Domains

A domain represents a set of values, such as Number or Boolean.

Domains are similar to types in programming languages. They are used to ensure that the arguments of functions are compatible. This information is also useful to select appropriate algorithms and to optimize compiled expressions.

Domains can be defined as a union or intersection of domains:

//  A number or a boolean.
["Union", "Number", "Boolean"]

// Any number except "1".
[("SetMinus", "Number", 1)]

Parametric Domains

Parametric domains are functions that define a domain:

// Range of non-negative integers
["Range", 0, "+Infinity"]

The ["Range", <min>, <max>] parametric domain defines a set of integers such that \( \mathord{min} \le n \le \mathord{max}, n \in \N \).

The ["Interval", <min>, <max>] parametric domain defines a set of real numbers such that \( \mathord{min} \le x \le \mathord{max}, n \in \R \).

To represent an open interval, use the Open function: ["Interval", ["Open", <min>], <max>] \( \operatorname{min} \lt x \le \operatorname{max}, n \in \R \) or \(x \in \rbrack \operatorname{min}, \operatorname{max} \rbrack \).

Domains Lattice

Domains are defined in a hierarchy (a lattice). The upper bound of the domain lattice is Anything and the lower bound is Nothing. The parent of a domain represent a is-a/subset-of relationship, for example, a List is-a Collection.

Anything domains Tensor domains Function domains Number domains

The implementation of the CortexJS domains is based on Weibel, Trudy & Gonnet, Gaston. (1991). An Algebra of Properties… 352-359. 10.1145/120694.120749.

Obtaining the Domain of an Expression

To obtain the domain of an expression, use the ce.domain() function.

const engine = new ComputeEngine();

engine.domain('Pi');
// ➔ "TranscendentalNumber"

engine.domain('Add');
// ➔ "Function": domain of the symbol "Add"

engine.domain(['Add', 5, 2]);
// ➔ "Number": the result of the "Add" function
// (its codomain) in general is a "Number"

engine.domain(engine.evaluate(['Add', 5, 2]));
// ➔ "PrimeNumber"

Converting Domains to a Canonical Form

To convert a domain to a canonical form, use ce.format(<domain>, 'canonical-domain').

format(['SetMinus', 'RealNumber', 'IrrationalNumber'], 'canonical-domain');
// ➔ "RationalNumber"

format(
  ['Union', ['Number', 0, '+Infinity'], ['Number', '-Infinity', 5]],
  'canonical-domain'
);
// ➔ "ExtendedRealNumber"

List of Domains

Boolean and MaybeBoolean

A Boolean is either True or False.

A MaybeBoolean is a True, False or Maybe.

String and Symbol

String is the domain of all the strings.

A Symbol is a string used to represent the name of a constant, variable or function in a MathJSON expression.

Number

Any numerical value.

  • PrimeNumber - An integer that cannot be produced as the product of 2 or more integers.
  • CompositeNumber - An integer that can be produced by the product of 2 or more integers.
  • "Integer" \(= \mathbb{Z}\). The set of whole numbers: \(0, 1, 2, 3\ldots\) and their additive inverse: \(-1, -2, -3, \ldots\).
  • "NumberZero" - The number \(0\): a composite number and an imaginary number.
  • "RationalNumber" \(= \mathbb{Q}\). A number which can be expressed as the quotient \(\frac{p}{q}\) of two integers \(p, q \in \mathbb{Z}\).
  • "RealNumber" \(= \mathbb{R}\).
  • "NaturalNumber" \(= \mathbb{N}\). Counting numbers, \(0, 1, 2, 3\ldots\) Note that \(0\) is included, following the convention from ISO/IEC 80000.
  • "IrrationalNumber" \(= \mathbb{Q^{\prime}}\). Numbers such as \(\sqrt{5}, \pi, \exponentialE\) that cannot be written as a quotient of two integers.
  • "ImaginaryNumber" - Any purely imaginary value (including zero).
  • "ComplexNumber" \(= \mathbb{C}\). A real or imaginary number

"Tensor"

  • "ComplexTensor" - A tensor whose elements are complex numbers.
  • "RealTensor" - A tensor whose elements are real numbers.
  • "IntegerTensor" - A tensor whose elements are integers.
  • "LogicalTensor" - A tensor whose elements are 0 or 1.
  • "Scalar" - A tensor of rank 0.
  • "Vector", "Row", "Column" - A tensor of rank 1. The argument of the parametric version specifies the number of elements in the vector.
  • "Matrix" - A tensor of rank 2. The argument of the parametric version specifies the number of rows and columns in the matrix.
  • "Quaternion" - A \(2\times2\) matrix of complex elements. Quaternions are commonly used to represent vectors in 3D space (\(\mathbb{R}^3\)).
  • "SquareMatrix" - A tensor with the same number of rows and columns.
  • "MonomialMatrix" - A square matrix with exactly one non-zero entry in each row and column.
  • "OrthogonalMatrix" - A real square matrix whose transpose is equal to its inverse: \(Q^{\mathrm{T}}=Q^{-1}\)
  • "PermutationMatrix" - A square matrix with with exactly one non-zero entry in each row and column.
  • "DiagonalMatrix" - A matrix in which the elements outside the main diagonal are zero.
  • IdentityMatrix - A diagonal matrix whose diagonal elements are 1.
  • ZeroMatrix a matrix whose elements are 0.
  • SymmetricMatrix - A real matrix that is equal to its transpose.
  • HermitianMatrix - A complex matrix that is equal to its conjugate transpose.

Function

A function maps some expressions, its arguments, to another expression.

  • ["Function", ...arg-domain, co-domain] is the parametric version of the Function domain. For example, ["Function", "Number", "Boolean"] is the domain of the functions that have a single argument, a number, and return a boolean (has a boolean codomain).
  • ContinuousFunction - A continuous function is a function that has no abrupt change in value (no discontinuity). The Weirestrass function is continuous, but not differentiable.
  • TranscendentalFunction - A function not expressible as a finite combination of the algebraic operations of addition, subtraction, multiplication, division, raising to a power, and extracting a root. Example: “Log”, “Sin”…
  • AlgebraicFunction - A function that can be defined as the root of a polynomial equation.
  • PeriodicFunction - A function that repeats its values at regular intervals.
  • TrigonometricFunction - Real functions which relate an angle of a right-angled triangle to ratios of two side lengths
  • HyperbolicFunction
  • MonotonicFunction - A function that is either entirely non-increasing, or entirely non-decreasing.
  • StrictMonotonicFunction
  • DifferentiableFunction - A function whose derivative exists at each point in its domain.
  • InfinitelyDifferentiableFunction
  • RationalFunction - A function that can be expressed as the ratio of two polynomials.
  • PolynomialFunction - A function expressed only with the operations of addition, subtraction, multiplication, and non-negative integer exponentiation.
  • "QuadraticFunction" - A function of the form \(f(x) = ax^2+ bx + c\)
  • "LinearFunction" = \(f(x) = ax+ b\) A function that is the product of an argument plus a constant.
  • ConstantFunction - A function that always return the same value.
  • MonotonicFunction
  • StrictMonotonicFunction

Predicate and LogicalFunction

A predicate is a function with a codomain of MaybeBoolean.

It may have one, two or more arguments of any domain.

A LogicalFunction is a predicate whose arguments are in the MaybeBoolean domain, for example the domain of And is LogicalFunction.

Domain and ParametricDomain

The domain of all the domains is Domain.

ParametricDomain is the domain of all the parametric domains, that is the functions that can define a domain.