cpsa-2.2.13: src/CPSA/DiffieHellman/Algebra.hs
-- Experimental Diffie-Hellman Algebra implementation
-- This module is an experimental version of Diffie-Hellman in which
-- exponents form an Abelian group. There are many issues to be
-- resolved before this version of Diffie-Hellman can be used.
-- The module implements a many-sorted algebra, but is used as an
-- order-sorted algebra. It exports a name, and the origin used to
-- generate variables.
-- Copyright (c) 2009 The MITRE Corporation
--
-- This program is free software: you can redistribute it and/or
-- modify it under the terms of the BSD License as published by the
-- University of California.
{-# LANGUAGE MultiParamTypeClasses, CPP #-}
module CPSA.DiffieHellman.Algebra (name, origin) where
import Control.Monad (foldM)
import qualified Data.List as L
import qualified Data.Set as S
import Data.Set (Set)
import qualified Data.Map as M
import Data.Map (Map)
import Data.Char (isDigit)
import qualified CPSA.Lib.CPSA as C
import CPSA.Lib.CPSA (SExpr(..), Pos, annotation)
import qualified CPSA.DiffieHellman.IntLinEq as I
-- Preserve the ability to use GHC 6.x
foldrWithKey :: (k -> a -> b -> b) -> b -> Map k a -> b
#if defined __GLASGOW_HASKELL__ && __GLASGOW_HASKELL__ < 700
foldrWithKey = M.foldWithKey
#else
foldrWithKey = M.foldrWithKey
#endif
{-- Debugging support
import System.IO.Unsafe
z :: Show a => a -> b -> b
z x y = unsafePerformIO (print x >> return y)
zz :: Show a => a -> a
zz x = z x x
zn :: Show a => a -> Maybe b -> Maybe b
zn x Nothing = z x Nothing
zn _ y = y
zf :: Show a => a -> Bool -> Bool
zf x False = z x False
zf _ y = y
zt :: Show a => a -> Bool -> Bool
zt x True = z x True
zt _ y = y
--}
{-- Export iUnify and iMatch for GHCi for debugging
iUnify :: String -> String -> String -> Maybe Subst
iUnify vars t t' =
iRun unify emptySubst vars t t'
iMatch :: String -> String -> String -> Maybe Env
iMatch vars t t' =
iRun match emptyEnv vars t t'
iRun :: (Term -> Term -> (Gen, a) -> Maybe (Gen, a)) -> a ->
String -> String -> String -> Maybe a
iRun f mt vars t t' =
do
vars <- C.load "" vars
[t] <- C.load "" t
[t'] <- C.load "" t'
(gen, vars) <- loadVars origin vars
t <- loadTerm vars t
t' <- loadTerm vars t'
(_, a) <- f t t' (gen, mt)
return a
gRun :: Gen -> Term -> a -> a
gRun (Gen n) t a =
foldVars f a t
where
f a t =
case varId t of
Id (m, _) | m >= n -> error ("Bad gen " ++ show n)
_ -> a
gMatch :: Term -> Term -> GenEnv -> Maybe GenEnv
gMatch t t' r@(g, _) = gRun g t' (match t t' r)
gUnify :: Term -> Term -> GenSubst -> Maybe GenSubst
gUnify t t' r@(g, _) = gRun g (F Cat [t, t']) (unify t t' r)
--}
name :: String
name = "experimental-diffie-hellman"
-- An identifier
newtype Id = Id (Integer, String) deriving Show
-- The integer distinguishes an identifier, the string is for printing.
instance Eq Id where
(Id (x, _)) == (Id (x', _)) = x == x'
instance Ord Id where
compare (Id (x, _)) (Id (x', _)) = compare x x'
idName :: Id -> String
idName (Id (_, name)) = name
-- Counter used for generating fresh identifiers.
newtype Gen = Gen (Integer) deriving Show
origin :: Gen
origin = Gen (0)
freshId :: Gen -> String -> (Gen, Id)
freshId (Gen (i)) name = (Gen (i + 1), Id (i, name))
cloneId :: Gen -> Id -> (Gen, Id)
cloneId gen x = freshId gen (idName x)
mash :: Gen -> Gen -> Gen
mash (Gen i) (Gen j) = Gen (max i j)
-- A term in an Abelian group is a map from identifiers to non-zero integers.
type Group = Map Id Int
isGroupVar :: Group -> Bool
isGroupVar t =
M.size t == 1 && head (M.elems t) == 1
groupVar :: Id -> Term
groupVar x = G $ M.singleton x 1
invert :: Group -> Group
invert t = M.map negate t
expg :: Group -> Int -> Group
expg _ 0 = M.empty
expg t 1 = t
expg t n = M.map (* n) t
mul :: Group -> Group -> Group
mul t t' =
foldrWithKey f t' t -- Fold over the mappings in t
where
f x c t = -- Alter the mapping of
M.alter (g c) x t -- variable x in t
g c Nothing = -- Variable x not currently mapped
Just c -- so add a mapping
g c (Just c') -- Variable x maps to c'
| c + c' == 0 = Nothing -- Delete the mapping
| otherwise = Just $ c + c' -- Adjust the mapping
group :: [(Id, Int)] -> Group
group assocs =
foldr f M.empty assocs
where
f (x, c) t = mul (expg (M.singleton x 1) c) t
-- Function symbols--see foldVar to see the arity of each symbol.
data Symbol
= Text -- Text atom
| Data -- Another text-like atom
| Name -- Principal atom
| Skey -- Symmetric key atom
| Base -- Base of an exponentiated atom
| Ltk -- Long term shared symmetric key
| Akey -- Asymmetric key atom
| Invk -- Inverse of asymmetric key
| Pubk -- Public asymmetric key of a principal
| Genr -- The generator constant for the group
| Exp -- Exponentiation function symbol
| Cat -- Term concatenation
| Enc -- Encryption
deriving (Show, Eq, Ord, Enum, Bounded)
-- A Basic Crypto Algebra Term
data Term
= I !Id
| C !String
| F !Symbol ![Term]
| G !Group -- An exponent, an Abelian group
deriving Show
equalTerm :: Term -> Term -> Bool
equalTerm (I x) (I y) = x == y
equalTerm (C c) (C c') = c == c'
equalTerm (F Invk [F Invk [t]]) t' = equalTerm t t'
equalTerm t (F Invk [F Invk [t']]) = equalTerm t t'
equalTerm (F Exp [t0, G t1]) t' | M.null t1 = equalTerm t0 t'
equalTerm t (F Exp [t0, G t1]) | M.null t1 = equalTerm t t0
equalTerm (F Exp [F Exp [t, G t0], G t1]) t' =
equalTerm (F Exp [t, G (mul t0 t1)]) t'
equalTerm t (F Exp [F Exp [t', G t0], G t1]) =
equalTerm t (F Exp [t', G (mul t0 t1)])
equalTerm (F s u) (F s' u') =
s == s' && equalTermLists u u'
equalTerm (G t) (G t') = t == t'
equalTerm _ _ = False
equalTermLists :: [Term] -> [Term] -> Bool
equalTermLists [] [] = True
equalTermLists (t : u) (t' : u') =
equalTerm t t' && equalTermLists u u'
equalTermLists _ _ = False
instance Eq Term where
(==) = equalTerm
-- Term comparison respecting the axiom
compareTerm :: Term -> Term -> Ordering
compareTerm (I x) (I y) = compare x y
compareTerm (C c) (C c') = compare c c'
compareTerm (F Invk [F Invk [t]]) t' = compareTerm t t'
compareTerm t (F Invk [F Invk [t']]) = compareTerm t t'
compareTerm (F Exp [t0, G t1]) t' | M.null t1 = compareTerm t0 t'
compareTerm t (F Exp [t0, G t1]) | M.null t1 = compareTerm t t0
compareTerm (F Exp [F Exp [t, G t0], G t1]) t' =
compareTerm (F Exp [t, G (mul t0 t1)]) t'
compareTerm t (F Exp [F Exp [t', G t0], G t1]) =
compareTerm t (F Exp [t', G (mul t0 t1)])
compareTerm (F s u) (F s' u') =
case compare s s' of
EQ -> compareTermLists u u'
o -> o
compareTerm (G t) (G t') = compare t t'
compareTerm (I _) (C _) = LT
compareTerm (C _) (I _) = GT
compareTerm (I _) (F _ _) = LT
compareTerm (F _ _) (I _) = GT
compareTerm (I _) (G _) = LT
compareTerm (G _) (I _) = GT
compareTerm (C _) (F _ _) = LT
compareTerm (F _ _) (C _) = GT
compareTerm (C _) (G _) = LT
compareTerm (G _) (C _) = GT
compareTerm (F _ _) (G _) = LT
compareTerm (G _) (F _ _) = GT
compareTermLists :: [Term] -> [Term] -> Ordering
compareTermLists [] [] = EQ
compareTermLists (t : u) (t' : u') =
case compareTerm t t' of
EQ -> compareTermLists u u'
o -> o
compareTermLists [] _ = LT
compareTermLists _ [] = GT
instance Ord Term where
compare = compareTerm
-- Basic terms are introduced by defining a function used to decide if
-- a term is well-formed. The context of an occurrence of an identifier
-- determines its sort. A term that contains just an identifier and its
-- sort information is called a variable. The sort of a variable is
-- one of mesg, text, data, name, skey, and akey.
