cpsa-2.2.5: src/CPSA/Lib/Strand.hs
-- Instance and preskeleton data structures and support functions.
-- 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.
module CPSA.Lib.Strand (Instance, mkInstance, bldInstance, mkListener,
role, env, trace, height, listenerTerm, Sid, Node, mkPreskel,
firstSkeleton, Pair, Preskel, gen, protocol, insts, orderings,
pov, knon, kunique, korig, kcomment, nstrands, kvars, kterms, uniqOrig,
preskelWellFormed, verbosePreskelWellFormed, Strand, GraphStrand,
inst, nodes, Vertex, GraphNode, preds, event, graphNode, strands,
vertex, Gist, gist, contract, augment, inheritRnon,
inheritRunique, addListener, Cause (..), Direction (..), Method
(..), Operation (..), operation, prob, homomorphism, toSkeleton,
generalize, collapse) where
import Control.Monad
import qualified Data.List as L
import qualified Data.Set as S
import Data.Set (Set)
import qualified Data.Maybe as M
import CPSA.Lib.Utilities
import CPSA.Lib.SExpr
import CPSA.Lib.Algebra
import CPSA.Lib.Protocol
{--
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
zb :: Show a => a -> Bool -> Bool
zb a False = z a False
zb _ b = b
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
zl :: Show a => [a] -> [a]
zl a = z (length a) a
zi :: Algebra t p g s e c => Instance t p g s e c -> String
zi inst =
show (map f e)
where
domain = rvars (role inst)
e = reify domain (env inst)
range = map snd e
f (x, t) = (displayTerm (context domain) x,
displayTerm (context range) t)
context ts = addToContext emptyContext ts
--}
-- Compile time switches for expermentation.
-- Sanity check: ensure no role variable occurs in a skeleton.
useCheckVars :: Bool
useCheckVars = False
useThinning :: Bool
useThinning = False -- True
usePruningDuringCollapsing :: Bool
usePruningDuringCollapsing = False -- True
usePruningWhileSolving :: Bool
usePruningWhileSolving = True -- False
usePrunedStrandNotInPOV :: Bool
usePrunedStrandNotInPOV = False -- True
useStrictPrecedesCheck :: Bool
useStrictPrecedesCheck = False -- True
usePruningUnboundCheck :: Bool
usePruningUnboundCheck = False -- True
useNoOrigPreservation :: Bool
useNoOrigPreservation = False -- True
useDeOrigination :: Bool
useDeOrigination = False -- True
-- Instances and Strand Identifiers
-- An Instance is an instance of a role, in the sense that each
-- variable in the role's trace has been replaced by a term. The
-- trace of the instance might be shorter that the role's trace, but
-- only truncating nodes from the end of the trace.
-- A preskeleton stores its strands as a ordered sequence of
-- instances. A strand is the position of an instance in the
-- sequence. Duplicates are allowed in the sequence, as two strands
-- can be instantiated from the same role in the same way.
type Sid = Int -- Strand Identifier
data Algebra t p g s e c => Instance t p g s e c = Instance
{ role :: Role t p g s e c, -- Role from which this was
-- instantiated
env :: !e, -- The environment used to build this
-- instance's trace from its role's
-- trace
trace :: ![Event t p g s e c], -- Instance's trace
height :: !Int } -- Height of the instance
deriving Show
-- Create a fresh instance of the given height. The environment
-- specifies how to map some variables in the role's trace. Unmapped
-- variables are instantiated with fresh variables to avoid naming
-- conflicts.
mkInstance :: Algebra t p g s e c => g -> Role t p g s e c ->
e -> Int -> (g, Instance t p g s e c)
mkInstance gen role env height =
let trace = rtrace role
rheight = length trace in
if height < 1 || height > rheight then
error "Strand.mkInstance: Bad strand height"
else
let (gen', env') = grow (rvars role) gen env
trace' = map (evtMap $ instantiate env') (take height trace) in
-- Ensure every variable in the range of the environment
-- occurs in the trace.
case bldInstance role trace' gen' of
(gen'', inst) : _ -> (gen'', inst)
[] -> error "Strand.mkInstance: Not an instance"
-- For each term that matches itself in the environment, extend the
-- mapping so that the term maps to one with a fresh set of variables.
-- It is an error if a variable in one of the terms is explicitly
-- mapped to itself in the initial environment.
grow :: Algebra t p g s e c => [t] -> g -> e -> (g, e)
grow [] gen env = (gen, env)
grow (t : ts) gen env =
case match t t (gen, env) of
[] -> grow ts gen env -- Term already mapped
_ -> -- Otherwise make a fresh mapping
let (gen', t') = clone gen t in
case match t t' (gen', env) of
(gen'', env') : _ -> grow ts gen'' env'
[] -> error "Strand.grow: Internal error"
-- Build an instance from a role and a trace. Returns the empty list
-- if the trace is not an instance of the given role.
bldInstance :: Algebra t p g s e c => Role t p g s e c ->
Trace t p g s e c -> g -> [(g, Instance t p g s e c)]
bldInstance _ [] _ = error "Strand.bldInstance: Bad trace"
bldInstance role trace gen =
loop (rtrace role) trace (gen, emptyEnv) -- Loop builds env
where
loop _ [] (gen, env) = -- Trace can be shorter than role's trace
[(gen, makeInstance role env trace)]
loop (In t : c) (In t' : c') ge =
do
env <- match t t' ge
loop c c' env
loop (Out t : c) (Out t' : c') ge =
do
env <- match t t' ge
loop c c' env
loop _ _ _ = []
makeInstance :: Algebra t p g s e c => Role t p g s e c -> e ->
Trace t p g s e c -> Instance t p g s e c
makeInstance role env trace =
Instance { role = role,
env = specialize env,
trace = trace,
height = length trace }
-- This is the only place a role is generated with an empty name.
-- This is what marks a strand as a listener.
mkListener :: Algebra t p g s e c => g -> t ->
(g, Instance t p g s e c)
mkListener gen term =
(gen'', Instance { role = mkRole "" vars [In rterm, Out rterm]
[] [] [] False,
env = env,
trace = [In term, Out term],
height = 2 })
where
(gen', rterm) = clone gen term -- Make role term
vars = addVars [] rterm -- Collect vars in role term
(gen'', env) =
case match rterm term (gen', emptyEnv) of
(gen'', env) : _ -> (gen'', env)
[] -> error msg
msg = "Strand.mkListener: cannot generate an environment."
-- Add to set s the variables that are in the range of instance i
addIvars :: Algebra t p g s e c => Set t -> Instance t p g s e c -> Set t
addIvars s i =
foldl g s (reify (rvars (role i)) (env i))
where
g s (_, t) = foldVars (flip S.insert) s t
listenerTerm :: Algebra t p g s e c => Instance t p g s e c -> Maybe t
listenerTerm inst =
if "" /= rname (role inst) then
Nothing -- Not a listener strand
else
Just $ evtTerm (trace inst !! 0)
-- Nodes, Pairs, and Graphs
-- A node is composed of two integers, a strand identifier and a
-- position. The position identifies an event in the strand's trace.
-- The second integer must be non-negative and less than the strand's
-- height
type Node = (Sid, Int)
-- A pair gives an ordering of two nodes, meaning the first node is
-- before the second one.
type Pair = (Node, Node)
-- Graphs of preskeletons
-- A strand is what is referred to by a strand ID.
data GraphStrand e i -- e for event, i for instance
= GraphStrand { inst :: i,
nodes :: [GraphNode e i],
sid :: Sid }
-- The Strand ID determines equality and orderings
instance Eq (GraphStrand e i) where
s0 == s1 = sid s0 == sid s1
instance Ord (GraphStrand e i) where
compare s0 s1 = compare (sid s0) (sid s1)
instance Show (GraphStrand e i) where
showsPrec _ s = shows (sid s)
-- A vertex is what is referred to by a node.
data GraphNode e i -- e for event, i for instance
= GraphNode { event :: e,
preds :: [GraphNode e i], -- Immediate preds including
strand :: GraphStrand e i, -- strand succession edges
pos :: Int } -- The position of the node in the strand
-- The node determines equality and orderings
instance Eq (GraphNode e i) where
n0 == n1 = (strand n0, pos n0) == (strand n1, pos n1)
instance Ord (GraphNode e i) where
compare n0 n1 = compare (strand n0, pos n0) (strand n1, pos n1)
instance Show (GraphNode e i) where
showsPrec _ n = let (s, p) = graphNode n in
showChar '(' . shows s . showString ", " .
