#include "storm/modelchecker/prctl/helper/SparseMdpPrctlHelper.h"

#include <boost/container/flat_map.hpp>


#include "storm/modelchecker/results/ExplicitQuantitativeCheckResult.h"
#include "storm/modelchecker/hints/ExplicitModelCheckerHint.h"

#include "storm/models/sparse/StandardRewardModel.h"

#include "storm/storage/MaximalEndComponentDecomposition.h"

#include "storm/utility/macros.h"
#include "storm/utility/vector.h"
#include "storm/utility/graph.h"

#include "storm/storage/expressions/Variable.h"
#include "storm/storage/expressions/Expression.h"
#include "storm/storage/Scheduler.h"

#include "storm/solver/MinMaxLinearEquationSolver.h"
#include "storm/solver/LpSolver.h"
 
#include "storm/settings/SettingsManager.h"
#include "storm/settings/modules/MinMaxEquationSolverSettings.h"

#include "storm/exceptions/InvalidStateException.h"
#include "storm/exceptions/InvalidPropertyException.h"
#include "storm/exceptions/InvalidSettingsException.h"
#include "storm/exceptions/IllegalFunctionCallException.h"
#include "storm/exceptions/IllegalArgumentException.h"

namespace storm {
    namespace modelchecker {
        namespace helper {

            template<typename ValueType>
            std::vector<ValueType> SparseMdpPrctlHelper<ValueType>::computeBoundedUntilProbabilities(OptimizationDirection dir, storm::storage::SparseMatrix<ValueType> const& transitionMatrix, storm::storage::SparseMatrix<ValueType> const& backwardTransitions, storm::storage::BitVector const& phiStates, storm::storage::BitVector const& psiStates, uint_fast64_t stepBound, storm::solver::MinMaxLinearEquationSolverFactory<ValueType> const& minMaxLinearEquationSolverFactory, ModelCheckerHint const& hint) {
                std::vector<ValueType> result(transitionMatrix.getRowGroupCount(), storm::utility::zero<ValueType>());
                
                // Determine the states that have 0 probability of reaching the target states.
                storm::storage::BitVector maybeStates;
                if (hint.isExplicitModelCheckerHint() && hint.template asExplicitModelCheckerHint<ValueType>().getComputeOnlyMaybeStates()) {
                    maybeStates = hint.template asExplicitModelCheckerHint<ValueType>().getMaybeStates();
                } else {
                    if (dir == OptimizationDirection::Minimize) {
                        maybeStates = storm::utility::graph::performProbGreater0A(transitionMatrix, transitionMatrix.getRowGroupIndices(), backwardTransitions, phiStates, psiStates, true, stepBound);
                    } else {
                        maybeStates = storm::utility::graph::performProbGreater0E(backwardTransitions, phiStates, psiStates, true, stepBound);
                    }
                    maybeStates &= ~psiStates;
                    STORM_LOG_INFO("Found " << maybeStates.getNumberOfSetBits() << " 'maybe' states.");
                }
                
                if (!maybeStates.empty()) {
                    // We can eliminate the rows and columns from the original transition probability matrix that have probability 0.
                    storm::storage::SparseMatrix<ValueType> submatrix = transitionMatrix.getSubmatrix(true, maybeStates, maybeStates, false);
                    std::vector<ValueType> b = transitionMatrix.getConstrainedRowGroupSumVector(maybeStates, psiStates);
                    
                    // Create the vector with which to multiply.
                    std::vector<ValueType> subresult(maybeStates.getNumberOfSetBits());
                    
                    std::unique_ptr<storm::solver::MinMaxLinearEquationSolver<ValueType>> solver = minMaxLinearEquationSolverFactory.create(std::move(submatrix));
                    solver->repeatedMultiply(dir, subresult, &b, stepBound);
                    
                    // Set the values of the resulting vector accordingly.
                    storm::utility::vector::setVectorValues(result, maybeStates, subresult);
                }
                storm::utility::vector::setVectorValues(result, psiStates, storm::utility::one<ValueType>());
                
                return result;
            }

            template<typename ValueType>
            std::vector<ValueType> SparseMdpPrctlHelper<ValueType>::computeNextProbabilities(OptimizationDirection dir, storm::storage::SparseMatrix<ValueType> const& transitionMatrix, storm::storage::BitVector const& nextStates, storm::solver::MinMaxLinearEquationSolverFactory<ValueType> const& minMaxLinearEquationSolverFactory) {

                // Create the vector with which to multiply and initialize it correctly.
                std::vector<ValueType> result(transitionMatrix.getRowGroupCount());
                storm::utility::vector::setVectorValues(result, nextStates, storm::utility::one<ValueType>());
                
                std::unique_ptr<storm::solver::MinMaxLinearEquationSolver<ValueType>> solver = minMaxLinearEquationSolverFactory.create(transitionMatrix);
                solver->repeatedMultiply(dir, result, nullptr, 1);
                
                return result;
            }
            
            template<typename ValueType>
            MDPSparseModelCheckingHelperReturnType<ValueType> SparseMdpPrctlHelper<ValueType>::computeUntilProbabilities(storm::solver::SolveGoal const& 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, bool produceScheduler, storm::solver::MinMaxLinearEquationSolverFactory<ValueType> const& minMaxLinearEquationSolverFactory, ModelCheckerHint const& hint) {
                STORM_LOG_THROW(!(qualitative && produceScheduler), storm::exceptions::InvalidSettingsException, "Cannot produce scheduler when performing qualitative model checking only.");
                     
                std::vector<ValueType> result(transitionMatrix.getRowGroupCount(), 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 probablity 1 and 0, respectively.
                storm::storage::BitVector maybeStates, statesWithProbability1, statesWithProbability0;
                
                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);
                    statesWithProbability0 = storm::storage::BitVector(maybeStates.size(), false);
                    storm::storage::BitVector nonMaybeStates = ~maybeStates;
                    for (auto const& 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");
                            statesWithProbability0.set(state, true);
                        }
                    }
                } else {
                    // Get all states that have probability 0 and 1 of satisfying the until-formula.
                     std::pair<storm::storage::BitVector, storm::storage::BitVector> statesWithProbability01;
                    if (goal.minimize()) {
                        statesWithProbability01 = storm::utility::graph::performProb01Min(transitionMatrix, transitionMatrix.getRowGroupIndices(), backwardTransitions, phiStates, psiStates);
                    } else {
                        statesWithProbability01 = storm::utility::graph::performProb01Max(transitionMatrix, transitionMatrix.getRowGroupIndices(), backwardTransitions, phiStates, psiStates);
                    }
                    statesWithProbability0 = std::move(statesWithProbability01.first);
                    statesWithProbability1 = std::move(statesWithProbability01.second);
                    maybeStates = ~(statesWithProbability0 | statesWithProbability1);
                    STORM_LOG_INFO("Found " << statesWithProbability0.getNumberOfSetBits() << " 'no' states.");
                    STORM_LOG_INFO("Found " << statesWithProbability1.getNumberOfSetBits() << " 'yes' states.");
                    STORM_LOG_INFO("Found " << maybeStates.getNumberOfSetBits() << " 'maybe' states.");
                
                    // 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>());
                }
                
                // If requested, we will produce a scheduler.
                std::unique_ptr<storm::storage::Scheduler<ValueType>> scheduler;
                if (produceScheduler) {
                    scheduler = std::make_unique<storm::storage::Scheduler<ValueType>>(transitionMatrix.getRowGroupCount());
                }
                
                // Check whether we need to compute exact probabilities for some states.
                if (qualitative) {
                    // 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 have to compute the probabilities.
                        