-- Terms that represent variables.
isVar :: Term -> Bool
isVar (I _) = True -- Sort: mesg
isVar (F s [I _]) =
-- Sorts: text, data, name, skey, and akey
s == Text || s == Data || s == Name || s == Skey || s == Akey || s == Base
isVar (G t) = isGroupVar t
isVar _ = False
-- Extract the identifier from a variable
varId :: Term -> Id
varId (I x) = x
varId (F Text [I x]) = x
varId (F Data [I x]) = x
varId (F Name [I x]) = x
varId (F Skey [I x]) = x
varId (F Akey [I x]) = x
varId (F Base [I x]) = x
varId (G t) | isGroupVar t = head $ M.keys t
varId _ = error "Algebra.varId: term not a variable with its sort"
-- A list of terms are well-formed if each one has the correct
-- structure and every occurrence of an identifier in a term has the
-- same sort. Variable environments are used to check the sort
-- condition. It maps an identifier to a variable that contains the
-- identifier.
-- termsWellFormed u ensures all terms in u use each identifier at the
-- same sort, and makes sure every term has the correct structure.
termsWellFormed :: [Term] -> Bool
termsWellFormed u =
loop emptyVarEnv u
where
loop _ [] = True
loop env (t : u) =
case termWellFormed env t of
Nothing -> False
Just env' -> loop env' u
newtype VarEnv = VarEnv (Map Id Term) deriving Show
emptyVarEnv :: VarEnv
emptyVarEnv = VarEnv M.empty
-- Check the structure and sort condition.
termWellFormed :: VarEnv -> Term -> Maybe VarEnv
termWellFormed xts t@(I x) =
extendVarEnv xts x t -- Mesg variable
termWellFormed xts t@(F Text [I x]) =
extendVarEnv xts x t -- Text variable
termWellFormed xts t@(F Data [I x]) =
extendVarEnv xts x t -- Data variable
termWellFormed xts t@(F Name [I x]) =
extendVarEnv xts x t -- Name variable
termWellFormed xts t@(F Skey [I x]) =
extendVarEnv xts x t -- Symmetric key variable
termWellFormed xts (F Skey [F Ltk [I x, I y]]) =
-- Long term shared symmetric key
doubleTermWellFormed xts (F Name [I x]) (F Name [I y])
termWellFormed xts (F Akey [t]) = -- Asymmetric key terms
case t of
I x -> extendVarEnv xts x (F Akey [I x])
F Invk [I x] -> extendVarEnv xts x (F Akey [I x])
F Pubk [I x] -> extendVarEnv xts x (F Name [I x])
F Pubk [C _, I x] -> extendVarEnv xts x (F Name [I x])
F Invk [F Pubk [I x]] -> extendVarEnv xts x (F Name [I x])
F Invk [F Pubk [C _, I x]] -> extendVarEnv xts x (F Name [I x])
_ -> Nothing
termWellFormed xts (F Base [t]) =
baseVarEnv xts t
where
baseVarEnv xts t@(I x) =
extendVarEnv xts x (F Base [t])
baseVarEnv xts (F Genr []) =
Just xts
baseVarEnv xts (F Exp [t0, G t1]) =
do
xts <- baseVarEnv xts t0
termWellFormed xts (G t1)
baseVarEnv _ _ = Nothing
termWellFormed xts (G t) =
foldM expnVarEnv xts (M.keys t)
where
expnVarEnv xts x =
extendVarEnv xts x (groupVar x)
termWellFormed xts (C _) =
Just xts -- Tags
termWellFormed xts (F Cat [t0, t1]) =
doubleTermWellFormed xts t0 t1 -- Concatenation
termWellFormed xts (F Enc [t0, t1]) =
doubleTermWellFormed xts t0 t1 -- Encryption
termWellFormed _ _ = Nothing
-- Extend when sorts agree
extendVarEnv :: VarEnv -> Id -> Term -> Maybe VarEnv
extendVarEnv (VarEnv env) x t =
case M.lookup x env of
Nothing -> Just $ VarEnv $ M.insert x t env
Just t' -> if t == t' then Just (VarEnv env) else Nothing
doubleTermWellFormed :: VarEnv -> Term -> Term -> Maybe VarEnv
doubleTermWellFormed xts t0 t1 =
do
xts <- termWellFormed xts t0
termWellFormed xts t1
-- Is the sort of the term a base sort?
isAtom :: Term -> Bool
isAtom (I _) = False
isAtom (C _) = False
isAtom (F s _) =
s == Text || s == Data || s == Name || s == Skey || s == Akey
isAtom (G _) = True
-- Does a term occur in another term?
occursIn :: Term -> Term -> Bool
occursIn (G t) (G t') =
all f (M.assocs t)
where
f (x, n) =
let n' = M.findWithDefault 0 x t' in
n > 0 && n' >= n || n < 0 && n' <= n
occursIn t t' =
t == t' ||
case t' of
F _ u -> any (occursIn t) u
_ -> False
-- Fold f through a term applying it to each variable in the term.
foldVars :: (a -> Term -> a) -> a -> Term -> a
foldVars f acc t@(I _) = f acc t -- Mesg variable
foldVars f acc t@(F Text [I _]) = f acc t -- Text variable
foldVars f acc t@(F Data [I _]) = f acc t -- Data variable
foldVars f acc t@(F Name [I _]) = f acc t -- Name variable
foldVars f acc t@(F Skey [I _]) =
f acc t -- Symmetric key variable
foldVars f acc (F Skey [F Ltk [I x, I y]]) =
-- Long term shared symmetric key
f (f acc (F Name [I x])) (F Name [I y])
foldVars f acc t@(F Akey [I _]) = f acc t -- Asymmetric keys
foldVars f acc (F Akey [F Invk [I x]]) = f acc (F Akey [I x])
foldVars f acc (F Akey [F Pubk [I x]]) = f acc (F Name [I x])
foldVars f acc (F Akey [F Pubk [C _, I x]]) = f acc (F Name [I x])
foldVars f acc (F Akey [F Invk [F Pubk [I x]]]) = f acc (F Name [I x])
foldVars f acc (F Akey [F Invk [F Pubk [C _, I x]]]) = f acc (F Name [I x])
foldVars f acc (F Base [t]) =
baseAddVars acc t
where
baseAddVars acc t@(I _) =
f acc (F Base [t])
baseAddVars acc (F Genr []) =
acc
baseAddVars acc (F Exp [t0, G t1]) =
foldVars f (baseAddVars acc t0) (G t1)
baseAddVars _ _ = error "Algebra.foldVars: Bad term"
foldVars f acc (G t) =
foldl expnAddVars acc (M.keys t)
where
expnAddVars acc x =
f acc (groupVar x)
foldVars _ acc (C _) = acc -- Tags
foldVars f acc (F Cat [t0, t1]) = -- Concatenation
foldVars f (foldVars f acc t0) t1
foldVars f acc (F Enc [t0, t1]) = -- Encryption
foldVars f (foldVars f acc t0) t1
foldVars _ _ t = error $ "Algebra.foldVars: Bad term " ++ show t
-- Fold f through a term applying it to each term that is carried by the term.
foldCarriedTerms :: (a -> Term -> a) -> a -> Term -> a
foldCarriedTerms f acc t@(F Cat [t0, t1]) = -- Concatenation
foldCarriedTerms f (foldCarriedTerms f (f acc t) t0) t1
foldCarriedTerms f acc t@(F Enc [t0, _]) = -- Encryption
foldCarriedTerms f (f acc t) t0
foldCarriedTerms f acc t@(F Base [F Exp [_, t1]]) = -- Exponents
f (f acc t) t1
foldCarriedTerms f acc t = f acc t -- atoms and tags
-- Is a term carried by another term?
carriedBy :: Term -> Term -> Bool
carriedBy t t' =
t == t' ||
case t' of
F Cat [t0, t1] -> carriedBy t t0 || carriedBy t t1
F Enc [t0, _] -> carriedBy t t0
_ -> False
-- Is a term held by another term?
heldBy :: Term -> Term -> Bool
heldBy t t' =
t == t' ||
case t' of
F Cat [t0, t1] -> heldBy t t0 || heldBy t t1
F Enc [t0, _] -> heldBy t t0
F Base [F Exp [_, t1]] -> t == t1 -- This can't be right.
