StrongConsensus.hs

{-# LANGUAGE FlexibleContexts #-}

module Solver.StrongConsensus
    (checkStrongConsensusSat,
     StrongConsensusCounterExample,
     findTrapConstraintsSat,
     findUTrapConstraintsSat,
     findUSiphonConstraintsSat,
     checkGeneralizedCoTrapSat,
     StableInequality)
where

import Data.SBV
import qualified Data.Map as M
import Data.List ((\\))

import Util
import PopulationProtocol
import Property
import Solver

type StrongConsensusCounterExample = (Configuration, Configuration, Configuration, FiringVector, FiringVector)

type StableInequality = (IMap State, Integer)

stateEquationConstraints :: PopulationProtocol -> SIMap State -> SIMap State -> SIMap Transition -> SBool
stateEquationConstraints pp m0 m x =
            bAnd $ map checkStateEquation $ states pp
        where checkStateEquation q =
                let incoming = map addTransition $ lpre pp q
                    outgoing = map addTransition $ lpost pp q
                in  val m0 q + sum incoming - sum outgoing .== val m q
              addTransition (t,w) = literal w * val x t

nonNegativityConstraints :: (Ord a, Show a) => SIMap a -> SBool
nonNegativityConstraints m =
            bAnd $ map checkVal $ vals m
        where checkVal x = x .>= 0

terminalConstraints :: PopulationProtocol -> SIMap State -> SBool
terminalConstraints pp m =
            bAnd $ map checkTransition $ transitions pp
        where checkTransition t = bOr $ map checkState $ lpre pp t
              checkState (q,w) = val m q .<= literal (fromInteger (w - 1))

initialConfiguration :: PopulationProtocol -> SIMap State -> SBool
initialConfiguration pp m0 =
        sum (mval m0 (states pp \\ initialStates pp)) .== 0

differentConsensusConstraints :: PopulationProtocol -> SIMap State -> SIMap State -> SBool
differentConsensusConstraints pp m1 m2 =
        (sum (mval m1 (yesStates pp)) .> 0 &&& sum (mval m2 (noStates pp)) .> 0)

unmarkedByConfiguration :: [State] -> SIMap State -> SBool
unmarkedByConfiguration r m = sum (mval m r) .== 0

markedByConfiguration :: [State] -> SIMap State -> SBool
markedByConfiguration r m = sum (mval m r) .> 0

sequenceNotIn :: [Transition] -> SIMap Transition -> SBool
sequenceNotIn u x = sum (mval x u) .== 0

sequenceIn :: [Transition] -> SIMap Transition -> SBool
sequenceIn u x = sum (mval x u) .> 0

checkUTrap :: PopulationProtocol -> SIMap State -> SIMap State -> SIMap State -> SIMap Transition -> SIMap Transition -> Trap -> SBool
checkUTrap pp m0 m1 m2 x1 x2 utrap =
            (((sequenceIn upre x1) &&& (sequenceNotIn uunmark x1)) ==> (markedByConfiguration utrap m1)) &&&
            (((sequenceIn upre x2) &&& (sequenceNotIn uunmark x2)) ==> (markedByConfiguration utrap m2))
        where upost = mpost pp utrap
              upre = mpre pp utrap
              uunmark = upost \\ upre

checkUTrapConstraints :: PopulationProtocol -> SIMap State -> SIMap State -> SIMap State -> SIMap Transition -> SIMap Transition -> [Trap] -> SBool
checkUTrapConstraints pp m0 m1 m2 x1 x2 traps =
        bAnd $ map (checkUTrap pp m0 m1 m2 x1 x2) traps

checkUSiphon :: PopulationProtocol -> SIMap State -> SIMap State -> SIMap State -> SIMap Transition -> SIMap Transition -> Siphon -> SBool
checkUSiphon pp m0 m1 m2 x1 x2 usiphon =
            (((sequenceIn upost x1) &&& (sequenceNotIn umark x1)) ==> (markedByConfiguration usiphon m0)) &&&
            (((sequenceIn upost x2) &&& (sequenceNotIn umark x2)) ==> (markedByConfiguration usiphon m0))
        where upost = mpost pp usiphon
              upre = mpre pp usiphon
              umark = upre \\ upost

checkUSiphonConstraints :: PopulationProtocol -> SIMap State -> SIMap State -> SIMap State -> SIMap Transition -> SIMap Transition -> [Siphon] -> SBool
checkUSiphonConstraints pp m0 m1 m2 x1 x2 siphons =
        bAnd $ map (checkUSiphon pp m0 m1 m2 x1 x2) siphons