                        // First, we can eliminate the rows and columns from the original transition probability matrix for states
                        // whose probabilities are already known.
                        storm::storage::SparseMatrix<ValueType> submatrix = transitionMatrix.getSubmatrix(true, maybeStates, maybeStates, false);
                        
                        // 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.getConstrainedRowGroupSumVector(maybeStates, statesWithProbability1);
                        
                        // obtain hint information if possible
                        bool skipEcWithinMaybeStatesCheck = goal.minimize() || (hint.isExplicitModelCheckerHint() && hint.asExplicitModelCheckerHint<ValueType>().getNoEndComponentsInMaybeStates());
                        std::pair<boost::optional<std::vector<ValueType>>, boost::optional<std::vector<uint_fast64_t>>> hintInformation = extractHintInformationForMaybeStates(transitionMatrix, backwardTransitions, maybeStates, boost::none, hint, skipEcWithinMaybeStatesCheck);
                        
                        // Now compute the results for the maybeStates
                        std::pair<std::vector<ValueType>, boost::optional<std::vector<uint_fast64_t>>> resultForMaybeStates = computeValuesOnlyMaybeStates(goal, submatrix, b, produceScheduler, minMaxLinearEquationSolverFactory, std::move(hintInformation.first), std::move(hintInformation.second), storm::utility::zero<ValueType>(), storm::utility::one<ValueType>());
                        
                        // Set values of resulting vector according to result.
                        storm::utility::vector::setVectorValues<ValueType>(result, maybeStates, resultForMaybeStates.first);

                        if (produceScheduler) {
                            std::vector<uint_fast64_t> const& subChoices = resultForMaybeStates.second.get();
                            auto subChoiceIt = subChoices.begin();
                            for (auto maybeState : maybeStates) {
                                scheduler->setChoice(*subChoiceIt, maybeState);
                                ++subChoiceIt;
                            }
                            assert(subChoiceIt == subChoices.end());
                        }
                    }
                }
                
                // Finally, if we need to produce a scheduler, we also need to figure out the parts of the scheduler for
                // the states with probability 1 or 0 (depending on whether we maximize or minimize).
                // We also need to define some arbitrary choice for the remaining states to obtain a fully defined scheduler.
                if (produceScheduler) {
                    if (goal.minimize()) {
                        storm::utility::graph::computeSchedulerProb0E(statesWithProbability0, transitionMatrix, *scheduler);
                        for (auto const& prob1State : statesWithProbability1) {
                            scheduler->setChoice(0, prob1State);
                        }
                    } else {
                        storm::utility::graph::computeSchedulerProb1E(statesWithProbability1, transitionMatrix, backwardTransitions, phiStates, psiStates, *scheduler);
                        for (auto const& prob0State : statesWithProbability0) {
                            scheduler->setChoice(0, prob0State);
                        }
                    }
                }
                
                STORM_LOG_ASSERT((!produceScheduler && !scheduler) || (!scheduler->isPartialScheduler() && scheduler->isDeterministicScheduler() && scheduler->isMemorylessScheduler()), "Unexpected format of obtained scheduler.");

                
                return MDPSparseModelCheckingHelperReturnType<ValueType>(std::move(result), std::move(scheduler));
            }
            
            template<typename ValueType>
            std::pair<boost::optional<std::vector<ValueType>>, boost::optional<std::vector<uint_fast64_t>>> SparseMdpPrctlHelper<ValueType>::extractHintInformationForMaybeStates(storm::storage::SparseMatrix<ValueType> const& transitionMatrix, storm::storage::SparseMatrix<ValueType> const& backwardTransitions, storm::storage::BitVector const& maybeStates, boost::optional<storm::storage::BitVector> const& selectedChoices, ModelCheckerHint const& hint, bool skipECWithinMaybeStatesCheck) {
                
                // Scheduler hint
                boost::optional<std::vector<uint_fast64_t>> subSchedulerChoices;
                if (hint.isExplicitModelCheckerHint() && hint.template asExplicitModelCheckerHint<ValueType>().hasSchedulerHint()) {
                    auto const& schedulerHint = hint.template asExplicitModelCheckerHint<ValueType>().getSchedulerHint();
                    std::vector<uint_fast64_t> hintChoices;
                    
                    // the scheduler hint is only applicable if it induces no BSCC consisting of maybestates
                    bool hintApplicable;
                    if (!skipECWithinMaybeStatesCheck) {
                        hintChoices.reserve(maybeStates.size());
                        for (uint_fast64_t state = 0; state < maybeStates.size(); ++state) {
                            hintChoices.push_back(schedulerHint.getChoice(state).getDeterministicChoice());
                        }
                        hintApplicable = storm::utility::graph::performProb1(transitionMatrix.transposeSelectedRowsFromRowGroups(hintChoices), maybeStates, ~maybeStates).full();
                    } else {
                        hintApplicable = true;
                    }
                    
                    if (hintApplicable) {
                        // Compute the hint w.r.t. the given subsystem
                        hintChoices.clear();
                        hintChoices.reserve(maybeStates.getNumberOfSetBits());
                        for (auto const& state : maybeStates) {
                            uint_fast64_t hintChoice = schedulerHint.getChoice(state).getDeterministicChoice();
                            if (selectedChoices) {
                                uint_fast64_t firstChoice = transitionMatrix.getRowGroupIndices()[state];
                                uint_fast64_t lastChoice = firstChoice + hintChoice;
                                hintChoice = 0;
                                for (uint_fast64_t choice = selectedChoices->getNextSetIndex(firstChoice); choice < lastChoice; choice = selectedChoices->getNextSetIndex(choice + 1)) {
                                    ++hintChoice;
                                }
                            }
                            hintChoices.push_back(hintChoice);
                        }
                        subSchedulerChoices = std::move(hintChoices);
                    }
                }
                
                // Solution value hint
                boost::optional<std::vector<ValueType>> subValues;
                // The result hint is only applicable if there are no End Components consisting of maybestates
                if (hint.isExplicitModelCheckerHint() && hint.template asExplicitModelCheckerHint<ValueType>().hasResultHint() &&
                   (skipECWithinMaybeStatesCheck || subSchedulerChoices ||
                    storm::utility::graph::performProb1A(transitionMatrix, transitionMatrix.getRowGroupIndices(), backwardTransitions, maybeStates, ~maybeStates).full())) {
                    subValues = storm::utility::vector::filterVector(hint.template asExplicitModelCheckerHint<ValueType>().getResultHint(), maybeStates);
                }
                return std::make_pair(std::move(subValues), std::move(subSchedulerChoices));
            }
            
            template<typename ValueType>
            std::pair<std::vector<ValueType>, boost::optional<std::vector<uint_fast64_t>>> SparseMdpPrctlHelper<ValueType>::computeValuesOnlyMaybeStates(storm::solver::SolveGoal const& goal, storm::storage::SparseMatrix<ValueType> const& submatrix, std::vector<ValueType> const& b, bool produceScheduler, storm::solver::MinMaxLinearEquationSolverFactory<ValueType> const& minMaxLinearEquationSolverFactory, boost::optional<std::vector<ValueType>>&& hintValues, boost::optional<std::vector<uint_fast64_t>>&& hintChoices, boost::optional<ValueType> const& lowerResultBound, boost::optional<ValueType> const& upperResultBound) {
                
                // Set up the solver
                std::unique_ptr<storm::solver::MinMaxLinearEquationSolver<ValueType>> solver = storm::solver::configureMinMaxLinearEquationSolver(goal, minMaxLinearEquationSolverFactory, submatrix);
                if (lowerResultBound) {
                    solver->setLowerBound(lowerResultBound.get());
                }
                if (upperResultBound) {
                    solver->setUpperBound(upperResultBound.get());
                }
                if (hintChoices) {
                    solver->setSchedulerHint(std::move(hintChoices.get()));
                }
                solver->setTrackScheduler(produceScheduler);
                
                // Initialize the solution vector.
                std::vector<ValueType> x = hintValues ? std::move(hintValues.get()) : std::vector<ValueType>(submatrix.getRowGroupCount(), lowerResultBound ? lowerResultBound.get() : storm::utility::zero<ValueType>());
                
                // Solve the corresponding system of equations.
                solver->solveEquations(x, b);
                
