> {-# OPTIONS_HADDOCK show-extensions #-}
> {-|
> Module    : LTK.Decide.GLT
> Copyright : (c) 2021 Dakotah Lambert
> License   : MIT

> This module implements an algorithm to decide whether a given FSA
> is generalized locally testable in the sense of Brzozowski and
> Fich (1984):
> https://doi.org/10.1016/0012-365X(84)90045-1
>
> @since 1.0
> -}
> module LTK.Decide.GLT (isGLT, isGLTM) where

> import qualified Data.Set as Set

> import LTK.FSA
> import LTK.Algebra

> -- |True iff the automaton recognizes a generalized locally-testable
> -- stringset.
> isGLT :: (Ord n, Ord e) => FSA n e -> Bool
> isGLT :: FSA n e -> Bool
isGLT = FSA ([Maybe n], [Symbol e]) e -> Bool
forall n e. (Ord n, Ord e) => FSA (n, [Symbol e]) e -> Bool
isGLTM (FSA ([Maybe n], [Symbol e]) e -> Bool)
-> (FSA n e -> FSA ([Maybe n], [Symbol e]) e) -> FSA n e -> Bool
forall b c a. (b -> c) -> (a -> b) -> a -> c
. FSA n e -> FSA ([Maybe n], [Symbol e]) e
forall e n.
(Ord e, Ord n) =>
FSA n e -> FSA ([Maybe n], [Symbol e]) e
syntacticMonoid

> -- |True iff the monoid satisfies the generalized local testabiltiy
> -- condition.
> isGLTM :: (Ord n, Ord e) => FSA (n, [Symbol e]) e -> Bool
> isGLTM :: FSA (n, [Symbol e]) e -> Bool
isGLTM FSA (n, [Symbol e]) e
f = (Set (State (n, [Symbol e])) -> Bool)
-> [Set (State (n, [Symbol e]))] -> Bool
forall (t :: * -> *) a. Foldable t => (a -> Bool) -> t a -> Bool
all Set (State (n, [Symbol e])) -> Bool
commutativeBand ([Set (State (n, [Symbol e]))] -> Bool)
-> ([State (n, [Symbol e])] -> [Set (State (n, [Symbol e]))])
-> [State (n, [Symbol e])]
-> Bool
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (State (n, [Symbol e]) -> Set (State (n, [Symbol e])))
-> [State (n, [Symbol e])] -> [Set (State (n, [Symbol e]))]
forall a b. (a -> b) -> [a] -> [b]
map (FSA (n, [Symbol e]) e
-> State (n, [Symbol e]) -> Set (State (n, [Symbol e]))
forall n e. (Ord n, Ord e) => FSA (S n e) e -> T n e -> Set (T n e)
emee FSA (n, [Symbol e]) e
f) ([State (n, [Symbol e])] -> Bool)
-> [State (n, [Symbol e])] -> Bool
forall a b. (a -> b) -> a -> b
$ Set (State (n, [Symbol e])) -> [State (n, [Symbol e])]
forall a. Set a -> [a]
Set.toList Set (State (n, [Symbol e]))
i
>     where i :: Set (State (n, [Symbol e]))
i = FSA (n, [Symbol e]) e -> Set (State (n, [Symbol e]))
forall n e. (Ord n, Ord e) => FSA (n, [Symbol e]) e -> Set (T n e)
idempotents FSA (n, [Symbol e]) e
f
>           commutativeBand :: Set (State (n, [Symbol e])) -> Bool
commutativeBand = (Set (State (n, [Symbol e])) -> Bool)
-> (Set (State (n, [Symbol e])) -> Bool)
-> Set (State (n, [Symbol e]))
-> Bool
forall a. (a -> Bool) -> (a -> Bool) -> a -> Bool
both (FSA (n, [Symbol e]) e -> Set (State (n, [Symbol e])) -> Bool
forall n e.
(Ord n, Ord e) =>
FSA (n, [Symbol e]) e -> Set (State (n, [Symbol e])) -> Bool
isCommutative FSA (n, [Symbol e]) e
f) (Set (State (n, [Symbol e])) -> Set (State (n, [Symbol e])) -> Bool
forall c a. (Container c a, Eq a) => c -> c -> Bool
isSubsetOf Set (State (n, [Symbol e]))
i)