SparseDtmcPrctlHelper.cpp

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#include "storm/modelchecker/prctl/helper/SparseDtmcPrctlHelper.h"
 
#include "storm/adapters/RationalFunctionAdapter.h"
#include "storm/environment/solver/SolverEnvironment.h"
#include "storm/exceptions/IllegalArgumentException.h"
#include "storm/exceptions/InvalidPropertyException.h"
#include "storm/exceptions/NotSupportedException.h"
#include "storm/exceptions/UncheckedRequirementException.h"
#include "storm/io/export.h"
#include "storm/modelchecker/csl/helper/SparseCtmcCslHelper.h"
#include "storm/modelchecker/helper/DiscountingHelper.h"
#include "storm/modelchecker/hints/ExplicitModelCheckerHint.h"
#include "storm/modelchecker/prctl/helper/DsMpiUpperRewardBoundsComputer.h"
#include "storm/modelchecker/prctl/helper/rewardbounded/MultiDimensionalRewardUnfolding.h"
#include "storm/modelchecker/results/ExplicitQuantitativeCheckResult.h"
#include "storm/settings/SettingsManager.h"
#include "storm/settings/modules/CoreSettings.h"
#include "storm/settings/modules/GeneralSettings.h"
#include "storm/settings/modules/IOSettings.h"
#include "storm/settings/modules/ModelCheckerSettings.h"
#include "storm/solver/LinearEquationSolver.h"
#include "storm/solver/multiplier/Multiplier.h"
#include "storm/storage/ConsecutiveUint64DynamicPriorityQueue.h"
#include "storm/storage/StronglyConnectedComponentDecomposition.h"
#include "storm/utility/ProgressMeasurement.h"
#include "storm/utility/SignalHandler.h"
#include "storm/utility/Stopwatch.h"
#include "storm/utility/graph.h"
#include "storm/utility/macros.h"
#include "storm/utility/vector.h"
 
namespace storm {
namespace modelchecker {
namespace helper {
 
template<>
std::map<storm::storage::sparse::state_type, storm::RationalFunction> SparseDtmcPrctlHelper<storm::RationalFunction>::computeRewardBoundedValues(
    Environment const& /*env*/, storm::models::sparse::Dtmc<storm::RationalFunction> const& /*model*/,
    std::shared_ptr<storm::logic::OperatorFormula const> /*rewardBoundedFormula*/) {
    STORM_LOG_THROW(false, storm::exceptions::NotSupportedException, "The specified property is not supported by this value type.");
    return std::map<storm::storage::sparse::state_type, storm::RationalFunction>();
}
 
template<typename ValueType, typename RewardModelType>
std::map<storm::storage::sparse::state_type, ValueType> SparseDtmcPrctlHelper<ValueType, RewardModelType>::computeRewardBoundedValues(
    Environment const& env, storm::models::sparse::Dtmc<ValueType> const& model, std::shared_ptr<storm::logic::OperatorFormula const> rewardBoundedFormula) {
    storm::utility::Stopwatch swAll(true), swBuild, swCheck;
 
    storm::modelchecker::helper::rewardbounded::MultiDimensionalRewardUnfolding<ValueType, true> rewardUnfolding(model, rewardBoundedFormula);
 
    // Get lower and upper bounds for the solution.
    auto lowerBound = rewardUnfolding.getLowerObjectiveBound();
    auto upperBound = rewardUnfolding.getUpperObjectiveBound();
 
    // Initialize epoch models
    auto initEpoch = rewardUnfolding.getStartEpoch();
    auto epochOrder = rewardUnfolding.getEpochComputationOrder(initEpoch);
 
    // initialize data that will be needed for each epoch
    std::vector<ValueType> x, b;
    std::unique_ptr<storm::solver::LinearEquationSolver<ValueType>> linEqSolver;
 
    Environment preciseEnv = env;
    ValueType precision = rewardUnfolding.getRequiredEpochModelPrecision(
        initEpoch, storm::utility::convertNumber<ValueType>(storm::settings::getModule<storm::settings::modules::GeneralSettings>().getPrecision()));
    preciseEnv.solver().setLinearEquationSolverPrecision(storm::utility::convertNumber<storm::RationalNumber>(precision));
 
    // In case of cdf export we store the necessary data.
    std::vector<std::vector<ValueType>> cdfData;
 
    // Set the correct equation problem format.
    storm::solver::GeneralLinearEquationSolverFactory<ValueType> linearEquationSolverFactory;
    rewardUnfolding.setEquationSystemFormatForEpochModel(linearEquationSolverFactory.getEquationProblemFormat(preciseEnv));
 
    storm::utility::ProgressMeasurement progress("epochs");
    progress.setMaxCount(epochOrder.size());
    progress.startNewMeasurement(0);
    uint64_t numCheckedEpochs = 0;
    for (auto const& epoch : epochOrder) {
        swBuild.start();
        auto& epochModel = rewardUnfolding.setCurrentEpoch(epoch);
        swBuild.stop();
        swCheck.start();
        rewardUnfolding.setSolutionForCurrentEpoch(epochModel.analyzeSingleObjective(preciseEnv, x, b, linEqSolver, lowerBound, upperBound));
        swCheck.stop();
        if (storm::settings::getModule<storm::settings::modules::IOSettings>().isExportCdfSet() &&
            !rewardUnfolding.getEpochManager().hasBottomDimension(epoch)) {
            std::vector<ValueType> cdfEntry;
            for (uint64_t i = 0; i < rewardUnfolding.getEpochManager().getDimensionCount(); ++i) {
                uint64_t offset = rewardUnfolding.getDimension(i).boundType == helper::rewardbounded::DimensionBoundType::LowerBound ? 1 : 0;
                cdfEntry.push_back(storm::utility::convertNumber<ValueType>(rewardUnfolding.getEpochManager().getDimensionOfEpoch(epoch, i) + offset) *
                                   rewardUnfolding.getDimension(i).scalingFactor);
            }
            cdfEntry.push_back(rewardUnfolding.getInitialStateResult(epoch));
            cdfData.push_back(std::move(cdfEntry));
        }
        ++numCheckedEpochs;
        progress.updateProgress(numCheckedEpochs);
        if (storm::utility::resources::isTerminate()) {
            break;
        }
    }
 
