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-- MIT License, Copyright (c) 2024 Marvin Borner
module Language.Mili.Analyzer
( linearity
) where
import Data.HashMap.Strict ( HashMap )
import qualified Data.HashMap.Strict as M
import Data.List ( intercalate
, isSuffixOf
)
import Data.Mili ( Nat(..)
, Term(..)
)
import Debug.Trace
-- | A trace specifies the exact path to any subterm
type Trace = [Breadcrumb]
data Breadcrumb = AbsD -- | Down into abstraction
| AppL -- | Left side of application
| AppR -- | Right side of application
| NumS -- | Successor of number
| RecT1 -- | Term 1 of rec
| RecT2 -- | Term 2 of rec
| RecU -- | End of rec
| RecV -- | Function 1 of rec
| RecW -- | Function 2 of rec
| Root -- | Root
deriving (Eq, Show)
-- | Map abstractions to a hashmap such that de Bruijn levels correspond to traces
-- | of abstractions at that level
traceAbs :: Term -> HashMap Int [Trace]
traceAbs = go 0 [Root]
where
go n t (Abs _ m) =
M.unionWith (++) (go (n + 1) (AbsD : t) m) (M.singleton n [t])
go n t (App a b) = M.unionWith (++) (go n (AppL : t) a) (go n (AppR : t) b)
go _ _ (Lvl _ ) = M.empty
go _ _ (Num Z ) = M.empty
go n t (Num (S i) ) = go n (NumS : t) i
go n t (Rec (t1, t2) u v w) = foldl1
(M.unionWith (++))
[ go n (RecT1 : t) t1
, go n (RecT2 : t) t2
, go n (RecU : t) u
, go n (RecV : t) v
, go n (RecW : t) w
]
-- TODO: Merge these two v^
-- | Map de Bruijn levels to a hashmap such that de Bruijn levels correspond to
-- | traces of de Bruijn levels at that level
traceLvl :: Term -> HashMap Int [Trace]
traceLvl = go [Root]
where
go t (Abs _ m ) = go (AbsD : t) m
go t (App a b) = M.unionWith (++) (go (AppL : t) a) (go (AppR : t) b)
go t (Lvl l ) = M.singleton l [t]
go _ (Num Z ) = M.empty
go t (Num (S i) ) = go (NumS : t) i
go t (Rec (t1, t2) u v w) = foldl1
(M.unionWith (++))
[ go (RecT1 : t) t1
, go (RecT2 : t) t2
, go (RecU : t) u
, go (RecV : t) v
, go (RecW : t) w
]
-- | Unify two two mapped traces to find levels at which the traces are no suffixes
-- | Level traces not being a suffix of abstraction traces implies nonlinearity
-- TODO: Proof?
unifyTraces :: HashMap Int [Trace] -> HashMap Int [Trace] -> HashMap Int [Bool]
unifyTraces traceA traceB = M.fromList $ map zipMap allKeys
where
allKeys = M.keys (traceA `M.union` traceB)
left key = M.lookupDefault [[]] key traceA
right key = M.lookupDefault [[]] key traceB
zipMap key
| length (left key) == length (right key)
= (key, zipWith isSuffixOf (left key) (right key))
| otherwise
= (key, [False])
linearityError
:: HashMap Int [Trace] -> HashMap Int [Trace] -> HashMap Int [Bool] -> String
linearityError absTrace lvlTrace = pretty . filter (any not . snd) . M.toList
where
pretty = intercalate "\n\n" . map
(\(depth, _) ->
"Linearity divergence at depth "
<> show depth
<> "\n\t"
<> show (M.lookupDefault [[]] depth absTrace)
<> "\nvs\n\t"
<> show (M.lookupDefault [[]] depth lvlTrace)
)
linearity :: Term -> Either String Term
linearity t =
let absTrace = traceAbs t
lvlTrace = traceLvl t
unified = unifyTraces absTrace lvlTrace
isLinear = all (all id) unified
in if isLinear
then Right t
else trace (show t <> "\n" <> show absTrace <> "\n" <> show lvlTrace)
(Left $ linearityError absTrace lvlTrace unified)
|
