orbital.logic.functor
Interface Functor

All Known Subinterfaces:

BinaryFunction, BinaryFunction, BinaryFunction.Composite, BinaryFunction.Composite, BinaryPredicate, BinaryPredicate.Composite, Formula, Formula.Composite, Function, Function, Function.Composite, Function.Composite, Functor.Composite, MathFunctor, MathFunctor.Composite, MutableFunction, Polynomial, Predicate, Predicate.Composite, Substitution, Type, Type.Composite, UnivariatePolynomial, VoidFunction, VoidFunction.Composite, VoidPredicate, VoidPredicate.Composite

All Known Implementing Classes:

Closer, Functionals.Anamorphism, Functionals.Catamorphism, Functionals.Hylomorphism, Functionals.Paramorphism, HeuristicAlgorithm.PatternDatabaseHeuristic, LogicBasis, MutableFunction.TableFunction

public interface Functor

An abstract base interface for all functors of an arity n applicable in any predicate-logic style. Usually denoted like "f(...)" or "P(...)". It provides a way for callers applying the functor, to callback callees implementing a derivative of Functor.

A Functor f/n is any function-like object (resembling function-pointers in C++) implementing Functor. Its signature Specification declares which methods are contained. The exact signature specification of a functor can either be defined explicitly with a sub interface of Functor that encapsulates those methods, or generically with an implicit interface accessed via Functor.Specification.

Types of functors:

predicates

the return-type will be modelled as a boolean-value, and P/n is interpreted as a relation, i.e. a subset of a cartesian product.

functions

the return-type will be any type and the symbol f/n is interpreted as a function-object.

functionals

(a higher-order function) are special functions where the return-type or any of the argument-types will be a kind of Functor.

Duality between functions and predicates:

Every function f/n:A→B induces an implicit predicate with the same extensional semantics

_fP/(n+1) := {(a₁,...,a_n,f(a₁,...,a_n)) ¦ (a₁,a₂,...,a_n)∈A}

If a predicate P/n⊆A is unique with respect to a certain parameter a_k it induces an implicit function with the same extensional semantics

_Pf/(n-1) := {f(a₁,...,a_k-1,a_k+1,...a_n):=a_k ¦ P(a₁,a₂,...,a_n) is true}

Also, whether the extension of a predicate is specified as a subset ρ∈℘(A), or with its characterisitic function χ_ρ with χ_ρ(x)=1 iff x∈ρ is a matter purely syntactic variant. Note however, that inspite of all this duality, functions and predicates can differ intensionally regardless of their extensional equality.

Finally, functions, predicates, relations, and graphs are all "isomorph" anyhow!

Author:

André Platzer

See Also:

Functor.Specification, Functor.Specification.getSpecification(Functor), Function, Predicate, Properties of Functions, Functors in the sense of category-theory

Nested Class Summary

static interface	Functor.Composite The base interface for all functors that are composed of other functors.
static class	Functor.Specification Represents a signature and type specification belonging to a functor.

Method Summary

boolean	equals(java.lang.Object o) .
int	hashCode()
java.lang.String	toString() Returns a string representation of the Functor.

Method Detail

equals

boolean equals(java.lang.Object o)

.

Note that functors will often provide intensional equality only, since the mathematical notion of extensional equality for functions and predicates is undecidable anyway (Proposition of Rice). Nevertheless implementations are encouraged to provide a larger subset of extensional equality as far as possible.

Overrides:

equals in class java.lang.Object

hashCode

int hashCode()

Overrides:

hashCode in class java.lang.Object

toString

java.lang.String toString()

Returns a string representation of the Functor.

This method is already provided in Object.toString(). If it is overwritten it should return a nice name for the functor.

Overrides:

toString in class java.lang.Object

Returns:

a nice name for the functor.