    std::map<storm::storage::sparse::state_type, ValueType> result;
    for (auto initState : model.getInitialStates()) {
        result[initState] = rewardUnfolding.getInitialStateResult(initEpoch, initState);
    }
 
    swAll.stop();
 
    if (storm::settings::getModule<storm::settings::modules::IOSettings>().isExportCdfSet()) {
        std::vector<std::string> headers;
        for (uint64_t i = 0; i < rewardUnfolding.getEpochManager().getDimensionCount(); ++i) {
            headers.push_back(rewardUnfolding.getDimension(i).formula->toString());
        }
        headers.push_back("Result");
        storm::io::exportDataToCSVFile<ValueType, std::string, std::string>(
            storm::settings::getModule<storm::settings::modules::IOSettings>().getExportCdfDirectory() + "cdf.csv", cdfData, headers);
    }
 
    if (storm::settings::getModule<storm::settings::modules::CoreSettings>().isShowStatisticsSet()) {
        STORM_PRINT_AND_LOG("---------------------------------\n");
        STORM_PRINT_AND_LOG("Statistics:\n");
        STORM_PRINT_AND_LOG("---------------------------------\n");
        STORM_PRINT_AND_LOG("          #checked epochs: " << epochOrder.size() << ".\n");
        STORM_PRINT_AND_LOG("             overall Time: " << swAll << ".\n");
        STORM_PRINT_AND_LOG("Epoch Model building Time: " << swBuild << ".\n");
        STORM_PRINT_AND_LOG("Epoch Model checking Time: " << swCheck << ".\n");
        STORM_PRINT_AND_LOG("---------------------------------\n");
    }
 
    return result;
}
 
template<typename ValueType, typename RewardModelType>
std::vector<ValueType> SparseDtmcPrctlHelper<ValueType, RewardModelType>::computeUntilProbabilities(
    Environment const& env, storm::solver::SolveGoal<ValueType>&& goal, storm::storage::SparseMatrix<ValueType> const& transitionMatrix,
    storm::storage::SparseMatrix<ValueType> const& backwardTransitions, storm::storage::BitVector const& phiStates, storm::storage::BitVector const& psiStates,
    bool qualitative, ModelCheckerHint const& hint) {
    std::vector<ValueType> result(transitionMatrix.getRowCount(), storm::utility::zero<ValueType>());
 
    // We need to identify the maybe states (states which have a probability for satisfying the until formula
    // that is strictly between 0 and 1) and the states that satisfy the formula with probability 1.
    storm::storage::BitVector maybeStates, statesWithProbability1;
 
    if (hint.isExplicitModelCheckerHint() && hint.template asExplicitModelCheckerHint<ValueType>().getComputeOnlyMaybeStates()) {
        maybeStates = hint.template asExplicitModelCheckerHint<ValueType>().getMaybeStates();
 
        // Treat the states with probability one
        std::vector<ValueType> const& resultsForNonMaybeStates = hint.template asExplicitModelCheckerHint<ValueType>().getResultHint();
        statesWithProbability1 = storm::storage::BitVector(maybeStates.size(), false);
        storm::storage::BitVector nonMaybeStates = ~maybeStates;
        for (auto state : nonMaybeStates) {
            if (storm::utility::isOne(resultsForNonMaybeStates[state])) {
                statesWithProbability1.set(state, true);
                result[state] = storm::utility::one<ValueType>();
            } else {
                STORM_LOG_THROW(storm::utility::isZero(resultsForNonMaybeStates[state]), storm::exceptions::IllegalArgumentException,
                                "Expected that the result hint specifies probabilities in {0,1} for non-maybe states");
            }
        }
 
        STORM_LOG_INFO("Preprocessing: " << statesWithProbability1.getNumberOfSetBits() << " states with probability 1 (" << maybeStates.getNumberOfSetBits()
                                         << " states remaining).");
    } else {
        // Get all states that have probability 0 and 1 of satisfying the until-formula.
        std::pair<storm::storage::BitVector, storm::storage::BitVector> statesWithProbability01 =
            storm::utility::graph::performProb01(backwardTransitions, phiStates, psiStates);
        storm::storage::BitVector statesWithProbability0 = std::move(statesWithProbability01.first);
        statesWithProbability1 = std::move(statesWithProbability01.second);
        maybeStates = ~(statesWithProbability0 | statesWithProbability1);
 
        STORM_LOG_INFO("Preprocessing: " << statesWithProbability1.getNumberOfSetBits() << " states with probability 1, "
                                         << statesWithProbability0.getNumberOfSetBits() << " with probability 0 (" << maybeStates.getNumberOfSetBits()
                                         << " states remaining).");
 
        // Set values of resulting vector that are known exactly.
        storm::utility::vector::setVectorValues<ValueType>(result, statesWithProbability0, storm::utility::zero<ValueType>());
        storm::utility::vector::setVectorValues<ValueType>(result, statesWithProbability1, storm::utility::one<ValueType>());
    }
 
    // Check if the values of the maybe states are relevant for the SolveGoal
    bool maybeStatesNotRelevant = goal.hasRelevantValues() && goal.relevantValues().isDisjointFrom(maybeStates);
 
    // Check whether we need to compute exact probabilities for some states.
    if (qualitative || maybeStatesNotRelevant) {
        // Set the values for all maybe-states to 0.5 to indicate that their probability values are neither 0 nor 1.
        storm::utility::vector::setVectorValues<ValueType>(result, maybeStates, storm::utility::convertNumber<ValueType>(0.5));
    } else {
        if (!maybeStates.empty()) {
            // In this case we have to compute the probabilities.
 