_ -> False
-- The key used to decrypt an encrypted term, otherwise Nothing.
decryptionKey :: Term -> Maybe Term
decryptionKey (F Enc [_, t]) = Just (inv t)
decryptionKey _ = Nothing
buildable :: Set Term -> Set Term -> Term -> Bool
buildable knowns unguessable term =
ba term
where
ba (I _) = True -- A mesg sorted variable is always buildable
ba (F Base [I _]) = True -- as is a base sorted variable
ba (C _) = True -- So is a tag
ba (F Cat [t0, t1]) =
ba t0 && ba t1
ba t@(F Enc [t0, t1]) =
S.member t knowns || ba t0 && ba t1
ba (F Base [t]) = bb t
ba t = isAtom t && not (S.member t unguessable)
-- Buildable base term
bb (I _) = True -- A variable of sort base is always buildable
bb (F Genr []) = True -- and so in the generator
bb t@(F Exp [t0, G t1]) =
S.member (F Base [t]) knowns || ba t0 && be t1
bb (_) = False
-- Buildable exponent
be t =
-- This is a hack. One should really solve a
-- linear equation here
any (instOf $ stripExpn ids t) kns
-- Exponent variables with origination assumptions
ids = getExpnOrigAssumptions unguessable
-- Known exponent without non-known variables
kns = map (stripExpn ids) (getExpns knowns)
getExpnOrigAssumptions :: Set Term -> [Id]
getExpnOrigAssumptions terms =
concatMap f $ S.elems terms
where
f (G t) = M.keys t -- This is an approximation
f _ = []
stripExpn :: [Id] -> Group -> Group
stripExpn ids grp =
foldl f grp $ M.keys grp
where
f grp key
| notElem key ids = M.delete key grp
| otherwise = grp
getExpns :: Set Term -> [Group]
getExpns terms =
foldl f [M.empty] $ S.elems terms
where
f a (G t) = t:a
f a _ = a
termGen :: Group -> Gen
termGen t =
Gen (1 + maxl (map idInt (M.keys t)))
where
idInt (Id (i, _)) = i
maxl [] = 0
maxl xs = maximum xs
instOf :: Group -> Group -> Bool
instOf grp pat =
maybe False (const True) (match (G pat) (G grp) (gen, emptyEnv))
where
gen = mash (termGen grp) (termGen pat)
-- Compute the decomposition given some known terms and some unguessable
-- atoms. The code is quite tricky. It iterates until the known
-- terms don't change. The known terms ends up with all the
-- encryptions that are known.
decompose :: Set Term -> Set Term -> (Set Term, Set Term)
decompose knowns unguessable =
loop unguessable knowns S.empty []
where
loop unguessable knowns old []
| old == knowns = (knowns, unguessable) -- Done
| otherwise = loop unguessable knowns knowns (S.elems knowns)
loop unguessable knowns old (t@(F Cat _) : todo) =
loop unguessable (decat t (S.delete t knowns)) old todo
loop unguessable knowns old ((F Enc [t0, t1]) : todo)
| buildable knowns unguessable (inv t1) = -- Add plaintext
loop unguessable (decat t0 knowns) old todo
| otherwise = loop unguessable knowns old todo
-- New case here: don't delete exponentiated values
loop unguessable knowns old (F Base [F Exp [_, _]] : todo) =
loop unguessable knowns old todo
-- New case here: don't delete exponents that
-- aren't in unguessable
loop unguessable knowns old (t@(G _) : todo)
| S.notMember t unguessable =
loop unguessable knowns old todo
loop unguessable knowns old (t : todo) =
loop (S.delete t unguessable) (S.delete t knowns) old todo
-- Decat
decat (F Cat [t0, t1]) s = decat t1 (decat t0 s)
decat t s = S.insert t s
-- Inverts an asymmetric key
inv :: Term -> Term
inv (F Akey [F Invk [t]]) = F Akey [t]
inv (F Akey [t]) = F Akey [F Invk [t]]
inv (I _) = error "Algebra.inv: Cannot invert a variable of sort mesg"
inv t = t
-- Extracts every encryption that is carried by a term along with its
-- encryption key.
encryptions :: Term -> [(Term, [Term])]
encryptions t =
reverse $ loop t []
where
loop (F Cat [t, t']) acc =
loop t' (loop t acc)
loop t@(F Enc [t', t'']) acc =
loop t' (adjoin (t, [t'']) acc)
loop t@(F Base [F Exp [_, t'']]) acc =
adjoin (t, [t'']) acc
loop _ acc = acc
adjoin x xs
| x `elem` xs = xs
| otherwise = x : xs
-- Returns the encryptions that carry the target. If the target is
-- carried outside all encryptions, or is exposed because a decription
-- key is derivable, Nothing is returned.
protectors :: (Term -> Bool) -> Term -> Term -> Maybe [Term]
protectors derivable target source =
do
ts <- bare source S.empty
return $ S.elems ts
where
bare source _
| source == target = Nothing
bare (F Cat [t, t']) acc =
maybe Nothing (bare t') (bare t acc)
bare t@(F Enc [t', key]) acc =
if target `heldBy` t' then
if derivable (inv key) then
bare t' acc
else
Just (S.insert t acc)
else
Just acc
bare t@(F Base [F Exp [_, t'']]) acc
| target == t'' = Just (S.insert t acc)
bare _ acc = Just acc
-- Support for data flow analysis of traces. A flow rule maps an
-- initial set of atoms and a set of available terms to sets of pairs
-- of the same sets.
type FlowRule = (Set Term, Set Term) -> Set (Set Term, Set Term)
-- Combine flow rules sequentially.
comb :: FlowRule -> FlowRule -> FlowRule
comb f g x =
S.fold h S.empty (g x)
where
h a s = S.union (f a) s
-- Analyze a term as a sent term.
outFlow :: Term -> FlowRule
outFlow t a@(_, available)
| S.member t available = S.singleton a
outFlow (I _) _ = S.empty
outFlow (C _) a = S.singleton a
outFlow (F Cat [t0, t1]) a = -- Construct non-atoms
comb (outFlow t1) (outFlow t0) a
outFlow (F Enc [t0, t1]) a =
comb (outFlow t1) (outFlow t0) a
outFlow t (initial, available) = -- Don't look inside atoms
S.singleton (S.insert t initial, S.insert t available)
-- Analyze a term as a received term.
inFlow :: Term -> FlowRule
inFlow (C _) a = S.singleton a
inFlow (F Cat [t0, t1]) a = -- Try to receive components
S.union -- in both orders
(comb (inFlow t1) (inFlow t0) a)
(comb (inFlow t0) (inFlow t1) a)
inFlow t@(F Enc [t0, t1]) (initial, available) =
S.union -- Encryption can be built
(outFlow t (initial, available)) -- or decrypted
(comb (inFlow t0) (outFlow (inv t1)) a)
where -- Derive decryption key first
a = (initial, S.insert t available)
inFlow t (initial, available) =
S.singleton (initial, S.insert t available)
instance C.Term Term where
isVar = isVar
isAtom = isAtom
termsWellFormed = termsWellFormed
occursIn = occursIn
foldVars = foldVars
foldCarriedTerms = foldCarriedTerms
carriedBy = carriedBy
heldBy = heldBy
decryptionKey = decryptionKey
decompose = decompose
buildable = buildable
encryptions = encryptions
protectors = protectors
outFlow = outFlow
inFlow = inFlow
loadTerm = loadTerm
-- Places
-- A place names a one subterm within a term. It is a list of
-- integers giving a path through a term to that named subterm. Each
-- integer in the list identifies the subterm in a function
-- application on the path to the named subterm. The integer is the
-- index of the subterm in the application's list of terms.