shows p . showChar ')'
-- The node of a vertex
graphNode :: GraphNode e i -> Node
graphNode n = (sid (strand n), pos n)
type GraphEdge e i = (GraphNode e i, GraphNode e i)
-- The pair of an edge
graphPair :: GraphEdge e i -> Pair
graphPair (n0, n1) = (graphNode n0, graphNode n1)
graphEdges :: [GraphStrand e i] -> [GraphEdge e i]
graphEdges strands =
[ (dst, src) | s <- strands, src <- nodes s, dst <- preds src ]
data Graph e i
= Graph { gstrands :: [GraphStrand e i],
gedges :: [GraphEdge e i] }
-- The graph associated with a preskeleton
graph :: (i -> [d]) -> (i -> Int) -> [i] -> [Pair] -> Graph d i
graph trace height insts pairs =
Graph { gstrands = strands,
gedges = map getEdge pairs }
where
strands = [ GraphStrand { inst = inst,
nodes = nodes !! sid,
sid = sid } |
(sid, inst) <- zip [0..] insts ]
nodes = [ [ GraphNode { event = trace (inst strand) !! pos,
preds = preds (sid, pos),
strand = strand,
pos = pos } |
pos <- nats (height (inst strand)) ] |
(sid, strand) <- zip [0..] strands ]
preds n = map getNode (entry n)
getNode (s, p) = nodes !! s !! p
getEdge (n0, n1) = (getNode n0, getNode n1)
entry n = enrich n [ n0 | (n0, n1) <- pairs, n1 == n ]
-- add strand succession edges
enrich (s, p) ns
| p > 0 = (s, p - 1) : ns
| otherwise = ns
-- Does start node precede end node?
graphPrecedes :: GraphNode e i -> GraphNode e i -> Bool
graphPrecedes start end =
let predecessors = preds end in
any (== start) predecessors || any (graphPrecedes start) predecessors
-- Compute the transitive reduction
graphReduce :: [GraphEdge e i] -> [GraphEdge e i]
graphReduce orderings =
filter essential orderings
where
essential (dst, src) =
loop dst (L.delete dst (preds src)) [src]
loop _ [] _ = True -- No other path found
loop dst (n : ns) seen
| n == dst = False -- There is another path
| elem n seen = loop dst ns seen
| otherwise = loop dst (preds n ++ ns) (n : seen)
-- Compute the transitive closure
-- This routine returns pairs that are not well ordered.
-- Deal with it!
graphClose :: [GraphEdge e i] -> [GraphEdge e i]
graphClose orderings =
filter (not . sameStrands) (loop orderings False orderings)
where
loop orderings False [] = orderings
loop orderings True [] =
loop orderings False orderings -- restart loop
loop orderings repeat ((n0, n1) : pairs) =
inner orderings repeat pairs [(n, n1) | n <- preds n0]
inner orderings repeat pairs [] =
loop orderings repeat pairs
inner orderings repeat pairs (p : rest)
| elem p orderings = inner orderings repeat pairs rest
| otherwise = inner (p : orderings) True pairs rest
sameStrands (n0, n1) = strand n0 == strand n1
-- Preskeltons
data Algebra t p g s e c => Preskel t p g s e c = Preskel
{ gen :: !g,
protocol :: Prot t p g s e c,
insts :: ![Instance t p g s e c],
strands :: ![Strand t p g s e c],
orderings :: ![Pair],
edges :: ![Edge t p g s e c],
knon :: ![t], -- A list of atoms
kunique :: ![t], -- A list of atoms
kcomment :: [SExpr ()], -- Comments from the input
korig :: ![(t, [Node])], -- This is an association list with a
-- pair for each element of kunique.
-- The value associated with a term
-- is a list of the nodes at which it
-- originates--the term's provenance.
pov :: Maybe (Preskel t p g s e c), -- Point of view, the
-- original problem statement.
strandids :: ![Sid],
tc :: [Pair], -- Transitive closure of orderings
-- Used only during generalization
operation :: Operation t p g s e c,
prob :: [Sid] } -- A map from the strands in the original
deriving Show -- problem statement, the pov, into
-- these strands.
-- The pov skeleton is the only skeleton that should have Nothing in
-- its pov field.
type Strand t p g s e c
= GraphStrand (Event t p g s e c) (Instance t p g s e c)
type Vertex t p g s e c
= GraphNode (Event t p g s e c) (Instance t p g s e c)
type Edge t p g s e c
= GraphEdge (Event t p g s e c) (Instance t p g s e c)
-- Data structure for tracking the causes for the creation of
-- preskeletons.
data Algebra t p g s e c => Cause t p g s e c
= Cause Direction Node t (Set t)
deriving Show
data Direction = Encryption | Nonce deriving Show
data Algebra t p g s e c => Method t p g s e c
= Deleted Node
| Weakened Pair
| Separated t
| Forgot t deriving Show
-- The operation used to generate the preskeleteton is either new via
-- the loader, a contraction, a regular augmentation, a listener
-- augmentation, or a mininization. The augmentation includes a role
-- name and instance height.
data Algebra t p g s e c => Operation t p g s e c
= New
| Contracted s (Cause t p g s e c)
| Displaced Int Int String Int (Cause t p g s e c)
| AddedStrand String Int (Cause t p g s e c)
| AddedListener t (Cause t p g s e c)
| Generalized (Method t p g s e c)
| Collapsed Int Int
deriving Show
-- Create a preskeleton. The point of view field is not filled in.
-- This version is exported for use by the loader. This preskeleton
-- must be consumed by firstSkeleton.
mkPreskel :: Algebra t p g s e c => g -> Prot t p g s e c ->
[Instance t p g s e c] -> [Pair] -> [t] -> [t] ->
[SExpr ()] -> Preskel t p g s e c
mkPreskel gen protocol insts orderings non unique comment =
k { kcomment = comment }
where
k = newPreskel gen protocol insts orderings non unique New prob Nothing
prob = strandids k -- Fixed point on k is okay.
-- Strand functions
strandInst :: Algebra t p g s e c => Preskel t p g s e c ->
Sid -> Instance t p g s e c
strandInst k strand = insts k !! strand
nstrands :: Algebra t p g s e c => Preskel t p g s e c -> Int
nstrands k = length (strandids k)
-- Convert the preskeleton made by the loader into the first skeleton
-- used in the search.
firstSkeleton :: Algebra t p g s e c => Preskel t p g s e c ->
[Preskel t p g s e c]
firstSkeleton k =
do
k <- wellFormedPreskel k
k' <- toSkeleton False k -- only k' should have pov = Nothing
return $ k' { prob = strandids k', pov = Just k' }
-- Create a preskeleton. The node ordering relation is put into the
-- preds field of each instance in this function. The maybe uniquely
-- originating term data is also filled in. This version is used
-- within this module.
newPreskel :: Algebra t p g s e c => g -> Prot t p g s e c ->
[Instance t p g s e c] -> [Pair] -> [t] -> [t] ->
Operation t p g s e c -> [Sid] ->
Maybe (Preskel t p g s e c) -> Preskel t p g s e c
newPreskel gen protocol insts orderings non unique oper prob pov =
let orderings' = L.nub orderings
unique' = L.nub unique
g = graph trace height insts orderings'
strands = gstrands g
edges = gedges g
orig = map (originationNodes strands) unique'
tc = filter pairWellOrdered (graphClose $ graphEdges strands)
k = Preskel { gen = gen,
protocol = protocol,
insts = insts,
strands = strands,
orderings = orderings',
edges = edges,
knon = L.nub non,
kunique = unique',
kcomment = [],
korig = orig,
tc = map graphPair tc,
strandids = nats (length insts),
operation = oper,
prob = prob,
pov = pov } in
if useCheckVars then
checkVars k
else k
checkVars :: Algebra t p g s e c => Preskel t p g s e c ->
Preskel t p g s e c
checkVars k =
foldl f k rolevars
where
skelvars = S.fromList $ kvars k
rolevars = concatMap (rvars . role) (insts k)
f k v
| S.member v skelvars =
error ("Strand.checkVars: role var in skel " ++ show k)
| otherwise = k
vertex :: Algebra t p g s e c => Preskel t p g s e c -> Node ->
Vertex t p g s e c
vertex k (s, p) =
nodes (strands k !! s) !! p
originationNodes :: Algebra t p g s e c => [Strand t p g s e c] ->
t -> (t, [Node])
originationNodes strands u =
(u, [ (sid strand, p) |
strand <- reverse strands,
p <- M.maybeToList $ originationPos u (trace (inst strand)) ])
uniqOrig :: Algebra t p g s e c => Preskel t p g s e c -> [t]
uniqOrig k =
do
(t, [_]) <- reverse (korig k)
return t
-- A preskeleton is well formed if the ordering relation is acyclic,
-- each atom declared to be uniquely-originating is carried in
-- some preskeleton term, and every variable that occurs in each base
-- term declared to be non-originating occurs in some preskeleton
-- term, and the atom must never be carried by any term, and
-- every uniquely originating role term mapped by an instance is
-- mapped to a term that originates on the instance's strand.
