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

> This module implements an algorithm to decide whether a given FSA
> is in CB, the subclass of PT where additionally all elements of
> the syntactic monoid are idempotent. This is additionally a subclass
> of LT, because every submonoid of a semilattice remains a semilattice.
>
> @since 1.0
> -}
> module LTK.Decide.CB (isCB, isCBM) where

> import qualified Data.Set as Set

> import LTK.FSA
> import LTK.Algebra

> -- |True iff the automaton recognizes a semilattice stringset.
> isCB :: (Ord n, Ord e) => FSA n e -> Bool
> isCB :: FSA n e -> Bool
isCB = SynMon n e -> Bool
forall n e. (Ord n, Ord e) => SynMon n e -> Bool
isCBM (SynMon n e -> Bool) -> (FSA n e -> SynMon n e) -> FSA n e -> Bool
forall b c a. (b -> c) -> (a -> b) -> a -> c
. FSA n e -> SynMon n e
forall e n.
(Ord e, Ord n) =>
FSA n e -> FSA ([Maybe n], [Symbol e]) e
syntacticMonoid

> -- |True iff the monoid is a semilattice.
> isCBM :: (Ord n, Ord e) => SynMon n e -> Bool
> isCBM :: SynMon n e -> Bool
isCBM SynMon n e
m = (SynMon n e -> Set (Set (State ([Maybe n], [Symbol e]))))
-> SynMon n e -> Bool
forall n e. (FSA n e -> Set (Set (State n))) -> FSA n e -> Bool
trivialUnder SynMon n e -> Set (Set (State ([Maybe n], [Symbol e])))
forall e n.
(Ord e, Ord n) =>
FSA ([Maybe n], [Symbol e]) e
-> Set (Set (State ([Maybe n], [Symbol e])))
jEquivalence SynMon n e
m Bool -> Bool -> Bool
&& (Set (State ([Maybe n], [Symbol e]))
i Set (State ([Maybe n], [Symbol e]))
-> Set (State ([Maybe n], [Symbol e])) -> Bool
forall a. Eq a => a -> a -> Bool
== SynMon n e -> Set (State ([Maybe n], [Symbol e]))
forall e n. (Ord e, Ord n) => FSA n e -> Set (State n)
states SynMon n e
m)
>     where i :: Set (State ([Maybe n], [Symbol e]))
i = Set (State ([Maybe n], [Symbol e]))
-> Set (State ([Maybe n], [Symbol e]))
-> Set (State ([Maybe n], [Symbol e]))
forall a. Ord a => Set a -> Set a -> Set a
Set.union (SynMon n e -> Set (State ([Maybe n], [Symbol e]))
forall n e. FSA n e -> Set (State n)
initials SynMon n e
m) (SynMon n e -> Set (State ([Maybe n], [Symbol e]))
forall n e. (Ord n, Ord e) => FSA (n, [Symbol e]) e -> Set (T n e)
idempotents SynMon n e
m)