checkInequalityConstraint :: PopulationProtocol -> SIMap State -> SIMap State -> SIMap State -> StableInequality -> SBool
checkInequalityConstraint pp m0 m1 m2 (k, c) =
            (checkInequality m0) ==> (checkInequality m1 &&& checkInequality m2)
        where checkInequality m = sum [ literal (val k q) * (val m q) | q <- states pp ] .>= literal c

checkInequalityConstraints :: PopulationProtocol -> SIMap State -> SIMap State -> SIMap State -> [StableInequality] -> SBool
checkInequalityConstraints pp m0 m1 m2 inequalities =
        bAnd [ checkInequalityConstraint pp m0 m1 m2 i | i <- inequalities ]

checkStrongConsensus :: PopulationProtocol -> SIMap State -> SIMap State -> SIMap State -> SIMap Transition -> SIMap Transition ->
        [Trap] -> [Siphon] -> [StableInequality] -> SBool
checkStrongConsensus pp m0 m1 m2 x1 x2 utraps usiphons inequalities =
        stateEquationConstraints pp m0 m1 x1 &&&
        stateEquationConstraints pp m0 m2 x2 &&&
        initialConfiguration pp m0 &&&
        nonNegativityConstraints m0 &&&
        nonNegativityConstraints m1 &&&
        nonNegativityConstraints m2 &&&
        nonNegativityConstraints x1 &&&
        nonNegativityConstraints x2 &&&
        terminalConstraints pp m1 &&&
        terminalConstraints pp m2 &&&
        differentConsensusConstraints pp m1 m2 &&&
        checkUTrapConstraints pp m0 m1 m2 x1 x2 utraps &&&
        checkUSiphonConstraints pp m0 m1 m2 x1 x2 usiphons &&&
        checkInequalityConstraints pp m0 m1 m2 inequalities

checkStrongConsensusSat :: PopulationProtocol -> [Trap] -> [Siphon] -> [StableInequality] -> ConstraintProblem Integer StrongConsensusCounterExample
checkStrongConsensusSat pp utraps usiphons inequalities =
        let m0 = makeVarMap $ states pp
            m1 = makeVarMapWith prime $ states pp
            m2 = makeVarMapWith (prime . prime) $ states pp
            x1 = makeVarMap $ transitions pp
            x2 = makeVarMapWith prime $ transitions pp
        in  ("strong consensus", "(c0, c1, c2, x1, x2)",
             getNames m0 ++ getNames m1 ++ getNames m2 ++ getNames x1 ++ getNames x2,
             \fm -> checkStrongConsensus pp (fmap fm m0) (fmap fm m1) (fmap fm m2) (fmap fm x1) (fmap fm x2) utraps usiphons inequalities,
             \fm -> configurationsFromAssignment (fmap fm m0) (fmap fm m1) (fmap fm m2) (fmap fm x1) (fmap fm x2))

configurationsFromAssignment :: IMap State -> IMap State -> IMap State -> IMap Transition -> IMap Transition -> StrongConsensusCounterExample
configurationsFromAssignment m0 m1 m2 x1 x2 =
        (makeVector m0, makeVector m1, makeVector m2, makeVector x1, makeVector x2)

-- trap and siphon refinement constraints

trapConstraint :: PopulationProtocol -> SIMap State -> Transition -> SBool
trapConstraint pp b t =
        sum (mval b $ pre pp t) .> 0 ==> sum (mval b $ post pp t) .> 0

siphonConstraint :: PopulationProtocol -> SIMap State -> Transition -> SBool
siphonConstraint pp b t =
        sum (mval b $ post pp t) .> 0 ==> sum (mval b $ pre pp t) .> 0

trapConstraints :: PopulationProtocol -> SIMap State -> SBool
trapConstraints pp b =
        bAnd $ map (trapConstraint pp b) $ transitions pp

uTrapConstraints :: PopulationProtocol -> FiringVector -> SIMap State -> SBool
uTrapConstraints pp x b =
        bAnd $ map (trapConstraint pp b) $ elems x

uSiphonConstraints :: PopulationProtocol -> FiringVector -> SIMap State -> SBool
uSiphonConstraints pp x b =
        bAnd $ map (siphonConstraint pp b) $ elems x

statesMarkedByConfiguration :: PopulationProtocol -> Configuration -> SIMap State -> SBool
statesMarkedByConfiguration pp m b = sum (mval b $ elems m) .> 0