                // If requested, a scheduler was produced
                if (produceScheduler) {
                   return std::pair<std::vector<ValueType>, boost::optional<std::vector<uint_fast64_t>>>(std::move(x), std::move(solver->getSchedulerChoices()));
                } else {
                    return std::pair<std::vector<ValueType>, boost::optional<std::vector<uint_fast64_t>>>(std::move(x), boost::none);
                }
            }

            template<typename ValueType>
            MDPSparseModelCheckingHelperReturnType<ValueType> SparseMdpPrctlHelper<ValueType>::computeUntilProbabilities(OptimizationDirection dir, storm::storage::SparseMatrix<ValueType> const& transitionMatrix, storm::storage::SparseMatrix<ValueType> const& backwardTransitions, storm::storage::BitVector const& phiStates, storm::storage::BitVector const& psiStates, bool qualitative, bool produceScheduler, storm::solver::MinMaxLinearEquationSolverFactory<ValueType> const& minMaxLinearEquationSolverFactory, ModelCheckerHint const& hint) {
                storm::solver::SolveGoal goal(dir);
                return std::move(computeUntilProbabilities(goal, transitionMatrix, backwardTransitions, phiStates, psiStates, qualitative, produceScheduler, minMaxLinearEquationSolverFactory, hint));
            }
           
            template<typename ValueType>
            std::vector<ValueType> SparseMdpPrctlHelper<ValueType>::computeGloballyProbabilities(OptimizationDirection dir, storm::storage::SparseMatrix<ValueType> const& transitionMatrix, storm::storage::SparseMatrix<ValueType> const& backwardTransitions, storm::storage::BitVector const& psiStates, bool qualitative, storm::solver::MinMaxLinearEquationSolverFactory<ValueType> const& minMaxLinearEquationSolverFactory, bool useMecBasedTechnique) {
                if (useMecBasedTechnique) {
                    storm::storage::MaximalEndComponentDecomposition<ValueType> mecDecomposition(transitionMatrix, backwardTransitions, psiStates);
                    storm::storage::BitVector statesInPsiMecs(transitionMatrix.getRowGroupCount());
                    for (auto const& mec : mecDecomposition) {
                        for (auto const& stateActionsPair : mec) {
                            statesInPsiMecs.set(stateActionsPair.first, true);
                        }
                    }
                    
                    return std::move(computeUntilProbabilities(dir, transitionMatrix, backwardTransitions, psiStates, statesInPsiMecs, qualitative, false, minMaxLinearEquationSolverFactory).values);
                } else {
                    std::vector<ValueType> result = computeUntilProbabilities(dir == OptimizationDirection::Minimize ? OptimizationDirection::Maximize : OptimizationDirection::Minimize, transitionMatrix, backwardTransitions, storm::storage::BitVector(transitionMatrix.getRowGroupCount(), true), ~psiStates, qualitative, false, minMaxLinearEquationSolverFactory).values;
                    for (auto& element : result) {
                        element = storm::utility::one<ValueType>() - element;
                    }
                    return std::move(result);
                }
            }
            
            template<typename ValueType>
            template<typename RewardModelType>
            std::vector<ValueType> SparseMdpPrctlHelper<ValueType>::computeInstantaneousRewards(OptimizationDirection dir, storm::storage::SparseMatrix<ValueType> const& transitionMatrix, RewardModelType const& rewardModel, uint_fast64_t stepCount, storm::solver::MinMaxLinearEquationSolverFactory<ValueType> const& minMaxLinearEquationSolverFactory) {

                // Only compute the result if the model has a state-based reward this->getModel().
                STORM_LOG_THROW(rewardModel.hasStateRewards(), storm::exceptions::InvalidPropertyException, "Missing reward model for formula. Skipping formula.");
                
                // Initialize result to state rewards of the this->getModel().
                std::vector<ValueType> result(rewardModel.getStateRewardVector());
                
                std::unique_ptr<storm::solver::MinMaxLinearEquationSolver<ValueType>> solver = minMaxLinearEquationSolverFactory.create(transitionMatrix);
                solver->repeatedMultiply(dir, result, nullptr, stepCount);
                
                return result;
            }
            
            template<typename ValueType>
            template<typename RewardModelType>
            std::vector<ValueType> SparseMdpPrctlHelper<ValueType>::computeCumulativeRewards(OptimizationDirection dir, storm::storage::SparseMatrix<ValueType> const& transitionMatrix, RewardModelType const& rewardModel, uint_fast64_t stepBound, storm::solver::MinMaxLinearEquationSolverFactory<ValueType> const& minMaxLinearEquationSolverFactory) {

                // 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>());
                
                std::unique_ptr<storm::solver::MinMaxLinearEquationSolver<ValueType>> solver = minMaxLinearEquationSolverFactory.create(transitionMatrix);
                solver->repeatedMultiply(dir, result, &totalRewardVector, stepBound);
                
                return result;
            }
            
            template<typename ValueType>
            template<typename RewardModelType>
            MDPSparseModelCheckingHelperReturnType<ValueType> SparseMdpPrctlHelper<ValueType>::computeReachabilityRewards(OptimizationDirection dir, storm::storage::SparseMatrix<ValueType> const& transitionMatrix, storm::storage::SparseMatrix<ValueType> const& backwardTransitions, RewardModelType const& rewardModel, storm::storage::BitVector const& targetStates, bool qualitative, bool produceScheduler, storm::solver::MinMaxLinearEquationSolverFactory<ValueType> const& minMaxLinearEquationSolverFactory, ModelCheckerHint const& hint) {
                // 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.");
                return computeReachabilityRewardsHelper(storm::solver::SolveGoal(dir), transitionMatrix, backwardTransitions,
                                                        [&rewardModel] (uint_fast64_t rowCount, storm::storage::SparseMatrix<ValueType> const& transitionMatrix, storm::storage::BitVector const& maybeStates) {
                                                            return rewardModel.getTotalRewardVector(rowCount, transitionMatrix, maybeStates);
                                                        },
                                                        targetStates, qualitative, produceScheduler, minMaxLinearEquationSolverFactory, hint);
            }
            
            template<typename ValueType>
            template<typename RewardModelType>
            MDPSparseModelCheckingHelperReturnType<ValueType> SparseMdpPrctlHelper<ValueType>::computeReachabilityRewards(storm::solver::SolveGoal const& goal, storm::storage::SparseMatrix<ValueType> const& transitionMatrix, storm::storage::SparseMatrix<ValueType> const& backwardTransitions, RewardModelType const& rewardModel, storm::storage::BitVector const& targetStates, bool qualitative, bool produceScheduler, storm::solver::MinMaxLinearEquationSolverFactory<ValueType> const& minMaxLinearEquationSolverFactory, ModelCheckerHint const& hint) {
                // 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.");
                return computeReachabilityRewardsHelper(goal, transitionMatrix, backwardTransitions,
                                                        [&rewardModel] (uint_fast64_t rowCount, storm::storage::SparseMatrix<ValueType> const& transitionMatrix, storm::storage::BitVector const& maybeStates) {
                                                            return rewardModel.getTotalRewardVector(rowCount, transitionMatrix, maybeStates);
                                                        },
                                                        targetStates, qualitative, produceScheduler, minMaxLinearEquationSolverFactory, hint);
            }
            
#ifdef STORM_HAVE_CARL
            template<typename ValueType>
            std::vector<ValueType> SparseMdpPrctlHelper<ValueType>::computeReachabilityRewards(OptimizationDirection dir, storm::storage::SparseMatrix<ValueType> const& transitionMatrix, storm::storage::SparseMatrix<ValueType> const& backwardTransitions, storm::models::sparse::StandardRewardModel<storm::Interval> const& intervalRewardModel, bool lowerBoundOfIntervals, storm::storage::BitVector const& targetStates, bool qualitative, storm::solver::MinMaxLinearEquationSolverFactory<ValueType> const& minMaxLinearEquationSolverFactory) {
                // Only compute the result if the reward model is not empty.
                STORM_LOG_THROW(!intervalRewardModel.empty(), storm::exceptions::InvalidPropertyException, "Missing reward model for formula. Skipping formula.");
                return computeReachabilityRewardsHelper(storm::solver::SolveGoal(dir), transitionMatrix, backwardTransitions, \
                                                        [&] (uint_fast64_t rowCount, storm::storage::SparseMatrix<ValueType> const& transitionMatrix, storm::storage::BitVector const& maybeStates) {
                                                            std::vector<ValueType> result;
                                                            result.reserve(rowCount);
                                                            std::vector<storm::Interval> subIntervalVector = intervalRewardModel.getTotalRewardVector(rowCount, transitionMatrix, maybeStates);
                                                            for (auto const& interval : subIntervalVector) {
                                                                result.push_back(lowerBoundOfIntervals ? interval.lower() : interval.upper());
                                                            }
                                                            return result;
                                                        }, \
                                                        targetStates, qualitative, false, minMaxLinearEquationSolverFactory).values;
            }
            
            template<>
            std::vector<storm::RationalNumber> SparseMdpPrctlHelper<storm::RationalNumber>::computeReachabilityRewards(OptimizationDirection, storm::storage::SparseMatrix<storm::RationalNumber> const&, storm::storage::SparseMatrix<storm::RationalNumber> const&, storm::models::sparse::StandardRewardModel<storm::Interval> const&, bool, storm::storage::BitVector const&, bool, storm::solver::MinMaxLinearEquationSolverFactory<storm::RationalNumber> const&) {
                STORM_LOG_THROW(false, storm::exceptions::IllegalFunctionCallException, "Computing reachability rewards is unsupported for this data type.");
            }
#endif
            
            template<typename ValueType>
            MDPSparseModelCheckingHelperReturnType<ValueType> SparseMdpPrctlHelper<ValueType>::computeReachabilityRewardsHelper(storm::solver::SolveGoal const& 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, bool produceScheduler, storm::solver::MinMaxLinearEquationSolverFactory<ValueType> const& minMaxLinearEquationSolverFactory, ModelCheckerHint const& hint) {
                
                std::vector<ValueType> result(transitionMatrix.getRowGroupCount(), storm::utility::zero<ValueType>());

                std::vector<uint_fast64_t> const& nondeterministicChoiceIndices = transitionMatrix.getRowGroupIndices();
                