-- The places and replace code fail to find the variable
-- (F Akey [I x]) in (F Akey [Invk [I x]]).
newtype Place = Place [Int] deriving Show
-- Returns the places a variable occurs within a term.
places :: Term -> Term -> [Place]
places var source =
f [] [] source
where
f paths path source
| var == source = Place (reverse path) : paths
f paths path (F _ u) =
g paths path 0 u
f paths path (G t) =
groupPlaces (varId var) paths path 0 (linearize t)
f paths _ _ = paths
g paths _ _ [] = paths
g paths path i (t : u) =
g (f paths (i: path) t) path (i + 1) u
linearize :: Group -> [Id]
linearize t =
do
(x, n) <- M.assocs t
replicate (if n >= 0 then n else negate n) x
groupPlaces :: Id -> [Place] -> [Int] -> Int -> [Id] -> [Place]
groupPlaces _ paths _ _ [] = paths
groupPlaces x paths path i (y:ys) =
let paths' = if x == y then
Place (reverse (i : path)) : paths
else paths in
groupPlaces x paths' path (i + 1) ys
-- Returns the places a term is carried by another term.
carriedPlaces :: Term -> Term -> [Place]
carriedPlaces target source =
f [] [] source
where
f paths path source
| target == source = Place (reverse path) : paths
f paths path (F Cat [t, t']) =
f (f paths (0 : path) t) (1 : path) t'
f paths path (F Enc [t, _]) =
f paths (0 : path) t
f paths path (F Base [F Exp [_, t]]) =
f paths (1 : 0 : path) t
f paths _ _ = paths
-- Replace a variable within a term at a given place.
replace :: Term -> Place -> Term -> Term
replace var (Place ints) source =
loop ints source
where
loop [] _ = var
loop (i : path) (F s u) =
F s (C.replaceNth (loop path (u !! i)) i u)
loop [i] (G t) =
groupReplace (varId var) i (factors t)
loop _ _ = error "Algebra.replace: Bad path to term"
factors :: Group -> [(Id, Int)]
factors t =
do
(x, n) <- M.assocs t
case n >= 0 of
True -> replicate n (x, 1)
False -> replicate (negate n) (x, -1)
groupReplace :: Id -> Int -> [(Id, Int)] -> Term
groupReplace x i factors =
let (_, n) = factors !! i in
G $ group $ C.replaceNth (x, n) i factors
-- Return the ancestors of the term at the given place.
ancestors :: Term -> Place -> [Term]
ancestors source (Place ints) =
loop [] ints source
where
loop ts [] _ = ts
loop ts (i: path) t@(F _ u) =
loop (t : ts) path (u !! i)
loop ts [_] t@(G _) = t : ts
loop _ _ _ = error "Algebra.ancestors: Bad path to term"
instance C.Place Term Place where
places = places
carriedPlaces = carriedPlaces
replace = replace
ancestors = ancestors
-- Rename the identifiers in a term. Gen keeps the state of the
-- renamer. (Question: should alist be replaced by a Map?)
clone :: Gen -> Term -> (Gen, Term)
clone gen t =
(gen', t')
where
(_, gen', t') = cloneTerm ([], gen) t
cloneTerm (alist, gen) t =
case t of -- The association list maps
I x -> -- identifiers to identifier.
case lookup x alist of
Just y -> (alist, gen, I y)
Nothing ->
let (gen', y) = cloneId gen x in
((x, y) : alist, gen', I y)
C c -> (alist, gen, C c)
F sym u ->
let (alist', gen', u') =
foldl cloneTermList (alist, gen, []) u in
(alist', gen', F sym $ reverse u')
G t ->
let (alist', gen', ts) =
foldl cloneGroupList (alist, gen, []) (M.assocs t) in
(alist', gen', G $ group ts)
cloneTermList (alist, gen, u) t =
let (alist', gen', t') = cloneTerm (alist, gen) t in
(alist', gen', t' : u)
cloneGroupList (alist, gen, ts) (x, n) =
case lookup x alist of
Just y -> (alist, gen, (y, n) : ts)
Nothing ->
let (gen', y) = cloneId gen x in
((x, y) : alist, gen', (y, n) : ts)
instance C.Gen Term Gen where
origin = origin
clone = clone
loadVars = loadVars
-- Functions used in both unification and matching
type IdMap = Map Id Term
emptyIdMap :: IdMap
emptyIdMap = M.empty
-- Apply a substitution to a term
idSubst :: IdMap -> Term -> Term
idSubst subst (I x) =
M.findWithDefault (I x) x subst
idSubst _ t@(C _) = t
idSubst subst (F Invk [t]) =
case idSubst subst t of
F Invk [t] -> t -- (invk (invk x)) = x
t -> F Invk [t]
idSubst subst (F Exp [t0, G t1]) =
case idSubst subst t0 of -- (exp (exp g x) y) = (exp g (mul x y))
F Exp [t0', G t1'] ->
idSubst subst (F Exp [t0', G $ mul t1' t1])
t -> expSubst subst t t1
idSubst subst (F s u) =
F s (map (idSubst subst) u)
idSubst subst (G t) =
G $ groupSubst subst t
expSubst :: IdMap -> Term -> Group -> Term
expSubst subst t0 t1 =
case groupSubst subst t1 of
t1' | M.null t1' -> t0 -- (exp g (one)) = g
| otherwise -> F Exp [t0, G t1']
groupSubst :: IdMap -> Group -> Group
groupSubst subst t =
foldrWithKey f M.empty t
where
f x n t =
mul (expg (groupLookup subst x) n) t
groupLookup :: IdMap -> Id -> Group
groupLookup subst x =
case M.findWithDefault (groupVar x) x subst of
G t -> t
w -> error ("Algebra.groupLookup: Bad substitution: " ++
show x ++ " -> " ++ show w)
showMap :: (Show a, Show b) => Map a b -> ShowS
showMap m =
showAssocs (M.assocs m)
where
showAssocs [] = id
showAssocs ((x,y):m) =
showString "\n " . shows x . showString " -> " .
shows y . showAssocs m
-- Unification and substitution
-- The rewrite rules used are:
--
-- (vars (h base) (x y expn))
--
-- 1. ((exp h x) y) ==> (exp h (mul x y))
-- 2. (exp h (one)) ==> h
-- 3. unify((exp(h, x)), (exp(h, y)), s) ==>
-- unify(x, y, s)
-- 4 unify((exp(h, x)), (exp((gen), y)), s) ==>
-- unify(h, (exp gen (mul y (rec x))), s)
-- 5. unify((exp((gen), x)), (exp(h, y)), s) ==>
-- unify((exp(h, x)), (exp((gen), y)), s)
newtype Subst = Subst IdMap deriving (Eq, Ord)
instance Show Subst where
showsPrec _ (Subst s) = showString "Subst (" . showMap s . showChar ')'
emptySubst :: Subst
emptySubst = Subst emptyIdMap
-- Apply a substitution created by unification
substitute :: Subst -> Term -> Term
substitute (Subst s) t =
idSubst s t
-- Composition of substitutions
-- substitute (compose s0 s1) t = substitute s0 (substitute s1 t)
-- 1. apply s0 to range of s1 to obtain s2;
-- 2. remove bindings is s0 where domains of s0 and s1 overlap to form s3;
-- 3. remove trivial bindings from s2 to form s4; and
-- 4. take the union of s4 and s3.
compose :: Subst -> Subst -> Subst
compose (Subst s0) (Subst s1) =
let s2 = M.map (substitute (Subst s0)) s1 -- Step 1
s4 = M.filterWithKey nonTrivialBinding s2 in -- Step 3
Subst (M.union s4 s0) -- Steps 2 and 4, union is left-biased
nonTrivialBinding :: Id -> Term -> Bool
nonTrivialBinding x (I y) = x /= y
nonTrivialBinding _ _ = True
-- During unification, variables determined to be equal are collected
-- into an equivalence class. Multiple lookups of each variable in
-- the internal representation of a substitution finds the canonical
-- representive of the class. The chase function finds the current
-- canonical representitive.