preskelWellFormed :: Algebra t p g s e c => Preskel t p g s e c -> Bool
preskelWellFormed k =
varSubset (knon k) terms &&
all nonCheck (knon k) &&
all uniqueCheck (kunique k) &&
wellOrdered k && acyclicOrder k &&
roleOrigCheck k
where
terms = kterms k
nonCheck t = all (not . carriedBy t) terms
uniqueCheck t = any (heldBy t) terms
-- Do notation friendly preskeleton well formed check.
wellFormedPreskel :: (Monad m, Algebra t p g s e c) =>
Preskel t p g s e c -> m (Preskel t p g s e c)
wellFormedPreskel k
| preskelWellFormed k = return k
| otherwise = fail "preskeleton not well formed"
-- A version of preskelWellFormed that explains why a preskeleton is
-- not well formed.
verbosePreskelWellFormed :: (Monad m, Algebra t p g s e c) =>
Preskel t p g s e c -> m ()
verbosePreskelWellFormed k =
do
failwith "a variable in non-orig is not in some trace"
$ varSubset (knon k) terms
mapM_ nonCheck $ knon k
mapM_ uniqueCheck $ kunique k
failwith "ordered pairs not well formed" $ wellOrdered k
failwith "cycle found in ordered pairs" $ acyclicOrder k
failwith "an inherited unique doesn't originate in its strand"
$ roleOrigCheck k
where
terms = kterms k
nonCheck t =
failwith (showString "non-orig " $ showst t " carried")
$ all (not . carriedBy t) terms
uniqueCheck t =
failwith (showString "uniq-orig " $ showst t " not held")
$ any (heldBy t) terms
failwith :: Monad m => String -> Bool -> m ()
failwith msg test =
case test of
True -> return ()
False -> fail msg
showst :: Algebra t p g s e c => t -> ShowS
showst t =
shows $ displayTerm (addToContext emptyContext [t]) t
-- Do the nodes in the orderings have the right direction?
wellOrdered :: Algebra t p g s e c => Preskel t p g s e c -> Bool
wellOrdered k =
all pairWellOrdered (edges k)
pairWellOrdered :: Algebra t p g s e c => Edge t p g s e c -> Bool
pairWellOrdered (n0, n1) =
case (event n0, event n1) of
(Out _, In _) -> True
_ -> False
-- The terms used in the strands in this preskeleton.
-- Should this return a set, or a multiset?
kterms :: Algebra t p g s e c => Preskel t p g s e c -> [t]
kterms k = iterms (insts k)
-- The terms used in a list of instances.
iterms :: Algebra t p g s e c => [Instance t p g s e c] -> [t]
iterms insts =
foldl addSTerms [] insts
where
addSTerms ts s =
foldl (flip $ adjoin . evtTerm) ts (trace s)
-- The node orderings form an acyclic order if there are no cycles.
-- Use depth first search to detect cycles. A graph with no node with
-- an indegree of zero is cyclic and must not be checked with depth
-- first search.
acyclicOrder :: Algebra t p g s e c => Preskel t p g s e c -> Bool
acyclicOrder k =
all (not . backEdge numbering) edges
where
edges = graphEdges (strands k)
-- The starting set contains the last node in every strand
start = map (last . nodes) (strands k)
-- Remove nodes that have non-zero indegree
start' = foldl (flip L.delete) start (map fst edges)
numbering = dfs preds start'
-- Variables in this preskeleton, excluding ones in roles, and ones
-- that only occur in a cause.
kvars :: Algebra t p g s e c => Preskel t p g s e c -> [t]
kvars k =
S.elems $ foldl addIvars S.empty (insts k)
-- Ensure each role unique origination assumption mapped by an
-- instance originates in the instance's strand.
roleOrigCheck :: Algebra t p g s e c => Preskel t p g s e c -> Bool
roleOrigCheck k =
all strandRoleOrig (strands k) -- Check each strand
where
strandRoleOrig strand = -- Check each role unique used in strand
all (uniqRoleOrig strand) $ ruorig $ role $ inst strand
uniqRoleOrig strand (ru, pos)
| pos < height (inst strand) =
case lookup (instantiate (env $ inst strand) ru) (korig k) of
Nothing -> True -- role term not mapped
Just ns -> any (\(s, p)-> sid strand == s && p == pos) ns
| otherwise = True
-- Isomorphism Check
-- Are two skeletons equivalent? Two skeletons are equivalent if they
-- are isomorphic. A key efficiency requirement in the implementation
-- of the cryptograhic protocol shapes analysis algorithm is to ensure
-- only one member of each equivalence class of skeletons is analyzed,
-- and the results of that analysis is immediately used for all other
-- members of the equivalence class.
-- To meet this requirement, a list of skeletons that have been seen
-- is maintained. Before a skeleton is put on a to do list for
-- analysis, it is checked to see if it is ismorphic to one already
-- seen. If so, the results of the analysis for the ismorphic
-- skeleton is used instead of putting the skeleton on the to do list.
-- Once a skeleton has been printed, the only reason for saving it is
-- for isomorphism checking. The isomorphism check is performed
-- frequently, so an specialized data structure is used. The gist of
-- a skeleton is all that is needed for the test for equivalence.
data Algebra t p g s e c => Gist t p g s e c = Gist
{ ggen :: g,
gtraces :: [(Int, Trace t p g s e c)],
gorderings :: [Pair],
gnon :: [t], -- A list of non-originating terms
gunique :: [t], -- A list of uniquely-originating terms
nvars :: !Int, -- Number of variables
ntraces :: !Int, -- Number of traces
norderings :: !Int, -- Number of orderings
nnon :: !Int, -- Number of non-originating terms
nunique :: !Int } -- Number of uniquely-originating terms
deriving Show
gist :: Algebra t p g s e c => Preskel t p g s e c ->
Gist t p g s e c
gist k =
Gist { ggen = gen k,
gtraces = gtraces,
gorderings = gorderings,
gnon = gnon,
gunique = gunique,
nvars = length (kvars k),
ntraces = length gtraces,
norderings = length gorderings,
nnon = length gnon,
nunique = length gunique }
where
gtraces = map (\i -> (height i, trace i)) (insts k)
gorderings = orderings k
gnon = knon k
gunique = kunique k
-- Test to see if two preskeletons are isomorphic
-- First, ensure the two preskeletons have:
-- 1. The same number of variables
-- 2. The same number of strands
-- 3. The same number of node orderings
-- 4. The same number of terms in knon and kunique
-- Next compute the plausible permutations and substitutions.
-- Next, for each permutation of the strands, eliminate the ones that
-- map a strand to another of a different length, and don't cause the
-- node orderings to be equal.
