0 0 1 1 0 1 1 0

1 1 0

0 0 0 0 0 0 0 0

0 0 0 0 0 1 0 1

0 1 1 0 0 1 1 1

0 0 0 0 1 1 1 0

flag1

flag2

unnamed holes

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Sets of Vectors Fixed-size vectors:

Arbitrary-size vectors:

XxixxXS inn .|,,1

i

iXXS

In PLs: arrays, strings, lists Issues:

– What are the index values?– Is the array size fixed at compile time?

What is the difference between this and tuples?

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Index values for array Java, C, C++:

– First element of an array a is a[0]– Indexes are always integers starting from 0

Pascal is more flexible:– integers, characters, enumerations, subranges– Starting/Ending index chosen by the programmer– size is fixed at compile time

type LetterCount = array['a'..'z'] of Integer;var Counts: LetterCount;

begin Counts['a'] = 1 ………

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Array designers need to decide:

1. What are the index values?2. Is array size fixed at compile time (static type)?3. What operations are supported?4. Are multiple dimensions allowed?5. Is a higher-dimensional array the same as an array of

arrays?6. Is re-dimensioning possible at runtime?7. What is the order of elements in memory?8. Is there a separate type for strings (not just array of

characters)?9. Is there a separate type for lists?

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Making Types by Union

C:

union element { int i; float f;};

ML:

datatype element = I of int | F of real;

Making Sets by Union: YXS

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Representing Union Values

Two (or more) representations overlap each other in memory.

This representation may or may not be exposed to the programmer

union element { int i; char *p;} u;

/* sizeof(u) is */ /* max(sizeof(u.i),sizeof(u.p)) */

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Loosely Typed Unions Some languages expose the details of union implementation

(i.e., the programmer knows the implementation)

union element { int i; float f;} e;

No type casting is involved

e

f

i

float x;int n;e.i = 100;e.f = 1.0;x = e.f;n = e.i;

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Strictly Typed Unions

In ML, an element type x is either I or F, thus, you have to say what to do with each type of value in the union:

datatype element = I of int | F of real;

Type casting is involved

fun getReal (F x) = x | getReal (I x) = real x;

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Variant Records Union where specific type is dynamically

determined by the value of a field (“discriminated union”)

Ada and Modula-2C: I don’t care what is it.

ML: I have to know what is it in advance

Ada: Let’s see what is it.

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Ada Variant Record Example

type DEVICE is (PRINTER, DISK);

type PERIPHERAL(Unit: DEVICE) is record HoursWorking: INTEGER;

case Unit is when PRINTER => Line_count: INTEGER;

when DISK => Cylinder: INTEGER; Track: INTEGER;

end case; end record;

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Subsets

We can define the subset selected by any predicate P:

)(| xPXxS

S X

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Making Subtypes

– Less general: Pascal subranges

type digit = 0..9;

– More general: Ada subtypes

subtype DIGIT is INTEGER range 0..9;subtype WEEKDAY is DAY range

MON..FRI;

– Most general: Lisp types with predicates

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Ada Subtypestype DEVICE is (PRINTER, DISK);

type PERIPHERAL(Unit: DEVICE) is record HoursWorking: INTEGER; case Unit is when PRINTER => Line_count: INTEGER; when DISK => Cylinder: INTEGER; Track: INTEGER; end case; end record;

subtype DISK_UNIT is PERIPHERAL(DISK);

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Types with Predicates in Lisp

(declare (type integer x))

(declare (type (or null cons) x))

(declare (type (and number (not integer)) x))

(declare (type (and integer (satisfies evenp)) x))

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Representing Subtype Values

Usually, we just use the same representation for the subtype as for the supertype

Questions:– Do we want to shorten it if we can?

I.e., does X: 1..9 take the same space as X: Integer?

– Do we enforce the subtyping?

Is X := 10 legal? What about X := X + 1?

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Type is a key idea of OOP

A class is a type: data and operations on that data, bundled together

A subclass is a subtype: it includes a subset of the objects, but supports a superset of the operations

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Making Sets of Functions

define a set of functions with given domain and range:

RfDffRDS ran dom|

A set can be a collection of functions

NNffNNS :|

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Making Types of Functions

Most languages have some notion of the type of a function:

int f(char a, char b) { return a==b;}

fun f(a:char, b:char) = (a = b);ML:

C:

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Operations on Function Values

call the function bound to variables passed as parameters Treated as value

Many languages support nothing beyond function call

(What can C do?)

ML supports many operations (all above and others)

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Type AnnotationsThe syntax for describing types

Some languages use naming conventions to declare the types of variables

– BASIC: S$ is a string – Fortran: I is an integer

Like explicit annotations, these conventions provide static type information

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Type Inference

Infers a static type for every expression and for every function, usually requires no annotations

Constants usually have static types– Java: 10 has type int, 10L has type long

Expressions may have static types inferred from operators and types of operands– Java: if a is double, a*0 is double

(0.0)

Simple

Complicate

Determining the types

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Static Type Checking

Static type checking determines a type for everything before running the program: variables, functions, expressions, everything

Compile-time error messages when static types are not consistent

– Operators: 1+"abc"– Functions: round("abc")– Statements: if "abc" then …

Most modern languages are statically typed

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Dynamic Typing

In some languages, programs are not statically type-checked before being run. they are dynamically type-checked during the runtime.

At runtime, the language system checks that operands are of suitable types for operations

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Example: Lisp

It won’t work if a or b is not a number An improper call, like (f nil nil), is

not caught at compile time It is caught at runtime – that is dynamic

typing

(defun f (a b) (+ a b))

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Explicit Runtime Type Tests

Some languages allow explicit runtime type tests:– Java: test object type with instanceof

operator Type information is known at runtime, even

when the language is mostly statically typed

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Strong Typing Weak Typing

The purpose of type-checking is to prevent the application of operations to incorrect types of operands

ML and Java, the type-checking is thorough enough to guarantee this—that’s strong typing

C has holes in the type system that add flexibility but weaken the guarantee

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Type Equivalence

What happens if apply f to an irpair2?

– Name equivalence does not permit this: irpair2 and irpair1 are different names

– Structural equivalence does permit this, since the types are constructed identically

ML uses Structural equivalence

type irpair1 = int * real;type irpair2 = int * real;fun f(x:irpair1) = #1 x;