-- Get the canonical representative of equivalent identifiers making use
-- of this algebra's axiom.
chase :: Subst -> Term -> Term
chase (Subst s) (I x) =
case M.lookup x s of
Nothing -> I x
Just t -> chase (Subst s) t
chase s (F Invk [t]) = chaseInvk s t
chase s (F Exp [t0, G t1]) = chaseExp s t0 t1
chase _ t = t
chaseInvk :: Subst -> Term -> Term
chaseInvk (Subst s) (I x) =
case M.lookup x s of
Nothing -> F Invk [I x]
Just t -> chaseInvk (Subst s) t
chaseInvk s (F Invk [t]) = chase s t
chaseInvk _ t = F Invk [t]
chaseExp :: Subst -> Term -> Group -> Term
chaseExp s t0 t1
| M.null t1 = chase s t0
chaseExp s (I x) t1 =
case chase s (I x) of
F Exp [t0', G t1'] -> chaseExp s t0' (mul t1 t1')
t0 -> F Exp [t0, G t1]
chaseExp s (F Exp [t0', G t1']) t1 =
chaseExp s t0' (mul t1 t1')
chaseExp _ t0 t1 = F Exp [t0, G t1]
-- Does x occur in t?
occurs :: Id -> Term -> Bool
occurs x (I y) = x == y
occurs _ (C _) = False
occurs x (F _ u) = any (occurs x) u
occurs x (G t) = elem x (M.keys t)
type GenSubst = (Gen, Subst)
unifyChase :: Term -> Term -> GenSubst -> Maybe GenSubst
unifyChase t t' (g, s) = unifyTerms (chase s t) (chase s t') (g, s)
unifyTerms :: Term -> Term -> GenSubst -> Maybe GenSubst
unifyTerms (I x) (I y) (g, Subst s)
| x == y = Just (g, Subst s)
| otherwise = Just (g, Subst $ M.insert x (I y) s)
unifyTerms (I x) t (g, Subst s)
| occurs x t = Nothing
| otherwise = Just (g, Subst $ M.insert x t s)
unifyTerms t (I x) s = unifyTerms (I x) t s
unifyTerms (C c) (C c') s
| c == c' = Just s
| otherwise = Nothing
unifyTerms (F Invk [I x]) (F Pubk [I y]) s =
unifyTerms (I x) (F Invk [F Pubk [I y]]) s
unifyTerms (F Invk [I x]) (F Pubk [C c, I y]) s =
unifyTerms (I x) (F Invk [F Pubk [C c, I y]]) s
unifyTerms (F Pubk [I x]) (F Invk [I y]) s =
unifyTerms (I y) (F Invk [F Pubk [I x]]) s
unifyTerms (F Pubk [C c, I x]) (F Invk [I y]) s =
unifyTerms (I y) (F Invk [F Pubk [C c, I x]]) s
unifyTerms (F Exp [t0, G t1]) (F Exp [t0', G t1']) s =
unifyExp t0 t1 t0' t1' s
unifyTerms (F sym u) (F sym' u') s
| sym == sym' = unifyTermLists u u' s
| otherwise = Nothing
unifyTerms (G t) (G t') s =
Just $ unifyGroup t t' s
unifyTerms _ _ _ = Nothing
unifyExp :: Term -> Group -> Term -> Group -> GenSubst -> Maybe GenSubst
unifyExp t0 t1 t0' t1' s
| t0 == t0' = Just $ unifyGroup t1 t1' s
unifyExp (I x) t1 (F Genr []) t1' (g, Subst s)
| t1 == t1' =
Just (g, Subst $ M.insert x (F Genr []) s)
| otherwise =
Just (g, Subst (M.insert
x
(F Exp [F Genr [], G $ mul t1' (invert t1)])
s))
unifyExp (F Genr []) t1 (I x) t1' s =
unifyExp (I x) t1' (F Genr []) t1 s
unifyExp _ _ _ _ _ = Nothing
unifyTermLists :: [Term] -> [Term] -> GenSubst -> Maybe GenSubst
unifyTermLists [] [] s = Just s
unifyTermLists (t : u) (t' : u') s =
maybe Nothing (unifyTermLists u u') (unifyChase t t' s)
unifyTermLists _ _ _ = Nothing
unifyGroup :: Group -> Group -> GenSubst -> GenSubst
unifyGroup t0 t1 (g, Subst s) =
case matchGroup (groupSubst s $ mul t0 $ invert t1) M.empty S.empty g of
Nothing -> error "Algebra.unifyGroup: unification failed"
Just (_, g', s') -> (g', Subst $ M.union s' s)
-- The exported unifier converts the internal representation of a
-- substitution into the external form using chaseMap.
unify :: Term -> Term -> GenSubst -> Maybe GenSubst
unify t t' s =
do
(g, s) <- unifyChase t t' s
return (g, chaseMap s)
-- Apply the chasing version of substitution to the range of s.
chaseMap :: Subst -> Subst
chaseMap (Subst s) =
Subst $ M.map (substChase (Subst s)) s
-- A chasing version of substitution.
substChase :: Subst -> Term -> Term
substChase subst t =
case chase subst t of
t@(I _) -> t
t@(C _) -> t
F Invk [t] ->
case substChase subst t of
F Invk [t] -> t -- Apply axiom
t -> F Invk [t]
F Exp [t0, G t1] ->
case substChase subst t0 of
F Exp [t0', G t1'] ->
substChase subst (F Exp [t0', G $ mul t1' t1])
t -> expChase subst t t1
F s u ->
F s (map (substChase subst) u)
G t -> G $ groupChase subst t
expChase :: Subst -> Term -> Group -> Term
expChase subst t0 t1 =
case groupChase subst t1 of
t1' | M.null t1' -> t0
| otherwise -> F Exp [t0, G t1']
groupChase :: Subst -> Group -> Group
groupChase (Subst subst) t = groupSubst subst t
-- more general than relation
-- s0 `lte` s1 if s1 = compose s2 s0 for some s2
moreGeneral :: (Gen, Subst) -> (Gen, Subst) -> Bool
moreGeneral (gen0, Subst s0) (gen1, Subst s1) =
let dom = S.elems $ foldl idSet (M.keysSet s0) (M.elems s0)
env = foldl (flip M.delete) s1 (M.keys s0) in
loop dom (mash gen0 gen1, Env (S.empty, env))
where
loop [] _ = True
loop (x : xs) env =
maybe False (loop xs) (match (get x s0) (get x s1) env)
get x env = M.findWithDefault (I x) x env
idSet :: Set Id -> Term -> Set Id
idSet set (I id) = S.insert id set
idSet set (C _) = set
idSet set (F _ u) = foldl idSet set u
idSet set (G t) = S.union (M.keysSet t) set
instance C.Subst Term Gen Subst where
emptySubst = emptySubst
substitute = substitute
unify t t' s = maybe [] (: []) $ unify t t' s
compose = compose
moreGeneral = moreGeneral
-- Matching and instantiation
newtype Env = Env (Set Id, IdMap) deriving (Eq, Ord)
instance Show Env where
showsPrec _ (Env (v, r)) =
showString "Env (\n " . shows v .
showChar ',' . showMap r . showChar ')'
-- An environment may contain an explicit identity mapping, whereas a
-- substitution is erroneous if it has one. The set of variables
-- associated with a map is the variables in the range that were
-- generated by matching and should be treated as variables when using
-- unification to perform matching. The other variables in the range
-- are treated as constants.
