cpsa-3.6.4: src/CPSA/Basic/Algebra.hs
-- Basic Crypto Algebra implementation
-- 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.
-- To support security goals, the message algebra has been augmented
-- with support for variables of sort node and pairs of integers. The
-- constructor D is used as N is taken for numbers in S-Expressions.
-- 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 #-}
#if !(MIN_VERSION_base(4,13,0))
#define MonadFail Monad
#endif
module CPSA.Basic.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.Algebra as C
import qualified CPSA.Lib.Utilities as C
import CPSA.Lib.SExpr (SExpr(..), Pos, annotation)
name :: String
name = "basic"
-- An identifier is a variable without any information about its sort
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)
gmerge :: Gen -> Gen -> Gen
gmerge (Gen i) (Gen j) = Gen $ max i j
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)
-- The Basic Crypto Order-Sorted Signature is
-- Sorts: mesg, text, data, name, skey, akey, and string
--
-- Subsorts: text, data, name, skey, akey < mesg
--
-- Operations:
-- cat : mesg X mesg -> mesg Pairing
-- enc : mesg X mesg -> mesg Encryption
-- hash : mesg -> mesg Hashing
-- string : mesg Tag constants
-- ltk : name X name -> skey Long term shared key
-- pubk : name -> akey Public key of principal
-- pubk : string X name -> akey Tagged public key of principal
-- invk : akey -> akey Inverse of asymmetric key
-- Variables of sort string are forbidden.
-- The implementation exploits the isomorphism between order-sorted
-- algebras and many-sorted algebras by adding inclusion operations to
-- produce an equivalent Basic Crypto Many-Sorted Signature. There is
-- an inclusion operation for each subsort of mesg.
-- Sorts: mesg, text, data, name, skey, akey, and string
--
-- Operations:
-- cat : mesg X mesg -> mesg Pairing
-- enc : mesg X mesg -> mesg Encryption
-- hash : mesg -> mesg Hashing
-- string : mesg Tag constants
-- ltk : name X name -> skey Long term shared key
-- pubk : name -> akey Public key of principal
-- pubk : string X name -> akey Tagged public key of principal
-- invk : akey -> akey Inverse of asymmetric key
-- text : text -> mesg Sort text inclusion
-- data : data -> mesg Sort date inclusion
-- name : name -> mesg Sort name inclusion
-- skey : skey -> mesg Sort skey inclusion
-- akey : akey -> mesg Sort akey inclusion
-- In both algebras, invk(invk(t)) = t for all t of sort akey
-- Operations other than the tag constant constructor
data Symbol
= Text -- Text
| Data -- Another text-like item
| Name -- Principal
| Skey -- Symmetric key
| Tag -- Tag atom
| Ltk -- Long term shared symmetric key
| Akey -- Asymmetric key
| Invk -- Inverse of asymmetric key
| Pubk -- Public asymmetric key of a principal
| Cat -- Term concatenation (Pairing really)
| Enc -- Encryption
| Hash -- Hashing
deriving (Show, Eq, Ord, Enum, Bounded)
-- A Basic Crypto Algebra Term
data Term
= I !Id
| C !String -- Tag constants
| F !Symbol ![Term]
| D !Id -- Strand variable
| Z Int -- Strand constant
deriving (Show, Eq, Ord)
-- In this algebra (F Invk [F Invk [t]]) == t is an axiom
-- 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 algebra 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 == Tag
isVar _ = False
-- Note that isVar of (D _) is false.
isStrdVar :: Term -> Bool
isStrdVar (D _) = True
isStrdVar _ = False
subNums :: Term -> Set Term
subNums _ = S.empty
indicator :: Term -> Term -> Maybe Int
indicator _ _ = Nothing
{-
forceVar :: Term -> Term
forceVar (I x) = (I x)
forceVar (C x) = (C x) -- no variable!
forceVar (F Text y) = (F Text y)
forceVar (F Data y) = (F Data y)
forceVar (F Name y) = (F Name y)
forceVar (F Tag [(I x)]) = (F Tag [(I x)])
forceVar (F Skey [(I x)]) = (F Skey [(I x)])
forceVar (F Skey [(F Ltk ns)]) = forceVar (head ns)
forceVar (F Akey [(I x)]) = (F Akey [(I x)])
forceVar (F Akey [F Invk ns]) = forceVar (head ns)
forceVar (F Akey [F Pubk ns]) = forceVar (head ns)
forceVar (F _ ns) = forceVar (head ns) -- for Cat, Enc.
forceVar (D x) = (D x)
forceVar (P x) = (P x) -- no variable!