-- For permutations that meet the previous conditions, see if there is
-- a renaming that maps every strand trace in one preskeleton into the
-- appropriate one in the other preskeleton. Finally, check to see if
-- the renaming works in the nr and ur terms.
instance Algebra t p g s e c => Eq (Gist t p g s e c) where
g == g' = isomorphic g g'
isomorphic :: Algebra t p g s e c => Gist t p g s e c ->
Gist t p g s e c -> Bool
isomorphic g g' =
nvars g == nvars g' &&
ntraces g == ntraces g' &&
norderings g == norderings g' &&
nnon g == nnon g' &&
nunique g == nunique g' &&
any (tryPerm g g') (permutations g g')
-- Extend a permutation while extending a substitution
-- Extend by matching later strands first
permutations :: Algebra t p g s e c => Gist t p g s e c ->
Gist t p g s e c -> [((g, e), [Sid])]
permutations g g' =
map rev $ perms (ggen g', emptyEnv)
(reverse $ gtraces g)
(reverse $ nats $ ntraces g)
where
perms env [] [] = [(env, [])]
perms env ((h, c):hcs) xs =
[ (env'', x:ys) |
x <- xs,
let (h', c') = gtraces g' !! x,
h == h',
env' <- jibeTraces c c' env,
(env'', ys) <- perms env' hcs (L.delete x xs) ]
perms _ _ _ = error "Strand.permutations: lists not same length"
rev (env, xs) = (env, reverse xs)
-- Length of matched traces must agree.
jibeTraces :: Algebra t p g s e c => Trace t p g s e c ->
Trace t p g s e c -> (g, e) -> [(g, e)]
jibeTraces [] [] ge = [ge]
jibeTraces (In t : c) (In t' : c') ge =
do
env <- match t t' ge
jibeTraces c c' env
jibeTraces (Out t : c) (Out t' : c') ge =
do
env <- match t t' ge
jibeTraces c c' env
jibeTraces _ _ _ = []
{-
-- Here is the permutation algorithm used
permutations :: Int -> [[Int]]
permutations n =
perms (nats n)
where
perms [] = [[]]
perms xs = [ x:ys | x <- xs, ys <- perms (L.delete x xs) ]
-- Here is the usual algorithm
-- Returns a list of all the permutations of the natural numbers less
-- that the argument. The identity permutation is the first one in
-- the returned list. The code is based on a function in Haskell 1.3.
permutations :: Int -> [[Int]]
permutations n =
perms (nats n)
where
perms [] = [[]]
perms (x:xs) = [zs | ys <- perms xs, zs <- interleave x ys]
interleave x [] = [[x]]
interleave x (y:ys) = (x:y:ys) : map (y:) (interleave x ys)
-}
tryPerm :: Algebra t p g s e c => Gist t p g s e c ->
Gist t p g s e c -> ((g, e), [Sid]) -> Bool
tryPerm g g' (env, perm) =
checkOrigs g g' env &&
containsMapped (permutePair perm) (gorderings g') (gorderings g)
-- containsMapped f xs ys is true when list xs contains each element
-- in ys after being mapped with function f.
containsMapped :: Eq a => (a -> a) -> [a] -> [a] -> Bool
containsMapped f xs ys =
all (flip elem xs) (map f ys)
permutePair :: [Sid] -> Pair -> Pair
permutePair perm (n, n') = (permuteNode perm n, permuteNode perm n')
permuteNode :: [Sid] -> Node -> Node
permuteNode perm (strand, pos) = (perm !! strand, pos)
checkOrigs :: Algebra t p g s e c => Gist t p g s e c ->
Gist t p g s e c -> (g, e) -> Bool
checkOrigs g g' env =
not (null
[ env'' |
env' <- checkOrig env (gnon g) (gnon g'),
env'' <- checkOrig env' (gunique g) (gunique g'),
matchRenaming env'' ])
-- Try all permutations as done above
checkOrig :: Algebra t p g s e c => (g, e) -> [t] -> [t] -> [(g, e)]
checkOrig env [] [] = [env]
checkOrig env (t:ts) ts' =
do
t' <- ts'
env' <- match t t' env
checkOrig env' ts (L.delete t' ts')
checkOrig _ _ _ = error "Strand.checkOrig: lists not same length"
-- Preskeleton Reduction System (PRS)
-- The PRS reduces a preskeleton to a list of skeletons. Along the way,
-- it applies the associtated homomorphism to a node and computes a
-- substitution. Thus if skel (k, n, empty) = [(k', n', sigma)], then
-- phi,sigma is a homomorphism from k to k', n' = phi(n).
type PRS t p g s e c = (Preskel t p g s e c, -- Parent
Preskel t p g s e c, -- Potential cohort member
Node, -- Image of test node in member
[Sid], -- Strand map part of homomorphism
s) -- Substition part of homomorphism
-- Extract the protential cohort member from a PRS.
skel :: Algebra t p g s e c => PRS t p g s e c -> Preskel t p g s e c
skel (_, k, _, _, _) = k
-- Returns the preskeletons that result from applying a substitution.
-- If validate is True, preskeletons that fail to preserve the nodes
-- at which each maybe uniquely originating term originates are filter
-- out.
ksubst :: Algebra t p g s e c => Bool -> PRS t p g s e c ->
(g, s) -> [PRS t p g s e c]
ksubst validate (k0, k, n, phi, hsubst) (gen, subst) =
do
(gen', insts') <- foldMapM (substInst subst) gen (insts k)
let non' = map (substitute subst) (knon k)
let unique' = map (substitute subst) (kunique k)
let operation' = substOper subst (operation k)
let k' = newPreskel gen' (protocol k) insts'
(orderings k) non' unique' operation' (prob k) (pov k)
k' <- wellFormedPreskel k'
case not validate || validateMappingSubst k0 phi subst k' of
True -> return (k0, k', n, phi, compose subst hsubst)
False -> fail ""
-- Monad version of mapAccumR
foldMapM :: Monad m => (a -> b -> m (a, c)) -> a -> [b] -> m (a, [c])
foldMapM _ acc [] = return (acc, [])
foldMapM f acc (x:xs) =
do
(acc', xs') <- foldMapM f acc xs
(acc'', x') <- f acc' x
return (acc'', x':xs')
substInst :: Algebra t p g s e c => s -> g -> Instance t p g s e c ->
[(g, Instance t p g s e c)]
substInst subst gen i =
bldInstance (role i) (map (evtMap $ substitute subst) (trace i)) gen
substOper :: Algebra t p g s e c => s ->
Operation t p g s e c ->
Operation t p g s e c
substOper _ New = New
substOper subst (Contracted s cause) =
Contracted (compose subst s) (substCause subst cause)
substOper _ m@(Displaced _ _ _ _ _) = m
substOper subst (AddedStrand role height cause) =
AddedStrand role height (substCause subst cause)
substOper subst (AddedListener t cause) =
AddedListener (substitute subst t) (substCause subst cause)
substOper _ m@(Generalized _) = m
substOper _ m@(Collapsed _ _) = m
substCause :: Algebra t p g s e c => s ->
Cause t p g s e c ->
Cause t p g s e c
substCause subst (Cause dir n t escape) =
Cause dir n (substitute subst t) (S.map (substitute subst) escape)
-- A compression (s is to be eliminated)
compress :: Algebra t p g s e c => Bool -> PRS t p g s e c ->
Sid -> Sid -> [PRS t p g s e c]
compress validate (k0, k, n, phi, hsubst) s s' =
do
let perm = updatePerm s s' (strandids k)
orderings' <- normalizeOrderings validate
(permuteOrderings perm (orderings k))
let k' =
newPreskel
(gen k)
(protocol k)
(deleteNth s (insts k))
orderings'
(knon k)
(kunique k)
(operation k)
(updateProb perm (prob k))
(pov k)
k'' <- wellFormedPreskel k'
return (k0, k'', permuteNode perm n, map (perm !!) phi, hsubst)
permuteOrderings :: [Sid] -> [Pair] -> [Pair]
permuteOrderings perm orderings = map (permutePair perm) orderings
updatePerm :: Int -> Int -> [Sid] -> [Sid]
updatePerm old new perm =
map f perm
where
f i =
let j = if old == i then new else i in
if j > old then j - 1 else j
-- Eliminates implied intrastrand orderings and fails if it finds a
-- reverse intrastrand ordering when flag is true.
