sat_solver.h

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// Copyright 2010-2021 Google LLC
// Licensed under the Apache License, Version 2.0 (the "License");
// you may not use this file except in compliance with the License.
// You may obtain a copy of the License at
//
//     http://www.apache.org/licenses/LICENSE-2.0
//
// Unless required by applicable law or agreed to in writing, software
// distributed under the License is distributed on an "AS IS" BASIS,
// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
// See the License for the specific language governing permissions and
// limitations under the License.
 
// This file implements a SAT solver.
// see http://en.wikipedia.org/wiki/Boolean_satisfiability_problem
// for more detail.
// TODO(user): Expand.
 
#ifndef OR_TOOLS_SAT_SAT_SOLVER_H_
#define OR_TOOLS_SAT_SAT_SOLVER_H_
 
#include <cstdint>
#include <functional>
#include <limits>
#include <memory>
#include <string>
#include <utility>
#include <vector>
 
#include "absl/container/flat_hash_map.h"
#include "absl/strings/string_view.h"
#include "absl/types/span.h"
#include "ortools/base/hash.h"
#include "ortools/base/integral_types.h"
#include "ortools/base/logging.h"
#include "ortools/base/macros.h"
#include "ortools/base/timer.h"
#include "ortools/sat/clause.h"
#include "ortools/sat/drat_proof_handler.h"
#include "ortools/sat/model.h"
#include "ortools/sat/pb_constraint.h"
#include "ortools/sat/restart.h"
#include "ortools/sat/sat_base.h"
#include "ortools/sat/sat_decision.h"
#include "ortools/sat/sat_parameters.pb.h"
#include "ortools/util/bitset.h"
#include "ortools/util/logging.h"
#include "ortools/util/stats.h"
#include "ortools/util/strong_integers.h"
#include "ortools/util/time_limit.h"
 
namespace operations_research {
namespace sat {
 
// A constant used by the EnqueueDecision*() API.
const int kUnsatTrailIndex = -1;
 
// The main SAT solver.
// It currently implements the CDCL algorithm. See
//    http://en.wikipedia.org/wiki/Conflict_Driven_Clause_Learning
class SatSolver {
 public:
  SatSolver();
  explicit SatSolver(Model* model);
  ~SatSolver();
 
  // TODO(user): Remove. This is temporary for accessing the model deep within
  // some old code that didn't use the Model object.
  Model* model() { return model_; }
 
  // Parameters management. Note that calling SetParameters() will reset the
  // value of many heuristics. For instance:
  // - The restart strategy will be reinitialized.
  // - The random seed and random generator will be reset to the value given in
  //   parameters.
  // - The global TimeLimit singleton will be reset and time will be
  //   counted from this call.
  void SetParameters(const SatParameters& parameters);
  const SatParameters& parameters() const;
 
  // Increases the number of variables of the current problem.
  //
  // TODO(user): Rename to IncreaseNumVariablesTo() until we support removing
  // variables...
  void SetNumVariables(int num_variables);
  int NumVariables() const { return num_variables_.value(); }
  BooleanVariable NewBooleanVariable() {
    const int num_vars = NumVariables();
 
    // We need to be able to encode the variable as a literal.
    CHECK_LT(2 * num_vars, std::numeric_limits<int32_t>::max());
    SetNumVariables(num_vars + 1);
    return BooleanVariable(num_vars);
  }
 
  // Fixes a variable so that the given literal is true. This can be used to
  // solve a subproblem where some variables are fixed. Note that it is more
  // efficient to add such unit clause before all the others.
  // Returns false if the problem is detected to be UNSAT.
  bool AddUnitClause(Literal true_literal);
 
  // Same as AddProblemClause() below, but for small clauses.
  bool AddBinaryClause(Literal a, Literal b);
  bool AddTernaryClause(Literal a, Literal b, Literal c);
 
  // Adds a clause to the problem. Returns false if the problem is detected to
  // be UNSAT.
  //
  // TODO(user): Rename this to AddClause(), also get rid of the specialized
  // AddUnitClause(), AddBinaryClause() and AddTernaryClause() since they
  // just end up calling this?
  bool AddProblemClause(absl::Span<const Literal> literals);
 
  // Adds a pseudo-Boolean constraint to the problem. Returns false if the
  // problem is detected to be UNSAT. If the constraint is always true, this
  // detects it and does nothing.
  //
  // Note(user): There is an optimization if the same constraint is added
  // consecutively (even if the bounds are different). This is particularly
  // useful for an optimization problem when we want to constrain the objective
  // of the problem more and more. Just re-adding such constraint is relatively
  // efficient.
  //
  // OVERFLOW: The sum of the absolute value of all the coefficients
  // in the constraint must not overflow. This is currently CHECKed().
  // TODO(user): Instead of failing, implement an error handling code.
  bool AddLinearConstraint(bool use_lower_bound, Coefficient lower_bound,
                           bool use_upper_bound, Coefficient upper_bound,
                           std::vector<LiteralWithCoeff>* cst);
 
  // Returns true if the model is UNSAT. Note that currently the status is
  // "sticky" and once this happen, nothing else can be done with the solver.
  //
  // Thanks to this function, a client can safely ignore the return value of any
  // Add*() functions. If one of them return false, then IsModelUnsat() will
  // return true.
  //
  // TODO(user): Rename to ModelIsUnsat().
  bool IsModelUnsat() const { return model_is_unsat_; }
 