-}
-- 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 Tag [I x]) = x
varId (D x) = x
varId _ = C.assertError "Algebra.varId: term not a variable with its sort"
isAcquiredVar :: Term -> Bool
isAcquiredVar (I _) = True
isAcquiredVar _ = False
isObtainedVar :: Term -> Bool
isObtainedVar _ = False
-- 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
-- A term is well-formed if it is in the language defined by the
-- following grammar, and variables occur at positions within the term
-- in a way that allows each variable to be assigned one sort.
--
-- The start symbol is T, for all well-formed terms. Terminal symbol
-- I is for variables and terminal symbol C is for quoted strings.
-- The non-terminal symbols are B, K, and T. Symbol B is for base
-- sorted terms, and K is for asymmetric keys.
--
-- T ::= I | C | B | cat(T, T) | enc(T, T) | hash(T)
--
-- B ::= text(I) | data(I) | name(I) | skey(I)
-- | skey(I) | skey(ltk(I, I)) | akey(K)
--
-- K ::= I | pubk(I) | invk(I) | invk(pubk(I))
-- termWellFormed checks 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 t@(F Tag [I x]) =
extendVarEnv xts x t -- Tag 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 Tag [(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 xts (F Hash [t]) =
termWellFormed xts t -- Hashing
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 _) = True
isAtom (F s _) =
s == Text || s == Data || s == Name || s == Skey || s == Akey || s == Tag
isAtom (D _) = False
isAtom (Z _) = False
-- Is the term numeric?
isNum :: Term -> Bool
isNum _ = False
-- Does a variable occur in a term?
occursIn :: Term -> Term -> Bool
occursIn t t' | isVar t = -- Does a variable occur in a term?
subterm (I $ varId t) t'
occursIn t _ =
C.assertError $ "Algebra.occursIn: Bad variable " ++ show t
subterm :: Term -> Term -> Bool
subterm t t' | t == t' =
True
subterm t (F _ u) =
any (subterm t) u
subterm _ _ =
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 Tag [I _]) = f acc t -- Tag 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 keys
foldVars f acc (F Skey [F Ltk [I x, I y]]) =
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 _ acc (F Tag [(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 f acc (F Hash [t]) = -- Hashing
foldVars f acc t
foldVars f acc t@(D _) = f acc t -- Strand variable
foldVars _ acc (Z _) = acc -- Strand constant
foldVars _ _ t = C.assertError $ "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 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 atom a constituent of a term? In other words, is atom among
-- the set of atoms required to construct the term?
constituent :: Term -> Term -> Bool
constituent t t' | isAtom t =
subterm t t'
constituent t _ =
C.assertError $ "Algebra.constituent: Bad atom " ++ show t
-- 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 Cat [t0, t1]) =
ba t0 && ba t1
ba t@(F Enc [t0, t1]) =
S.member t knowns || ba t0 && ba t1
ba t@(F Hash [t1]) =
S.member t knowns || ba t1
ba t = isAtom t && not (S.member t unguessable)
-- Penetrator derivable predicate and checking for unrealized skeletons.
derivable :: Set Term -> Set Term -> Term -> Bool
derivable avoid sent term =
let (knowns, unknowns) = decompose sent avoid in
buildable knowns unknowns term
-- 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
loop unguessable knowns old ((F Hash [_]) : todo) =
loop unguessable knowns old todo -- Hash can't be decomposed
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 t@(F _ _) = t
inv (I _) = error "Algebra.inv: Cannot invert a variable of sort mesg"
inv (C _) = error "Algebra.inv: Cannot invert a tag constant"
inv (D _) = error "Algebra.inv: Cannot invert a variable of sort strd"
inv (Z _) = error "Algebra.inv: Cannot invert a strand constant"
-- Extracts every encryption that is carried by a term along with its
-- encryption key. Note that a hash is treated as a kind of
-- encryption in which the term that is hashed is the 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 Hash [t']) acc =
adjoin (t, [t']) acc
loop _ acc = acc
adjoin x xs
| x `elem` xs = xs
| otherwise = x : xs
escapeSet :: Set Term -> Set Term -> Term -> Maybe (Set Term)
escapeSet ts a ct =
if buildable ts a ct then
Nothing
else
Just $ S.filter f ts
where
f (F Enc [t, key]) =
carriedBy ct t &&
not (buildable ts a (inv key))
f _ = False
isBase :: Term -> Bool
isBase _ = False
isExpr :: Term -> Bool
isExpr _ = False
{-
isExpn :: Term -> Bool
isExpn _ = False
-}
expnInExpr :: Term -> Term -> Bool
expnInExpr _ _ = False
consts :: Term -> [Term] -> [Term]
consts (F Tag _) ts =
loop [] ts
where
loop acc [] = acc
loop acc ((C str): ts) = loop ((C str): acc) ts
loop acc ((F _ subts): ts) = loop acc (subts ++ ts)
loop acc (_ : ts) = loop acc ts
consts _ _ = []
instance C.Term Term where
derivable = derivable
isNum = isNum
subNums = subNums
indicator = indicator
isVar = isVar
isAcquiredVar = isAcquiredVar
isObtainedVar = isObtainedVar
isAtom = isAtom
isStrdVar = isStrdVar
termsWellFormed = termsWellFormed
occursIn = occursIn
foldVars = foldVars
foldCarriedTerms = foldCarriedTerms
carriedBy = carriedBy
constituent = constituent
decryptionKey = decryptionKey
decompose = decompose
buildable = buildable
encryptions = encryptions
escapeSet = escapeSet
loadTerm = loadTerm2
isBase = isBase
isExpr = isExpr
expnInExpr = expnInExpr
consts = consts
-- 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.