            // Check whether we need to convert the input to equation system format.
            storm::solver::GeneralLinearEquationSolverFactory<ValueType> linearEquationSolverFactory;
            bool convertToEquationSystem =
                linearEquationSolverFactory.getEquationProblemFormat(env) == storm::solver::LinearEquationSolverProblemFormat::EquationSystem;
 
            // We can eliminate the rows and columns from the original transition probability matrix.
            storm::storage::SparseMatrix<ValueType> submatrix = transitionMatrix.getSubmatrix(true, maybeStates, maybeStates, convertToEquationSystem);
            if (convertToEquationSystem) {
                // Converting the matrix from the fixpoint notation to the form needed for the equation
                // system. That is, we go from x = A*x + b to (I-A)x = b.
                submatrix.convertToEquationSystem();
            }
 
            // Initialize the x vector with the hint (if available) or with 0.5 for each element.
            // This is the initial guess for the iterative solvers. It should be safe as for all
            // 'maybe' states we know that the probability is strictly larger than 0.
            std::vector<ValueType> x;
            if (hint.isExplicitModelCheckerHint() && hint.template asExplicitModelCheckerHint<ValueType>().hasResultHint()) {
                x = storm::utility::vector::filterVector(hint.template asExplicitModelCheckerHint<ValueType>().getResultHint(), maybeStates);
            } else {
                x = std::vector<ValueType>(maybeStates.getNumberOfSetBits(), storm::utility::convertNumber<ValueType>(0.5));
            }
 
            // Prepare the right-hand side of the equation system. For entry i this corresponds to
            // the accumulated probability of going from state i to some 'yes' state.
            std::vector<ValueType> b = transitionMatrix.getConstrainedRowSumVector(maybeStates, statesWithProbability1);
 
            // Now solve the created system of linear equations.
            goal.restrictRelevantValues(maybeStates);
            std::unique_ptr<storm::solver::LinearEquationSolver<ValueType>> solver =
                storm::solver::configureLinearEquationSolver(env, std::move(goal), linearEquationSolverFactory, std::move(submatrix));
            solver->setBounds(storm::utility::zero<ValueType>(), storm::utility::one<ValueType>());
            solver->solveEquations(env, x, b);
 
            // Set values of resulting vector according to result.
            storm::utility::vector::setVectorValues<ValueType>(result, maybeStates, x);
        }
    }
    return result;
}
 
template<typename ValueType, typename RewardModelType>
std::vector<ValueType> SparseDtmcPrctlHelper<ValueType, RewardModelType>::computeAllUntilProbabilities(
    Environment const& env, storm::solver::SolveGoal<ValueType>&& goal, storm::storage::SparseMatrix<ValueType> const& transitionMatrix,
    storm::storage::BitVector const& initialStates, storm::storage::BitVector const& phiStates, storm::storage::BitVector const& psiStates) {
    uint_fast64_t numberOfStates = transitionMatrix.getRowCount();
    std::vector<ValueType> result(numberOfStates, storm::utility::zero<ValueType>());
 
    // All states are relevant
    storm::storage::BitVector relevantStates(numberOfStates, true);
 
    // Compute exact probabilities for some states.
    if (!relevantStates.empty()) {
        // Check whether we need to convert the input to equation system format.
        storm::solver::GeneralLinearEquationSolverFactory<ValueType> linearEquationSolverFactory;
        bool convertToEquationSystem =
            linearEquationSolverFactory.getEquationProblemFormat(env) == storm::solver::LinearEquationSolverProblemFormat::EquationSystem;
 
        storm::storage::SparseMatrix<ValueType> submatrix(transitionMatrix);
        submatrix.makeRowsAbsorbing(phiStates);
        submatrix.makeRowsAbsorbing(psiStates);
        // submatrix.deleteDiagonalEntries(psiStates);
        // storm::storage::BitVector failState(numberOfStates, false);
        // failState.set(0, true);
        submatrix.deleteDiagonalEntries();
        submatrix = submatrix.transpose();
        submatrix = submatrix.getSubmatrix(true, relevantStates, relevantStates, convertToEquationSystem);
 
        if (convertToEquationSystem) {
            // Converting the matrix from the fixpoint notation to the form needed for the equation
            // system. That is, we go from x = A*x + b to (I-A)x = b.
            submatrix.convertToEquationSystem();
        }
 
        // Initialize the x vector with 0.5 for each element.
        // This is the initial guess for the iterative solvers. It should be safe as for all
        // 'maybe' states we know that the probability is strictly larger than 0.
        std::vector<ValueType> x = std::vector<ValueType>(relevantStates.getNumberOfSetBits(), storm::utility::convertNumber<ValueType>(0.5));
 
        // Prepare the right-hand side of the equation system.
        std::vector<ValueType> b(relevantStates.getNumberOfSetBits(), storm::utility::zero<ValueType>());
        // Set initial states
        size_t i = 0;
        ValueType initDist = storm::utility::one<ValueType>() / storm::utility::convertNumber<ValueType>(initialStates.getNumberOfSetBits());
        for (auto state : relevantStates) {
            if (initialStates.get(state)) {
                b[i] = initDist;
            }
            ++i;
        }
 
        // Now solve the created system of linear equations.
        goal.restrictRelevantValues(relevantStates);
        std::unique_ptr<storm::solver::LinearEquationSolver<ValueType>> solver =
            storm::solver::configureLinearEquationSolver(env, std::move(goal), linearEquationSolverFactory, std::move(submatrix));
        solver->setBounds(storm::utility::zero<ValueType>(), storm::utility::one<ValueType>());
        solver->solveEquations(env, x, b);
 
        // Set values of resulting vector according to result.
        storm::utility::vector::setVectorValues<ValueType>(result, relevantStates, x);
    }
    return result;
}
 
template<typename ValueType, typename RewardModelType>
std::vector<ValueType> SparseDtmcPrctlHelper<ValueType, RewardModelType>::computeGloballyProbabilities(
    Environment const& env, storm::solver::SolveGoal<ValueType>&& goal, storm::storage::SparseMatrix<ValueType> const& transitionMatrix,
    storm::storage::SparseMatrix<ValueType> const& backwardTransitions, storm::storage::BitVector const& psiStates, bool qualitative) {
    goal.oneMinus();
    std::vector<ValueType> result = computeUntilProbabilities(env, std::move(goal), transitionMatrix, backwardTransitions,
                                                              storm::storage::BitVector(transitionMatrix.getRowCount(), true), ~psiStates, qualitative);
    for (auto& entry : result) {
        entry = storm::utility::one<ValueType>() - entry;
    }
    return result;
}
 
template<typename ValueType, typename RewardModelType>
std::vector<ValueType> SparseDtmcPrctlHelper<ValueType, RewardModelType>::computeNextProbabilities(
    Environment const& env, storm::storage::SparseMatrix<ValueType> const& transitionMatrix, storm::storage::BitVector const& nextStates) {
    // Create the vector with which to multiply and initialize it correctly.
    std::vector<ValueType> result(transitionMatrix.getRowCount());
    storm::utility::vector::setVectorValues(result, nextStates, storm::utility::one<ValueType>());
 