-- An environment contains an IdMap and the set of variables
-- generated while matching.
emptyEnv :: Env
emptyEnv = Env (S.empty, emptyIdMap)
-- Apply a substitution created my matching
instantiate :: Env -> Term -> Term
instantiate (Env (_, r)) t = idSubst r t
-- Matching
type GenEnv = (Gen, Env)
-- The matcher has the property that when pattern P and term T match
-- then instantiate (match P T emptyEnv) P = T.
match :: Term -> Term -> GenEnv -> Maybe GenEnv
match (I x) t (g, Env (v, r)) =
case M.lookup x r of
Nothing -> Just (g, Env (v, M.insert x t r))
Just t' -> if t == t' then Just (g, Env (v, r)) else Nothing
match (C c) (C c') r = if c == c' then Just r else Nothing
match (F Exp [t0, G t1]) (F Exp [t0', G t1']) r
= matchExp t0 t1 t0' t1' r
match (F s u) (F s' u') r
| s == s' = matchLists u u' r
match (F Invk [t]) t' r =
match t (F Invk [t']) r
match (G t) (G t') (g, Env (v, r)) =
do
let (t0, t0') = merge t t' r
(v, g, r') <- matchGroup t0 t0' v g
return (g, Env (v, M.union r' r))
match _ _ _ = Nothing
matchExp :: Term -> Group -> Term -> Group -> GenEnv -> Maybe GenEnv
matchExp (I x) t1 t0' t1' r@(_, Env (_, e)) =
case M.lookup x e of
Just (F Exp [t0'', G t1''])
| t0' == t0'' -> match (G t1) (G (mul t1' (invert t1''))) r
_ -> matchLists [I x, G t1] [t0', G t1'] r
matchExp (F Genr []) t1 t0' t1' r =
matchLists [F Genr [], G t1] [t0', G t1'] r
matchExp _ _ _ _ _ = error "Algebra.matchExp: Bad match term"
matchLists :: [Term] -> [Term] -> GenEnv -> Maybe GenEnv
matchLists [] [] r = Just r
matchLists (t : u) (t' : u') r =
maybe Nothing (matchLists u u') (match t t' r)
matchLists _ _ _ = Nothing
merge :: Group -> Group -> IdMap -> (Group, Group)
merge t t' r =
(group t0, t0')
where
(t0, t0') = loop (M.assocs t) ([], t')
loop [] acc = acc
loop (p@(x, n) : t0) (t1, t1') =
case M.lookup x r of
Nothing -> loop t0 (p : t1, t1')
Just (G t) ->
loop t0 (t1, mul (expg t (negate n)) t1')
Just t ->
error $ "Algebra.merge: expecting an expn but got " ++ show t
matchGroup :: Group -> Group -> Set Id -> Gen -> Maybe (Set Id, Gen, IdMap)
matchGroup t0 t1 v g =
case partition t0 t1 v of
(_, [], []) -> return (v, g, emptyIdMap)
(_, [], _) -> Nothing
(n, t0, t1) ->
do
subst <- I.intLinEq (map snd t0, map snd t1)
return $ mgu v n (map fst t0) (map fst t1) subst g
type Coeff = [(Id, Int)]
-- Move variables on the RHS of the equation to the LHS
partition :: Group -> Group -> Set Id -> (Int, Coeff, Coeff)
partition t0 t1 v =
(length v0, v0 ++ map (\(x,n) -> (x, negate n)) v1, c1)
where
v0 = M.assocs t0
(v1, c1) = L.partition f (M.assocs t1)
f (x, _) = S.member x v
mgu :: Set Id -> Int -> [Id] -> [Id] -> I.Subst -> Gen -> (Set Id, Gen, IdMap)
mgu v _ _ _ [] gen = (v, gen, emptyIdMap)
mgu v n vars syms subst gen =
(v', gen', foldl f emptyIdMap (zip vars [0..(n - 1)]))
where
(gen', genSyms) = genVars vars (length (fst (snd (head subst)))) gen
v' = foldl (flip S.insert) v genSyms
f s (x, n) =
case lookup n subst of
Just (factors, consts) ->
M.insert x (g factors consts) s
Nothing ->
M.insert x (groupVar $ genSyms !! n) s
g factors consts =
G $ group (zip genSyms factors ++ zip syms consts)
genVars :: [Id] -> Int -> Gen -> (Gen, [Id])
genVars [] _ _ =
error "Algebra.genVars: no variables to clone"
-- let (g', id) = freshId g "dh" in
-- genVars [id] n g'
genVars vars n g =
loop n vars (g, [])
where
loop n [] (g, ids) =
loop n vars (g, ids)
loop n (v:vs) (g, ids)
| n <= 0 = (g, ids)
| otherwise =
case cloneId g v of
(g, id) -> loop (n - 1) vs (g, id : ids)
-- Does every varible in ts not occur in the domain of e?
-- Trivial bindings in e are ignored.
identityEnvFor :: GenEnv -> [Term] -> Maybe GenEnv
identityEnvFor ge ts =
let env@(_, Env (_, r)) = nonTrivialEnv ge in
if all (allId $ flip S.notMember $ M.keysSet r) ts then
Just env
else
Nothing
allId :: (Id -> Bool) -> Term -> Bool
allId f (I x) = f x
allId _ (C _) = True
allId f (F _ u) = all (allId f) u
allId f (G t) = all f (M.keys t)
-- Eliminate all trivial bindings so that an environment can be used
-- as a substitution.
nonTrivialEnv :: GenEnv -> GenEnv
nonTrivialEnv (g, Env (v, r)) =
nonGroupEnv (M.assocs r) M.empty []
where
nonGroupEnv [] env grp =
groupEnv g v env grp grp
nonGroupEnv ((x, I y):r) env grp
| x == y = nonGroupEnv r env grp
nonGroupEnv ((x, G y):r) env grp
| isGroupVar y && varId (G y) == x =
nonGroupEnv r env grp
| otherwise = nonGroupEnv r env ((x, y):grp)
nonGroupEnv ((x, y):r) env grp = nonGroupEnv r (M.insert x y env) grp
groupEnv :: Gen -> Set Id -> IdMap -> [(Id, Group)] -> [(Id, Group)] -> GenEnv
groupEnv g v env grp [] =
(g, Env (v, foldl (\env (x, y) -> M.insert x (G y) env) env grp))
groupEnv g v env grp ((x, t):map)
| M.lookup x t /= Just 1 = groupEnv g v env grp map
| otherwise =
let (_, t0, t1) = partition M.empty (mul t (M.singleton x (-1))) v in
case matchGroup (group t0) (group t1) S.empty g of
Nothing -> groupEnv g v env grp map
Just (v', g', subst) ->
let grp' = L.delete (x, t) grp
grp'' = L.map (\(x, t) -> (x, groupSubst subst t)) grp' in
groupEnv g' (S.union v' v) env grp'' grp''
-- Specialize an environment by mapping the generated variables to one.
specialize :: Env -> Env
specialize (Env (v, r)) =
Env (S.empty, foldrWithKey f M.empty r)
where
f x t r = M.insert x (instantiate special t) r
-- Environment mapping generated variables to one
special = Env (S.empty, S.fold g M.empty v)
g x r = M.insert x (G M.empty) r
-- Cast an environment into a substitution by filtering out trivial
-- bindings.
substitution :: Env -> Subst
substitution (Env (_, r)) =
Subst $ M.filterWithKey nonTrivialBinding r
-- Add type information to an environment, and return it as a list of
-- associations.
reify :: [Term] -> Env -> [(Term, Term)]
reify domain (Env (_, env)) =
map (loop domain) $ M.assocs env
where
loop [] (x, _) =
error $ "Algebra.reify: variable missing from domain " ++ idName x
loop (I x : _) (y, t)
| x == y = (I x, t)
loop (F Text [I x] : _) (y, t)
| x == y = (F Text [I x], F Text [t])
loop (F Data [I x] : _) (y, t)
| x == y = (F Data [I x], F Data [t])
loop (F Name [I x] : _) (y, t)
| x == y = (F Name [I x], F Name [t])
loop (F Skey [I x] : _) (y, t)
| x == y = (F Skey [I x], F Skey [t])
loop (F Akey [I x] : _) (y, t)
| x == y = (F Akey [I x], F Akey [t])
loop (F Base [I x] : _) (y, t)
| x == y = (F Base [I x], F Base [t])
loop (G x : _) (y, G t)
| isGroupVar x && varId (G x) == y = (G x, G t)
loop (_ : domain) pair = loop domain pair
-- Ensure the range of an environment contains only variables and that
-- the environment is injective.