normalizeOrderings :: Bool -> [Pair] -> [[Pair]]
normalizeOrderings True orderings =
loop [] orderings
where
loop acc [] = [acc]
loop acc (p@((s0, p0), (s1, p1)) : ps)
| s0 /= s1 = loop (p : acc) ps
| p0 < p1 = loop acc ps
| otherwise = []
normalizeOrderings False orderings =
[filter (\ ((s0, _), (s1, _)) -> s0 /= s1) orderings]
updateProb :: [Sid] -> [Sid] -> [Sid]
updateProb mapping prob =
map (mapping !!) prob
-- Hulling or Ensuring Unique Origination
-- This is the starting point of the Preskeleton Reduction System
hull :: Algebra t p g s e c => Bool -> PRS t p g s e c ->
[PRS t p g s e c]
hull prune prs =
loop (korig $ skel prs)
where
-- No uniques originate on more than one strand
loop [] = enrich prune prs
-- Found a pair that needs hulling
loop ((u, (s, p) : (s', p') : _) : _)
| useDeOrigination =
hullByCompressing prune prs s s' ++
hullByDeOrigination prune prs u (s, p) (s', p')
| otherwise =
hullByCompressing prune prs s s'
loop(_ : orig) = loop orig
hullByCompressing :: Algebra t p g s e c => Bool -> PRS t p g s e c ->
Sid -> Sid -> [PRS t p g s e c]
hullByCompressing prune prs s s' =
do
(s'', s''', subst) <- unifyStrands (skel prs) s s'
prs <- ksubst False prs subst
prs' <- compress True prs s'' s'''
hull prune prs'
-- See if two strands unify. They can be of differing heights. The
-- second strand returned may be longer.
unifyStrands :: Algebra t p g s e c => Preskel t p g s e c ->
Sid -> Sid -> [(Sid, Sid, (g, s))]
unifyStrands k s s' =
let i = strandInst k s
i' = strandInst k s' in
if height i > height i' then
unifyStrands k s' s
else
do
(gen', subst) <- unifyTraces (trace i) (trace i') (gen k, emptySubst)
return (s, s', (gen', subst))
-- Unify traces where the first trace is allowed to be shorter than
-- the second trace.
unifyTraces :: Algebra t p g s e c => Trace t p g s e c ->
Trace t p g s e c -> (g, s) -> [(g, s)]
unifyTraces [] _ subst = [subst]
unifyTraces (In t : c) (In t' : c') subst =
do
s <- unify t t' subst
unifyTraces c c' s
unifyTraces (Out t : c) (Out t' : c') subst =
do
s <- unify t t' subst
unifyTraces c c' s
unifyTraces _ _ _ = []
-- De-Origination
hullByDeOrigination :: Algebra t p g s e c => Bool -> PRS t p g s e c ->
t -> Node -> Node -> [PRS t p g s e c]
hullByDeOrigination prune prs u (s, p) (s', p') =
do
subst <- deOrig (skel prs) u (s, p) ++ deOrig (skel prs) u (s', p')
prs <- ksubst False prs subst
hull prune prs
deOrig :: Algebra t p g s e c => Preskel t p g s e c -> t -> Node -> [(g, s)]
deOrig k u (s, p) =
[ (g, s) |
let tr = trace $ strandInst k s,
e <- take p tr,
t <- inbnd e,
subterm <- S.toList $ foldCarriedTerms (flip S.insert) S.empty t,
(g, s) <- unify u subterm (gen k, emptySubst),
not $ originates (substitute s u) (map (evtMap $ substitute s) tr) ]
-- Consider inbound messages only
inbnd :: Algebra t p g s e c => Event t p g s e c -> [t]
inbnd (In t) = [t]
inbnd (Out _) = []
-- Order Enrichment
-- Adds orderings so that a skeleton respects origination.
enrich :: Algebra t p g s e c => Bool -> PRS t p g s e c ->
[PRS t p g s e c]
enrich prune (k0, k, n, phi, hsubst) =
let o = foldl (addOrderings k) (orderings k) (kunique k) in
if length o == length (orderings k) then
maybePrune prune (k0, k, n, phi, hsubst) -- Nothing to add
else
do
let k' =
newPreskel
(gen k)
(protocol k)
(insts k)
o
(knon k)
(kunique k)
(operation k)
(prob k)
(pov k)
k' <- wellFormedPreskel k'
maybePrune prune (k0, k', n, phi, hsubst)
maybePrune :: Algebra t p g s e c => Bool -> PRS t p g s e c ->
[PRS t p g s e c]
maybePrune True prs
| useThinning = thin prs
| otherwise = prune prs
maybePrune False prs = reduce prs
origNode :: Algebra t p g s e c => Preskel t p g s e c ->
t -> Maybe Node
origNode k t =
case lookup t (korig k) of
Nothing -> error "Strand.origNode: term not in kunique"
Just [] -> Nothing
Just [n] -> Just n
Just _ -> error "Strand.origNode: not a hulled skeleton"
addOrderings :: Algebra t p g s e c => Preskel t p g s e c ->
[Pair] -> t -> [Pair]
addOrderings k orderings t =
case origNode k t of
Nothing -> orderings
Just n@(s, _) ->
foldl f orderings (L.delete s (strandids k))
where
f orderings s =
case gainedPos t (trace (strandInst k s)) of
Nothing -> orderings
Just pos -> adjoin (n, (s, pos)) orderings
-- At what position is a term gained in a trace?
gainedPos :: Algebra t p g s e c => t ->
Trace t p g s e c -> Maybe Int
gainedPos t c =
loop 0 c
where
loop _ [] = Nothing -- Term is not carried
loop pos (Out t' : c)
| t `heldBy` t' = Nothing -- Term is not gained
| otherwise = loop (pos + 1) c
loop pos (In t' : c)
| t `heldBy` t' = Just pos -- Found it
| otherwise = loop (pos + 1) c
-- Redundant Strand Elimination (also known as pruning)
prune :: Algebra t p g s e c => PRS t p g s e c -> [PRS t p g s e c]
prune prs =
pruneStrands prs (reverse $ strandids $ skel prs) (strandids $ skel prs)
pruneStrands :: Algebra t p g s e c => PRS t p g s e c ->
[Sid] -> [Sid] -> [PRS t p g s e c]
pruneStrands prs [] _ =
reduce prs -- No redundant strands found
pruneStrands prs (_:ss) [] =
pruneStrands prs ss (strandids $ skel prs)
pruneStrands prs (s:ss) (s':ss')
| s == s' = pruneStrands prs (s:ss) ss'
| otherwise =
case pruneStrand prs s s' of
[] -> pruneStrands prs (s:ss) ss' -- Try next pair
prss -> -- Success
do
prs <- prss
prune prs
-- Strand s is redundant if there is an environment that maps
-- the trace of s into a prefix of the trace of s', but changes
-- no other traces in the preskeleton.
