Safe Haskell	None
Language	Haskell2010

SubHask.Algebra.Logic

Synopsis

Documentation

class (Ord r, Ring r) => OrdRing_ r Source

Instances

(Ord r, Ring r) => OrdRing_ r Source

type Goedel = Goedel_ Rational Source

The Goedel fuzzy logic is one of the simpler fuzzy logics. In particular, it is an example of a Heyting algebra that is not also a Boolean algebra.

See the standford encyclopedia of logic

newtype Goedel_ r Source

Constructors

Goedel_ r

Instances

Read r0 => Read (Goedel_ r) Source
Show r0 => Show (Goedel_ r) Source
(OrdRing_ r, Arbitrary r) => Arbitrary (Goedel_ r) Source
NFData r0 => NFData (Goedel_ r) Source
IsMutable r0 => IsMutable (Goedel_ r) Source
OrdRing_ r => Heyting (Goedel_ r) Source
OrdRing_ r => Bounded (Goedel_ r) Source
OrdRing_ r => Ord_ (Goedel_ r) Source
OrdRing_ r => Lattice_ (Goedel_ r) Source
OrdRing_ r => MinBound_ (Goedel_ r) Source
OrdRing_ r => POrd_ (Goedel_ r) Source
Eq_ r0 => Eq_ (Goedel_ r) Source
data Mutable m (Goedel_ r0) = Mutable_Goedel_ (Mutable m r) Source
type Elem (Goedel_ r0) = Elem r0 Source
type Elem (Goedel_ r0) = Elem r0 Source
type Elem (Goedel_ r0) = Elem r0 Source
type Elem (Goedel_ r0) = Elem r0 Source
type Scalar (Goedel_ r0) = Scalar r0 Source
type Scalar (Goedel_ r0) = Scalar r0 Source
type Scalar (Goedel_ r0) = Scalar r0 Source
type Scalar (Goedel_ r0) = Scalar r0 Source
type Actor (Goedel_ r0) = Actor r0 Source
type Actor (Goedel_ r0) = Actor r0 Source
type Actor (Goedel_ r0) = Actor r0 Source
type Actor (Goedel_ r0) = Actor r0 Source
type Logic (Goedel_ r0) = Logic r0 Source
type Logic (Goedel_ r0) = Logic r0 Source
type Logic (Goedel_ r0) = Logic r0 Source
type Logic (Goedel_ r0) = Logic r0 Source

data H3 Source

H3 is the smallest Heyting algebra that is not also a boolean algebra. In addition to true and false, there is a value to represent whether something's truth is unknown. AFAIK it has no real applications.

See wikipedia

Constructors

HTrue
HFalse
HUnknown

Instances

Read H3 Source
Show H3 Source
Arbitrary H3 Source
NFData H3 Source
Heyting H3 Source
Bounded H3 Source
Ord_ H3 Source
Lattice_ H3 Source
MinBound_ H3 Source
POrd_ H3 Source
Eq_ H3 Source
type Logic H3 = Bool Source

data K3 Source

K3 stands for Kleene's 3-valued logic. In addition to true and false, there is a value to represent whether something's truth is unknown. K3 is an example of a logic that is neither Boolean nor Heyting.

See wikipedia.

FIXME: We need a way to represent implication and negation for logics outside of the Lattice hierarchy.

Constructors

KTrue
KFalse
KUnknown

Instances

Read K3 Source
Show K3 Source
Arbitrary K3 Source
NFData K3 Source
Bounded K3 Source
Ord_ K3 Source
Lattice_ K3 Source
MinBound_ K3 Source
POrd_ K3 Source
Eq_ K3 Source
type Logic K3 = Bool Source

newtype Boolean2Ring b Source

A Boolean algebra is a special type of Ring. Their applications (set-like operations) tend to be very different than Rings, so it makes sense for the class hierarchies to be completely unrelated. The Boolean2Ring type, however, provides the correct transformation.