statesUnmarkedByConfiguration :: PopulationProtocol -> Configuration -> SIMap State -> SBool
statesUnmarkedByConfiguration pp m b = sum (mval b $ elems m) .== 0

statesPostsetOfSequence :: PopulationProtocol -> FiringVector -> SIMap State -> SBool
statesPostsetOfSequence pp x b = sum (mval b $ mpost pp $ elems x) .> 0

statesPresetOfSequence :: PopulationProtocol -> FiringVector -> SIMap State -> SBool
statesPresetOfSequence pp x b = sum (mval b $ mpre pp $ elems x) .> 0

noOutputTransitionEnabled :: PopulationProtocol -> Configuration -> SIMap State -> SBool
noOutputTransitionEnabled pp m b =
            bAnd $ map outputTransitionNotEnabled $ transitions pp
        where
            outputTransitionNotEnabled t = outputTransitionOfB t ==> transitionNotEnabledInB t
            outputTransitionOfB t = sum [val b q | (q, w) <- lpre pp t, val m q >= w] .> 0
            transitionNotEnabledInB t = sum [val b q | (q, w) <- lpre pp t, val m q < w] .> 0

nonemptySet :: (Ord a, Show a) => SIMap a -> SBool
nonemptySet b = sum (vals b) .> 0

checkBinary :: SIMap State -> SBool
checkBinary = bAnd . map (\x -> x .== 0 ||| x .== 1) . vals

checkSizeLimit :: SIMap State -> Maybe (Int, Integer) -> SBool
checkSizeLimit _ Nothing = true
checkSizeLimit b (Just (1, curSize)) = (.< literal curSize) $ sumVal b
checkSizeLimit b (Just (2, curSize)) = (.> literal curSize) $ sumVal b
checkSizeLimit _ (Just (_, _)) = error "minimization method not supported"

minimizeMethod :: Int -> Integer -> String
minimizeMethod 1 curSize = "size smaller than " ++ show curSize
minimizeMethod 2 curSize = "size larger than " ++ show curSize
minimizeMethod _ _ = error "minimization method not supported"

findTrap :: PopulationProtocol -> Configuration -> Configuration -> Configuration -> FiringVector -> FiringVector -> SIMap State -> Maybe (Int, Integer) -> SBool
findTrap pp m0 m1 m2 x1 x2 b sizeLimit =
        checkSizeLimit b sizeLimit &&&
        checkBinary b &&&
        trapConstraints pp b &&&
        (
            (statesPostsetOfSequence pp x1 b &&& statesUnmarkedByConfiguration pp m1 b) |||
            (statesPostsetOfSequence pp x2 b &&& statesUnmarkedByConfiguration pp m2 b)
        )

findTrapConstraintsSat :: PopulationProtocol -> Configuration -> Configuration -> Configuration -> FiringVector -> FiringVector -> MinConstraintProblem Integer Trap Integer
findTrapConstraintsSat pp m0 m1 m2 x1 x2 =
        let b = makeVarMap $ states pp
        in  (minimizeMethod, \sizeLimit ->
            ("trap marked by x1 or x2 and not marked in m1 or m2", "trap",
             getNames b,
             \fm -> findTrap pp m0 m1 m2 x1 x2 (fmap fm b) sizeLimit,
             \fm -> statesFromAssignment (fmap fm b)))

findUTrapConstraints :: PopulationProtocol -> Configuration -> Configuration -> Configuration -> FiringVector -> FiringVector -> SIMap State -> Maybe (Int, Integer) -> SBool
findUTrapConstraints pp m0 m1 m2 x1 x2 b sizeLimit =
        checkSizeLimit b sizeLimit &&&
        checkBinary b &&&
        (
            (statesPostsetOfSequence pp x1 b &&& uTrapConstraints pp x1 b &&& statesUnmarkedByConfiguration pp m1 b) |||
            (statesPostsetOfSequence pp x2 b &&& uTrapConstraints pp x2 b &&& statesUnmarkedByConfiguration pp m2 b)
        )

findUTrapConstraintsSat :: PopulationProtocol -> Configuration -> Configuration -> Configuration -> FiringVector -> FiringVector -> MinConstraintProblem Integer Trap Integer
findUTrapConstraintsSat pp m0 m1 m2 x1 x2 =
        let b = makeVarMap $ states pp
        in  (minimizeMethod, \sizeLimit ->
            ("u-trap (w.r.t. x1 or x2) marked by x1 or x2 and not marked in m1 or m2", "u-trap",
             getNames b,
             \fm -> findUTrapConstraints pp m0 m1 m2 x1 x2 (fmap fm b) sizeLimit,
             \fm -> statesFromAssignment (fmap fm b)))