  // Adds and registers the given propagator with the sat solver. Note that
  // during propagation, they will be called in the order they were added.
  void AddPropagator(SatPropagator* propagator);
  void AddLastPropagator(SatPropagator* propagator);
  void TakePropagatorOwnership(std::unique_ptr<SatPropagator> propagator) {
    owned_propagators_.push_back(std::move(propagator));
  }
 
  // Wrapper around the same functions in SatDecisionPolicy.
  //
  // TODO(user): Clean this up by making clients directly talk to
  // SatDecisionPolicy.
  void SetAssignmentPreference(Literal literal, double weight) {
    decision_policy_->SetAssignmentPreference(literal, weight);
  }
  std::vector<std::pair<Literal, double>> AllPreferences() const {
    return decision_policy_->AllPreferences();
  }
  void ResetDecisionHeuristic() {
    return decision_policy_->ResetDecisionHeuristic();
  }
  void ResetDecisionHeuristicAndSetAllPreferences(
      const std::vector<std::pair<Literal, double>>& prefs) {
    decision_policy_->ResetDecisionHeuristic();
    for (const std::pair<Literal, double>& p : prefs) {
      decision_policy_->SetAssignmentPreference(p.first, p.second);
    }
  }
 
  // Solves the problem and returns its status.
  // An empty problem is considered to be SAT.
  //
  // Note that the conflict limit applies only to this function and starts
  // counting from the time it is called.
  //
  // This will restart from the current solver configuration. If a previous call
  // to Solve() was interrupted by a conflict or time limit, calling this again
  // will resume the search exactly as it would have continued.
  //
  // Note that this will use the TimeLimit singleton, so the time limit
  // will be counted since the last time TimeLimit was reset, not from
  // the start of this function.
  enum Status {
    ASSUMPTIONS_UNSAT,
    INFEASIBLE,
    FEASIBLE,
    LIMIT_REACHED,
  };
  Status Solve();
 
  // Same as Solve(), but with a given time limit. Note that this will not
  // update the TimeLimit singleton, but only the passed object instead.
  Status SolveWithTimeLimit(TimeLimit* time_limit);
 
  // Simple interface to solve a problem under the given assumptions. This
  // simply ask the solver to solve a problem given a set of variables fixed to
  // a given value (the assumptions). Compared to simply calling AddUnitClause()
  // and fixing the variables once and for all, this allow to backtrack over the
  // assumptions and thus exploit the incrementally between subsequent solves.
  //
  // This function backtrack over all the current decision, tries to enqueue the
  // given assumptions, sets the assumption level accordingly and finally calls
  // Solve().
  //
  // If, given these assumptions, the model is UNSAT, this returns the
  // ASSUMPTIONS_UNSAT status. INFEASIBLE is reserved for the case where the
  // model is proven to be unsat without any assumptions.
  //
  // If ASSUMPTIONS_UNSAT is returned, it is possible to get a "core" of unsat
  // assumptions by calling GetLastIncompatibleDecisions().
  Status ResetAndSolveWithGivenAssumptions(
      const std::vector<Literal>& assumptions);
 
  // Changes the assumption level. All the decisions below this level will be
  // treated as assumptions by the next Solve(). Note that this may impact some
  // heuristics, like the LBD value of a clause.
  void SetAssumptionLevel(int assumption_level);
 
  // Returns the current assumption level. Note that if a solve was done since
  // the last SetAssumptionLevel(), then the returned level may be lower than
  // the one that was set. This is because some assumptions may now be
  // consequences of others before them due to the newly learned clauses.
  int AssumptionLevel() const { return assumption_level_; }
 
  // This can be called just after SolveWithAssumptions() returned
  // ASSUMPTION_UNSAT or after EnqueueDecisionAndBacktrackOnConflict() leaded
  // to a conflict. It returns a subsequence (in the correct order) of the
  // previously enqueued decisions that cannot be taken together without making
  // the problem UNSAT.
  std::vector<Literal> GetLastIncompatibleDecisions();
 
  // Advanced usage. The next 3 functions allow to drive the search from outside
  // the solver.
 
  // Takes a new decision (the given true_literal must be unassigned) and
  // propagates it. Returns the trail index of the first newly propagated
  // literal. If there is a conflict and the problem is detected to be UNSAT,
  // returns kUnsatTrailIndex.
  //
  // Important: In the presence of assumptions, this also returns
  // kUnsatTrailIndex on ASSUMPTION_UNSAT. One can know the difference with
  // IsModelUnsat().
  //
  // A client can determine if there is a conflict by checking if the
  // CurrentDecisionLevel() was increased by 1 or not.
  //
  // If there is a conflict, the given decision is not applied and:
  // - The conflict is learned.
  // - The decisions are potentially backtracked to the first decision that
  //   propagates more variables because of the newly learned conflict.
  // - The returned value is equal to trail_->Index() after this backtracking
  //   and just before the new propagation (due to the conflict) which is also
  //   performed by this function.
  int EnqueueDecisionAndBackjumpOnConflict(Literal true_literal);
 
  // This function starts by calling EnqueueDecisionAndBackjumpOnConflict(). If
  // there is no conflict, it stops there. Otherwise, it tries to reapply all
  // the decisions that were backjumped over until the first one that can't be
  // taken because it is incompatible. Note that during this process, more
  // conflicts may happen and the trail may be backtracked even further.
  //
  // In any case, the new decisions stack will be the largest valid "prefix"
  // of the old stack. Note that decisions that are now consequence of the ones
  // before them will no longer be decisions.
  //
  // Note(user): This function can be called with an already assigned literal.
  int EnqueueDecisionAndBacktrackOnConflict(Literal true_literal);
 