    // Perform one single matrix-vector multiplication.
    auto multiplier = storm::solver::MultiplierFactory<ValueType>().create(env, transitionMatrix);
    multiplier->multiply(env, result, nullptr, result);
    return result;
}
 
template<typename ValueType, typename RewardModelType>
std::vector<ValueType> SparseDtmcPrctlHelper<ValueType, RewardModelType>::computeCumulativeRewards(
    Environment const& env, storm::solver::SolveGoal<ValueType>&& goal, storm::storage::SparseMatrix<ValueType> const& transitionMatrix,
    RewardModelType const& rewardModel, uint_fast64_t stepBound) {
    // Initialize result to the null vector.
    std::vector<ValueType> result(transitionMatrix.getRowCount());
 
    // Compute the reward vector to add in each step based on the available reward models.
    std::vector<ValueType> totalRewardVector = rewardModel.getTotalRewardVector(transitionMatrix);
 
    // Perform the matrix vector multiplication as often as required by the formula bound.
    auto multiplier = storm::solver::MultiplierFactory<ValueType>().create(env, transitionMatrix);
    multiplier->repeatedMultiply(env, result, &totalRewardVector, stepBound);
 
    return result;
}
 
template<typename ValueType, typename RewardModelType>
std::vector<ValueType> SparseDtmcPrctlHelper<ValueType, RewardModelType>::computeInstantaneousRewards(
    Environment const& env, storm::solver::SolveGoal<ValueType>&& goal, storm::storage::SparseMatrix<ValueType> const& transitionMatrix,
    RewardModelType const& rewardModel, uint_fast64_t stepCount) {
    // Only compute the result if the model has a state-based reward this->getModel().
    STORM_LOG_THROW(rewardModel.hasStateRewards(), storm::exceptions::InvalidPropertyException,
                    "Computing instantaneous rewards for a reward model that does not define any state-rewards. The result is trivially 0.");
 
    // Initialize result to state rewards of the model.
    std::vector<ValueType> result = rewardModel.getStateRewardVector();
 
    // Perform the matrix vector multiplication as often as required by the formula bound.
    auto multiplier = storm::solver::MultiplierFactory<ValueType>().create(env, transitionMatrix);
    multiplier->repeatedMultiply(env, result, nullptr, stepCount);
 
    return result;
}
 
template<typename ValueType, typename RewardModelType>
std::vector<ValueType> SparseDtmcPrctlHelper<ValueType, RewardModelType>::computeTotalRewards(
    Environment const& env, storm::solver::SolveGoal<ValueType>&& goal, storm::storage::SparseMatrix<ValueType> const& transitionMatrix,
    storm::storage::SparseMatrix<ValueType> const& backwardTransitions, RewardModelType const& rewardModel, bool qualitative, ModelCheckerHint const& hint) {
    // Identify the states from which only states with zero reward are reachable.
    // We can then compute reachability rewards assuming these states as target set.
    storm::storage::BitVector statesWithoutReward = rewardModel.getStatesWithZeroReward(transitionMatrix);
    storm::storage::BitVector rew0States = storm::utility::graph::performProbGreater0(backwardTransitions, statesWithoutReward, ~statesWithoutReward);
    rew0States.complement();
    return computeReachabilityRewards(env, std::move(goal), transitionMatrix, backwardTransitions, rewardModel, rew0States, qualitative, hint);
}
 
template<>
std::vector<storm::RationalFunction> SparseDtmcPrctlHelper<storm::RationalFunction>::computeDiscountedCumulativeRewards(
    Environment const& env, storm::solver::SolveGoal<storm::RationalFunction>&& goal,
    storm::storage::SparseMatrix<storm::RationalFunction> const& transitionMatrix,
    storm::models::sparse::StandardRewardModel<storm::RationalFunction> const& rewardModel, uint_fast64_t stepBound, storm::RationalFunction discountFactor) {
    STORM_LOG_THROW(false, storm::exceptions::NotSupportedException, "The specified property is not supported by this value type.");
    return {};
}
 
template<typename ValueType, typename RewardModelType>
std::vector<ValueType> SparseDtmcPrctlHelper<ValueType, RewardModelType>::computeDiscountedCumulativeRewards(
    Environment const& env, storm::solver::SolveGoal<ValueType>&& goal, storm::storage::SparseMatrix<ValueType> const& transitionMatrix,
    RewardModelType const& rewardModel, uint_fast64_t stepBound, ValueType discountFactor) {
    // Only compute the result if the model has at least one reward this->getModel().
    STORM_LOG_THROW(!rewardModel.empty(), storm::exceptions::InvalidPropertyException, "Missing reward model for formula. Skipping formula.");
 
    // Compute the reward vector to add in each step based on the available reward models.
    std::vector<ValueType> totalRewardVector = rewardModel.getTotalRewardVector(transitionMatrix);
 
