> {-# OPTIONS_HADDOCK show-extensions #-}
> {-|
> Module    : LTK.Decide.Definite
> Copyright : (c) 2021-2024 Dakotah Lambert
> License   : MIT

> This module implements an algorithm to decide whether a given FSA
> is Definite (Def) or Reverse Definite (RDef) based on the classic
> semigroup characterizations summarized by Brzozowski and Fich in
> their 1984 work "On Generalized Locally Testable Languages".
>
> @since 1.0
> -}
> module LTK.Decide.Definite
>     ( -- *Plain
>       isDef
>     , isDefM
>     , isDefs
>     , isRDef
>     , isRDefM
>     , isRDefs
>       -- *Tier-Based
>     , isTDef
>     , isTDefM
>     , isTDefs
>     , isTRDef
>     , isTRDefM
>     , isTRDefs
>     ) where

> import Data.Representation.FiniteSemigroup

> import LTK.FSA
> import LTK.Algebra(SynMon)
> import LTK.Tiers (project)

> -- |True iff the automaton recognizes a definite stringset,
> -- characterized by a set of permitted suffixes.
> isDef :: (Ord n, Ord e) => FSA n e -> Bool
> isDef :: forall n e. (Ord n, Ord e) => FSA n e -> Bool
isDef = GeneratedAction -> Bool
forall s. FiniteSemigroupRep s => s -> Bool
isDefs (GeneratedAction -> Bool)
-> (FSA n e -> GeneratedAction) -> FSA n e -> Bool
forall b c a. (b -> c) -> (a -> b) -> a -> c
. FSA n e -> GeneratedAction
forall n e. (Ord n, Ord e) => FSA n e -> GeneratedAction
syntacticSemigroup

> -- |True iff \(Se=e\) for idempotents \(e\).
> isDefM :: (Ord n, Ord e) => SynMon n e -> Bool
> isDefM :: forall n e. (Ord n, Ord e) => SynMon n e -> Bool
isDefM = FSA ([Maybe n], [Symbol e]) e -> Bool
forall n e. (Ord n, Ord e) => FSA n e -> Bool
isDef

> -- |True iff \(Se=e\) for idempotents \(e\).
> --
> -- @since 1.2
> isDefs :: FiniteSemigroupRep s => s -> Bool
> isDefs :: forall s. FiniteSemigroupRep s => s -> Bool
isDefs = FSMult -> Bool
forall s. FiniteSemigroupRep s => s -> Bool
isRDefs (FSMult -> Bool) -> (s -> FSMult) -> s -> Bool
forall b c a. (b -> c) -> (a -> b) -> a -> c
. s -> FSMult
forall s. FiniteSemigroupRep s => s -> FSMult
dual

> -- |True iff the automaton recognizes a reverse definite stringset,
> -- characterized by a set of permitted prefixes.
> isRDef :: (Ord n, Ord e) => FSA n e -> Bool
> isRDef :: forall n e. (Ord n, Ord e) => FSA n e -> Bool
isRDef = GeneratedAction -> Bool
forall s. FiniteSemigroupRep s => s -> Bool
isRDefs (GeneratedAction -> Bool)
-> (FSA n e -> GeneratedAction) -> FSA n e -> Bool
forall b c a. (b -> c) -> (a -> b) -> a -> c
. FSA n e -> GeneratedAction
forall n e. (Ord n, Ord e) => FSA n e -> GeneratedAction
syntacticSemigroup

> -- |True iff \(eS=e\) for idempotents \(e\).
> isRDefM :: (Ord n, Ord e) => SynMon n e -> Bool
> isRDefM :: forall n e. (Ord n, Ord e) => SynMon n e -> Bool
isRDefM = FSA ([Maybe n], [Symbol e]) e -> Bool
forall n e. (Ord n, Ord e) => FSA n e -> Bool
isRDef