  // Tries to enqueue the given decision and performs the propagation.
  // Returns true if no conflict occurred. Otherwise, returns false and restores
  // the solver to the state just before this was called.
  //
  // Note(user): With this function, the solver doesn't learn anything.
  bool EnqueueDecisionIfNotConflicting(Literal true_literal);
 
  // Restores the state to the given target decision level. The decision at that
  // level and all its propagation will not be undone. But all the trail after
  // this will be cleared. Calling this with 0 will revert all the decisions and
  // only the fixed variables will be left on the trail.
  void Backtrack(int target_level);
 
  // Advanced usage. This is meant to restore the solver to a "proper" state
  // after a solve was interrupted due to a limit reached.
  //
  // Without assumption (i.e. if AssumptionLevel() is 0), this will revert all
  // decisions and make sure that all the fixed literals are propagated. In
  // presence of assumptions, this will either backtrack to the assumption level
  // or re-enqueue any assumptions that may have been backtracked over due to
  // conflits resolution. In both cases, the propagation is finished.
  //
  // Note that this may prove the model to be UNSAT or ASSUMPTION_UNSAT in which
  // case it will return false.
  bool RestoreSolverToAssumptionLevel();
 
  // Advanced usage. Finish the progation if it was interrupted. Note that this
  // might run into conflict and will propagate again until a fixed point is
  // reached or the model was proven UNSAT. Returns IsModelUnsat().
  bool FinishPropagation();
 
  // Like Backtrack(0) but make sure the propagation is finished and return
  // false if unsat was detected. This also removes any assumptions level.
  bool ResetToLevelZero();
 
  // Changes the assumptions level and the current solver assumptions. Returns
  // false if the model is UNSAT or ASSUMPTION_UNSAT, true otherwise.
  //
  // This uses the "new" assumptions handling, where all assumptions are
  // enqueued at once at decision level 1 before we start to propagate. This has
  // many advantages. In particular, because we propagate with the binary
  // implications first, if we ever have assumption => not(other_assumptions) we
  // are guaranteed to find it and returns a core of size 2.
  //
  // Paper: "Speeding Up Assumption-Based SAT", Randy Hickey and Fahiem Bacchus
  // http://www.maxhs.org/docs/Hickey-Bacchus2019_Chapter_SpeedingUpAssumption-BasedSAT.pdf
  bool ResetWithGivenAssumptions(const std::vector<Literal>& assumptions);
 
  // Advanced usage. If the decision level is smaller than the assumption level,
  // this will try to reapply all assumptions. Returns true if this was doable,
  // otherwise returns false in which case the model is either UNSAT or
  // ASSUMPTION_UNSAT.
  bool ReapplyAssumptionsIfNeeded();
 
  // Helper functions to get the correct status when one of the functions above
  // returns false.
  Status UnsatStatus() const {
    return IsModelUnsat() ? INFEASIBLE : ASSUMPTIONS_UNSAT;
  }
 
  // Extract the current problem clauses. The Output type must support the two
  // functions:
  //  - void AddBinaryClause(Literal a, Literal b);
  //  - void AddClause(absl::Span<const Literal> clause);
  //
  // TODO(user): also copy the removable clauses?
  template <typename Output>
  void ExtractClauses(Output* out) {
    CHECK(!IsModelUnsat());
    Backtrack(0);
    if (!FinishPropagation()) return;
 
    // It is important to process the newly fixed variables, so they are not
    // present in the clauses we export.
    if (num_processed_fixed_variables_ < trail_->Index()) {
      ProcessNewlyFixedVariables();
    }
    clauses_propagator_->DeleteRemovedClauses();
 
    // Note(user): Putting the binary clauses first help because the presolver
    // currently process the clauses in order.
    out->SetNumVariables(NumVariables());
    binary_implication_graph_->ExtractAllBinaryClauses(out);
    for (SatClause* clause : clauses_propagator_->AllClausesInCreationOrder()) {
      if (!clauses_propagator_->IsRemovable(clause)) {
        out->AddClause(clause->AsSpan());
      }
    }
  }
 
  // Functions to manage the set of learned binary clauses.
  // Only clauses added/learned when TrackBinaryClause() is true are managed.
  void TrackBinaryClauses(bool value) { track_binary_clauses_ = value; }
  bool AddBinaryClauses(const std::vector<BinaryClause>& clauses);
  const std::vector<BinaryClause>& NewlyAddedBinaryClauses();
  void ClearNewlyAddedBinaryClauses();
 
  struct Decision {
    Decision() {}
    Decision(int i, Literal l) : trail_index(i), literal(l) {}
    int trail_index = 0;
    Literal literal;
  };
 
  // Note that the Decisions() vector is always of size NumVariables(), and that
  // only the first CurrentDecisionLevel() entries have a meaning.
  const std::vector<Decision>& Decisions() const { return decisions_; }
  int CurrentDecisionLevel() const { return current_decision_level_; }
  const Trail& LiteralTrail() const { return *trail_; }
  const VariablesAssignment& Assignment() const { return trail_->Assignment(); }
 
  // Some statistics since the creation of the solver.
  int64_t num_branches() const;
  int64_t num_failures() const;
  int64_t num_propagations() const;
 
  // Note that we count the number of backtrack to level zero from a positive
  // level. Those can corresponds to actual restarts, or conflicts that learn
  // unit clauses or any other reason that trigger such backtrack.
  int64_t num_restarts() const;
 
  // A deterministic number that should be correlated with the time spent in
  // the Solve() function. The order of magnitude should be close to the time
  // in seconds.
  double deterministic_time() const;
 
