Type Systems

Typing

Michal Pıse (CTU in Prague) Object Programming Lect. 1: Type Systems September 21, 2010 2 / 30

The Purpose of Typing

”Well-typed programs never go wrong.”

What Does ”Go Wrong” Entail?

Execution errors.

Program crash.

Divergence.

The Purpose of Typing (II)

As a side effect, typed programs are also more likely to actually do whatthey are supposed to do.

Two Kinds of Typing

Static typing.

Dynamic typing (more precisely: dynamic checking).

Why Do They Both Exist?

It is provenly impossible to create a type system that rejects allincorrect programs (written in a real-world programming language)and accepts all correct ones.

Benefits vs. costs trade-off: static typing rejects at least someincorrect programs at the expense of not accepting certain correctones.

Why Do They Both Exist?

It is provenly impossible to create a type system that rejects allincorrect programs (written in a real-world programming language)and accepts all correct ones.

Benefits vs. costs trade-off: static typing rejects at least someincorrect programs at the expense of not accepting certain correctones.

Example

class Foo { public void urgh() { ...} }class Bar { public void urgh() { ...} }

void main() {Object var;

if ( something()) var = new Foo();

else var = new Bar();

var.urgh();

It is obvious that an object pointed at by var has method urgh, however,many type systems don’t provide any way of expressing it withoutmodifying (and recompiling) definitions of Foo and Bar.

Example

class Foo { public void urgh() { ...} }class Bar { public void urgh() { ...} }

void main() {Object var;

if ( something()) var = new Foo();

else var = new Bar();

var.urgh();

It is obvious that an object pointed at by var has method urgh, however,many type systems don’t provide any way of expressing it withoutmodifying (and recompiling) definitions of Foo and Bar.

Which Is Better?

Debating which is better may (and probably will) get you into aviolent flamewar.

The real question is whether there will ever be a type system that doesnot stand in programmers’ way and rejects all incorrect programs.

Relaxed Notions of Typing

Untrapped errors vs. trapped errors, forbidden errors.

Safe languages disallow untrapped errors (usually by a mixture ofstatic and runtime checks).

Strongly typed languages disallow forbidden errors (ditto).

Weakly typed languages allow even for untrapped errors.

Type Algebra

Function Type

A→ B

Product Type

Intersection Type

A ∩ B

Union Type

A ∪ B

Type System

Two Components of Type Control

Type system—a specification.

Type checking algorithm—an algorithm.

There may be zero or more than one distinct type checkingalgorithms for a particular type system.

Type systems described in a formal language have lowerprobability of ambiguities, unconsidered corner cases etc.

Example

string s;

s = "abc";

string t;

t = s + "def";

Grammar

P → begin CL end programCL→ C ; CL sequential composition

εC → T I declaration

I = E assignmentT → string types

intE → I identifier

N numeralS string literalE1 + E2 sum of two subexpressions

Key Components of a Type System Specification

Judgements.

Type rules.

Environment.

Type System

Empty Environment

∅ ` ◦

Type System (II)

Variable Creation

I : string 6∈ Γ I : int 6∈ Γ Γ ` ◦Γ ∪ {I : string} ` ◦ Γ ∪ {I : int} ` ◦

Type System (III)

Literals

Γ ` ◦Γ ` S : string

Γ ` ◦Γ ` N : int

Type System (IV)

Variables in Environment

I : string ∈ Γ

Γ ` I : string

I : int ∈ Γ

Γ ` I : int

Type System (V)

Γ ` E : string Γ ` F : string

Γ ` E + F : string

Γ ` E : int Γ ` F : int

Γ ` E + F : int

Type System (VI)

Assignment

Γ ` I : string Γ ` E : string

Γ ` I = E : �Γ ` I : int Γ ` E : int

Γ ` I = E : �

Type System (VII)

Sequential Composition

Γ ` ◦Γ ` ε : �

Γ ∪ {I : string} ` ◦ Γ ∪ {I : string} ` C : �Γ ` string I ;C : �

Γ ∪ {I : int} ` ◦ Γ ∪ {I : int} ` C : �Γ ` int I ;C : �

Γ ` C : � Γ ` CL : �Γ ` C ;CL : �

Type System (VIII)

Program

∅ ` CL : �` begin CL end : �

Other Uses of Type Information

Semantics.

Optimization.

Static analysis.

Luca Cardelli. Type Systems. ACM Computing Surveys 28, 1 (March1996), 263-264. http://doi.acm.org/10.1145/234313.234418

Type Systems

Education

Modelling Shape Languages with Type Constraint Systems3. Modelling the Urform Grammar as Type Con-straint System 3.1 Type constraint systems The implementation of Type Constraint Systems

˜LX˚ LIGHTING TYPE BUSBAR TRUNKING SYSTEMS

Domain-Specific Type Systems

Type systems for programming languages - Cristal

Type Systems, Type Inference, and Polymorphismdstefan/cse130-winter17/book/ch06.pdf · Type Systems, Type Inference, and Polymorphism Programming involves a wide range of computational

More type systems

Negations - University of TokyoContribution Development of type systems refuting derivability in some type systems such as • a basic type system for the 𝜆𝜆-calculus • a type

TYPE FMS - TROX › en › monitoring-systems › type-fms... · MONITORING SYSTEMS TYPE FMS EFFECTIVE PRESSURE SENSOR TYPE FMS FOR THE MONITORING OF VOLUME FLOWS Electronic, self-powered

Type B Gas Vent Systems

Type Systems, Type Inference, and Polymorphismmsagiv/courses/pl14/chap6-1.pdf · 2012. 5. 1. · 6 Type Systems, Type Inference, and Polymorphism Programming involves a wide range

Type Systems 11 - Recursive Types

Type and Eﬀect Systems

Next Generation Type Systems

Type Systems and Semantics

8 Type Systems - Poster Design

Type Systems For Distributed Data Structures

Extending Type Systems in a Library

Type Systems & Checking - Computer Sciencebbb/comp524/doc/11TypeSystems.pdf · UNC Chapel Hill Brandenburg — Spring 2010 11: Type Systems & Type Checking COMP 524: Programming Language

Dependent Type Systems - Electrical Engineering and

Monomorphic Type Systems