matchRenaming :: GenEnv -> Bool
matchRenaming (gen, Env (v, e)) =
nonGrp S.empty (M.elems e) &&
groupMatchRenaming v gen (foldrWithKey grp M.empty e)
where
nonGrp _ [] = True
nonGrp s (I x:e) =
not (S.member x s) && nonGrp (S.insert x s) e
nonGrp s (G _:e) = nonGrp s e -- Check group bindings elsewhere
nonGrp _ _ = False
grp x (G t) map = M.insert x t map
grp _ _ map = map
groupMatchRenaming :: Set Id -> Gen -> Map Id Group -> Bool
groupMatchRenaming v gen map =
loop S.empty $ M.elems map
where
loop _ [] = True
loop s (t:ge)
| M.null t = False
| isGroupVar t =
let x = varId (G t) in
not (S.member x s) && loop (S.insert x s) ge
| otherwise = any (groupMatchElim v gen map t) (M.assocs t)
groupMatchElim :: Set Id -> Gen -> Map Id Group -> Group -> (Id, Int) -> Bool
groupMatchElim v gen ge t (x, 1) =
let (_, t0, t1) = partition M.empty (mul t (M.singleton x (-1))) v in
case matchGroup (group t0) (group t1) S.empty gen of
Nothing -> False
Just (v', gen', subst) ->
groupMatchRenaming (S.union v' v) gen' $ M.map (groupSubst subst) ge
groupMatchElim _ _ _ _ _ = False
instance C.Env Term Gen Subst Env where
emptyEnv = emptyEnv
instantiate = instantiate
match t t' e = maybe [] (: []) $ match t t' e
identityEnvFor e ts = maybe [] (: []) $ identityEnvFor e ts
specialize = specialize
substitution = substitution
reify = reify
matchRenaming = matchRenaming
-- Term specific loading functions
loadVars :: Monad m => Gen -> [SExpr Pos] -> m (Gen, [Term])
loadVars gen sexprs =
do
pairs <- mapM loadVarPair sexprs
(g, vars) <- foldM loadVar (gen, []) (concat pairs)
return (g, reverse vars)
loadVarPair :: Monad m => SExpr Pos -> m [(SExpr Pos, SExpr Pos)]
loadVarPair (L _ (x:xs)) =
let (t:vs) = reverse (x:xs) in
return [(v,t) | v <- reverse vs]
loadVarPair x = fail (shows (annotation x) "Bad variable declaration")
loadVar :: Monad m => (Gen, [Term]) -> (SExpr Pos, SExpr Pos) ->
m (Gen, [Term])
loadVar (gen, vars) (S pos name, S pos' sort) =
case loadLookup pos vars name of
Right _ ->
fail (shows pos "Duplicate variable declaration for " ++ name)
Left _ ->
do
let (gen', x) = freshId gen name
p <- mkVar x
return (gen', p : vars)
where
mkVar x =
let t = I x in
case sort of
"mesg" -> return t
"text" -> return $ F Text [t]
"data" -> return $ F Data [t]
"name" -> return $ F Name [t]
"skey" -> return $ F Skey [t]
"akey" -> return $ F Akey [t]
"base" -> return $ F Base [t]
"expn" -> return $ groupVar x
_ -> fail (shows pos' "Sort " ++ sort ++ " not recognized")
loadVar _ (x,_) = fail (shows (annotation x) "Bad variable syntax")
loadLookup :: Pos -> [Term] -> String -> Either String Term
loadLookup pos [] name = Left (shows pos $ "Identifier " ++ name ++ " unknown")
loadLookup pos (t : u) name =
let name' = idName (varId t) in
if name' == name then Right t else loadLookup pos u name
loadLookupName :: Monad m => Pos -> [Term] -> String -> m Term
loadLookupName pos vars name =
either fail f (loadLookup pos vars name)
where
f t@(F Name [I _]) = return t
f _ = fail (shows pos $ "Expecting " ++ name ++ " to be a name")
loadLookupAkey :: Monad m => Pos -> [Term] -> String -> m Term
loadLookupAkey pos vars name =
either fail f (loadLookup pos vars name)
where
f t@(F Akey [I _]) = return t
f _ = fail (shows pos $ "Expecting " ++ name ++ " to be an akey")
-- Load term and check that it is well-formed.
loadTerm :: Monad m => [Term] -> SExpr Pos -> m Term
loadTerm vars (S pos s) =
either fail return (loadLookup pos vars s)
loadTerm _ (Q _ t) =
return (C t)
loadTerm vars (L pos (S _ s : l)) =
case lookup s loadDispatch of
Nothing -> fail (shows pos "Keyword " ++ s ++ " unknown")
Just f -> f pos vars l
loadTerm _ x = fail (shows (annotation x) "Malformed term")
type LoadFunction m = Pos -> [Term] -> [SExpr Pos] -> m Term
loadDispatch :: Monad m => [(String, LoadFunction m)]
loadDispatch =
[("pubk", loadPubk)
,("privk", loadPrivk)
,("invk", loadInvk)
,("ltk", loadLtk)
,("gen", loadGen)
,("exp", loadExp)
,("one", loadOne)
,("rec", loadRec)
,("mul", loadMul)
,("cat", loadCat)
,("enc", loadEnc)
]
-- Atom constructors: pubk privk invk ltk
loadPubk :: Monad m => LoadFunction m
loadPubk _ vars [S pos s] =
do
t <- loadLookupName pos vars s
return $ F Akey [F Pubk [I $ varId t]]
loadPubk _ vars [Q _ c, S pos s] =
do
t <- loadLookupName pos vars s
return $ F Akey [F Pubk [C c, I $ varId t]]
loadPubk pos _ _ = fail (shows pos "Malformed pubk")
loadPrivk :: Monad m => LoadFunction m
loadPrivk _ vars [S pos s] =
do
t <- loadLookupName pos vars s
return $ F Akey [F Invk [F Pubk [I $ varId t]]]
loadPrivk _ vars [Q _ c, S pos s] =
do
t <- loadLookupName pos vars s
return $ F Akey [F Invk [F Pubk [C c, I $ varId t]]]
loadPrivk pos _ _ = fail (shows pos "Malformed privk")
loadInvk :: Monad m => LoadFunction m
loadInvk _ vars [S pos s] =
do
t <- loadLookupAkey pos vars s
return $ F Akey [F Invk [I $ varId t]]
loadInvk pos _ _ = fail (shows pos "Malformed invk")
loadLtk :: Monad m => LoadFunction m
loadLtk _ vars [S pos s, S pos' s'] =
do
t <- loadLookupName pos vars s
t' <- loadLookupName pos' vars s'
return $ F Skey [F Ltk [I $ varId t, I $ varId t']]
loadLtk pos _ _ = fail (shows pos "Malformed ltk")
-- Base and exponents
loadGen :: Monad m => LoadFunction m
loadGen _ _ [] =
return $ F Base [F Genr []]
loadGen pos _ _ = fail (shows pos "Malformed gen")
loadExp :: Monad m => LoadFunction m
loadExp _ vars [x, x'] =
do
t <- loadBase vars x
t' <- loadExpn vars x'
return $ F Base [idSubst emptyIdMap $ F Exp [t, G t']]
loadExp pos _ _ = fail (shows pos "Malformed exp")
loadBase :: Monad m => [Term] -> SExpr Pos -> m Term
loadBase vars x =
do
t <- loadTerm vars x
case t of
F Base [t] -> return t
_ -> fail (shows (annotation x) "Malformed base")
loadExpn :: Monad m => [Term] -> SExpr Pos -> m Group
loadExpn vars x =
do
t <- loadTerm vars x
case t of
G t -> return t
_ -> fail (shows (annotation x) "Malformed expn")
loadOne :: Monad m => LoadFunction m
loadOne _ _ [] =
return $ G M.empty
loadOne pos _ _ = fail (shows pos "Malformed one")
loadRec :: Monad m => LoadFunction m
loadRec _ vars [x] =
do
t <- loadExpn vars x
return $ G $ invert t
loadRec pos _ _ = fail (shows pos "Malformed rec")
loadMul :: Monad m => LoadFunction m
loadMul _ vars xs =
do
t <- foldM f M.empty xs
return $ G t
where
f acc x =
do
t <- loadExpn vars x
return $ mul t acc
-- Term constructors: cat enc
loadCat :: Monad m => LoadFunction m
loadCat _ vars (l : ls) =
do
ts <- mapM (loadTerm vars) (l : ls)
