Relation Type Theory

Foundations & Applications

Overview

• Introduction• Type Theory• Object Type Theory– Critique of Object Type Theory– Patches for Object Type Theory

• Relation Type Theory– Applications• With Object Type Theory (Co-existence)• Without Object Type Theory (Standalone)

Introduction

• Methodology of Science• Contradictory Manifestos• Programming Paradigm Proliferation

Methodology of Science

Contradictory Manifestos

Programming Paradigm Proliferation

• Declarative– Functional– Logic– Constraint

• Imperative– Procedural– Object Oriented (and Subject Oriented)– Aspect Oriented– …

• But they are all based on Object Type Theory!

Type Theory

• What is a type?• Why do we need types?• What kind of types?• How are things bound to types?• How do types relate to each other?

What is a Type? (wiktionary)• Type

– A grouping based on shared characteristics; a class. – […]– (computing theory) A tag attached to variables and values used

in determining what values may be assigned to what variables.

• Class– A group, collection, category or set sharing characteristics or

attributes. [..] Often used to imply membership of a large class.– […]– (computing) A set of objects possibly differing in state but not

behavior.

What is a Type?

• A type is – a grouping – of things – based on some kind of commonality.

Why do we need Types?

• To define behavior for the– Creation– Destruction– Mutation– Assignment (special kind of Mutation)of the things we are grouping.

• It is all about SHARED BEHAVIOR!

What kinds of types?

• Differentiate on the things that get grouped– things can be objects– things can be relations

• This gives us two kinds of types– Object Types– Relation Types

How are things bound to types?

• Objects are a priori bound to types– objects are bound to one type– objects remain bound to this same type

• Relations are a posteriori bound to Types– relations are bound to a type based on rules– relations can be rebound if relations/rules change

How do types relate to each other?

• Two kinds of type inheritance– Derivation by Extension

(OO Languages , XML Schema)– Derivation by Restriction

(Relational Model, XML Schema)

Kinds of Type Theory

• There are 2 kinds of Type Theory– Object Type Theory– Relation Type Theory

Object Type Theory

• No consensus, at least 2 competing camps– Object Oriented Model (OODBS Manifesto)– Relational Model (Third Manifesto)

• But from a Type Theory perspective, they are actually not that different!

Object Oriented Model (OODBS Manifesto)

• Defines mandatory and optional features that an OO Database System must support

• Most important are– Complex Objects– Object Identity

Complex Objects

Object Identity

Relational Model(Third Manifesto)

Relation-Valued Attributes

• The value of an attribute can be any type– So it can also be a Relation Type

• Closure of Relation Composition– Relations can contain other Relations (Relation-

Valued attributes) or scalar values.

Relational Algebra and Assignment

• Relations are immutable• Relation variables are mutable

• So all operations on relations are set based

Critique of Object Type Theory

• Object Types determine behavior– But relations actually determine behavior!

• Relations have different meaning– Relations are either set based or object based

• Type binding is too static• Relations are second (or third) rate citizens– Objects are ‘owners’ of relation endpoints

Object Types determine behavior

• But relations actually determine behavior!

Relations have different meaning

• Relations are either set based or object based

Type binding is too static

Relations are second rate citizens

Patches for Object Type Theory

• Design by Contract• State Machines (Finite Automata)• Aspect Oriented Programming• Subject Oriented Programming

Design By Contract

• Class invariants using Pre- and Post conditions

State Machines

Aspect Oriented Programming

Subject Oriented Programming

Relation Type Theory• A Relation Type– is a type– that is a priori– statically bound– to a relation

• A Relations Type (Is this necessary?)– is a type – that is a posteriori – dynamically bound – to relations – based on rules about those relations– and that defines behavior for those relations

Applications

• Applications of Relations Type Theory– With Object Type Theory (Co-existence)• Relation Oriented Method Overloading• Multiple Type Declarations• 3rd Generation Object Oriented Programming (OOP)

– Without Object Type Theory (Standalone)• Relation Oriented Programming (ROP)

Relation Oriented Method Overloading

• Same signature, Different behavior

RO Method Overloading - Advantages

• Extra level of polymorphism• More robust (contrary to Contract Based)

Multiple Type Declarations

• Definition• Advantages• Example• Implementation

Muldecs - Definition• A Multiple Type Declaration (muldec) is the

specification of a variables types– the objects assigned to the variable should implement

all types (usually interfaces)

• Muldecs can be supported at– The platform level (JVM, CLR, C++ linker …)– The language level (Java, C#, C++, …)– The framework level (Allors, …)

• Muldecs can be overloaded

Muldecs - Advantages

• Reduces the gap between statically and dynamically typed languages

• Less casting • Higher level of modularization– Looser coupling is possible

• Better evolution– More additive

• Especially suited for 3rd generation OOP

Muldecs - Example

Muldecs – Implementation in Allors

Muldecs - Notations

• C derivatives– type1 type2 … variable– methodName( type1 type2 … argument)

• Uml

3rd Generation Object Oriented Programming (OOP)

• Can be seen as a evolution from– 1st Generation OOP (Inheritance)– 2nd Generation OOP (Delegation)

• Builds upon– Relation Oriented Method Overloading– Multiple Type Declaration

1st Generation OOP

2nd Generation OOP

3rd Generation OO

Relation Oriented Programming