  // Only used for debugging. Save the current assignment in debug_assignment_.
  // The idea is that if we know that a given assignment is satisfiable, then
  // all the learned clauses or PB constraints must be satisfiable by it. In
  // debug mode, and after this is called, all the learned clauses are tested to
  // satisfy this saved assignment.
  void SaveDebugAssignment();
 
  // Returns true iff the loaded problem only contains clauses.
  bool ProblemIsPureSat() const { return problem_is_pure_sat_; }
 
  void SetDratProofHandler(DratProofHandler* drat_proof_handler) {
    drat_proof_handler_ = drat_proof_handler;
    clauses_propagator_->SetDratProofHandler(drat_proof_handler_);
    binary_implication_graph_->SetDratProofHandler(drat_proof_handler_);
  }
 
  // This function is here to deal with the case where a SAT/CP model is found
  // to be trivially UNSAT while the user is constructing the model. Instead of
  // having to test the status of all the lines adding a constraint, one can
  // just check if the solver is not UNSAT once the model is constructed. Note
  // that we usually log a warning on the first constraint that caused a
  // "trival" unsatisfiability.
  void NotifyThatModelIsUnsat() { model_is_unsat_ = true; }
 
  // Adds a clause at any level of the tree and propagate any new deductions.
  // Returns false if the model becomes UNSAT. Important: We currently do not
  // support adding a clause that is already falsified at a positive decision
  // level. Doing that will cause a check fail.
  //
  // TODO(user): Backjump and propagate on a falsified clause? this is currently
  // not needed.
  bool AddClauseDuringSearch(absl::Span<const Literal> literals);
 
  // Performs propagation of the recently enqueued elements.
  // Mainly visible for testing.
  bool Propagate();
 
  // This must be called at level zero. It will spend the given num decision and
  // use propagation to try to minimize some clauses from the database.
  void MinimizeSomeClauses(int decisions_budget);
 
  // Sets the export function to the shared clauses manager.
  void SetShareBinaryClauseCallback(const std::function<void(Literal, Literal)>&
                                        shared_binary_clauses_callback) {
    shared_binary_clauses_callback_ = shared_binary_clauses_callback;
  }
 
  // Advance the given time limit with all the deterministic time that was
  // elapsed since last call.
  void AdvanceDeterministicTime(TimeLimit* limit) {
    const double current = deterministic_time();
    limit->AdvanceDeterministicTime(
        current - deterministic_time_at_last_advanced_time_limit_);
    deterministic_time_at_last_advanced_time_limit_ = current;
  }
 
  // Simplifies the problem when new variables are assigned at level 0.
  void ProcessNewlyFixedVariables();
 
  int64_t NumFixedVariables() const {
    if (!decisions_.empty()) return decisions_[0].trail_index;
    CHECK_EQ(CurrentDecisionLevel(), 0);
    return trail_->Index();
  }
 
 private:
  // Calls Propagate() and returns true if no conflict occurred. Otherwise,
  // learns the conflict, backtracks, enqueues the consequence of the learned
  // conflict and returns false.
  //
  // When handling assumptions, this might return false without backtracking
  // in case of ASSUMPTIONS_UNSAT.
  bool PropagateAndStopAfterOneConflictResolution();
 
  // All Solve() functions end up calling this one.
  Status SolveInternal(TimeLimit* time_limit);
 
  // Adds a binary clause to the BinaryImplicationGraph and to the
  // BinaryClauseManager when track_binary_clauses_ is true.
  //
  // If export_clause is true, then we will also export_clause that to a
  // potential shared_binary_clauses_callback_.
  void AddBinaryClauseInternal(Literal a, Literal b, bool export_clause);
 
  // See SaveDebugAssignment(). Note that these functions only consider the
  // variables at the time the debug_assignment_ was saved. If new variables
  // were added since that time, they will be considered unassigned.
  bool ClauseIsValidUnderDebugAssignment(
      const std::vector<Literal>& clause) const;
  bool PBConstraintIsValidUnderDebugAssignment(
      const std::vector<LiteralWithCoeff>& cst, const Coefficient rhs) const;
 
  // Logs the given status if parameters_.log_search_progress() is true.
  // Also returns it.
  Status StatusWithLog(Status status);
 
  // Main function called from SolveWithAssumptions() or from Solve() with an
  // assumption_level of 0 (meaning no assumptions).
  Status SolveInternal(int assumption_level);
 
  // Applies the previous decisions (which are still on decisions_), in order,
  // starting from the one at the current decision level. Stops at the one at
  // decisions_[level] or on the first decision already propagated to "false"
  // and thus incompatible.
  //
  // Note that during this process, conflicts may arise which will lead to
  // backjumps. In this case, we will simply keep reapplying decisions from the
  // last one backtracked over and so on.
  //
  // Returns FEASIBLE if no conflict occurred, INFEASIBLE if the model was
  // proven unsat and ASSUMPTION_UNSAT otherwise. In the last case the first non
  // taken old decision will be propagated to false by the ones before.
  //
  // first_propagation_index will be filled with the trail index of the first
  // newly propagated literal, or with -1 if INFEASIBLE is returned.
  Status ReapplyDecisionsUpTo(int level, int* first_propagation_index);
 
  // Returns false if the thread memory is over the limit.
  bool IsMemoryLimitReached() const;
 
  // Sets model_is_unsat_ to true and return false.
  bool SetModelUnsat();
 
  // Returns the decision level of a given variable.
  int DecisionLevel(BooleanVariable var) const {
    return trail_->Info(var).level;
  }
 