    // Initialize result to the zero vector.
    std::vector<ValueType> result(transitionMatrix.getRowGroupCount(), storm::utility::zero<ValueType>());
 
    auto multiplier = storm::solver::MultiplierFactory<ValueType>().create(env, transitionMatrix);
    multiplier->repeatedMultiplyWithFactor(env, result, &totalRewardVector, stepBound, discountFactor);
 
    return result;
}
 
template<>
std::vector<storm::RationalFunction> SparseDtmcPrctlHelper<storm::RationalFunction>::computeDiscountedTotalRewards(
    Environment const& env, storm::solver::SolveGoal<storm::RationalFunction>&& goal,
    storm::storage::SparseMatrix<storm::RationalFunction> const& transitionMatrix,
    storm::storage::SparseMatrix<storm::RationalFunction> const& backwardTransitions,
    storm::models::sparse::StandardRewardModel<storm::RationalFunction> const& rewardModel, bool qualitative, storm::RationalFunction discountFactor,
    ModelCheckerHint const& hint) {
    STORM_LOG_THROW(false, storm::exceptions::NotSupportedException, "The specified property is not supported by this value type.");
    return {};
}
 
template<typename ValueType, typename RewardModelType>
std::vector<ValueType> SparseDtmcPrctlHelper<ValueType, RewardModelType>::computeDiscountedTotalRewards(
    Environment const& env, storm::solver::SolveGoal<ValueType>&& goal, storm::storage::SparseMatrix<ValueType> const& transitionMatrix,
    storm::storage::SparseMatrix<ValueType> const& backwardTransitions, RewardModelType const& rewardModel, bool qualitative, ValueType discountFactor,
    ModelCheckerHint const& hint) {
    // If the solver is set to force exact results, throw an error if the method is not explicitly set to a value iteration type.
    STORM_LOG_THROW(!env.solver().isForceExact(), storm::exceptions::NotSupportedException,
                    "Exact solving of discounted total reward objectives is currently not supported.");
 
    // Reduce to reachability rewards
    std::vector<ValueType> b;
 
    std::vector<ValueType> x = std::vector<ValueType>(transitionMatrix.getRowGroupCount(), storm::utility::zero<ValueType>());
    b = rewardModel.getTotalRewardVector(transitionMatrix);
    storm::modelchecker::helper::DiscountingHelper<ValueType, true> discountingHelper(transitionMatrix, discountFactor);
    discountingHelper.solveWithDiscountedValueIteration(env, std::nullopt, x, b);
    return std::vector<ValueType>(std::move(x));
}
 
template<typename ValueType, typename RewardModelType>
std::vector<ValueType> SparseDtmcPrctlHelper<ValueType, RewardModelType>::computeReachabilityRewards(
    Environment const& env, storm::solver::SolveGoal<ValueType>&& goal, storm::storage::SparseMatrix<ValueType> const& transitionMatrix,
    storm::storage::SparseMatrix<ValueType> const& backwardTransitions, RewardModelType const& rewardModel, storm::storage::BitVector const& targetStates,
    bool qualitative, ModelCheckerHint const& hint) {
    return computeReachabilityRewards(
        env, std::move(goal), transitionMatrix, backwardTransitions,
        [&](uint_fast64_t numberOfRows, storm::storage::SparseMatrix<ValueType> const& transitionMatrix, storm::storage::BitVector const& maybeStates) {
            return rewardModel.getTotalRewardVector(numberOfRows, transitionMatrix, maybeStates);
        },
        targetStates, qualitative, [&]() { return rewardModel.getStatesWithZeroReward(transitionMatrix); }, hint);
}
 
template<typename ValueType, typename RewardModelType>
std::vector<ValueType> SparseDtmcPrctlHelper<ValueType, RewardModelType>::computeReachabilityRewards(
    Environment const& env, storm::solver::SolveGoal<ValueType>&& goal, storm::storage::SparseMatrix<ValueType> const& transitionMatrix,
    storm::storage::SparseMatrix<ValueType> const& backwardTransitions, std::vector<ValueType> const& totalStateRewardVector,
    storm::storage::BitVector const& targetStates, bool qualitative, ModelCheckerHint const& hint) {
    return computeReachabilityRewards(
        env, std::move(goal), transitionMatrix, backwardTransitions,
        [&](uint_fast64_t numberOfRows, storm::storage::SparseMatrix<ValueType> const&, storm::storage::BitVector const& maybeStates) {
            std::vector<ValueType> result(numberOfRows);
            storm::utility::vector::selectVectorValues(result, maybeStates, totalStateRewardVector);
            return result;
        },
        targetStates, qualitative, [&]() { return storm::utility::vector::filterZero(totalStateRewardVector); }, hint);
}
 
template<typename ValueType, typename RewardModelType>
std::vector<ValueType> SparseDtmcPrctlHelper<ValueType, RewardModelType>::computeReachabilityTimes(
    Environment const& env, storm::solver::SolveGoal<ValueType>&& goal, storm::storage::SparseMatrix<ValueType> const& transitionMatrix,
    storm::storage::SparseMatrix<ValueType> const& backwardTransitions, storm::storage::BitVector const& targetStates, bool qualitative,
    ModelCheckerHint const& hint) {
    return computeReachabilityRewards(
        env, std::move(goal), transitionMatrix, backwardTransitions,
        [&](uint_fast64_t numberOfRows, storm::storage::SparseMatrix<ValueType> const&, storm::storage::BitVector const&) {
            return std::vector<ValueType>(numberOfRows, storm::utility::one<ValueType>());
        },
        targetStates, qualitative, [&]() { return storm::storage::BitVector(transitionMatrix.getRowGroupCount(), false); }, hint);
}
 
// This function computes an upper bound on the reachability rewards (see Baier et al, CAV'17).
template<typename ValueType>
std::vector<ValueType> computeUpperRewardBounds(storm::storage::SparseMatrix<ValueType> const& transitionMatrix, std::vector<ValueType> const& rewards,
                                                std::vector<ValueType> const& oneStepTargetProbabilities) {
    DsMpiDtmcUpperRewardBoundsComputer<ValueType> dsmpi(transitionMatrix, rewards, oneStepTargetProbabilities);
    std::vector<ValueType> bounds = dsmpi.computeUpperBounds();
    return bounds;
}
 
template<>
std::vector<storm::RationalFunction> computeUpperRewardBounds(storm::storage::SparseMatrix<storm::RationalFunction> const& /*transitionMatrix*/,
                                                              std::vector<storm::RationalFunction> const& /*rewards*/,
                                                              std::vector<storm::RationalFunction> const& /*oneStepTargetProbabilities*/) {
    STORM_LOG_THROW(false, storm::exceptions::NotSupportedException, "Computing upper reward bounds is not supported for rational functions.");
}
 
template<typename ValueType, typename RewardModelType>
std::vector<ValueType> SparseDtmcPrctlHelper<ValueType, RewardModelType>::computeReachabilityRewards(
    Environment const& env, storm::solver::SolveGoal<ValueType>&& goal, storm::storage::SparseMatrix<ValueType> const& transitionMatrix,
    storm::storage::SparseMatrix<ValueType> const& backwardTransitions,
    std::function<std::vector<ValueType>(uint_fast64_t, storm::storage::SparseMatrix<ValueType> const&, storm::storage::BitVector const&)> const&
        totalStateRewardVectorGetter,
    storm::storage::BitVector const& targetStates, bool qualitative, std::function<storm::storage::BitVector()> const& zeroRewardStatesGetter,
    ModelCheckerHint const& hint) {
    std::vector<ValueType> result(transitionMatrix.getRowCount(), storm::utility::zero<ValueType>());
 