                
                // Determine which states have a reward that is infinity or less than infinity.
                storm::storage::BitVector maybeStates, infinityStates;
                if (hint.isExplicitModelCheckerHint() && hint.template asExplicitModelCheckerHint<ValueType>().getComputeOnlyMaybeStates()) {
                    maybeStates = hint.template asExplicitModelCheckerHint<ValueType>().getMaybeStates();
                    infinityStates = ~(maybeStates | targetStates);
                } else {
                    storm::storage::BitVector trueStates(transitionMatrix.getRowGroupCount(), true);
                    if (goal.minimize()) {
                        STORM_LOG_WARN("Results of reward computation may be too low, because of zero-reward loops.");
                        infinityStates = storm::utility::graph::performProb1E(transitionMatrix, nondeterministicChoiceIndices, backwardTransitions, trueStates, targetStates);
                    } else {
                        infinityStates = storm::utility::graph::performProb1A(transitionMatrix, nondeterministicChoiceIndices, backwardTransitions, trueStates, targetStates);
                    }
                    infinityStates.complement();
                    maybeStates = ~(targetStates | infinityStates);
                    STORM_LOG_INFO("Found " << infinityStates.getNumberOfSetBits() << " 'infinity' states.");
                    STORM_LOG_INFO("Found " << targetStates.getNumberOfSetBits() << " 'target' states.");
                    STORM_LOG_INFO("Found " << maybeStates.getNumberOfSetBits() << " 'maybe' states.");
                }
                storm::utility::vector::setVectorValues(result, infinityStates, storm::utility::infinity<ValueType>());
                
                // If requested, we will produce a scheduler.
                std::unique_ptr<storm::storage::Scheduler<ValueType>> scheduler;
                if (produceScheduler) {
                    scheduler = std::make_unique<storm::storage::Scheduler<ValueType>>(transitionMatrix.getRowGroupCount());
                }
                
                // Check whether we need to compute exact rewards for some states.
                if (qualitative) {
                    STORM_LOG_INFO("The rewards for the initial states were determined in a preprocessing step. No exact rewards were computed.");
                    // 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()) {
                        // In this case we have to compute the reward values for the remaining states.
                        
                        // Prepare matrix and vector for the equation system.
                        storm::storage::SparseMatrix<ValueType> submatrix;
                        std::vector<ValueType> b;
                        // Remove rows and columns from the original transition probability matrix for states whose reward values are already known.
                        // If there are infinity states, we additionaly have to remove choices of maybeState that lead to infinity
                        boost::optional<storm::storage::BitVector> selectedChoices; // if not given, all maybeState choices are selected
                        if (infinityStates.empty()) {
                            submatrix = transitionMatrix.getSubmatrix(true, maybeStates, maybeStates, false);
                            b = totalStateRewardVectorGetter(submatrix.getRowCount(), transitionMatrix, maybeStates);
                        } else {
                            selectedChoices = transitionMatrix.getRowFilter(maybeStates, ~infinityStates);
                            submatrix = transitionMatrix.getSubmatrix(false, *selectedChoices, maybeStates, false);
                            b = totalStateRewardVectorGetter(transitionMatrix.getRowCount(), transitionMatrix, storm::storage::BitVector(transitionMatrix.getRowGroupCount(), true));
                            storm::utility::vector::filterVectorInPlace(b, *selectedChoices);
                        }
                        
                        // obtain hint information if possible
                        bool skipEcWithinMaybeStatesCheck = !goal.minimize() || (hint.isExplicitModelCheckerHint() && hint.asExplicitModelCheckerHint<ValueType>().getNoEndComponentsInMaybeStates());
                        std::pair<boost::optional<std::vector<ValueType>>, boost::optional<std::vector<uint_fast64_t>>> hintInformation = extractHintInformationForMaybeStates(transitionMatrix, backwardTransitions, maybeStates, selectedChoices, hint, skipEcWithinMaybeStatesCheck);
                        
                        // Now compute the results for the maybeStates
                        std::pair<std::vector<ValueType>, boost::optional<std::vector<uint_fast64_t>>> resultForMaybeStates = computeValuesOnlyMaybeStates(goal, submatrix, b, produceScheduler, minMaxLinearEquationSolverFactory, std::move(hintInformation.first), std::move(hintInformation.second), storm::utility::zero<ValueType>());

                        // Set values of resulting vector according to result.
                        storm::utility::vector::setVectorValues<ValueType>(result, maybeStates, resultForMaybeStates.first);
                        
                        if (produceScheduler) {
                            std::vector<uint_fast64_t> const& subChoices = resultForMaybeStates.second.get();
                            auto subChoiceIt = subChoices.begin();
                            if (selectedChoices) {
                                for (auto maybeState : maybeStates) {
                                    // find the rowindex that corresponds to the selected row of the submodel
                                    uint_fast64_t firstRowIndex = transitionMatrix.getRowGroupIndices()[maybeState];
                                    uint_fast64_t selectedRowIndex = selectedChoices->getNextSetIndex(firstRowIndex);
                                    for (uint_fast64_t choice = 0; choice < *subChoiceIt; ++choice) {
                                        selectedRowIndex = selectedChoices->getNextSetIndex(selectedRowIndex + 1);
                                    }
                                    scheduler->setChoice(selectedRowIndex - firstRowIndex, maybeState);
                                    ++subChoiceIt;
                                }
                            } else {
                                for (auto maybeState : maybeStates) {
                                    scheduler->setChoice(*subChoiceIt, maybeState);
                                    ++subChoiceIt;
                                }
                            }
                            assert(subChoiceIt == subChoices.end());
                        }
                    }
                }
                
                // Finally, if we need to produce a scheduler, we also need to figure out the parts of the scheduler for
                // the states with reward infinity. Moreover, we have to set some arbitrary choice for the remaining states
                // to obtain a fully defined scheduler
                if (produceScheduler) {
                    if (!goal.minimize()) {
                        storm::utility::graph::computeSchedulerProb0E(infinityStates, transitionMatrix, *scheduler);
                    } else {
                        for (auto const& state : infinityStates) {
                            scheduler->setChoice(0, state);
                        }
                    }
                    for (auto const& state : targetStates) {
                        scheduler->setChoice(0, state);
                    }
                }
                STORM_LOG_ASSERT((!produceScheduler && !scheduler) || (!scheduler->isPartialScheduler() && scheduler->isDeterministicScheduler() && scheduler->isMemorylessScheduler()), "Unexpected format of obtained scheduler.");

                
                return MDPSparseModelCheckingHelperReturnType<ValueType>(std::move(result), std::move(scheduler));
            }
            
            template<typename ValueType>
            std::vector<ValueType> SparseMdpPrctlHelper<ValueType>::computeLongRunAverageProbabilities(OptimizationDirection dir, storm::storage::SparseMatrix<ValueType> const& transitionMatrix, storm::storage::SparseMatrix<ValueType> const& backwardTransitions, storm::storage::BitVector const& psiStates, storm::solver::MinMaxLinearEquationSolverFactory<ValueType> const& minMaxLinearEquationSolverFactory) {
                
                // If there are no goal states, we avoid the computation and directly return zero.
                if (psiStates.empty()) {
                    return std::vector<ValueType>(transitionMatrix.getRowGroupCount(), storm::utility::zero<ValueType>());
                }
                
                // Likewise, if all bits are set, we can avoid the computation and set.
                if (psiStates.full()) {
                    return std::vector<ValueType>(transitionMatrix.getRowGroupCount(), storm::utility::one<ValueType>());
                }
                