> -- |True iff \(eS=e\) for idempotents \(e\).
> --
> -- @since 1.2
> isRDefs :: FiniteSemigroupRep s => s -> Bool
> isRDefs :: forall s. FiniteSemigroupRep s => s -> Bool
isRDefs = (s -> Bool) -> (s -> Bool) -> s -> Bool
forall a. (a -> Bool) -> (a -> Bool) -> a -> Bool
both s -> Bool
forall s. FiniteSemigroupRep s => s -> Bool
isRTrivial ((FSMult -> Bool) -> s -> Bool
forall s. FiniteSemigroupRep s => (FSMult -> Bool) -> s -> Bool
locally FSMult -> Bool
forall s. FiniteSemigroupRep s => s -> Bool
isTrivial)

> -- |Definite on some tier.
> isTDef :: (Ord n, Ord e) => FSA n e -> Bool
> isTDef :: forall n e. (Ord n, Ord e) => FSA n e -> Bool
isTDef = FSA n e -> Bool
forall n e. (Ord n, Ord e) => FSA n e -> Bool
isDef (FSA n e -> Bool) -> (FSA n e -> FSA n e) -> FSA n e -> Bool
forall b c a. (b -> c) -> (a -> b) -> a -> c
. FSA n e -> FSA n e
forall n e. (Ord n, Ord e) => FSA n e -> FSA n e
project

> -- |Definite on the projected subsemigroup.
> isTDefM :: (Ord n, Ord e) => SynMon n e -> Bool
> isTDefM :: forall n e. (Ord n, Ord e) => SynMon n e -> Bool
isTDefM = SynMon n e -> Bool
forall n e. (Ord n, Ord e) => SynMon n e -> Bool
isDefM (SynMon n e -> Bool)
-> (SynMon n e -> SynMon n e) -> SynMon n e -> Bool
forall b c a. (b -> c) -> (a -> b) -> a -> c
. SynMon n e -> SynMon n e
forall n e. (Ord n, Ord e) => FSA n e -> FSA n e
project

> -- |Definite on the projected subsemigroup.
> --
> -- @since 1.2
> isTDefs :: FiniteSemigroupRep s => s -> Bool
> isTDefs :: forall s. FiniteSemigroupRep s => s -> Bool
isTDefs = FSMult -> Bool
forall s. FiniteSemigroupRep s => s -> Bool
isDefs (FSMult -> Bool) -> (s -> FSMult) -> s -> Bool
forall b c a. (b -> c) -> (a -> b) -> a -> c
. s -> FSMult
forall s. FiniteSemigroupRep s => s -> FSMult
projectedSubsemigroup

> -- |Reverse definite on some tier.
> isTRDef :: (Ord n, Ord e) => FSA n e -> Bool
> isTRDef :: forall n e. (Ord n, Ord e) => FSA n e -> Bool
isTRDef = FSA n e -> Bool
forall n e. (Ord n, Ord e) => FSA n e -> Bool
isRDef (FSA n e -> Bool) -> (FSA n e -> FSA n e) -> FSA n e -> Bool
forall b c a. (b -> c) -> (a -> b) -> a -> c
. FSA n e -> FSA n e
forall n e. (Ord n, Ord e) => FSA n e -> FSA n e
project

> -- |Reverse definite on the projected subsemigroup.
> isTRDefM :: (Ord n, Ord e) => SynMon n e -> Bool
> isTRDefM :: forall n e. (Ord n, Ord e) => SynMon n e -> Bool
isTRDefM = SynMon n e -> Bool
forall n e. (Ord n, Ord e) => SynMon n e -> Bool
isRDefM (SynMon n e -> Bool)
-> (SynMon n e -> SynMon n e) -> SynMon n e -> Bool
forall b c a. (b -> c) -> (a -> b) -> a -> c
. SynMon n e -> SynMon n e
forall n e. (Ord n, Ord e) => FSA n e -> FSA n e
project

> -- |Reverse definite on the projected subsemigroup.
> --
> -- @since 1.2
> isTRDefs :: FiniteSemigroupRep s => s -> Bool
> isTRDefs :: forall s. FiniteSemigroupRep s => s -> Bool
isTRDefs = FSMult -> Bool
forall s. FiniteSemigroupRep s => s -> Bool
isRDefs (FSMult -> Bool) -> (s -> FSMult) -> s -> Bool
forall b c a. (b -> c) -> (a -> b) -> a -> c
. s -> FSMult
forall s. FiniteSemigroupRep s => s -> FSMult
projectedSubsemigroup