    // Determine which states have reward zero
    storm::storage::BitVector rew0States;
    if (storm::settings::getModule<storm::settings::modules::ModelCheckerSettings>().isFilterRewZeroSet()) {
        rew0States = storm::utility::graph::performProb1(backwardTransitions, zeroRewardStatesGetter(), targetStates);
    } else {
        rew0States = targetStates;
    }
 
    // Determine which states have a reward that is less than infinity.
    storm::storage::BitVector maybeStates;
    if (hint.isExplicitModelCheckerHint() && hint.template asExplicitModelCheckerHint<ValueType>().getComputeOnlyMaybeStates()) {
        maybeStates = hint.template asExplicitModelCheckerHint<ValueType>().getMaybeStates();
        storm::utility::vector::setVectorValues(result, ~(maybeStates | rew0States), storm::utility::infinity<ValueType>());
 
        STORM_LOG_INFO("Preprocessing: " << rew0States.getNumberOfSetBits() << " States with reward zero (" << maybeStates.getNumberOfSetBits()
                                         << " states remaining).");
    } else {
        storm::storage::BitVector trueStates(transitionMatrix.getRowCount(), true);
        storm::storage::BitVector infinityStates = storm::utility::graph::performProb1(backwardTransitions, trueStates, rew0States);
        infinityStates.complement();
        maybeStates = ~(rew0States | infinityStates);
 
        STORM_LOG_INFO("Preprocessing: " << infinityStates.getNumberOfSetBits() << " states with reward infinity, " << rew0States.getNumberOfSetBits()
                                         << " states with reward zero (" << maybeStates.getNumberOfSetBits() << " states remaining).");
 
        storm::utility::vector::setVectorValues(result, infinityStates, storm::utility::infinity<ValueType>());
    }
 
    // Check if the values of the maybe states are relevant for the SolveGoal
    bool maybeStatesNotRelevant = goal.hasRelevantValues() && goal.relevantValues().isDisjointFrom(maybeStates);
 
    // Check whether we need to compute exact rewards for some states.
    if (qualitative || maybeStatesNotRelevant) {
        // Set the values for all maybe-states to 1 to indicate that their reward values
        // are neither 0 nor infinity.
        storm::utility::vector::setVectorValues<ValueType>(result, maybeStates, storm::utility::one<ValueType>());
    } else {
        if (!maybeStates.empty()) {
            // Check whether we need to convert the input to equation system format.
            storm::solver::GeneralLinearEquationSolverFactory<ValueType> linearEquationSolverFactory;
            bool convertToEquationSystem =
                linearEquationSolverFactory.getEquationProblemFormat(env) == storm::solver::LinearEquationSolverProblemFormat::EquationSystem;
 
            // In this case we have to compute the reward values for the remaining states.
            // We can eliminate the rows and columns from the original transition probability matrix.
            storm::storage::SparseMatrix<ValueType> submatrix = transitionMatrix.getSubmatrix(true, maybeStates, maybeStates, convertToEquationSystem);
 
            // Initialize the x vector with the hint (if available) or with 1 for each element.
            // This is the initial guess for the iterative solvers.
            std::vector<ValueType> x;
            if (hint.isExplicitModelCheckerHint() && hint.template asExplicitModelCheckerHint<ValueType>().hasResultHint()) {
                x = storm::utility::vector::filterVector(hint.template asExplicitModelCheckerHint<ValueType>().getResultHint(), maybeStates);
            } else {
                x = std::vector<ValueType>(submatrix.getColumnCount(), storm::utility::one<ValueType>());
            }
 
            // Prepare the right-hand side of the equation system.
            std::vector<ValueType> b = totalStateRewardVectorGetter(submatrix.getRowCount(), transitionMatrix, maybeStates);
 
            storm::solver::LinearEquationSolverRequirements requirements = linearEquationSolverFactory.getRequirements(env);
            boost::optional<std::vector<ValueType>> upperRewardBounds;
            requirements.clearLowerBounds();
            if (requirements.upperBounds()) {
                upperRewardBounds = computeUpperRewardBounds(submatrix, b, transitionMatrix.getConstrainedRowSumVector(maybeStates, rew0States));
                requirements.clearUpperBounds();
            }
            STORM_LOG_THROW(!requirements.hasEnabledCriticalRequirement(), storm::exceptions::UncheckedRequirementException,
                            "Solver requirements " + requirements.getEnabledRequirementsAsString() + " not checked.");
 
            // If necessary, convert the matrix from the fixpoint notation to the form needed for the equation system.
            if (convertToEquationSystem) {
                // go from x = A*x + b to (I-A)x = b.
                submatrix.convertToEquationSystem();
            }
 
            // Create the solver.
            goal.restrictRelevantValues(maybeStates);
            std::unique_ptr<storm::solver::LinearEquationSolver<ValueType>> solver =
                storm::solver::configureLinearEquationSolver(env, std::move(goal), linearEquationSolverFactory, std::move(submatrix));
            solver->setLowerBound(storm::utility::zero<ValueType>());
            if (upperRewardBounds) {
                solver->setUpperBounds(std::move(upperRewardBounds.get()));
            }
 
            // Now solve the resulting equation system.
            solver->solveEquations(env, x, b);
 
