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DeMorgan.scala
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DeMorgan.scala
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/*
* Copyright (c) 2015 Typelevel
*
* Permission is hereby granted, free of charge, to any person obtaining a copy of
* this software and associated documentation files (the "Software"), to deal in
* the Software without restriction, including without limitation the rights to
* use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of
* the Software, and to permit persons to whom the Software is furnished to do so,
* subject to the following conditions:
*
* The above copyright notice and this permission notice shall be included in all
* copies or substantial portions of the Software.
*
* THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
* IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS
* FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR
* COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER
* IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN
* CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
*/
package algebra
package lattice
import scala.{specialized => sp}
/**
* De Morgan algebras are bounded lattices that are also equipped with
* a De Morgan involution.
*
* De Morgan involution obeys the following laws:
*
* - ¬¬a = a
* - ¬(x∧y) = ¬x∨¬y
*
* However, in De Morgan algebras this involution does not necessarily
* provide the law of the excluded middle. This means that there is no
* guarantee that (a ∨ ¬a) = 1. De Morgan algebra do not not necessarily
* provide the law of non contradiction either. This means that there is
* no guarantee that (a ∧ ¬a) = 0.
*
* De Morgan algebras are useful to model fuzzy logic. For a model of
* classical logic, see the boolean algebra type class implemented as
* [[Bool]].
*/
trait DeMorgan[@sp(Int, Long) A] extends Any with Logic[A] { self =>
def meet(a: A, b: A): A = and(a, b)
def join(a: A, b: A): A = or(a, b)
def imp(a: A, b: A): A = or(not(a), b)
}
trait DeMorganFunctions[H[A] <: DeMorgan[A]]
extends BoundedMeetSemilatticeFunctions[H]
with BoundedJoinSemilatticeFunctions[H]
with LogicFunctions[H]
object DeMorgan extends DeMorganFunctions[DeMorgan] {
/**
* Access an implicit `DeMorgan[A]`.
*/
@inline final def apply[@sp(Int, Long) A](implicit ev: DeMorgan[A]): DeMorgan[A] = ev
/**
* Turn a [[Bool]] into a `DeMorgan`
* Used for binary compatibility.
*/
final def fromBool[@sp(Int, Long) A](bool: Bool[A]): DeMorgan[A] =
new DeMorgan[A] {
def and(a: A, b: A): A = bool.and(a, b)
def or(a: A, b: A): A = bool.or(a, b)
def not(a: A): A = bool.complement(a)
def one: A = bool.one
def zero: A = bool.zero
}
}