pruneStrand :: Algebra t p g s e c => PRS t p g s e c ->
Sid -> Sid -> [PRS t p g s e c]
pruneStrand prs s s' =
do
let k = skel prs
case usePrunedStrandNotInPOV && elem s (prob k) && elem s' (prob k) of
True -> fail "" -- Strands s and s' in image of POV skeleton
False -> return ()
env <- matchTraces
(trace (strandInst k s))
(trace (strandInst k s'))
(gen k, emptyEnv)
(gen', env') <- idempotentEnvFor env
(concatMap (tterms . trace) $ deleteNth s $ insts k)
case matchRenaming (gen', env') of
True -> return ()
False -> fail ""
case origCheck k env' of
True -> return ()
False -> fail ""
case all (precedesCheck k s s') (edges k) of
True -> return ()
False -> fail ""
case not useStrictPrecedesCheck ||
strictPrecedesCheck s s' (orderings k) &&
strictPrecedesCheck s' s (orderings k) of
True -> return ()
False -> fail ""
case not usePruningUnboundCheck || unboundCheck k s s' env' of
True -> return ()
False -> fail ""
prs <- ksubst True prs (gen', substitution env')
compress True prs s s'
matchTraces :: Algebra t p g s e c => Trace t p g s e c ->
Trace t p g s e c -> (g, e) -> [(g, e)]
matchTraces [] _ env = [env] -- Pattern can be shorter
matchTraces (In t : c) (In t' : c') env =
do
e <- match t t' env
matchTraces c c' e
matchTraces (Out t : c) (Out t' : c') env =
do
e <- match t t' env
matchTraces c c' e
matchTraces _ _ _ = []
origCheck :: Algebra t p g s e c => Preskel t p g s e c -> e -> Bool
origCheck k env =
check (kunique k) && check (knon k)
where
check orig =
all (pred orig) orig
pred set item =
elem (instantiate env item) set
precedesCheck :: Algebra t p g s e c => Preskel t p g s e c ->
Sid -> Sid -> Edge t p g s e c -> Bool
precedesCheck k s s' (gn0, gn1)
| s == sid (strand gn0) = graphPrecedes (vertex k (s', pos gn0)) gn1
| s == sid (strand gn1) = graphPrecedes gn0 (vertex k (s', pos gn1))
| otherwise = True
strictPrecedesCheck :: Sid -> Sid -> [Pair] -> Bool
strictPrecedesCheck s s' pairs =
all (flip elem pairs) before &&
all (flip elem pairs) after
where
before = [ (n0, (s', i1)) |
(n0, (s1, i1)) <- pairs,
s1 == s ]
after = [ ((s', i0), n1) |
((s0, i0), n1) <- pairs,
s0 == s ]
unboundCheck :: Algebra t p g s e c => Preskel t p g s e c ->
Sid -> Sid -> e -> Bool
unboundCheck k s s' env =
all (flip S.member unbound) (map (instantiate env) (S.elems unbound))
where
-- The vars in s and s'
this = addIvars (addIvars S.empty (strandInst k s)) (strandInst k s')
-- The vars in strands other than s and s'
others = kvarsBut k s s'
unbound = S.difference this others
kvarsBut :: Algebra t p g s e c => Preskel t p g s e c ->
Sid -> Sid -> Set t
kvarsBut k s s' =
loop S.empty (insts k) 0
where
loop xs [] _ = xs
loop xs (i:is) s''
| s'' == s || s'' == s' =
loop xs is (s'' + 1)
| otherwise =
loop (addIvars xs i) is (s'' + 1)
-- Thinning
thin :: Algebra t p g s e c => PRS t p g s e c -> [PRS t p g s e c]
thin prs =
thinStrands prs $ reverse ss
where -- Remove strands in image of POV
ss = filter (\s -> notElem s (prob $ skel prs)) (strandids $ skel prs)
thinStrands :: Algebra t p g s e c => PRS t p g s e c ->
[Sid] -> [PRS t p g s e c]
thinStrands prs [] =
reduce prs
thinStrands prs (s:ss) =
thinStrandPairs prs s ss ss
thinStrandPairs :: Algebra t p g s e c => PRS t p g s e c ->
Sid -> [Sid] -> [Sid] -> [PRS t p g s e c]
thinStrandPairs prs _ ss [] =
thinStrands prs ss
thinStrandPairs prs s ss (s':ss') =
case thinStrand prs s s' of
[] -> thinStrandPairs prs s ss ss' -- Try next pair
prss -> -- Success
do
prs <- prss
thin prs
thinStrand :: Algebra t p g s e c => PRS t p g s e c ->
Sid -> Sid -> [PRS t p g s e c]
thinStrand prs s s' =
do
let k = skel prs
let i = strandInst k s
let i' = strandInst k s'
case height i /= height i' of
True -> fail ""
False -> return ()
env <- matchTraces (trace i) (trace i') (gen k, emptyEnv)
(gen', env') <- idempotentEnvFor env
(concatMap (tterms . trace) $ deleteNth s $ insts k)
case matchRenaming (gen', env') of
True -> return ()
False -> fail ""
case origCheck k env' of
True -> return ()
False -> fail ""
[ prs' | prs <- ksubst False prs (gen', substitution env'),
prs' <- compress False prs s s',
prs'' <- compress False prs s' s,
gist (skel prs') == gist (skel prs'')]
-- Transitive Reduction
-- An edge is essential if its removal eliminates all paths from its
-- source to its destination. This function removes all non-essential
-- edges from the ordering relation.
reduce :: Algebra t p g s e c => PRS t p g s e c -> [PRS t p g s e c]
reduce (k0, k, n, phi, hsubst) =
let o = map graphPair (graphReduce (edges k)) in
if length o == length (orderings k) then
return (k0, k, n, phi, hsubst) -- Nothing to take away
else
do
let k' =
newPreskel
(gen k)
(protocol k)
(insts k)
o
(knon k)
(kunique k)
(operation k)
(prob k)
(pov k)
k'' <- wellFormedPreskel k'
return (k0, k'', n, phi, hsubst)
-- Answers for cohorts
type Ans t p g s e c = (Preskel t p g s e c, Node, s)
ans :: Algebra t p g s e c => PRS t p g s e c -> Ans t p g s e c
ans (_, k, n, _, subst) = (k, n, subst)
-- Homomorphism Filter
homomorphismFilter :: Algebra t p g s e c => PRS t p g s e c ->
[Ans t p g s e c]
homomorphismFilter prs@(k0, k, _, phi, subst)
| validateMappingSubst k0 phi subst k = [ans prs]
| otherwise = []
-- Ensure origination nodes a preserved as required to be a homomorphism
validateMappingSubst :: Algebra t p g s e c => Preskel t p g s e c ->
[Sid] -> s -> Preskel t p g s e c -> Bool
validateMappingSubst k phi subst k' =
useNoOrigPreservation || all check (korig k)
where
check (u, ns) =
case lookup (substitute subst u) (korig k') of
Nothing -> False
Just ns' -> all (flip elem ns') (map (permuteNode phi) ns)
-- Returns the skeleton associated with a preskeleton or nothing when
-- there isn't one. Manufacture a node and a term, and then drop them
-- afterwards.
toSkeleton :: Algebra t p g s e c => Bool -> Preskel t p g s e c ->
[Preskel t p g s e c]
toSkeleton prune k =
do
prs <- hull prune (k, k, (0, 0), strandids k, emptySubst)
(k', _, _) <- homomorphismFilter prs
return k'
-- Contraction
contract :: Algebra t p g s e c => Preskel t p g s e c -> Node ->
Cause t p g s e c -> (g, s) -> [Ans t p g s e c]
contract k n cause subst =
do
prs <- ksubst False (k, k { operation = Contracted emptySubst cause },
n, strandids k, emptySubst) subst
prs' <- hull usePruningWhileSolving prs
homomorphismFilter prs'
-- Regular Augmentation
-- First argument determines if displacement is used.
augment :: Algebra t p g s e c => Bool -> Preskel t p g s e c ->
Node -> Cause t p g s e c -> Role t p g s e c ->
(g, s) -> Instance t p g s e c -> [Ans t p g s e c]
augment True k0 n cause role subst inst =
do
prs <- augmentAndDisplace k0 n cause role subst inst
homomorphismFilter prs
augment False k0 n cause role subst inst =
do
prs <- augmentOnly k0 n cause role subst inst
homomorphismFilter prs
-- Apply a substitution, and then augment and displace. Augmentations
-- add an instance and one ordered pair. Displacements add an ordered
-- pair and may add nodes.
augmentAndDisplace :: Algebra t p g s e c => Preskel t p g s e c ->
Node -> Cause t p g s e c -> Role t p g s e c ->
(g, s) -> Instance t p g s e c -> [PRS t p g s e c]
augmentAndDisplace k0 n cause role subst inst =
do
prs <- substAndAugment k0 n cause role subst inst
hull usePruningWhileSolving prs ++ augDisplace prs
-- Apply a substitution and then augment. Augmentations add an
-- instance and one ordered pair.
augmentOnly :: Algebra t p g s e c =>Preskel t p g s e c ->
Node -> Cause t p g s e c -> Role t p g s e c ->
(g, s) -> Instance t p g s e c -> [PRS t p g s e c]
augmentOnly k0 n cause role subst inst =
do
prs <- substAndAugment k0 n cause role subst inst
hull usePruningWhileSolving prs
-- Apply the substitution and apply augmentation operator.
substAndAugment :: Algebra t p g s e c => Preskel t p g s e c ->
Node -> Cause t p g s e c -> Role t p g s e c ->
(g, s) -> Instance t p g s e c -> [PRS t p g s e c]
substAndAugment k n cause role subst inst =
do
let operation' = AddedStrand (rname role) (height inst) cause
prs <- ksubst False (k, k { operation = operation' }, n,
strandids k, emptySubst) subst
aug prs inst
-- Apply the augmentation operator by adding an instance and one
-- ordered pair.