            // Set values of resulting vector according to result.
            storm::utility::vector::setVectorValues<ValueType>(result, maybeStates, x);
        }
    }
    return result;
}
 
template<typename ValueType, typename RewardModelType>
typename SparseDtmcPrctlHelper<ValueType, RewardModelType>::BaierTransformedModel SparseDtmcPrctlHelper<ValueType, RewardModelType>::computeBaierTransformation(
    Environment const& env, storm::storage::SparseMatrix<ValueType> const& transitionMatrix, storm::storage::SparseMatrix<ValueType> const& backwardTransitions,
    storm::storage::BitVector const& targetStates, storm::storage::BitVector const& conditionStates,
    boost::optional<std::vector<ValueType>> const& stateRewards) {
    BaierTransformedModel result;
 
    // Start by computing all 'before' states, i.e. the states for which the conditional probability is defined.
    std::vector<ValueType> probabilitiesToReachConditionStates =
        computeUntilProbabilities(env, storm::solver::SolveGoal<ValueType>(), transitionMatrix, backwardTransitions,
                                  storm::storage::BitVector(transitionMatrix.getRowCount(), true), conditionStates, false);
 
    result.beforeStates = storm::storage::BitVector(targetStates.size(), true);
    uint_fast64_t state = 0;
    uint_fast64_t beforeStateIndex = 0;
    for (auto const& value : probabilitiesToReachConditionStates) {
        if (value == storm::utility::zero<ValueType>()) {
            result.beforeStates.set(state, false);
        } else {
            probabilitiesToReachConditionStates[beforeStateIndex] = value;
            ++beforeStateIndex;
        }
        ++state;
    }
    probabilitiesToReachConditionStates.resize(beforeStateIndex);
 
    if (targetStates.empty()) {
        result.noTargetStates = true;
        return result;
    } else if (!result.beforeStates.empty()) {
        // If there are some states for which the conditional probability is defined and there are some
        // states that can reach the target states without visiting condition states first, we need to
        // do more work.
 
        // First, compute the relevant states and some offsets.
        storm::storage::BitVector allStates(targetStates.size(), true);
        std::vector<uint_fast64_t> numberOfBeforeStatesUpToState = result.beforeStates.getNumberOfSetBitsBeforeIndices();
        storm::storage::BitVector statesWithProbabilityGreater0 = storm::utility::graph::performProbGreater0(backwardTransitions, allStates, targetStates);
        statesWithProbabilityGreater0 &= storm::utility::graph::getReachableStates(transitionMatrix, conditionStates, allStates, targetStates);
        uint_fast64_t normalStatesOffset = result.beforeStates.getNumberOfSetBits();
        std::vector<uint_fast64_t> numberOfNormalStatesUpToState = statesWithProbabilityGreater0.getNumberOfSetBitsBeforeIndices();
 
        // All transitions going to states with probability zero, need to be redirected to a deadlock state.
        bool addDeadlockState = false;
        uint_fast64_t deadlockState = normalStatesOffset + statesWithProbabilityGreater0.getNumberOfSetBits();
 
        // Now, we create the matrix of 'before' and 'normal' states.
        storm::storage::SparseMatrixBuilder<ValueType> builder;
 
        // Start by creating the transitions of the 'before' states.
        uint_fast64_t currentRow = 0;
        for (auto beforeState : result.beforeStates) {
            if (conditionStates.get(beforeState)) {
                // For condition states, we move to the 'normal' states.
                ValueType zeroProbability = storm::utility::zero<ValueType>();
                for (auto const& successorEntry : transitionMatrix.getRow(beforeState)) {
                    if (statesWithProbabilityGreater0.get(successorEntry.getColumn())) {
                        builder.addNextValue(currentRow, normalStatesOffset + numberOfNormalStatesUpToState[successorEntry.getColumn()],
                                             successorEntry.getValue());
                    } else {
                        zeroProbability += successorEntry.getValue();
                    }
                }
                if (!storm::utility::isZero(zeroProbability)) {
                    builder.addNextValue(currentRow, deadlockState, zeroProbability);
                }
            } else {
                // For non-condition states, we scale the probabilities going to other before states.
                for (auto const& successorEntry : transitionMatrix.getRow(beforeState)) {
                    if (result.beforeStates.get(successorEntry.getColumn())) {
                        builder.addNextValue(currentRow, numberOfBeforeStatesUpToState[successorEntry.getColumn()],
                                             successorEntry.getValue() *
                                                 probabilitiesToReachConditionStates[numberOfBeforeStatesUpToState[successorEntry.getColumn()]] /
                                                 probabilitiesToReachConditionStates[currentRow]);
                    }
                }
            }
            ++currentRow;
        }
 
        // Then, create the transitions of the 'normal' states.
        for (auto state : statesWithProbabilityGreater0) {
            ValueType zeroProbability = storm::utility::zero<ValueType>();
            for (auto const& successorEntry : transitionMatrix.getRow(state)) {
                if (statesWithProbabilityGreater0.get(successorEntry.getColumn())) {
                    builder.addNextValue(currentRow, normalStatesOffset + numberOfNormalStatesUpToState[successorEntry.getColumn()], successorEntry.getValue());
                } else {
                    zeroProbability += successorEntry.getValue();
                }
            }
            if (!storm::utility::isZero(zeroProbability)) {
                addDeadlockState = true;
                builder.addNextValue(currentRow, deadlockState, zeroProbability);
            }
            ++currentRow;
        }
        if (addDeadlockState) {
            builder.addNextValue(deadlockState, deadlockState, storm::utility::one<ValueType>());
        }
 
        // Build the new transition matrix and the new targets.
        result.transitionMatrix = builder.build(addDeadlockState ? (deadlockState + 1) : deadlockState);
        storm::storage::BitVector newTargetStates = targetStates % result.beforeStates;
        newTargetStates.resize(result.transitionMatrix.get().getRowCount());
        for (auto state : targetStates % statesWithProbabilityGreater0) {
            newTargetStates.set(normalStatesOffset + state, true);
        }
        result.targetStates = std::move(newTargetStates);
 