Constructors

Boolean2Ring b

Instances

Read b0 => Read (Boolean2Ring b) Source
Show b0 => Show (Boolean2Ring b) Source
Arbitrary b0 => Arbitrary (Boolean2Ring b) Source
NFData b0 => NFData (Boolean2Ring b) Source
IsMutable b0 => IsMutable (Boolean2Ring b) Source
(IsMutable b, Boolean b, ValidLogic b) => Ring (Boolean2Ring b) Source
(IsMutable b, Boolean b, ValidLogic b) => Rig (Boolean2Ring b) Source
(IsMutable b, Boolean b, ValidLogic b) => Rg (Boolean2Ring b) Source
(IsMutable b, Boolean b, ValidLogic b) => Abelian (Boolean2Ring b) Source
(IsMutable b, Boolean b, ValidLogic b) => Group (Boolean2Ring b) Source
(IsMutable b, Boolean b, ValidLogic b) => Cancellative (Boolean2Ring b) Source
(IsMutable b, Boolean b, ValidLogic b) => Monoid (Boolean2Ring b) Source
(IsMutable b, Boolean b, ValidLogic b) => Semigroup (Boolean2Ring b) Source
(Boolean b0, Complemented b0, Heyting b0) => Boolean (Boolean2Ring b) Source
(Heyting b0, Bounded b0) => Heyting (Boolean2Ring b) Source
(Complemented b0, Bounded b0) => Complemented (Boolean2Ring b) Source
(Bounded b0, Lattice_ b0, MinBound_ b0) => Bounded (Boolean2Ring b) Source
(Lattice_ b0, POrd_ b0) => Lattice_ (Boolean2Ring b) Source
(MinBound_ b0, POrd_ b0) => MinBound_ (Boolean2Ring b) Source
(POrd_ b0, Eq_ b0) => POrd_ (Boolean2Ring b) Source
Eq_ b0 => Eq_ (Boolean2Ring b) Source
data Mutable m (Boolean2Ring b0) = Mutable_Boolean2Ring (Mutable m b) Source
type Elem (Boolean2Ring b0) = Elem b0 Source
type Elem (Boolean2Ring b0) = Elem b0 Source
type Elem (Boolean2Ring b0) = Elem b0 Source
type Elem (Boolean2Ring b0) = Elem b0 Source
type Elem (Boolean2Ring b0) = Elem b0 Source
type Elem (Boolean2Ring b0) = Elem b0 Source
type Elem (Boolean2Ring b0) = Elem b0 Source
type Elem (Boolean2Ring b0) = Elem b0 Source
type Elem (Boolean2Ring b0) = Elem b0 Source
type Elem (Boolean2Ring b0) = Elem b0 Source
type Elem (Boolean2Ring b0) = Elem b0 Source
type Elem (Boolean2Ring b0) = Elem b0 Source
type Scalar (Boolean2Ring b0) = Scalar b0 Source
type Scalar (Boolean2Ring b0) = Scalar b0 Source
type Scalar (Boolean2Ring b0) = Scalar b0 Source
type Scalar (Boolean2Ring b0) = Scalar b0 Source
type Scalar (Boolean2Ring b0) = Scalar b0 Source
type Scalar (Boolean2Ring b0) = Scalar b0 Source
type Scalar (Boolean2Ring b0) = Scalar b0 Source
type Scalar (Boolean2Ring b0) = Scalar b0 Source
type Scalar (Boolean2Ring b0) = Scalar b0 Source
type Scalar (Boolean2Ring b0) = Scalar b0 Source
type Scalar (Boolean2Ring b0) = Scalar b0 Source
type Scalar (Boolean2Ring b0) = Scalar b0 Source
type Actor (Boolean2Ring b0) = Actor b0 Source
type Actor (Boolean2Ring b0) = Actor b0 Source
type Actor (Boolean2Ring b0) = Actor b0 Source
type Actor (Boolean2Ring b0) = Actor b0 Source
type Actor (Boolean2Ring b0) = Actor b0 Source
type Actor (Boolean2Ring b0) = Actor b0 Source
type Actor (Boolean2Ring b0) = Actor b0 Source
type Actor (Boolean2Ring b0) = Actor b0 Source
type Actor (Boolean2Ring b0) = Actor b0 Source
type Actor (Boolean2Ring b0) = Actor b0 Source
type Actor (Boolean2Ring b0) = Actor b0 Source
type Actor (Boolean2Ring b0) = Actor b0 Source
type Logic (Boolean2Ring b0) = Logic b0 Source
type Logic (Boolean2Ring b0) = Logic b0 Source
type Logic (Boolean2Ring b0) = Logic b0 Source
type Logic (Boolean2Ring b0) = Logic b0 Source
type Logic (Boolean2Ring b0) = Logic b0 Source
type Logic (Boolean2Ring b0) = Logic b0 Source
type Logic (Boolean2Ring b0) = Logic b0 Source
type Logic (Boolean2Ring b0) = Logic b0 Source
type Logic (Boolean2Ring b0) = Logic b0 Source
type Logic (Boolean2Ring b0) = Logic b0 Source
type Logic (Boolean2Ring b0) = Logic b0 Source
type Logic (Boolean2Ring b0) = Logic b0 Source

mkBoolean2Ring :: Boolean b => b -> Boolean2Ring b Source