  // Returns the relevant pointer if the given variable was propagated by the
  // constraint in question. This is used to bump the activity of the learned
  // clauses or pb constraints.
  SatClause* ReasonClauseOrNull(BooleanVariable var) const;
  UpperBoundedLinearConstraint* ReasonPbConstraintOrNull(
      BooleanVariable var) const;
 
  // This does one step of a pseudo-Boolean resolution:
  // - The variable var has been assigned to l at a given trail_index.
  // - The reason for var propagates it to l.
  // - The conflict propagates it to not(l)
  // The goal of the operation is to combine the two constraints in order to
  // have a new conflict at a lower trail_index.
  //
  // Returns true if the reason for var was a normal clause. In this case,
  // the *slack is updated to its new value.
  bool ResolvePBConflict(BooleanVariable var,
                         MutableUpperBoundedLinearConstraint* conflict,
                         Coefficient* slack);
 
  // Returns true iff the clause is the reason for an assigned variable.
  //
  // TODO(user): With our current data structures, we could also return true
  // for clauses that were just used as a reason (like just before an untrail).
  // This may be beneficial, but should properly be defined so that we can
  // have the same behavior if we change the implementation.
  bool ClauseIsUsedAsReason(SatClause* clause) const {
    const BooleanVariable var = clause->PropagatedLiteral().Variable();
    return trail_->Info(var).trail_index < trail_->Index() &&
           (*trail_)[trail_->Info(var).trail_index].Variable() == var &&
           ReasonClauseOrNull(var) == clause;
  }
 
  // Add a problem clause. The clause is assumed to be "cleaned", that is no
  // duplicate variables (not strictly required) and not empty.
  bool AddProblemClauseInternal(absl::Span<const Literal> literals);
 
  // This is used by all the Add*LinearConstraint() functions. It detects
  // infeasible/trivial constraints or clause constraints and takes the proper
  // action.
  bool AddLinearConstraintInternal(const std::vector<LiteralWithCoeff>& cst,
                                   Coefficient rhs, Coefficient max_value);
 
  // Makes sure a pseudo boolean constraint is in canonical form.
  void CanonicalizeLinear(std::vector<LiteralWithCoeff>* cst,
                          Coefficient* bound_shift, Coefficient* max_value);
 
  // Adds a learned clause to the problem. This should be called after
  // Backtrack(). The backtrack is such that after it is applied, all the
  // literals of the learned close except one will be false. Thus the last one
  // will be implied True. This function also Enqueue() the implied literal.
  //
  // Returns the LBD of the clause.
  int AddLearnedClauseAndEnqueueUnitPropagation(
      const std::vector<Literal>& literals, bool is_redundant);
 
  // Creates a new decision which corresponds to setting the given literal to
  // True and Enqueue() this change.
  void EnqueueNewDecision(Literal literal);
 
  // Returns true if everything has been propagated.
  //
  // TODO(user): This test is fast but not exhaustive, especially regarding the
  // integer propagators. Fix.
  bool PropagationIsDone() const;
 
  // Update the propagators_ list with the relevant propagators.
  void InitializePropagators();
 
  // Unrolls the trail until a given point. This unassign the assigned variables
  // and add them to the priority queue with the correct weight.
  void Untrail(int target_trail_index);
 
  // Output to the DRAT proof handler any newly fixed variables.
  void ProcessNewlyFixedVariablesForDratProof();
 
  // Returns the maximum trail_index of the literals in the given clause.
  // All the literals must be assigned. Returns -1 if the clause is empty.
  int ComputeMaxTrailIndex(absl::Span<const Literal> clause) const;
 
  // Computes what is known as the first UIP (Unique implication point) conflict
  // clause starting from the failing clause. For a definition of UIP and a
  // comparison of the different possible conflict clause computation, see the
  // reference below.
  //
  // The conflict will have only one literal at the highest decision level, and
  // this literal will always be the first in the conflict vector.
  //
  // L Zhang, CF Madigan, MH Moskewicz, S Malik, "Efficient conflict driven
  // learning in a boolean satisfiability solver" Proceedings of the 2001
  // IEEE/ACM international conference on Computer-aided design, Pages 279-285.
  // http://www.cs.tau.ac.il/~msagiv/courses/ATP/iccad2001_final.pdf
  void ComputeFirstUIPConflict(
      int max_trail_index, std::vector<Literal>* conflict,
      std::vector<Literal>* reason_used_to_infer_the_conflict,
      std::vector<SatClause*>* subsumed_clauses);
 
  // Fills literals with all the literals in the reasons of the literals in the
  // given input. The output vector will have no duplicates and will not contain
  // the literals already present in the input.
  void ComputeUnionOfReasons(const std::vector<Literal>& input,
                             std::vector<Literal>* literals);
 
  // Do the full pseudo-Boolean constraint analysis. This calls multiple
  // time ResolvePBConflict() on the current conflict until we have a conflict
  // that allow us to propagate more at a lower decision level. This level
  // is the one returned in backjump_level.
  void ComputePBConflict(int max_trail_index, Coefficient initial_slack,
                         MutableUpperBoundedLinearConstraint* conflict,
                         int* backjump_level);
 
  // Applies some heuristics to a conflict in order to minimize its size and/or
  // replace literals by other literals from lower decision levels. The first
  // function choose which one of the other functions to call depending on the
  // parameters.
  //
  // Precondidtion: is_marked_ should be set to true for all the variables of
  // the conflict. It can also contains false non-conflict variables that
  // are implied by the negation of the 1-UIP conflict literal.
  void MinimizeConflict(
      std::vector<Literal>* conflict,
      std::vector<Literal>* reason_used_to_infer_the_conflict);
  void MinimizeConflictExperimental(std::vector<Literal>* conflict);
  void MinimizeConflictSimple(std::vector<Literal>* conflict);
  void MinimizeConflictRecursively(std::vector<Literal>* conflict);
 
