formal language derivations

Logic/Language/Impl/M.hs

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+module Logic.Language.Impl.M where
+
+import Logic.Language (Language(..), ConcatShowList(..))
+import Logic.Language.Derivation (Derivation(..))
+
+-- The language M
+-- (from "Gödel, Escher, Bach: An Eternal Golden Braid" by Douglas Hofstadter)
+data AlphaM
+  = M
+  | I
+  | U
+  deriving (Eq, Show)
+
+type StringM = [AlphaM]
+
+instance Language AlphaM where
+  isWellFormed (M:_) = True
+  isWellFormed _ = False
+
+  axiom0 = [[M, I]]
+  infer1 =
+    [ mRule1
+    , mRule2
+    , mRule3
+    , mRule4
+    ]
+
+-- RULE I: If you possess a string whose last letter is I, you can add on a U at the end.
+mRule1 :: StringM -> [StringM]
+mRule1 [I] = [[I, U]]
+mRule1 (x:xs) = (x:) <$> mRule1 xs
+mRule1 _ = []
+
+-- RULE II: Suppose you have Mx. Then you may add Mxx to your collection.
+mRule2 :: StringM -> [StringM]
+mRule2 string@(M:xs) = [string ++ xs]
+mRule2 _ = []
+
+-- RULE III: If III occurs in one of the strings in your collection, you may
+-- make a new string with U in place of III.
+mRule3 :: StringM -> [StringM]
+mRule3 string@(M:xs) = (M:) <$> aux xs
+  where
+    aux (x@I:xs@(I:I:xs')) = (U:xs'):((x:) <$> aux xs)
+    aux (x:xs) = (x:) <$> aux xs
+    aux _  = []
+mRule3 _ = []
+
+-- RULE IV: If UU occurs inside one of your strings, you can drop it.
+mRule4 :: StringM -> [StringM]
+mRule4 string@(M:xs) = (M:) <$> aux xs
+  where
+    aux (x@U:xs@(U:xs')) = xs':((x:) <$> aux xs)
+    aux (x:xs) = (x:) <$> aux xs
+    aux _ = []
+mRule4 _ = []
+
+{-
+ghci> map ConcatShowList infer0 :: [ConcatShowList AlphaM]
+[MI]
+ghci> map ConcatShowList $ concat $ map ($ [M, I, I, I, I, U, U, I]) infer1
+[MIIIIUUIU,MIIIIUUIIIIIUUI,MUIUUI,MIUUUI,MIIIII]
+-}
+
+deriveMIIUII :: Derivation AlphaM
+deriveMIIUII =
+  Infer1 3 0 $
+  Infer1 2 2 $
+  Infer1 0 0 $
+  Infer1 3 0 $
+  Infer1 3 0 $
+  Infer1 2 2 $
+  Infer1 1 0 $
+  Infer1 2 5 $
+  Infer1 0 0 $
+  Infer1 1 0 $
+  Infer1 1 0 $
+  Infer1 1 0 $
+  Axiom0 0
+
+{-
+ghci> import Logic.Language.Derivation (resolveDerivation)
+ghci> resolveDerivation deriveMIIUII
+Right [M,I,I,U,I,I]
+-}