newtype Place = Place [Int] deriving (Eq, Show)
-- Returns the places a variable occurs within another term.
places :: Term -> Term -> [Place]
places target source =
f [] [] source
where
f paths path source
| target == source = Place (reverse path) : paths
f paths path (F _ u) =
g paths path 0 u
f paths _ _ = paths
g paths _ _ [] = paths
g paths path i (t : u) =
g (f paths (i: path) t) path (i + 1) u
-- Replace a variable within a term at a given place.
replace :: Term -> Place -> Term -> Term
replace target (Place ints) source =
loop ints source
where
loop [] _ = target
loop (i : path) (F s u) =
if length u <= i then
C.assertError ("Bogus i " ++ show i)
else
F s (C.replaceNth (loop path (u !! i)) i u)
loop _ _ = C.assertError "Algebra.replace: Bad path to variable"
-- 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 _ _ = paths
-- 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 _ _ _ = C.assertError "Algebra.ancestors: Bad path to term"
prefix :: Place -> Place -> Bool
prefix (Place l) (Place l') = L.isPrefixOf l l'
strip :: Place -> Place -> Maybe Place
strip (Place l) (Place l') =
loop l l'
where
loop [] l' = Just $ Place l'
loop (i : l) (i' : l') | i == i' = loop l l'
loop _ _ = Nothing
instance C.Place Term Place where
places = places
carriedPlaces = carriedPlaces
carriedRelPlaces x y _ = carriedPlaces x y
replace = replace
ancestors = ancestors
placeIsPrefixOf = prefix
placeStripPrefix = strip
-- 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')
D x -> -- identifiers to identifier.
case lookup x alist of
Just y -> (alist, gen, D y)
Nothing ->
let (gen', y) = cloneId gen x in
((x, y) : alist, gen', D y)
Z p -> (alist, gen, Z p)
cloneTermList (alist, gen, u) t =
let (alist', gen', t') = cloneTerm (alist, gen) t in
(alist', gen', t' : u)
basePrecursor :: Gen -> Term -> (Gen, Term)
basePrecursor _ _ =
error "Algebra.basePrecursor: Not implemented for the basic algebra"
{-
mostGenPrecursors :: Gen -> Term -> [(Gen, Term)]
mostGenPrecursors _ _ = []
-}
instance C.Gen Term Gen where
origin = origin
gmerge = gmerge
-- genericize = genericize
-- genericVersion = genericize
clone = clone
loadVars = loadVars
basePrecursor = basePrecursor
-- 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 -- Apply axiom
t -> F Invk [t]
idSubst subst (F s u) =
F s (map (idSubst subst) u)
idSubst subst (D x) =
M.findWithDefault (D x) x subst
idSubst _ t@(Z _) = t
-- Is every variable in a term a key in the map?