                // Reduce long run average probabilities to long run average rewards by
                // building a reward model assigning one reward to every psi state
                std::vector<ValueType> stateRewards(psiStates.size(), storm::utility::zero<ValueType>());
                storm::utility::vector::setVectorValues(stateRewards, psiStates, storm::utility::one<ValueType>());
                storm::models::sparse::StandardRewardModel<ValueType> rewardModel(std::move(stateRewards));
                return computeLongRunAverageRewards(dir, transitionMatrix, backwardTransitions, rewardModel, minMaxLinearEquationSolverFactory);
            }
            
            template<typename ValueType>
            template<typename RewardModelType>
            std::vector<ValueType> SparseMdpPrctlHelper<ValueType>::computeLongRunAverageRewards(OptimizationDirection dir, storm::storage::SparseMatrix<ValueType> const& transitionMatrix, storm::storage::SparseMatrix<ValueType> const& backwardTransitions, RewardModelType const& rewardModel, storm::solver::MinMaxLinearEquationSolverFactory<ValueType> const& minMaxLinearEquationSolverFactory) {
                
                uint64_t numberOfStates = transitionMatrix.getRowGroupCount();

                // Start by decomposing the MDP into its MECs.
                storm::storage::MaximalEndComponentDecomposition<ValueType> mecDecomposition(transitionMatrix, backwardTransitions);
                
                // Get some data members for convenience.
                std::vector<uint_fast64_t> const& nondeterministicChoiceIndices = transitionMatrix.getRowGroupIndices();
                ValueType zero = storm::utility::zero<ValueType>();
                
                //first calculate LRA for the Maximal End Components.
                storm::storage::BitVector statesInMecs(numberOfStates);
                std::vector<uint_fast64_t> stateToMecIndexMap(transitionMatrix.getColumnCount());
                std::vector<ValueType> lraValuesForEndComponents(mecDecomposition.size(), zero);
                
                for (uint_fast64_t currentMecIndex = 0; currentMecIndex < mecDecomposition.size(); ++currentMecIndex) {
                    storm::storage::MaximalEndComponent const& mec = mecDecomposition[currentMecIndex];
                    
                    lraValuesForEndComponents[currentMecIndex] = computeLraForMaximalEndComponent(dir, transitionMatrix, rewardModel, mec, minMaxLinearEquationSolverFactory);
                    
                    // Gather information for later use.
                    for (auto const& stateChoicesPair : mec) {
                        statesInMecs.set(stateChoicesPair.first);
                        stateToMecIndexMap[stateChoicesPair.first] = currentMecIndex;
                    }
                }
                
                // For fast transition rewriting, we build some auxiliary data structures.
                storm::storage::BitVector statesNotContainedInAnyMec = ~statesInMecs;
                uint_fast64_t firstAuxiliaryStateIndex = statesNotContainedInAnyMec.getNumberOfSetBits();
                uint_fast64_t lastStateNotInMecs = 0;
                uint_fast64_t numberOfStatesNotInMecs = 0;
                std::vector<uint_fast64_t> statesNotInMecsBeforeIndex;
                statesNotInMecsBeforeIndex.reserve(numberOfStates);
                for (auto state : statesNotContainedInAnyMec) {
                    while (lastStateNotInMecs <= state) {
                        statesNotInMecsBeforeIndex.push_back(numberOfStatesNotInMecs);
                        ++lastStateNotInMecs;
                    }
                    ++numberOfStatesNotInMecs;
                }
                
                // Finally, we are ready to create the SSP matrix and right-hand side of the SSP.
                std::vector<ValueType> b;
                uint64_t numberOfSspStates = numberOfStatesNotInMecs + mecDecomposition.size();

                typename storm::storage::SparseMatrixBuilder<ValueType> sspMatrixBuilder(0, numberOfSspStates, 0, false, true, numberOfSspStates);
                
                // If the source state is not contained in any MEC, we copy its choices (and perform the necessary modifications).
                uint_fast64_t currentChoice = 0;
                for (auto state : statesNotContainedInAnyMec) {
                    sspMatrixBuilder.newRowGroup(currentChoice);
                    
                    for (uint_fast64_t choice = nondeterministicChoiceIndices[state]; choice < nondeterministicChoiceIndices[state + 1]; ++choice, ++currentChoice) {
                        std::vector<ValueType> auxiliaryStateToProbabilityMap(mecDecomposition.size());
                        b.push_back(storm::utility::zero<ValueType>());
                        
                        for (auto element : transitionMatrix.getRow(choice)) {
                            if (statesNotContainedInAnyMec.get(element.getColumn())) {
                                // If the target state is not contained in an MEC, we can copy over the entry.
                                sspMatrixBuilder.addNextValue(currentChoice, statesNotInMecsBeforeIndex[element.getColumn()], element.getValue());
                            } else {
                                // If the target state is contained in MEC i, we need to add the probability to the corresponding field in the vector
                                // so that we are able to write the cumulative probability to the MEC into the matrix.
                                auxiliaryStateToProbabilityMap[stateToMecIndexMap[element.getColumn()]] += element.getValue();
                            }
                        }
                        
                        // Now insert all (cumulative) probability values that target an MEC.
                        for (uint_fast64_t mecIndex = 0; mecIndex < auxiliaryStateToProbabilityMap.size(); ++mecIndex) {
                            if (auxiliaryStateToProbabilityMap[mecIndex] != 0) {
                                sspMatrixBuilder.addNextValue(currentChoice, firstAuxiliaryStateIndex + mecIndex, auxiliaryStateToProbabilityMap[mecIndex]);
                            }
                        }
                    }
                }
                
                // Now we are ready to construct the choices for the auxiliary states.
                for (uint_fast64_t mecIndex = 0; mecIndex < mecDecomposition.size(); ++mecIndex) {
                    storm::storage::MaximalEndComponent const& mec = mecDecomposition[mecIndex];
                    sspMatrixBuilder.newRowGroup(currentChoice);
                    
                    for (auto const& stateChoicesPair : mec) {
                        uint_fast64_t state = stateChoicesPair.first;
                        boost::container::flat_set<uint_fast64_t> const& choicesInMec = stateChoicesPair.second;
                        
                        for (uint_fast64_t choice = nondeterministicChoiceIndices[state]; choice < nondeterministicChoiceIndices[state + 1]; ++choice) {
                            // If the choice is not contained in the MEC itself, we have to add a similar distribution to the auxiliary state.
                            if (choicesInMec.find(choice) == choicesInMec.end()) {
                                std::vector<ValueType> auxiliaryStateToProbabilityMap(mecDecomposition.size());
                                b.push_back(storm::utility::zero<ValueType>());
                                
                                for (auto element : transitionMatrix.getRow(choice)) {
                                    if (statesNotContainedInAnyMec.get(element.getColumn())) {
                                        // If the target state is not contained in an MEC, we can copy over the entry.
                                        sspMatrixBuilder.addNextValue(currentChoice, statesNotInMecsBeforeIndex[element.getColumn()], element.getValue());
                                    } else {
                                        // If the target state is contained in MEC i, we need to add the probability to the corresponding field in the vector
                                        // so that we are able to write the cumulative probability to the MEC into the matrix.
                                        auxiliaryStateToProbabilityMap[stateToMecIndexMap[element.getColumn()]] += element.getValue();
                                    }
                                }
                                
                                // Now insert all (cumulative) probability values that target an MEC.
                                for (uint_fast64_t targetMecIndex = 0; targetMecIndex < auxiliaryStateToProbabilityMap.size(); ++targetMecIndex) {
                                    if (auxiliaryStateToProbabilityMap[targetMecIndex] != 0) {
                                        sspMatrixBuilder.addNextValue(currentChoice, firstAuxiliaryStateIndex + targetMecIndex, auxiliaryStateToProbabilityMap[targetMecIndex]);
                                    }
                                }
                                
                                ++currentChoice;
                            }
                        }
                    }
                    
                    // For each auxiliary state, there is the option to achieve the reward value of the LRA associated with the MEC.
                    ++currentChoice;
                    b.push_back(lraValuesForEndComponents[mecIndex]);
                }
                
                // Finalize the matrix and solve the corresponding system of equations.
                storm::storage::SparseMatrix<ValueType> sspMatrix = sspMatrixBuilder.build(currentChoice, numberOfSspStates, numberOfSspStates);
                
                std::vector<ValueType> sspResult(numberOfSspStates);
                std::unique_ptr<storm::solver::MinMaxLinearEquationSolver<ValueType>> solver = minMaxLinearEquationSolverFactory.create(std::move(sspMatrix));
                solver->solveEquations(dir, sspResult, b);
                