findUSiphonConstraints :: PopulationProtocol -> Configuration -> Configuration -> Configuration -> FiringVector -> FiringVector -> SIMap State -> Maybe (Int, Integer) -> SBool
findUSiphonConstraints pp m0 m1 m2 x1 x2 b sizeLimit =
        checkSizeLimit b sizeLimit &&&
        checkBinary b &&&
        statesUnmarkedByConfiguration pp m0 b &&&
        (
            (statesPresetOfSequence pp x1 b &&& uSiphonConstraints pp x1 b) |||
            (statesPresetOfSequence pp x2 b &&& uSiphonConstraints pp x2 b)
        )

findUSiphonConstraintsSat :: PopulationProtocol -> Configuration -> Configuration -> Configuration -> FiringVector -> FiringVector -> MinConstraintProblem Integer Siphon Integer
findUSiphonConstraintsSat pp m0 m1 m2 x1 x2 =
        let b = makeVarMap $ states pp
        in  (minimizeMethod, \sizeLimit ->
            ("u-siphon (w.r.t. x1 or x2) used by x1 or x2 and unmarked in m0", "u-siphon",
             getNames b,
             \fm -> findUSiphonConstraints pp m0 m1 m2 x1 x2 (fmap fm b) sizeLimit,
             \fm -> statesFromAssignment (fmap fm b)))

statesFromAssignment :: IMap State -> ([State], Integer)
statesFromAssignment b = (M.keys (M.filter (> 0) b), sum (M.elems b))

-- stable linear inequalities

checkStableInequalityForConfiguration :: PopulationProtocol -> Configuration -> SIMap State -> SInteger -> SBool
checkStableInequalityForConfiguration pp m k c =
        sum [ (val k q) * literal (val m q) | q <- states pp ] .>= c

checkSemiPositiveConstraints :: (Ord a, Show a) => SIMap a -> SBool
checkSemiPositiveConstraints k =
            bAnd [ x .>= 0 | x <- vals k ]

checkSemiNegativeConstraints :: (Ord a, Show a) => SIMap a -> SBool
checkSemiNegativeConstraints k =
            bAnd [ x .<= 0 | x <- vals k ]

checkGeneralizedTCoTrap :: PopulationProtocol -> SIMap State -> SInteger -> Transition -> SBool
checkGeneralizedTCoTrap pp k c t =
            checkTSurInvariant ||| checkTDisabled
        where checkTSurInvariant = sum output - sum input .>= c
              checkTDisabled = sum input .< c
              input = map addState $ lpre pp t
              output = map addState $ lpost pp t
              addState (q, w) = literal w * val k q

checkGeneralizedCoTrap :: PopulationProtocol -> SIMap State -> SInteger -> SBool
checkGeneralizedCoTrap pp k c =
        bAnd [ checkGeneralizedTCoTrap pp k c t | t <- transitions pp ]

checkGeneralizedCoTrapConstraints :: PopulationProtocol -> Configuration -> Configuration -> Configuration -> FiringVector -> FiringVector -> SIMap State -> SInteger -> SBool
checkGeneralizedCoTrapConstraints pp m0 m1 m2 x1 x2 k c =
        checkSemiNegativeConstraints k &&&
        checkGeneralizedCoTrap pp k c &&&
        checkStableInequalityForConfiguration pp m0 k c &&&
        ((bnot (checkStableInequalityForConfiguration pp m1 k c)) ||| (bnot (checkStableInequalityForConfiguration pp m2 k c)))

checkGeneralizedCoTrapSat :: PopulationProtocol -> Configuration -> Configuration -> Configuration -> FiringVector -> FiringVector -> ConstraintProblem Integer StableInequality
checkGeneralizedCoTrapSat pp m0 m1 m2 x1 x2 =
        let k = makeVarMap $ states pp
            c = "'c"
        in  ("generalized co-trap (stable inequality) holding m but not in m1 or m2", "stable inequality",
             c : getNames k,
             \fm -> checkGeneralizedCoTrapConstraints pp m0 m1 m2 x1 x2 (fmap fm k) (fm c),
             \fm -> stableInequalityFromAssignment (fmap fm k) (fm c))

stableInequalityFromAssignment :: IMap State -> Integer -> StableInequality
stableInequalityFromAssignment k c = (k, c)