idMapped :: IdMap -> Term -> Bool
idMapped subst (I x) = M.member x subst
idMapped _ (C _) = True
idMapped subst (F _ u) =
all (idMapped subst) u
idMapped subst (D x) = M.member x subst
idMapped _ (Z _) = True
-- Unification and substitution
newtype Subst = Subst IdMap deriving (Eq, Ord, Show)
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 (Subst s) (D x) =
case M.lookup x s of
Nothing -> D x
Just t -> chase (Subst s) t
chase s (F Invk [t]) = chaseInvk s t
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]
-- 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 (D y) = x == y
occurs _ (Z _) = False
unifyChase :: Term -> Term -> Subst -> Maybe Subst
unifyChase t t' s = unifyTerms (chase s t) (chase s t') s
unifyTerms :: Term -> Term -> Subst -> Maybe Subst
unifyTerms (I x) (I y) (Subst s)
| x == y = Just (Subst s)
| otherwise = Just (Subst $ M.insert x (I y) s)
unifyTerms (I x) t (Subst s)
| occurs x t = Nothing
| otherwise = Just (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 sym u) (F sym' u') s
| sym == sym' = unifyTermLists u u' s
| otherwise = Nothing
unifyTerms (D x) (D y) (Subst s)
| x == y = Just (Subst s)
| otherwise = Just (Subst $ M.insert x (D y) s)
unifyTerms (D x) (Z p) (Subst s) =
Just (Subst $ M.insert x (Z p) s)
unifyTerms t (D x) s = unifyTerms (D x) t s
unifyTerms (Z p) (Z p') s
| p == p' = Just s
| otherwise = Nothing
unifyTerms _ _ _ = Nothing
unifyTermLists :: [Term] -> [Term] -> Subst -> Maybe Subst
unifyTermLists [] [] s = Just s
unifyTermLists (t : u) (t' : u') s =
maybe Nothing (unifyTermLists u u') (unifyChase t t' s)
unifyTermLists _ _ _ = Nothing
-- The exported unifier converts the internal representation of a
-- substitution into the external form using chaseMap.
unify :: Term -> Term -> Subst -> Maybe Subst
unify t t' s =
do
s <- unifyChase t t' s
return $ 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 s u ->
F s (map (substChase subst) u)
t@(D _) -> t
t@(Z _) -> t
destroyer :: Term -> Maybe Subst
destroyer _ = Nothing
-- Stub absence substitution generator.
absenceSubst :: Gen -> [(Term, Term)] -> [(Gen, Subst)]
absenceSubst g [] = [(g, emptySubst)]
absenceSubst _ _ = error "Algebra.absenceSubst: not supported"
instance C.Subst Term Gen Subst where
emptySubst = emptySubst
destroyer = destroyer
substitute = substitute
unify t t' (g, s) =
maybe [] (\s -> [(g, s)]) $ unify t t' s
compose = compose
absenceSubst = absenceSubst
-- Matching and instantiation
newtype Env = Env IdMap deriving (Eq, Ord, Show)
-- An environment may contain an explicit identity mapping, whereas a
-- substitution is erroneous if it has one.
emptyEnv :: Env
emptyEnv = Env emptyIdMap
-- Apply a substitution created my matching
instantiate :: Env -> Term -> Term
instantiate (Env r) t = idSubst r t
-- Is every variable in t in the domain of r?
matched :: Env -> Term -> Bool
matched (Env r) t = idMapped r t
-- Apply a substitution to the range of an environment
substUpdate :: Env -> Subst -> Env
substUpdate (Env r) s =
Env $ M.map (substitute s) r
-- Matching
-- The matcher has the property that when pattern P and term T match
-- then instantiate (match P T emptyEnv) P = T.
match :: Term -> Term -> Env -> Maybe Env
match (I x) t (Env r) =
case M.lookup x r of
Nothing -> Just $ Env $ M.insert x t r
Just t' -> if t == t' then Just $ Env r else Nothing
match (C c) (C c') r = if c == c' then Just r else Nothing
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 (D x) t (Env r) =
case M.lookup x r of
Nothing -> Just $ Env $ M.insert x t r
Just t' -> if t == t' then Just $ Env r else Nothing
match (Z p) (Z p') r = if p == p' then Just r else Nothing
match _ _ _ = Nothing
matchLists :: [Term] -> [Term] -> Env -> Maybe Env
matchLists [] [] r = Just r
matchLists (t : u) (t' : u') r =
maybe Nothing (matchLists u u') (match t t' r)
matchLists _ _ _ = Nothing
-- 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, _) =
C.assertError $ "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 Tag [I x] : _) (y, t)
| x == y = (F Tag [I x], F Tag [t])
loop (D x : _) (y, t)
| x == y = (D x, t)
loop (_ : domain) pair = loop domain pair
strdMatch :: Term -> Int -> Env -> Maybe Env
strdMatch t p env = match t (Z p) env
strdLookup :: Env -> Term -> Maybe Int
strdLookup env t =
case instantiate env t of
Z p -> Just p
_ -> Nothing
strdUpdate :: Env -> (Int -> Int) -> Env
strdUpdate (Env e) f =
Env $ M.map h e
where
h (Z z) = Z $ f z
h t = t