                // Prepare result vector.
                std::vector<ValueType> result(numberOfStates, zero);
                
                // Set the values for states not contained in MECs.
                storm::utility::vector::setVectorValues(result, statesNotContainedInAnyMec, sspResult);
                
                // Set the values for all states in MECs.
                for (auto state : statesInMecs) {
                    result[state] = sspResult[firstAuxiliaryStateIndex + stateToMecIndexMap[state]];
                }
                
                return result;
            }
            
            template<typename ValueType>
            template<typename RewardModelType>
            ValueType SparseMdpPrctlHelper<ValueType>::computeLraForMaximalEndComponent(OptimizationDirection dir, storm::storage::SparseMatrix<ValueType> const& transitionMatrix, RewardModelType const& rewardModel, storm::storage::MaximalEndComponent const& mec, storm::solver::MinMaxLinearEquationSolverFactory<ValueType> const& minMaxLinearEquationSolverFactory) {
                
                // If the mec only consists of a single state, we compute the LRA value directly
                if (++mec.begin() == mec.end()) {
                    uint64_t state = mec.begin()->first;
                    auto choiceIt = mec.begin()->second.begin();
                    ValueType result = rewardModel.getTotalStateActionReward(state, *choiceIt, transitionMatrix);
                    for (++choiceIt; choiceIt != mec.begin()->second.end(); ++choiceIt) {
                        if (storm::solver::minimize(dir)) {
                            result = std::min(result, rewardModel.getTotalStateActionReward(state, *choiceIt, transitionMatrix));
                        } else {
                            result = std::max(result, rewardModel.getTotalStateActionReward(state, *choiceIt, transitionMatrix));
                        }
                    }
                    return result;
                }
                
                // Solve MEC with the method specified in the settings
                storm::solver::LraMethod method = storm::settings::getModule<storm::settings::modules::MinMaxEquationSolverSettings>().getLraMethod();
                if (method == storm::solver::LraMethod::LinearProgramming) {
                    return computeLraForMaximalEndComponentLP(dir, transitionMatrix, rewardModel, mec);
                } else if (method == storm::solver::LraMethod::ValueIteration) {
                    return computeLraForMaximalEndComponentVI(dir, transitionMatrix, rewardModel, mec, minMaxLinearEquationSolverFactory);
                } else {
                    STORM_LOG_THROW(false, storm::exceptions::InvalidSettingsException, "Unsupported technique.");
                }
            }
            
            template<typename ValueType>
            template<typename RewardModelType>
            ValueType SparseMdpPrctlHelper<ValueType>::computeLraForMaximalEndComponentVI(OptimizationDirection dir, storm::storage::SparseMatrix<ValueType> const& transitionMatrix, RewardModelType const& rewardModel, storm::storage::MaximalEndComponent const& mec, storm::solver::MinMaxLinearEquationSolverFactory<ValueType> const& minMaxLinearEquationSolverFactory) {
                
                // Initialize data about the mec
                storm::storage::BitVector mecStates(transitionMatrix.getRowGroupCount(), false);
                storm::storage::BitVector mecChoices(transitionMatrix.getRowCount(), false);
                for (auto const& stateChoicesPair : mec) {
                    mecStates.set(stateChoicesPair.first);
                    for (auto const& choice : stateChoicesPair.second) {
                        mecChoices.set(choice);
                    }
                }
                
                boost::container::flat_map<uint64_t, uint64_t> toSubModelStateMapping;
                uint64_t currState = 0;
                toSubModelStateMapping.reserve(mecStates.getNumberOfSetBits());
                for (auto const& mecState : mecStates) {
                    toSubModelStateMapping.insert(std::pair<uint64_t, uint64_t>(mecState, currState));
                    ++currState;
                }
                
                // Get a transition matrix that only considers the states and choices within the MEC
                storm::storage::SparseMatrixBuilder<ValueType> mecTransitionBuilder(mecChoices.getNumberOfSetBits(), mecStates.getNumberOfSetBits(), 0, true, true, mecStates.getNumberOfSetBits());
                std::vector<ValueType> choiceRewards;
                choiceRewards.reserve(mecChoices.getNumberOfSetBits());
                uint64_t currRow = 0;
                ValueType selfLoopProb = storm::utility::convertNumber<ValueType>(0.1); // todo try other values
                ValueType scalingFactor = storm::utility::one<ValueType>() - selfLoopProb;
                for (auto const& mecState : mecStates) {
                    mecTransitionBuilder.newRowGroup(currRow);
                    uint64_t groupStart = transitionMatrix.getRowGroupIndices()[mecState];
                    uint64_t groupEnd = transitionMatrix.getRowGroupIndices()[mecState + 1];
                    for (uint64_t choice = mecChoices.getNextSetIndex(groupStart); choice < groupEnd; choice = mecChoices.getNextSetIndex(choice + 1)) {
                        bool insertedDiagElement = false;
                        for (auto const& entry : transitionMatrix.getRow(choice)) {
                            uint64_t column = toSubModelStateMapping[entry.getColumn()];
                            if (!insertedDiagElement && entry.getColumn() > mecState) {
                                mecTransitionBuilder.addNextValue(currRow, toSubModelStateMapping[mecState], selfLoopProb);
                                insertedDiagElement = true;
                            }
                            if (!insertedDiagElement && entry.getColumn() == mecState) {
                                mecTransitionBuilder.addNextValue(currRow, column, selfLoopProb + scalingFactor * entry.getValue());
                                insertedDiagElement = true;
                            } else {
                                mecTransitionBuilder.addNextValue(currRow, column,  scalingFactor * entry.getValue());
                            }
                        }
                        if (!insertedDiagElement) {
                            mecTransitionBuilder.addNextValue(currRow, toSubModelStateMapping[mecState], selfLoopProb);
                        }
                        
                        // Compute the rewards obtained for this choice
                        choiceRewards.push_back(scalingFactor * rewardModel.getTotalStateActionReward(mecState, choice, transitionMatrix));
                        
                        ++currRow;
                    }
                }
                auto mecTransitions = mecTransitionBuilder.build();
                STORM_LOG_ASSERT(mecTransitions.isProbabilistic(), "The MEC-Matrix is not probabilistic.");
                
                // start the iterations
                ValueType precision = storm::utility::convertNumber<ValueType>(storm::settings::getModule<storm::settings::modules::MinMaxEquationSolverSettings>().getPrecision());
                std::vector<ValueType> x(mecTransitions.getRowGroupCount(), storm::utility::zero<ValueType>());
                std::vector<ValueType> xPrime = x;
                auto solver = minMaxLinearEquationSolverFactory.create(std::move(mecTransitions));
                solver->setCachingEnabled(true);
                ValueType maxDiff, minDiff;
                while (true) {
                    // Compute the obtained rewards for the next step
                    solver->repeatedMultiply(dir, x, &choiceRewards, 1);
                    
                    // update xPrime and check for convergence
                    // to avoid large (and numerically unstable) x-values, we substract a reference value.
                    auto xIt = x.begin();
                    auto xPrimeIt = xPrime.begin();
                    ValueType refVal = *xIt;
                    maxDiff = *xIt - *xPrimeIt;
                    minDiff = maxDiff;
                    *xIt -= refVal;
                    *xPrimeIt = *xIt;
                    for (++xIt, ++xPrimeIt; xIt != x.end(); ++xIt, ++xPrimeIt) {
                        ValueType diff = *xIt - *xPrimeIt;
                        maxDiff = std::max(maxDiff, diff);
                        minDiff = std::min(minDiff, diff);
                        *xIt -= refVal;
                        *xPrimeIt = *xIt;
                    }

                    if ((maxDiff - minDiff) < precision) {
                        break;
                    }
                }
                return (maxDiff + minDiff) / (storm::utility::convertNumber<ValueType>(2.0) * scalingFactor);
            }
            
            template<typename ValueType>
            template<typename RewardModelType>
            ValueType SparseMdpPrctlHelper<ValueType>::computeLraForMaximalEndComponentLP(OptimizationDirection dir, storm::storage::SparseMatrix<ValueType> const& transitionMatrix, RewardModelType const& rewardModel, storm::storage::MaximalEndComponent const& mec) {
                std::shared_ptr<storm::solver::LpSolver> solver = storm::utility::solver::getLpSolver("LRA for MEC");
                solver->setOptimizationDirection(invert(dir));
                