aug :: Algebra t p g s e c => PRS t p g s e c ->
Instance t p g s e c -> [PRS t p g s e c]
aug (k0, k, n, phi, hsubst) inst =
do
let insts' = (insts k) ++ [inst]
let pair = ((length (insts k), height inst - 1), n)
let orderings' = pair : orderings k
let non' = inheritRnon inst ++ (knon k)
let unique' = inheritRunique inst ++ (kunique k)
let k' = newPreskel (gen k) (protocol k) insts'
orderings' non' unique' (operation k) (prob k) (pov k)
k' <- wellFormedPreskel k'
return (k0, k', n, phi, hsubst)
-- Inherit non-originating atoms if the traces is long enough
inheritRnon :: Algebra t p g s e c => Instance t p g s e c -> [t]
inheritRnon i =
inherit i (rnorig (role i))
-- Inherit uniquely originating atoms if the traces is long enough
inheritRunique :: Algebra t p g s e c => Instance t p g s e c -> [t]
inheritRunique i =
inherit i (ruorig (role i))
inherit :: Algebra t p g s e c => Instance t p g s e c -> [(t, Int)] -> [t]
inherit i rorigs =
map (instantiate (env i) . fst) $ filter f rorigs
where
f (_, pos) = pos < height i
-- Add all displacements
augDisplace :: Algebra t p g s e c => PRS t p g s e c -> [PRS t p g s e c]
augDisplace prs =
do
let s = nstrands (skel prs) - 1
s' <- nats s
augDisplaceStrands prs s s'
-- Try to displace with strand s'
augDisplaceStrands :: Algebra t p g s e c => PRS t p g s e c ->
Sid -> Sid -> [PRS t p g s e c]
augDisplaceStrands (k0, k, n, phi, hsubst) s s' =
do
(s, s', subst) <- unifyStrands k s s'
let op = addedToDisplaced (operation k) s s'
prs <- ksubst False (k0, k { operation = op}, n, phi, hsubst) subst
prs <- compress True prs s s'
hull usePruningWhileSolving prs
addedToDisplaced :: Algebra t p g s e c => Operation t p g s e c ->
Int -> Int -> Operation t p g s e c
addedToDisplaced (AddedStrand role height cause) s s' =
Displaced s s' role height cause
addedToDisplaced _ _ _ = error "Strand.addedToDisplaced: Bad operation"
-- Listener Augmentation
addListener :: Algebra t p g s e c => Preskel t p g s e c -> Node ->
Cause t p g s e c -> t -> [Ans t p g s e c]
addListener k n cause t =
do
k' <- wellFormedPreskel k'
prs <- hull usePruningWhileSolving (k, k', n, strandids k, emptySubst)
homomorphismFilter prs
where
k' = newPreskel gen' (protocol k) insts' orderings'
(knon k) (kunique k) (AddedListener t cause) (prob k) (pov k)
(gen', inst) = mkListener (gen k) t
insts' = insts k ++ [inst]
pair = ((length (insts k), 1), n)
orderings' = pair : orderings k
-- Homomorphisms
-- Find a substitution that demonstrates the existence of a
-- homomorphism between the two skeletons using the given
-- strand mapping function.
homomorphism :: Algebra t p g s e c => Preskel t p g s e c ->
Preskel t p g s e c -> [Sid] -> [e]
homomorphism k k' mapping =
do
(_, env) <- findReplacement k k' mapping
case validateEnv k k' mapping env of
True -> [env]
False -> []
findReplacement :: Algebra t p g s e c => Preskel t p g s e c ->
Preskel t p g s e c -> [Sid] -> [(g, e)]
findReplacement k k' mapping =
foldM (matchStrand k k' mapping) (gen k', emptyEnv) (strandids k)
matchStrand :: Algebra t p g s e c => Preskel t p g s e c ->
Preskel t p g s e c -> [Sid] -> (g, e) -> Sid -> [(g, e)]
matchStrand k k' mapping env s =
matchTraces (trace (strandInst k s)) (trace (strandInst k' s')) env
where
s' = mapping !! s
validateEnv :: Algebra t p g s e c => Preskel t p g s e c ->
Preskel t p g s e c -> [Sid] -> e -> Bool
validateEnv k k' mapping env =
all (flip elem (knon k')) (map (instantiate env) (knon k)) &&
all (flip elem (kunique k')) (map (instantiate env) (kunique k)) &&
validateEnvOrig k k' mapping env &&
all (flip elem (tc k')) (permuteOrderings mapping (orderings k))
validateEnvOrig :: Algebra t p g s e c => Preskel t p g s e c ->
Preskel t p g s e c -> [Sid] -> e -> Bool
validateEnvOrig k k' mapping env =
useNoOrigPreservation || all check (korig k)
where
check (u, ns) =
case lookup (instantiate env u) (korig k') of
Nothing -> error "Strand.validateEnv: term not in kunique"
Just ns' -> all (flip elem ns') (map (permuteNode mapping) ns)
-- Given a realized skeleton k, generate candidates for minimization.
-- A candidate is a preskeleton and a strand mapping from the
-- candidate to k. The preskeleton need not be well formed, as that
-- test is applied elsewhere.
type Candidate t p g s e c = (Preskel t p g s e c, [Sid])
addIdentity :: Algebra t p g s e c => Preskel t p g s e c ->
Candidate t p g s e c
addIdentity k = (k, strandids k)
generalize :: Algebra t p g s e c => Preskel t p g s e c ->
[Candidate t p g s e c]
generalize k = deleteNodes k ++
weakenOrderings k ++
forgetAssumption k ++
separateVariables k
-- Node deletion
{-
delete node n in k
1. if (s, 0) part of prob return Nothing
2. if not initial node, truncate instance of node else delete instance
3. weaken ordering when filtering it (see shortenOrdering)
4. drop nons that aren't mentioned anywhere
5. drop uniques that aren't carried anywhere
6. update prob upon instance deletion
-}
deleteNodes :: Algebra t p g s e c => Preskel t p g s e c ->
[Candidate t p g s e c]
deleteNodes k =
do
strand <- strands k
node <- nodes strand
deleteNode k node
deleteNode :: Algebra t p g s e c => Preskel t p g s e c ->
Vertex t p g s e c -> [(Preskel t p g s e c, [Sid])]
deleteNode k n
| p == 0 && elem s (prob k) = []
| p == 0 =
do
let mapping = deleteNth s (strandids k)
let k' = deleteNodeRest k (gen k) (s, p) (deleteNth s (insts k))
(deleteOrderings s (tc k)) (updatePerm s s (prob k))
return (k', mapping)
| otherwise =
do
let mapping = strandids k
let i = inst (strand n)
(gen', i') <- bldInstance (role i) (take p (trace i)) (gen k)
let k' = deleteNodeRest k gen' (s, p) (replaceNth i' s (insts k))
(shortenOrderings (s, p) (tc k)) (prob k)
return (k', mapping)
where
p = pos n
s = sid (strand n)
-- Update orderings when a strand is eliminated (p == 0)
deleteOrderings :: Sid -> [Pair] -> [Pair]
deleteOrderings s ps =
do
p <- ps
deleteOrdering p
where
deleteOrdering (n0@(s0, _), n1@(s1, _))
| s == s0 || s == s1 = []
| otherwise = [(adjust n0, adjust n1)]
adjust n@(s', p')
| s' > s = (s' - 1, p')
| otherwise = n
-- Update orderings when a strand is shortened (p > 0)
shortenOrderings :: Node -> [Pair] -> [Pair]
shortenOrderings (s, p) ps =
do
pair <- ps
shortenOrdering pair
where
shortenOrdering (n0@(s0, p0), n1@(s1, p1))
| s == s0 && p <= p0 = []
| s == s1 && p <= p1 = []
| otherwise = [(n0, n1)]
deleteNodeRest :: Algebra t p g s e c => Preskel t p g s e c ->
g -> Node -> [Instance t p g s e c] -> [Pair] ->
[Sid] -> Preskel t p g s e c
deleteNodeRest k gen n insts' orderings prob =
newPreskel gen (protocol k) insts'
orderings non' unique' (Generalized (Deleted n)) prob (pov k)
where
-- Drop nons that aren't mentioned anywhere
non' = filter mentionedIn (knon k)
mentionedIn t = varSubset [t] terms
terms = iterms insts'
-- Drop uniques that aren't held anywhere
unique' = filter heldIn (kunique k)
heldIn t = any (heldBy t) terms
-- Node ordering weakening
-- To weaken, create a candidate for each element in the current
-- ordering, which is already the result of a transitive reduction.