  // Utility function used by MinimizeConflictRecursively().
  bool CanBeInferedFromConflictVariables(BooleanVariable variable);
 
  // To be used in DCHECK(). Verifies some property of the conflict clause:
  // - There is an unique literal with the highest decision level.
  // - This literal appears in the first position.
  // - All the other literals are of smaller decision level.
  // - Ther is no literal with a decision level of zero.
  bool IsConflictValid(const std::vector<Literal>& literals);
 
  // Given the learned clause after a conflict, this computes the correct
  // backtrack level to call Backtrack() with.
  int ComputeBacktrackLevel(const std::vector<Literal>& literals);
 
  // The LBD (Literal Blocks Distance) is the number of different decision
  // levels at which the literals of the clause were assigned. Note that we
  // ignore the decision level 0 whereas the definition in the paper below
  // doesn't:
  //
  // G. Audemard, L. Simon, "Predicting Learnt Clauses Quality in Modern SAT
  // Solver" in Twenty-first International Joint Conference on Artificial
  // Intelligence (IJCAI'09), july 2009.
  // http://www.ijcai.org/papers09/Papers/IJCAI09-074.pdf
  //
  // IMPORTANT: All the literals of the clause must be assigned, and the first
  // literal must be of the highest decision level. This will be the case for
  // all the reason clauses.
  template <typename LiteralList>
  int ComputeLbd(const LiteralList& literals);
 
  // Checks if we need to reduce the number of learned clauses and do
  // it if needed. Also updates the learned clause limit for the next cleanup.
  void CleanClauseDatabaseIfNeeded();
 
  // Activity management for clauses. This work the same way at the ones for
  // variables, but with different parameters.
  void BumpReasonActivities(const std::vector<Literal>& literals);
  void BumpClauseActivity(SatClause* clause);
  void RescaleClauseActivities(double scaling_factor);
  void UpdateClauseActivityIncrement();
 
  std::string DebugString(const SatClause& clause) const;
  std::string StatusString(Status status) const;
  std::string RunningStatisticsString() const;
 
  // Marks as "non-deletable" all clauses that were used to infer the given
  // variable. The variable must be currently assigned.
  void KeepAllClauseUsedToInfer(BooleanVariable variable);
 
  // Use propagation to try to minimize the given clause. This is really similar
  // to MinimizeCoreWithPropagation(). It must be called when the current
  // decision level is zero. Note that because this do a small tree search, it
  // will impact the variable/clauses activities and may add new conflicts.
  void TryToMinimizeClause(SatClause* clause);
 
  // This is used by the old non-model constructor.
  Model* model_;
  std::unique_ptr<Model> owned_model_;
 
  BooleanVariable num_variables_ = BooleanVariable(0);
 
  // Internal propagators. We keep them here because we need more than the
  // SatPropagator interface for them.
  BinaryImplicationGraph* binary_implication_graph_;
  LiteralWatchers* clauses_propagator_;
  PbConstraints* pb_constraints_;
 
  // Ordered list of propagators used by Propagate()/Untrail().
  std::vector<SatPropagator*> propagators_;
  std::vector<SatPropagator*> non_empty_propagators_;
 
  // Ordered list of propagators added with AddPropagator().
  std::vector<SatPropagator*> external_propagators_;
  SatPropagator* last_propagator_ = nullptr;
 
  // For the old, non-model interface.
  std::vector<std::unique_ptr<SatPropagator>> owned_propagators_;
 
  // Keep track of all binary clauses so they can be exported.
  bool track_binary_clauses_;
  BinaryClauseManager binary_clauses_;
 
  // Pointers to singleton Model objects.
  Trail* trail_;
  TimeLimit* time_limit_;
  SatParameters* parameters_;
  RestartPolicy* restart_;
  SatDecisionPolicy* decision_policy_;
  SolverLogger* logger_;
 
  // Used for debugging only. See SaveDebugAssignment().
  VariablesAssignment debug_assignment_;
 
  // The stack of decisions taken by the solver. They are stored in [0,
  // current_decision_level_). The vector is of size num_variables_ so it can
  // store all the decisions. This is done this way because in some situation we
  // need to remember the previously taken decisions after a backtrack.
  int current_decision_level_ = 0;
  std::vector<Decision> decisions_;
 
  // The trail index after the last Backtrack() call or before the last
  // EnqueueNewDecision() call.
  int last_decision_or_backtrack_trail_index_ = 0;
 
  // The assumption level. See SolveWithAssumptions().
  int assumption_level_ = 0;
  std::vector<Literal> assumptions_;
 
  // The size of the trail when ProcessNewlyFixedVariables() was last called.
  // Note that the trail contains only fixed literals (that is literals of
  // decision levels 0) before this point.
  int num_processed_fixed_variables_ = 0;
  double deterministic_time_of_last_fixed_variables_cleanup_ = 0.0;
 
  // Used in ProcessNewlyFixedVariablesForDratProof().
  int drat_num_processed_fixed_variables_ = 0;
 
  // Tracks various information about the solver progress.
  struct Counters {
    int64_t num_branches = 0;
    int64_t num_failures = 0;
    int64_t num_restarts = 0;
 
    // Minimization stats.
    int64_t num_minimizations = 0;
    int64_t num_literals_removed = 0;
 
    // PB constraints.
    int64_t num_learned_pb_literals = 0;
 
