Type System

Finitio's type system is different from those you can find in a programming language. The aim here is to capture information, not software behavior. Therefore, the definition of type differs. In Finitio, a type is a set of values, a subtype is a subset, a supertype is a superset. That's it.

However, the aim here is not to define yet another type system with a fixed set of available types such as boolean, string and integer, but rather to provide an abstract way of building information types and to 'connect' them to the types available in a host programming language, or data exchange language.

For this, a Finitio implementation has to define a representation function that maps, for each Finitio type, a type of the host language that will represent values of the information type. This representation function is host/implementation-specific; see the documentation of the binding you use.

Rep(FinitioType) -> HostType

Any type

Any type captures the set of all possible 'values' representable in the host language. It is mostly provided for the sake of completeness and some implementation needs by specific bindings. It is occasionally useful in "don't care" situations too. Any is captured by a single dot:

Any = .

Builtin types

A builtin type starts with a dot followed by the name of an abstraction in the host language, a Ruby class for instance. The set of values captured by the Finitio type is defined the same set as the host abstraction. For instance,

.Integer  # The set of values captured by the Integer class

To avoid builtins being spread everywhere, it is usual to define type aliases and build higher-level types with those aliases instead. This also provides better host-language independence and interoperability. For instance, the so-called default system in Finitio-rb includes the following definitions:

Integer = .Integer
Nil     = .NilClass

Sub types

Sub types are subsets of values. Finitio uses so-called 'specialization by constraint' to define sub types. E.g., the set of positive integers can be defined as follows:

Posint = Integer( i | i >= 0 )

Multiple constraints can be distinguished by name:

Evens = Integer( i | positive: i >= 0, even: i%2 == 0 )

All types can be sub-typed through constraints. In addition, Finitio uses structural type equivalence, which means that the type captured by the definition of Evens above is actually equivalent to the following one:

Evens = PosInt( i | i % 2 == 0 )

Union types

In some respect, union types are the dual of subtypes. They allow defining new types by generalization, through the union of the sets of values defined by other types. For instance, the missing Boolean type of Ruby is simply captured as:

Boolean = .TrueClass|.FalseClass

Union types are also very useful for capturing possibly missing information (aka NULL/nil). For instance, the following type will capture either an integer or nil:

MaybeInt = Integer|Nil

Seq types

Capturing sequences (aka arrays) of values is straightforward. Sequences are ordered and may contain duplicates:

Measures = [Posint]

Set types

Capturing sets of values is straightforward too. Set are unordered and may not contain duplicates:

Hobbies = {String}

Struct types

Capturing structs is straightforward too. Structs can be used to capture ordered pairs, triples, and so forth. Each struct component has its own type:

Signature = <String, Posint, Boolean>

Tuple & Multi-Tuple types

Tuples capture information facts. Unlike structs, tuples have named components called 'attributes'. Attributes must all have different names and are not particularly ordered. A set of such (name,Type) pairs is called a heading.

ProgrammingLanguage = { name: String, author: String, since: Date }

By default, all attributes are mandatory. It is very useful in practice to allow optional attributes too. Multi-tuples provide such support. Multi-tuples are a very convenient shorthand over unions of tuple types. For instance, the multi-tuple type below allows since to be omitted:

{ name: String, author: String, since :? Date }

Relation & Multi-relation types

Relations are sets of tuples, all of which have the same heading. The notation for defining relation types naturally follows:

Languages = {{ name: String, author: String, since: Date }}

Relation types and their syntax are first-class in Finitio, most notably because of the availability of relational algebra for them, unlike pure sets of tuples.

Note that relations do not allow duplicates and have no significant ordering of their tuples. If the ordering is significant, you should consider a sequence of tuples instead:

Preferences = [{ lang: String, reason: String }]

Similarly to tuples, multi-relations allow optional attributes. For instance,

Languages = {{ name: String, author: String, since :? Date }}

Abstract Data types

Abstract data types, also called user-defined types, provide the way to define higher level abstractions easily and to optionally connect them to types of the host language (e.g. a Ruby class). For instance, a Color abstraction can be defined as follows:

Color = <rgb> {r: Byte, g: Byte, b: Byte},
        <hex> String( s | s =~ /^#[0-9a-f]{6}$/i )

The Color definition above shows that a color can be represented either by a RGB triple (through a tuple type), or by a hexadecimal string (e.g. #8a2be2). rgb and hex are called the information representations of the Color abstraction.

Binding an ADT to the host language

Defined as above, the type will behave as a union type, i.e. it will let pass valid RGB triples and hexadecimal strings. Now, representations can be complemented to connect the Color abstraction to a host language type, e.g. a Color Ruby class, and raise the level of discourse on the programming side. This amounts to providing one information contract per representation.

Suppose for example that the following Color class has been defined:

class Color

  def initialize(r, g, b)
    @r, @g, @b = r, g, b
  end
  attr_reader :r, :g, :b

end

Connecting our information ADT to this Color class can be done through a builtin type and two explicit converters, called the dresser and the undresser: (We only show the rgb case here; the hex one is defined in a similar way)

Color = .Color <rgb> {r: Byte, g: Byte, b: Byte}
                     \( tuple | Color.new(tuple.r, tuple.g, tuple.b) )
                     \( color | {r: color.r, g: color.g, b: color.b} )

The converters provide load/dump code to convert from information types to the code abstraction and vice versa, thereby complementing a representation with a so-called information contract. A binding of Finitio, e.g. Finitio-rb, guarantees that the dresser will only be executed on valid representations of the corresponding information type. As the dresser tends to be exposed to an unsafe world, however, it should always be kept pure and safe (no side effects, no metaprogramming, no code evaluation, etc.).

In order to keep Finitio schemas as clean as possible, implementations may provide conventions-over-configuration protocols for automating information contracts. Bindings typically come with so-called 'internal' and 'external' ADT protocols.

Internal ADT protocol

For instance, Finitio-rb provides a more idiomatic way of connecting Ruby classes to information types. The information contracts may indeed be moved to the class itself, as one would probably do it, e.g. for testing purpose.

class Color

  def initialize(r, g, b)
    @r, @g, @b = r, g, b
  end
  attr_reader :r, :g, :b

  def self.rgb(tuple)
    Color.new(tuple[:r], tuple[:g], tuple[:b])
  end

  def to_rgb
    {r: @r, g: @g, b: @b}
  end

end

In Finitio-rb, the following definition, that refers to the builtin type but has no dresser/undresser, makes the assumption that the convention is met and will use the Color.rgb(...) and Color#to_rgb methods:

Color = .Color <rgb> {r: Byte, g: Byte, b: Byte}

External ADT protocol

Sometimes, relying on methods provided by the abstraction itself is impossible or not wanted (e.g. because it would lead to core extensions considered intrusive). For this reason, Finitio bindings may also support so called 'external' protocols.

Support for example that the rgb information contract is not provided by the Color class itself and that it's not possible to implement it there. Finitio-rb also allows the developer to define the contract methods in a specific module/class:

module RgbContract

  def self.dress(tuple)
    Color.new(tuple[:r], tuple[:g], tuple[:b])
  end

  def self.undress(color)
    { r: color.r, g: color.g, b: color.b }
  end

end

Then, in Finitio, the external contract can be specified as follows:

Color = .Color <rgb> {r: Byte, g: Byte, b: Byte} .RgbContract

The mechanism described here for abstract data types is actually more general and applies to most of Finitio's work. The next section describes information contracts in more details.