-- Weaken by computing the transitive closure of the ordering, and
-- then remove the selected element from the current ordering. After
-- computing the transitive closure, filter out the edges that are not
-- well ordered, i.e. originate at a reception node or terminate at a
-- transmission node. Also, filter out edges that link nodes in the
-- same strand. The preskeleton constructor function performs a
-- transitive reduction on the generated ordering.
weakenOrderings :: Algebra t p g s e c => Preskel t p g s e c ->
[Candidate t p g s e c]
weakenOrderings k =
map (weakenOrdering k) (orderings k)
weakenOrdering :: Algebra t p g s e c => Preskel t p g s e c ->
Pair -> Candidate t p g s e c
weakenOrdering k p =
weaken k p (L.delete p (tc k))
weaken :: Algebra t p g s e c => Preskel t p g s e c ->
Pair -> [Pair] -> Candidate t p g s e c
weaken k p orderings =
addIdentity k'
where
k' = newPreskel (gen k) (protocol k) (insts k)
orderings (knon k) (kunique k)
(Generalized (Weakened p)) (prob k) (pov k)
-- Origination assumption forgetting
-- Delete each non-originating term that is not specified by a
-- role. Do the same for each uniquely-originating term.
forgetAssumption :: Algebra t p g s e c => Preskel t p g s e c ->
[Candidate t p g s e c]
forgetAssumption k =
forgetNonTerm k ++ forgetUniqueTerm k
-- Non-originating terms
forgetNonTerm :: Algebra t p g s e c => Preskel t p g s e c ->
[Candidate t p g s e c]
forgetNonTerm k =
map (addIdentity . delNon) (skelNons k)
where
delNon t =
k { knon = L.delete t (knon k),
operation = Generalized (Forgot t) }
skelNons :: Algebra t p g s e c => Preskel t p g s e c -> [t]
skelNons k =
filter (flip notElem ru) (knon k)
where
ru = [u | i <- insts k, u <- inheritRnon i]
-- Uniquely-originating terms
forgetUniqueTerm :: Algebra t p g s e c => Preskel t p g s e c ->
[Candidate t p g s e c]
forgetUniqueTerm k =
map (addIdentity . delUniq) (skelUniques k)
where
delUniq t =
k { kunique = L.delete t (kunique k),
operation = Generalized (Forgot t) }
skelUniques :: Algebra t p g s e c => Preskel t p g s e c -> [t]
skelUniques k =
filter (flip notElem ru) (kunique k)
where
ru = [u | i <- insts k, u <- inheritRunique i]
-- Variable separation
-- A location is a strand, a role variable, and a position in the term
-- associated with the role variable.
-- step one: extract places
--
-- for each maplet in each environment in each strand
-- for each variable V in the range of the maplet
-- let P be the places at which V occurs
-- associate V with the location described by each element in P
--
-- step two: generate preskeletons
--
-- for each variable V in the skeleton K
-- let V' be a clone of V
-- let L be the set of locations associated with V from above
-- for each subset L' of L
-- for each instance I in K
-- update the environment of I by replacing V' at the locations
-- given by L' that refer to I, and use the modified environment
-- to update the instance
-- let K' be the result of updating instances in K as above
-- if V occurs in non, add terms with V replaced by V' to
-- the non's of K'
-- if V occurs in unique, add terms with V replaced by V'
-- to the unique's of K'
-- add K' to the list of generated preskeletons
separateVariables :: Algebra t p g s e c => Preskel t p g s e c ->
[Candidate t p g s e c]
separateVariables k =
do
v <- kvars k
separateVariable k (extractPlaces k) v
-- A location is a strand, a role variable, and a position in the term
-- associated with the role variable.
type Location t p g s e c = (Sid, t, p)
-- Returns a list of pairs. For each occurrence of a preskeleton
-- variable in every instance, there is a pair where the first element
-- is the variable, and the second as the location at which it occurs.
extractPlaces :: Algebra t p g s e c => Preskel t p g s e c ->
[(t, Location t p g s e c)]
extractPlaces k =
[ (var, (sid s, v, p)) |
s <- strands k,
(v, t) <- instAssocs (inst s),
var <- foldVars (flip adjoin) [] t,
p <- places var t ]
instAssocs :: Algebra t p g s e c => Instance t p g s e c -> [(t, t)]
instAssocs i =
reify (rvars (role i)) (env i)
-- For each variable, generate candidates by generating a fresh
-- variable for subsets of the locations associated with the variable.
separateVariable :: Algebra t p g s e c => Preskel t p g s e c ->
[(t, Location t p g s e c)] -> t ->
[Candidate t p g s e c]
separateVariable k ps t =
sepVar (locsFor ps t)
where
sepVar [] = []
sepVar [_] = []
sepVar locs =
do
locs' <- parts locs
changeLocations k env gen'' t' locs'
(gen'', env) = matchAlways t t' (gen', emptyEnv)
parts locs = map (map (locs !!)) (subsets (length locs))
(gen', t') = clone (gen k) t
-- Extract the locations for a given variable
locsFor :: Algebra t p g s e c => [(t, Location t p g s e c)] ->
t -> [Location t p g s e c]
locsFor ps t =
map snd (filter (\(t', _) -> t == t') (reverse ps)) -- Why reverse?
matchAlways :: Algebra t p g s e c => t -> t -> (g, e) -> (g, e)
matchAlways t t' env =
case match t t' env of
e : _ -> e
[] -> error "Strand.matchAlways: bad match"
-- Change the given locations and create the resulting preskeleton
changeLocations :: Algebra t p g s e c => Preskel t p g s e c ->
e -> g -> t -> [Location t p g s e c] ->
[Candidate t p g s e c]
changeLocations k env gen t locs =
[addIdentity k0, addIdentity k1]
where
k0 = newPreskel gen' (protocol k) insts' (orderings k)
non unique0 (Generalized (Separated t)) (prob k) (pov k)
k1 = newPreskel gen' (protocol k) insts' (orderings k)
non unique1 (Generalized (Separated t)) (prob k) (pov k)
(gen', insts') = changeStrands locs t gen (strands k)
non = knon k ++ map (instantiate env) (knon k)
unique0 = kunique k ++ unique'
unique1 = map (instantiate env) (kunique k) ++ unique'
-- Ensure all role unique assumptions are in.
unique' = concatMap inheritRunique insts'
changeStrands :: Algebra t p g s e c => [Location t p g s e c] -> t ->
g -> [Strand t p g s e c] -> (g, [Instance t p g s e c])
changeStrands locs copy gen strands =
case foldMapM (changeStrand locs copy) gen strands of
i : _ -> i
[] -> error "Strand.changeStrands: bad strand build"
-- Create an new environment incorporating changes, and from that,
-- create the new strand.
changeStrand :: Algebra t p g s e c => [Location t p g s e c] ->
t -> g -> Strand t p g s e c -> [(g, Instance t p g s e c)]
changeStrand locs copy gen s =
bldInstance (role i) trace' gen'
where
i = inst s
(gen', env') = foldl f (gen, emptyEnv)
(map (changeMaplet locs copy (sid s)) (instAssocs i))
f env (v, t) = matchAlways v t env
trace' = map (evtMap $ instantiate env') trace
trace = take (height i) (rtrace (role i))
-- Change a maplet
changeMaplet :: Algebra t p g s e c => [Location t p g s e c] ->
t -> Sid -> (t, t) -> (t, t)
changeMaplet [] _ _ maplet = maplet
changeMaplet ((s', v', p) : locs) copy s (v, t) =
changeMaplet locs copy s (v, t')
where
t' = if s' == s && v' == v then replace copy p t else t
-- Return the set of subsets of natural numbers less than n
subsets :: Int -> [[Int]]
subsets n
| n < 0 = error $ "Utilities.subsets: Bad argument " ++ show n
| n == 0 = []
| otherwise =
[n - 1] : subset ++ map (n - 1 :) subset
where
subset = subsets (n - 1)
-- Collapse a shape by unifying strands.
collapse :: Algebra t p g s e c => Preskel t p g s e c ->
[Preskel t p g s e c]
collapse k =
[k' | s <- strandids k, s' <- nats s,
k' <- collapseStrands k s s']
collapseStrands :: Algebra t p g s e c => Preskel t p g s e c ->
Sid -> Sid -> [Preskel t p g s e c]
collapseStrands k s s' =
do
(s, s', subst) <- unifyStrands k s s'
prs <- ksubst False (k, k { operation = Collapsed s s' },
(0, 0), strandids k, emptySubst) subst
prs <- compress True prs s s'
prs <- hull usePruningDuringCollapsing prs
return $ skel prs