return $ foldr1 (\a b -> F Cat [a, b]) ts
loadCat pos _ _ = fail (shows pos "Malformed cat")
loadEnc :: Monad m => LoadFunction m
loadEnc pos vars (l : l' : ls) =
do
let (butLast, last) = splitLast l (l' : ls)
t <- loadCat pos vars butLast
t' <- loadTerm vars last
return $ F Enc [t, t']
loadEnc pos _ _ = fail (shows pos "Malformed enc")
splitLast :: a -> [a] -> ([a], a)
splitLast x xs =
loop [] x xs
where
loop z x [] = (reverse z, x)
loop z x (y : ys) = loop (x : z) y ys
-- Term specific displaying functions
newtype Context = Context [(Id, String)] deriving Show
displayVars :: Context -> [Term] -> [SExpr ()]
displayVars _ [] = []
displayVars ctx vars =
let (v,t):pairs = map (displayVar ctx) vars in
loop t [v] pairs
where
loop t vs [] = [L () (reverse (t:vs))]
loop t vs ((v',t'):xs)
| t == t' = loop t (v':vs) xs
| otherwise = L () (reverse (t:vs)):loop t' [v'] xs
displayVar :: Context -> Term -> (SExpr (), SExpr ())
displayVar ctx (I x) = displaySortId "mesg" ctx x
displayVar ctx (F Text [I x]) = displaySortId "text" ctx x
displayVar ctx (F Data [I x]) = displaySortId "data" ctx x
displayVar ctx (F Name [I x]) = displaySortId "name" ctx x
displayVar ctx (F Skey [I x]) = displaySortId "skey" ctx x
displayVar ctx (F Akey [I x]) = displaySortId "akey" ctx x
displayVar ctx (F Base [I x]) = displaySortId "base" ctx x
displayVar ctx t@(G x)
| isGroupVar x = displaySortId "expn" ctx (varId t)
displayVar _ _ =
error "Algebra.displayVar: term not a variable with its sort"
displaySortId :: String -> Context -> Id -> (SExpr (), SExpr ())
displaySortId sort ctx x = (displayId ctx x, S () sort)
displayId :: Context -> Id -> SExpr ()
displayId (Context ctx) x =
case lookup x ctx of
Nothing ->
let msg = idName x ++ " in a display context" in
error $ "Algebra.displayId: Cannot find variable " ++ msg
Just name -> S () name
displayTerm :: Context -> Term -> SExpr ()
displayTerm ctx (I x) = displayId ctx x
displayTerm ctx (F Text [I x]) = displayId ctx x
displayTerm ctx (F Data [I x]) = displayId ctx x
displayTerm ctx (F Name [I x]) = displayId ctx x
displayTerm ctx (F Skey [I x]) = displayId ctx x
displayTerm ctx (F Skey [F Ltk [I x, I y]]) =
L () [S () "ltk", displayId ctx x, displayId ctx y]
displayTerm ctx (F Akey [t]) =
case t of
I x -> displayId ctx x
F Invk [I x] -> L () [S () "invk", displayId ctx x]
F Pubk [I x] -> L () [S () "pubk", displayId ctx x]
F Pubk [C c, I x] -> L () [S () "pubk", Q () c, displayId ctx x]
F Invk [F Pubk [I x]] -> L () [S () "privk", displayId ctx x]
F Invk [F Pubk [C c, I x]] ->
L () [S () "privk", Q () c, displayId ctx x]
_ -> error ("Algebra.displayAkey: Bad term " ++ show t)
displayTerm ctx (F Base [t]) =
displayBase t
where
displayBase (I x) = displayId ctx x
displayBase (F Genr []) =
L () [S () "gen"]
displayBase (F Exp [t0, G t1]) =
L () [S () "exp", displayBase t0, displayTerm ctx (G t1)]
displayBase t = error ("Algebra.displayBase: Bad term " ++ show t)
displayTerm ctx (G t) =
displayExpn t
where
displayExpn t
| M.null t = L () [S () "one"]
| otherwise =
case factors t of
[f] -> displayFactor f
fs -> L () (S () "mul" : map displayFactor fs)
displayFactor (x, n)
| n >= 0 = displayId ctx x
| otherwise = L () [S () "rec", displayId ctx x]
displayTerm _ (C t) = Q () t
displayTerm ctx (F Cat [t0, t1]) =
L () (S () "cat" : displayTerm ctx t0 : displayList ctx t1)
displayTerm ctx (F Enc [t0, t1]) =
L () (S () "enc" : displayEnc ctx t0 t1)
displayTerm _ t = error ("Algebra.displayTerm: Bad term " ++ show t)
displayList :: Context -> Term -> [SExpr ()]
displayList ctx (F Cat [t0, t1]) = displayTerm ctx t0 : displayList ctx t1
displayList ctx t = [displayTerm ctx t]
displayEnc :: Context -> Term -> Term -> [SExpr ()]
displayEnc ctx (F Cat [t0, t1]) t = displayTerm ctx t0 : displayEnc ctx t1 t
displayEnc ctx t0 t1 = [displayTerm ctx t0, displayTerm ctx t1]
displayEnv :: Context -> Context -> Env -> [SExpr ()]
displayEnv ctx ctx' (Env (_, r)) =
map (\(x, t) -> L () [displayTerm ctx x, displayTerm ctx' t]) r'
where
r' = map (\(x, t) -> (I x, inferSort t)) $ M.assocs r
-- displaySubst c s displays a substitution s in context c, where some
-- variables that occur in s might not be in c. Enough sort
-- inference is performed so as to allow the extension of the context.
displaySubst :: Context -> Subst -> [SExpr ()]
displaySubst ctx s@(Subst r) =
map (\(x, t) -> L () [displayTerm ctx' x, displayTerm ctx' t]) r'
where
r' = map (\(x, t) -> (I x, inferSort (substitute s t))) $ M.assocs r
ctx' = foldl (\ctx (x, t) -> addToContext ctx [x, t]) ctx r'
inferSort :: Term -> Term
inferSort t@(F Invk _) = F Akey [t]
inferSort t@(F Pubk _) = F Akey [t]
inferSort t@(F Ltk _) = F Skey [t]
inferSort t@(F Genr _) = F Base [t]
inferSort t@(F Exp _) = F Base [t]
inferSort t = t
emptyContext :: Context
emptyContext = Context []
-- Generate names for output renaming as necessary.
-- Assumes the input is a list of term that are well-formed
addToContext :: Context -> [Term] -> Context
addToContext ctx u =
foldl (foldVars varContext) ctx u
varContext :: Context -> Term -> Context
varContext ctx t =
let x = varId t
name = rootName $ idName x in
if hasId ctx x then
ctx
else
if hasName ctx name then
extendContext ctx x (genName ctx name)
else
extendContext ctx x name
hasId :: Context -> Id -> Bool
hasId (Context ctx) id =
maybe False (const True) (lookup id ctx)
hasName :: Context -> String -> Bool
hasName (Context ctx) name =
maybe False (const True) (L.find ((name ==) . snd) ctx)
extendContext :: Context -> Id -> String -> Context
extendContext (Context ctx) x name =
Context $ (x, name) : ctx
genName :: Context -> String -> String
genName ctx name =
loop 0
where
root = '-' : reverse name
loop :: Int -> String
loop n =
let name' = revapp root (show n) in
if hasName ctx name' then
loop (n + 1)
else
name'
revapp [] s = s
revapp (c : cs) s = revapp cs (c : s)
rootName :: String -> String
rootName name =
noHyphen 0 name
where
noHyphen _ [] = name
noHyphen i (c : s)
| c == '-' = hyphen i (i + 1) s
| otherwise = noHyphen (i + 1) s
hyphen i _ [] = rootName $ take i name
hyphen i j (c : s)
| isDigit c = hyphen i (j + 1) s
| otherwise = noHyphen j (c : s)
instance C.Context Term Gen Subst Env Context where
emptyContext = emptyContext
addToContext = addToContext
displayVars = displayVars
displayTerm = displayTerm
displayEnv = displayEnv
displaySubst = displaySubst
instance C.Algebra Term Place Gen Subst Env Context