instance C.Env Term Gen Subst Env where
emptyEnv = emptyEnv
instantiate = instantiate
match t t' (g, e) =
maybe [] (\e -> [(g, e)]) $ match t t' e
substitution = substitution
reify = reify
substUpdate = substUpdate
matched = matched
strdMatch t t' (g, e) =
maybe [] (\e -> [(g, e)]) $ strdMatch t t' e
strdLookup = strdLookup
strdUpdate = strdUpdate
-- Term specific loading functions
loadVars :: MonadFail 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 :: MonadFail m => SExpr Pos -> m [(SExpr Pos, SExpr Pos)]
loadVarPair (L _ (x:y:xs)) =
let (t:vs) = reverse (x:y:xs) in
return [(v,t) | v <- reverse vs]
loadVarPair x = fail (shows (annotation x) "Bad variable declaration")
loadVar :: MonadFail 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 =
case sort of
"mesg" -> return (I x)
"text" -> return $ F Text [I x]
"data" -> return $ F Data [I x]
"name" -> return $ F Name [I x]
"skey" -> return $ F Skey [I x]
"akey" -> return $ F Akey [I x]
"tag" -> return $ F Tag [I x]
"strd" -> return (D x)
_ -> fail (shows pos' "Sort " ++ sort ++ " not recognized")
loadVar _ (x,_) = fail (shows (annotation x) "Malformed vars declaration")
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 :: MonadFail 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 :: MonadFail 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")
loadTerm2 :: MonadFail m => [Term] -> Bool -> SExpr Pos -> m Term
loadTerm2 vars _ pos = loadTerm vars pos
-- Load term and check that it is well-formed.
loadTerm :: MonadFail m => [Term] -> SExpr Pos -> m Term
loadTerm vars (S pos s) =
either fail return (loadLookup pos vars s)
loadTerm _ (Q _ t) =
return (F Tag [(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 :: MonadFail m => [(String, LoadFunction m)]
loadDispatch =
[("pubk", loadPubk)
,("privk", loadPrivk)
,("invk", loadInvk)
,("ltk", loadLtk)
,("cat", loadCat)
,("enc", loadEnc)
,("hash", loadHash)
]
-- Atom constructors: pubk privk invk ltk
loadPubk :: MonadFail 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 :: MonadFail 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 :: MonadFail m => LoadFunction m
loadInvk _ vars [S pos s] =
do
t <- loadLookupAkey pos vars s
return $ F Akey [F Invk [I $ varId t]]
loadInvk _ vars [L _ [S _ pubk, S pos s]]
| pubk == "pubk" =
do
t <- loadLookupName pos vars s
return $ F Akey [F Invk [F Pubk [I $ varId t]]]
loadInvk _ vars [L _ [S _ pubk, Q _ c, S pos s]]
| pubk == "pubk" =
do
t <- loadLookupName pos vars s
return $ F Akey [F Invk [F Pubk [C c, I $ varId t]]]
loadInvk _ vars [L _ [S _ privk, S pos s]]
| privk == "privk" =
do
t <- loadLookupName pos vars s
return $ F Akey [F Pubk [I $ varId t]]
loadInvk _ vars [L _ [S _ privk, Q _ c, S pos s]]
| privk == "privk" =
do
t <- loadLookupName pos vars s
return $ F Akey [F Pubk [C c, I $ varId t]]
loadInvk _ vars [L _ [S _ invk, t]]
| invk == "invk" =
do
a <- loadTerm vars t
case a of
F Akey _ -> return a
_ -> fail (shows (annotation t) "Expecting an akey")
loadInvk pos _ _ = fail (shows pos "Malformed invk")
loadLtk :: MonadFail 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")
-- Term constructors: cat enc
loadCat :: MonadFail 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 :: MonadFail 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
loadHash :: MonadFail m => LoadFunction m
loadHash _ vars (l : ls) =
do
ts <- mapM (loadTerm vars) (l : ls)
return $ F Hash [foldr1 (\a b -> F Cat [a, b]) ts]
loadHash pos _ _ = fail (shows pos "Malformed hash")
-- 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 Tag [I x]) = displaySortId "tag" ctx x
displayVar ctx (D x) = displaySortId "strd" ctx x
displayVar _ _ =
C.assertError "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
C.assertError $ "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 Tag [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]
_ -> C.assertError ("Algebra.displayAkey: Bad term " ++ show t)
displayTerm _ (F Tag [(C t)]) = Q () t
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 ctx (F Hash [t]) =
L () (S () "hash" : displayList ctx t)
displayTerm ctx (D x) = displayId ctx x
displayTerm _ (Z z) = N () z
displayTerm _ t = C.assertError ("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 (Subst r) =
map (\(x, t) -> L () [displayTerm ctx' x, displayTerm ctx' t]) r'
where
r' = map (\(x, t) -> (I x, inferSort 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 = 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