        // If a reward model was given, we need to compute the rewards for the transformed model.
        if (stateRewards) {
            std::vector<ValueType> newStateRewards(result.beforeStates.getNumberOfSetBits());
            storm::utility::vector::selectVectorValues(newStateRewards, result.beforeStates, stateRewards.get());
 
            newStateRewards.reserve(result.transitionMatrix.get().getRowCount());
            for (auto state : statesWithProbabilityGreater0) {
                newStateRewards.push_back(stateRewards.get()[state]);
            }
            // Add a zero reward to the deadlock state.
            if (addDeadlockState) {
                newStateRewards.push_back(storm::utility::zero<ValueType>());
            }
            result.stateRewards = std::move(newStateRewards);
        }
    }
 
    return result;
}
 
template<typename ValueType, typename RewardModelType>
storm::storage::BitVector SparseDtmcPrctlHelper<ValueType, RewardModelType>::BaierTransformedModel::getNewRelevantStates() const {
    storm::storage::BitVector newRelevantStates(transitionMatrix.get().getRowCount());
    for (uint64_t i = 0; i < this->beforeStates.getNumberOfSetBits(); ++i) {
        newRelevantStates.set(i);
    }
    return newRelevantStates;
}
 
template<typename ValueType, typename RewardModelType>
storm::storage::BitVector SparseDtmcPrctlHelper<ValueType, RewardModelType>::BaierTransformedModel::getNewRelevantStates(
    storm::storage::BitVector const& oldRelevantStates) const {
    storm::storage::BitVector result = oldRelevantStates % this->beforeStates;
    result.resize(transitionMatrix.get().getRowCount());
    return result;
}
 
template<typename ValueType, typename RewardModelType>
std::vector<ValueType> SparseDtmcPrctlHelper<ValueType, RewardModelType>::computeConditionalProbabilities(
    Environment const& env, storm::solver::SolveGoal<ValueType>&& goal, storm::storage::SparseMatrix<ValueType> const& transitionMatrix,
    storm::storage::SparseMatrix<ValueType> const& backwardTransitions, storm::storage::BitVector const& targetStates,
    storm::storage::BitVector const& conditionStates, bool qualitative) {
    // Prepare result vector.
    std::vector<ValueType> result(transitionMatrix.getRowCount(), storm::utility::infinity<ValueType>());
 
    if (!conditionStates.empty()) {
        BaierTransformedModel transformedModel =
            computeBaierTransformation(env, transitionMatrix, backwardTransitions, targetStates, conditionStates, boost::none);
 
        if (transformedModel.noTargetStates) {
            storm::utility::vector::setVectorValues(result, transformedModel.beforeStates, storm::utility::zero<ValueType>());
        } else {
            // At this point, we do not need to check whether there are 'before' states, since the condition
            // states were non-empty so there is at least one state with a positive probability of satisfying
            // the condition.
 
            // Now compute reachability probabilities in the transformed model.
            storm::storage::SparseMatrix<ValueType> const& newTransitionMatrix = transformedModel.transitionMatrix.get();
            storm::storage::BitVector newRelevantValues;
            if (goal.hasRelevantValues()) {
                newRelevantValues = transformedModel.getNewRelevantStates(goal.relevantValues());
            } else {
                newRelevantValues = transformedModel.getNewRelevantStates();
            }
            goal.setRelevantValues(std::move(newRelevantValues));
            std::vector<ValueType> conditionalProbabilities =
                computeUntilProbabilities(env, std::move(goal), newTransitionMatrix, newTransitionMatrix.transpose(),
                                          storm::storage::BitVector(newTransitionMatrix.getRowCount(), true), transformedModel.targetStates.get(), qualitative);
 
            storm::utility::vector::setVectorValues(result, transformedModel.beforeStates, conditionalProbabilities);
        }
    }
 
    return result;
}
 
template<typename ValueType, typename RewardModelType>
std::vector<ValueType> SparseDtmcPrctlHelper<ValueType, RewardModelType>::computeConditionalRewards(
    Environment const& env, storm::solver::SolveGoal<ValueType>&& goal, storm::storage::SparseMatrix<ValueType> const& transitionMatrix,
    storm::storage::SparseMatrix<ValueType> const& backwardTransitions, RewardModelType const& rewardModel, storm::storage::BitVector const& targetStates,
    storm::storage::BitVector const& conditionStates, bool qualitative) {
    // Prepare result vector.
    std::vector<ValueType> result(transitionMatrix.getRowCount(), storm::utility::infinity<ValueType>());
 
    if (!conditionStates.empty()) {
        BaierTransformedModel transformedModel = computeBaierTransformation(env, transitionMatrix, backwardTransitions, targetStates, conditionStates,
                                                                            rewardModel.getTotalRewardVector(transitionMatrix));
 
        if (transformedModel.noTargetStates) {
            storm::utility::vector::setVectorValues(result, transformedModel.beforeStates, storm::utility::zero<ValueType>());
        } else {
            // At this point, we do not need to check whether there are 'before' states, since the condition
            // states were non-empty so there is at least one state with a positive probability of satisfying
            // the condition.
 
            // Now compute reachability probabilities in the transformed model.
            storm::storage::SparseMatrix<ValueType> const& newTransitionMatrix = transformedModel.transitionMatrix.get();
            storm::storage::BitVector newRelevantValues;
            if (goal.hasRelevantValues()) {
                newRelevantValues = transformedModel.getNewRelevantStates(goal.relevantValues());
            } else {
                newRelevantValues = transformedModel.getNewRelevantStates();
            }
            goal.setRelevantValues(std::move(newRelevantValues));
            std::vector<ValueType> conditionalRewards =
                computeReachabilityRewards(env, std::move(goal), newTransitionMatrix, newTransitionMatrix.transpose(), transformedModel.stateRewards.get(),
                                           transformedModel.targetStates.get(), qualitative);
            storm::utility::vector::setVectorValues(result, transformedModel.beforeStates, conditionalRewards);
        }
    }
 
    return result;
}
 
template class SparseDtmcPrctlHelper<double>;
 
template class SparseDtmcPrctlHelper<storm::RationalNumber>;
template class SparseDtmcPrctlHelper<storm::RationalFunction>;
}  // namespace helper
}  // namespace modelchecker
}  // namespace storm