    // Clause learning /deletion stats.
    int64_t num_literals_learned = 0;
    int64_t num_literals_forgotten = 0;
    int64_t num_subsumed_clauses = 0;
 
    // TryToMinimizeClause() stats.
    int64_t minimization_num_clauses = 0;
    int64_t minimization_num_decisions = 0;
    int64_t minimization_num_true = 0;
    int64_t minimization_num_subsumed = 0;
    int64_t minimization_num_removed_literals = 0;
  };
  Counters counters_;
 
  // Solver information.
  WallTimer timer_;
 
  // This is set to true if the model is found to be UNSAT when adding new
  // constraints.
  bool model_is_unsat_ = false;
 
  // Increment used to bump the variable activities.
  double clause_activity_increment_;
 
  // This counter is decremented each time we learn a clause that can be
  // deleted. When it reaches zero, a clause cleanup is triggered.
  int num_learned_clause_before_cleanup_ = 0;
 
  // Temporary members used during conflict analysis.
  SparseBitset<BooleanVariable> is_marked_;
  SparseBitset<BooleanVariable> is_independent_;
  SparseBitset<BooleanVariable> tmp_mark_;
  std::vector<int> min_trail_index_per_level_;
 
  // Temporary members used by CanBeInferedFromConflictVariables().
  std::vector<BooleanVariable> dfs_stack_;
  std::vector<BooleanVariable> variable_to_process_;
 
  // Temporary member used when adding clauses.
  std::vector<Literal> literals_scratchpad_;
 
  // A boolean vector used to temporarily mark decision levels.
  DEFINE_STRONG_INDEX_TYPE(SatDecisionLevel);
  SparseBitset<SatDecisionLevel> is_level_marked_;
 
  // Temporary vectors used by EnqueueDecisionAndBackjumpOnConflict().
  std::vector<Literal> learned_conflict_;
  std::vector<Literal> reason_used_to_infer_the_conflict_;
  std::vector<Literal> extra_reason_literals_;
  std::vector<SatClause*> subsumed_clauses_;
 
  // When true, temporarily disable the deletion of clauses that are not needed
  // anymore. This is a hack for TryToMinimizeClause() because we use
  // propagation in this function which might trigger a clause database
  // deletion, but we still want the pointer to the clause we wants to minimize
  // to be valid until the end of that function.
  bool block_clause_deletion_ = false;
 
  // "cache" to avoid inspecting many times the same reason during conflict
  // analysis.
  VariableWithSameReasonIdentifier same_reason_identifier_;
 
  // Boolean used to include/exclude constraints from the core computation.
  bool is_relevant_for_core_computation_;
 
  // The current pseudo-Boolean conflict used in PB conflict analysis.
  MutableUpperBoundedLinearConstraint pb_conflict_;
 
  // The deterministic time when the time limit was updated.
  // As the deterministic time in the time limit has to be advanced manually,
  // it is necessary to keep track of the last time the time was advanced.
  double deterministic_time_at_last_advanced_time_limit_ = 0;
 
  // This is true iff the loaded problem only contains clauses.
  bool problem_is_pure_sat_;
 
  DratProofHandler* drat_proof_handler_;
 
  mutable StatsGroup stats_;
 
  std::function<void(Literal, Literal)> shared_binary_clauses_callback_ =
      nullptr;
 
  DISALLOW_COPY_AND_ASSIGN(SatSolver);
};
 
// Tries to minimize the given UNSAT core with a really simple heuristic.
// The idea is to remove literals that are consequences of others in the core.
// We already know that in the initial order, no literal is propagated by the
// one before it, so we just look for propagation in the reverse order.
//
// Important: The given SatSolver must be the one that just produced the given
// core.
//
// TODO(user): One should use MinimizeCoreWithPropagation() instead.
void MinimizeCore(SatSolver* solver, std::vector<Literal>* core);
 
// ============================================================================
// Model based functions.
//
// TODO(user): move them in another file, and unit-test them.
// ============================================================================
 
inline std::function<void(Model*)> BooleanLinearConstraint(
    int64_t lower_bound, int64_t upper_bound,
    std::vector<LiteralWithCoeff>* cst) {
  return [=](Model* model) {
    model->GetOrCreate<SatSolver>()->AddLinearConstraint(
        /*use_lower_bound=*/true, Coefficient(lower_bound),
        /*use_upper_bound=*/true, Coefficient(upper_bound), cst);
  };
}
 
inline std::function<void(Model*)> CardinalityConstraint(
    int64_t lower_bound, int64_t upper_bound,
    const std::vector<Literal>& literals) {
  return [=](Model* model) {
    std::vector<LiteralWithCoeff> cst;
    cst.reserve(literals.size());
    for (int i = 0; i < literals.size(); ++i) {
      cst.emplace_back(literals[i], 1);
    }
    model->GetOrCreate<SatSolver>()->AddLinearConstraint(
        /*use_lower_bound=*/true, Coefficient(lower_bound),
        /*use_upper_bound=*/true, Coefficient(upper_bound), &cst);
  };
}
 
inline std::function<void(Model*)> ExactlyOneConstraint(
    const std::vector<Literal>& literals) {
  return [=](Model* model) {
    std::vector<LiteralWithCoeff> cst;
    cst.reserve(literals.size());
    for (const Literal l : literals) {
      cst.emplace_back(l, Coefficient(1));
    }
    model->GetOrCreate<SatSolver>()->AddLinearConstraint(
        /*use_lower_bound=*/true, Coefficient(1),
        /*use_upper_bound=*/true, Coefficient(1), &cst);
  };
}
 