                // First, we need to create the variables for the problem.
                std::map<uint_fast64_t, storm::expressions::Variable> stateToVariableMap;
                for (auto const& stateChoicesPair : mec) {
                    std::string variableName = "h" + std::to_string(stateChoicesPair.first);
                    stateToVariableMap[stateChoicesPair.first] = solver->addUnboundedContinuousVariable(variableName);
                }
                storm::expressions::Variable lambda = solver->addUnboundedContinuousVariable("L", 1);
                solver->update();
                
                // Now we encode the problem as constraints.
                for (auto const& stateChoicesPair : mec) {
                    uint_fast64_t state = stateChoicesPair.first;
                    
                    // Now, based on the type of the state, create a suitable constraint.
                    for (auto choice : stateChoicesPair.second) {
                        storm::expressions::Expression constraint = -lambda;
                        
                        for (auto element : transitionMatrix.getRow(choice)) {
                            constraint = constraint + stateToVariableMap.at(element.getColumn()) * solver->getConstant(storm::utility::convertNumber<double>(element.getValue()));
                        }
                        typename RewardModelType::ValueType r = rewardModel.getTotalStateActionReward(state, choice, transitionMatrix);
                        constraint = solver->getConstant(storm::utility::convertNumber<double>(r)) + constraint;
                        
                        if (dir == OptimizationDirection::Minimize) {
                            constraint = stateToVariableMap.at(state) <= constraint;
                        } else {
                            constraint = stateToVariableMap.at(state) >= constraint;
                        }
                        solver->addConstraint("state" + std::to_string(state) + "," + std::to_string(choice), constraint);
                    }
                }
                
                solver->optimize();
                return storm::utility::convertNumber<ValueType>(solver->getContinuousValue(lambda));
            }
            
            template<typename ValueType>
            std::unique_ptr<CheckResult> SparseMdpPrctlHelper<ValueType>::computeConditionalProbabilities(OptimizationDirection dir, storm::storage::sparse::state_type initialState, storm::storage::SparseMatrix<ValueType> const& transitionMatrix, storm::storage::SparseMatrix<ValueType> const& backwardTransitions, storm::storage::BitVector const& targetStates, storm::storage::BitVector const& conditionStates, storm::solver::MinMaxLinearEquationSolverFactory<ValueType> const& minMaxLinearEquationSolverFactory) {
                
                // For the max-case, we can simply take the given target states. For the min-case, however, we need to
                // find the MECs of non-target states and make them the new target states.
                storm::storage::BitVector fixedTargetStates;
                if (dir == OptimizationDirection::Maximize) {
                    fixedTargetStates = targetStates;
                } else {
                    fixedTargetStates = storm::storage::BitVector(targetStates.size());
                    storm::storage::MaximalEndComponentDecomposition<ValueType> mecDecomposition(transitionMatrix, backwardTransitions, ~targetStates);
                    for (auto const& mec : mecDecomposition) {
                        for (auto const& stateActionsPair : mec) {
                            fixedTargetStates.set(stateActionsPair.first);
                        }
                    }
                }
                
                // We solve the max-case and later adjust the result if the optimization direction was to minimize.
                storm::storage::BitVector initialStatesBitVector(transitionMatrix.getRowGroupCount());
                initialStatesBitVector.set(initialState);
                
                storm::storage::BitVector allStates(initialStatesBitVector.size(), true);
                std::vector<ValueType> conditionProbabilities = std::move(computeUntilProbabilities(OptimizationDirection::Maximize, transitionMatrix, backwardTransitions, allStates, conditionStates, false, false, minMaxLinearEquationSolverFactory).values);
                
                // If the conditional probability is undefined for the initial state, we return directly.
                if (storm::utility::isZero(conditionProbabilities[initialState])) {
                    return std::unique_ptr<CheckResult>(new ExplicitQuantitativeCheckResult<ValueType>(initialState, storm::utility::infinity<ValueType>()));
                }
                
                std::vector<ValueType> targetProbabilities = std::move(computeUntilProbabilities(OptimizationDirection::Maximize, transitionMatrix, backwardTransitions, allStates, fixedTargetStates, false, false, minMaxLinearEquationSolverFactory).values);

                storm::storage::BitVector statesWithProbabilityGreater0E(transitionMatrix.getRowGroupCount(), true);
                storm::storage::sparse::state_type state = 0;
                for (auto const& element : conditionProbabilities) {
                    if (storm::utility::isZero(element)) {
                        statesWithProbabilityGreater0E.set(state, false);
                    }
                    ++state;
                }

                // Determine those states that need to be equipped with a restart mechanism.
                storm::storage::BitVector problematicStates = storm::utility::graph::performProb0E(transitionMatrix, transitionMatrix.getRowGroupIndices(), backwardTransitions, allStates, conditionStates | fixedTargetStates);

                // Otherwise, we build the transformed MDP.
                storm::storage::BitVector relevantStates = storm::utility::graph::getReachableStates(transitionMatrix, initialStatesBitVector, allStates, conditionStates | fixedTargetStates);
                std::vector<uint_fast64_t> numberOfStatesBeforeRelevantStates = relevantStates.getNumberOfSetBitsBeforeIndices();
                storm::storage::sparse::state_type newGoalState = relevantStates.getNumberOfSetBits();
                storm::storage::sparse::state_type newStopState = newGoalState + 1;
                storm::storage::sparse::state_type newFailState = newStopState + 1;
                
                // Build the transitions of the (relevant) states of the original model.
                storm::storage::SparseMatrixBuilder<ValueType> builder(0, newFailState + 1, 0, true, true);
                uint_fast64_t currentRow = 0;
                for (auto state : relevantStates) {
                    builder.newRowGroup(currentRow);
                    if (fixedTargetStates.get(state)) {
                        builder.addNextValue(currentRow, newGoalState, conditionProbabilities[state]);
                        if (!storm::utility::isZero(conditionProbabilities[state])) {
                            builder.addNextValue(currentRow, newFailState, storm::utility::one<ValueType>() - conditionProbabilities[state]);
                        }
                        ++currentRow;
                    } else if (conditionStates.get(state)) {
                        builder.addNextValue(currentRow, newGoalState, targetProbabilities[state]);
                        if (!storm::utility::isZero(targetProbabilities[state])) {
                            builder.addNextValue(currentRow, newStopState, storm::utility::one<ValueType>() - targetProbabilities[state]);
                        }
                        ++currentRow;
                    } else {
                        for (uint_fast64_t row = transitionMatrix.getRowGroupIndices()[state]; row < transitionMatrix.getRowGroupIndices()[state + 1]; ++row) {
                            for (auto const& successorEntry : transitionMatrix.getRow(row)) {
                                builder.addNextValue(currentRow, numberOfStatesBeforeRelevantStates[successorEntry.getColumn()], successorEntry.getValue());
                            }
                            ++currentRow;
                        }
                        if (problematicStates.get(state)) {
                            builder.addNextValue(currentRow, numberOfStatesBeforeRelevantStates[initialState], storm::utility::one<ValueType>());
                            ++currentRow;
                        }
                    }
                }
                
                // Now build the transitions of the newly introduced states.
                builder.newRowGroup(currentRow);
                builder.addNextValue(currentRow, newGoalState, storm::utility::one<ValueType>());
                ++currentRow;
                builder.newRowGroup(currentRow);
                builder.addNextValue(currentRow, newStopState, storm::utility::one<ValueType>());
                ++currentRow;
                builder.newRowGroup(currentRow);
                builder.addNextValue(currentRow, numberOfStatesBeforeRelevantStates[initialState], storm::utility::one<ValueType>());
                ++currentRow;
                
                // Finally, build the matrix and dispatch the query as a reachability query.
                storm::storage::BitVector newGoalStates(newFailState + 1);
                newGoalStates.set(newGoalState);
                storm::storage::SparseMatrix<ValueType> newTransitionMatrix = builder.build();
                storm::storage::SparseMatrix<ValueType> newBackwardTransitions = newTransitionMatrix.transpose(true);
                std::vector<ValueType> goalProbabilities = std::move(computeUntilProbabilities(OptimizationDirection::Maximize, newTransitionMatrix, newBackwardTransitions, storm::storage::BitVector(newFailState + 1, true), newGoalStates, false, false, minMaxLinearEquationSolverFactory).values);
                
                return std::unique_ptr<CheckResult>(new ExplicitQuantitativeCheckResult<ValueType>(initialState, dir == OptimizationDirection::Maximize ? goalProbabilities[numberOfStatesBeforeRelevantStates[initialState]] : storm::utility::one<ValueType>() - goalProbabilities[numberOfStatesBeforeRelevantStates[initialState]]));
            }
            
            template class SparseMdpPrctlHelper<double>;
            template std::vector<double> SparseMdpPrctlHelper<double>::computeInstantaneousRewards(OptimizationDirection dir, storm::storage::SparseMatrix<double> const& transitionMatrix, storm::models::sparse::StandardRewardModel<double> const& rewardModel, uint_fast64_t stepCount, storm::solver::MinMaxLinearEquationSolverFactory<double> const& minMaxLinearEquationSolverFactory);
            template std::vector<double> SparseMdpPrctlHelper<double>::computeCumulativeRewards(OptimizationDirection dir, storm::storage::SparseMatrix<double> const& transitionMatrix, storm::models::sparse::StandardRewardModel<double> const& rewardModel, uint_fast64_t stepBound, storm::solver::MinMaxLinearEquationSolverFactory<double> const& minMaxLinearEquationSolverFactory);
            template MDPSparseModelCheckingHelperReturnType<double> SparseMdpPrctlHelper<double>::computeReachabilityRewards(OptimizationDirection dir, storm::storage::SparseMatrix<double> const& transitionMatrix, storm::storage::SparseMatrix<double> const& backwardTransitions, storm::models::sparse::StandardRewardModel<double> const& rewardModel, storm::storage::BitVector const& targetStates, bool qualitative, bool produceScheduler, storm::solver::MinMaxLinearEquationSolverFactory<double> const& minMaxLinearEquationSolverFactory, ModelCheckerHint const& hint);
            template MDPSparseModelCheckingHelperReturnType<double> SparseMdpPrctlHelper<double>::computeReachabilityRewards(storm::solver::SolveGoal const& goal, storm::storage::SparseMatrix<double> const& transitionMatrix, storm::storage::SparseMatrix<double> const& backwardTransitions, storm::models::sparse::StandardRewardModel<double> const& rewardModel, storm::storage::BitVector const& targetStates, bool qualitative, bool produceScheduler, storm::solver::MinMaxLinearEquationSolverFactory<double> const& minMaxLinearEquationSolverFactory, ModelCheckerHint const& hint);
            template std::vector<double> SparseMdpPrctlHelper<double>::computeLongRunAverageRewards(OptimizationDirection dir, storm::storage::SparseMatrix<double> const& transitionMatrix, storm::storage::SparseMatrix<double> const& backwardTransitions, storm::models::sparse::StandardRewardModel<double> const& rewardModel, storm::solver::MinMaxLinearEquationSolverFactory<double> const& minMaxLinearEquationSolverFactory);
            template double SparseMdpPrctlHelper<double>::computeLraForMaximalEndComponent(OptimizationDirection dir, storm::storage::SparseMatrix<double> const& transitionMatrix, storm::models::sparse::StandardRewardModel<double> const& rewardModel, storm::storage::MaximalEndComponent const& mec, storm::solver::MinMaxLinearEquationSolverFactory<double> const& minMaxLinearEquationSolverFactory);
            template double SparseMdpPrctlHelper<double>::computeLraForMaximalEndComponentVI(OptimizationDirection dir, storm::storage::SparseMatrix<double> const& transitionMatrix, storm::models::sparse::StandardRewardModel<double> const& rewardModel, storm::storage::MaximalEndComponent const& mec, storm::solver::MinMaxLinearEquationSolverFactory<double> const& minMaxLinearEquationSolverFactory);
            template double SparseMdpPrctlHelper<double>::computeLraForMaximalEndComponentLP(OptimizationDirection dir, storm::storage::SparseMatrix<double> const& transitionMatrix, storm::models::sparse::StandardRewardModel<double> const& rewardModel, storm::storage::MaximalEndComponent const& mec);

#ifdef STORM_HAVE_CARL
            template class SparseMdpPrctlHelper<storm::RationalNumber>;
            template std::vector<storm::RationalNumber> SparseMdpPrctlHelper<storm::RationalNumber>::computeInstantaneousRewards(OptimizationDirection dir, storm::storage::SparseMatrix<storm::RationalNumber> const& transitionMatrix, storm::models::sparse::StandardRewardModel<storm::RationalNumber> const& rewardModel, uint_fast64_t stepCount, storm::solver::MinMaxLinearEquationSolverFactory<storm::RationalNumber> const& minMaxLinearEquationSolverFactory);
            template std::vector<storm::RationalNumber> SparseMdpPrctlHelper<storm::RationalNumber>::computeCumulativeRewards(OptimizationDirection dir, storm::storage::SparseMatrix<storm::RationalNumber> const& transitionMatrix, storm::models::sparse::StandardRewardModel<storm::RationalNumber> const& rewardModel, uint_fast64_t stepBound, storm::solver::MinMaxLinearEquationSolverFactory<storm::RationalNumber> const& minMaxLinearEquationSolverFactory);
            template MDPSparseModelCheckingHelperReturnType<storm::RationalNumber> SparseMdpPrctlHelper<storm::RationalNumber>::computeReachabilityRewards(OptimizationDirection dir, storm::storage::SparseMatrix<storm::RationalNumber> const& transitionMatrix, storm::storage::SparseMatrix<storm::RationalNumber> const& backwardTransitions, storm::models::sparse::StandardRewardModel<storm::RationalNumber> const& rewardModel, storm::storage::BitVector const& targetStates, bool qualitative, bool produceScheduler, storm::solver::MinMaxLinearEquationSolverFactory<storm::RationalNumber> const& minMaxLinearEquationSolverFactory, ModelCheckerHint const& hint);
            template MDPSparseModelCheckingHelperReturnType<storm::RationalNumber> SparseMdpPrctlHelper<storm::RationalNumber>::computeReachabilityRewards(storm::solver::SolveGoal const& goal, storm::storage::SparseMatrix<storm::RationalNumber> const& transitionMatrix, storm::storage::SparseMatrix<storm::RationalNumber> const& backwardTransitions, storm::models::sparse::StandardRewardModel<storm::RationalNumber> const& rewardModel, storm::storage::BitVector const& targetStates, bool qualitative, bool produceScheduler, storm::solver::MinMaxLinearEquationSolverFactory<storm::RationalNumber> const& minMaxLinearEquationSolverFactory, ModelCheckerHint const& hint);
            template std::vector<storm::RationalNumber> SparseMdpPrctlHelper<storm::RationalNumber>::computeLongRunAverageRewards(OptimizationDirection dir, storm::storage::SparseMatrix<storm::RationalNumber> const& transitionMatrix, storm::storage::SparseMatrix<storm::RationalNumber> const& backwardTransitions, storm::models::sparse::StandardRewardModel<storm::RationalNumber> const& rewardModel, storm::solver::MinMaxLinearEquationSolverFactory<storm::RationalNumber> const& minMaxLinearEquationSolverFactory);
            template storm::RationalNumber SparseMdpPrctlHelper<storm::RationalNumber>::computeLraForMaximalEndComponent(OptimizationDirection dir, storm::storage::SparseMatrix<storm::RationalNumber> const& transitionMatrix, storm::models::sparse::StandardRewardModel<storm::RationalNumber> const& rewardModel, storm::storage::MaximalEndComponent const& mec, storm::solver::MinMaxLinearEquationSolverFactory<storm::RationalNumber> const& minMaxLinearEquationSolverFactory);
            template storm::RationalNumber SparseMdpPrctlHelper<storm::RationalNumber>::computeLraForMaximalEndComponentVI(OptimizationDirection dir, storm::storage::SparseMatrix<storm::RationalNumber> const& transitionMatrix, storm::models::sparse::StandardRewardModel<storm::RationalNumber> const& rewardModel, storm::storage::MaximalEndComponent const& mec, storm::solver::MinMaxLinearEquationSolverFactory<storm::RationalNumber> const& minMaxLinearEquationSolverFactory);
            template storm::RationalNumber SparseMdpPrctlHelper<storm::RationalNumber>::computeLraForMaximalEndComponentLP(OptimizationDirection dir, storm::storage::SparseMatrix<storm::RationalNumber> const& transitionMatrix, storm::models::sparse::StandardRewardModel<storm::RationalNumber> const& rewardModel, storm::storage::MaximalEndComponent const& mec);

#endif
        }
    }
}