inline std::function<void(Model*)> AtMostOneConstraint(
    const std::vector<Literal>& literals) {
  return [=](Model* model) {
    std::vector<LiteralWithCoeff> cst;
    cst.reserve(literals.size());
    for (const Literal l : literals) {
      cst.emplace_back(l, Coefficient(1));
    }
    model->GetOrCreate<SatSolver>()->AddLinearConstraint(
        /*use_lower_bound=*/false, Coefficient(0),
        /*use_upper_bound=*/true, Coefficient(1), &cst);
  };
}
 
inline std::function<void(Model*)> ClauseConstraint(
    absl::Span<const Literal> literals) {
  return [=](Model* model) {
    std::vector<LiteralWithCoeff> cst;
    cst.reserve(literals.size());
    for (const Literal l : literals) {
      cst.emplace_back(l, Coefficient(1));
    }
    model->GetOrCreate<SatSolver>()->AddLinearConstraint(
        /*use_lower_bound=*/true, Coefficient(1),
        /*use_upper_bound=*/false, Coefficient(1), &cst);
  };
}
 
// a => b.
inline std::function<void(Model*)> Implication(Literal a, Literal b) {
  return [=](Model* model) {
    model->GetOrCreate<SatSolver>()->AddBinaryClause(a.Negated(), b);
  };
}
 
// a == b.
inline std::function<void(Model*)> Equality(Literal a, Literal b) {
  return [=](Model* model) {
    model->GetOrCreate<SatSolver>()->AddBinaryClause(a.Negated(), b);
    model->GetOrCreate<SatSolver>()->AddBinaryClause(a, b.Negated());
  };
}
 
// r <=> (at least one literal is true). This is a reified clause.
inline std::function<void(Model*)> ReifiedBoolOr(
    const std::vector<Literal>& literals, Literal r) {
  return [=](Model* model) {
    std::vector<Literal> clause;
    for (const Literal l : literals) {
      model->Add(Implication(l, r));  // l => r.
      clause.push_back(l);
    }
 
    // All false => r false.
    clause.push_back(r.Negated());
    model->Add(ClauseConstraint(clause));
  };
}
 
// enforcement_literals => clause.
inline std::function<void(Model*)> EnforcedClause(
    absl::Span<const Literal> enforcement_literals,
    absl::Span<const Literal> clause) {
  return [=](Model* model) {
    std::vector<Literal> tmp;
    for (const Literal l : enforcement_literals) {
      tmp.push_back(l.Negated());
    }
    for (const Literal l : clause) {
      tmp.push_back(l);
    }
    model->Add(ClauseConstraint(tmp));
  };
}
 
// r <=> (all literals are true).
//
// Note(user): we could have called ReifiedBoolOr() with everything negated.
inline std::function<void(Model*)> ReifiedBoolAnd(
    const std::vector<Literal>& literals, Literal r) {
  return [=](Model* model) {
    std::vector<Literal> clause;
    for (const Literal l : literals) {
      model->Add(Implication(r, l));  // r => l.
      clause.push_back(l.Negated());
    }
 
    // All true => r true.
    clause.push_back(r);
    model->Add(ClauseConstraint(clause));
  };
}
 
// r <=> (a <= b).
inline std::function<void(Model*)> ReifiedBoolLe(Literal a, Literal b,
                                                 Literal r) {
  return [=](Model* model) {
    // r <=> (a <= b) is the same as r <=> not(a=1 and b=0).
    // So r <=> a=0 OR b=1.
    model->Add(ReifiedBoolOr({a.Negated(), b}, r));
  };
}
 
// This checks that the variable is fixed.
inline std::function<int64_t(const Model&)> Value(Literal l) {
  return [=](const Model& model) {
    const Trail* trail = model.Get<Trail>();
    CHECK(trail->Assignment().VariableIsAssigned(l.Variable()));
    return trail->Assignment().LiteralIsTrue(l);
  };
}
 
// This checks that the variable is fixed.
inline std::function<int64_t(const Model&)> Value(BooleanVariable b) {
  return [=](const Model& model) {
    const Trail* trail = model.Get<Trail>();
    CHECK(trail->Assignment().VariableIsAssigned(b));
    return trail->Assignment().LiteralIsTrue(Literal(b, true));
  };
}
 
// This can be used to enumerate all the solutions. After each SAT call to
// Solve(), calling this will reset the solver and exclude the current solution
// so that the next call to Solve() will give a new solution or UNSAT is there
// is no more new solutions.
inline std::function<void(Model*)> ExcludeCurrentSolutionAndBacktrack() {
  return [=](Model* model) {
    SatSolver* sat_solver = model->GetOrCreate<SatSolver>();
 
    // Note that we only exclude the current decisions, which is an efficient
    // way to not get the same SAT assignment.
    const int current_level = sat_solver->CurrentDecisionLevel();
    std::vector<Literal> clause_to_exclude_solution;
    clause_to_exclude_solution.reserve(current_level);
    for (int i = 0; i < current_level; ++i) {
      clause_to_exclude_solution.push_back(
          sat_solver->Decisions()[i].literal.Negated());
    }
    sat_solver->Backtrack(0);
    model->Add(ClauseConstraint(clause_to_exclude_solution));
  };
}
 
// Returns a string representation of a SatSolver::Status.
std::string SatStatusString(SatSolver::Status status);
inline std::ostream& operator<<(std::ostream& os, SatSolver::Status status) {
  os << SatStatusString(status);
  return os;
}
 
}  // namespace sat
}  // namespace operations_research
 
#endif  // OR_TOOLS_SAT_SAT_SOLVER_H_