sat/table.cc

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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.
 
#include "ortools/sat/table.h"
 
#include <algorithm>
#include <cstdint>
#include <limits>
#include <memory>
#include <set>
#include <utility>
 
#include "absl/container/flat_hash_map.h"
#include "absl/container/flat_hash_set.h"
#include "absl/strings/str_cat.h"
#include "absl/strings/str_join.h"
#include "ortools/base/int_type.h"
#include "ortools/base/logging.h"
#include "ortools/base/map_util.h"
#include "ortools/base/stl_util.h"
#include "ortools/sat/sat_base.h"
#include "ortools/sat/sat_solver.h"
#include "ortools/sat/util.h"
#include "ortools/util/sorted_interval_list.h"
 
namespace operations_research {
namespace sat {
 
namespace {
 
// Converts the vector representation returned by FullDomainEncoding() to a map.
absl::flat_hash_map<IntegerValue, Literal> GetEncoding(IntegerVariable var,
                                                       Model* model) {
  absl::flat_hash_map<IntegerValue, Literal> encoding;
  IntegerEncoder* encoder = model->GetOrCreate<IntegerEncoder>();
  for (const auto& entry : encoder->FullDomainEncoding(var)) {
    encoding[entry.value] = entry.literal;
  }
  return encoding;
}
 
// Add the implications and clauses to link one column of a table to the Literal
// controling if the lines are possible or not. The column has the given values,
// and the Literal of the column variable can be retrieved using the encoding
// map. Thew tuples_with_any vector provides a list of line_literals that will
// support any value.
void ProcessOneColumn(
    const std::vector<Literal>& line_literals,
    const std::vector<IntegerValue>& values,
    const absl::flat_hash_map<IntegerValue, Literal>& encoding,
    const std::vector<Literal>& tuples_with_any, Model* model) {
  CHECK_EQ(line_literals.size(), values.size());
  std::vector<std::pair<IntegerValue, Literal>> pairs;
 
  // If a value is false (i.e not possible), then the tuple with this value
  // is false too (i.e not possible). Conversely, if the tuple is selected,
  // the value must be selected.
  for (int i = 0; i < values.size(); ++i) {
    const IntegerValue v = values[i];
    if (!encoding.contains(v)) {
      model->Add(ClauseConstraint({line_literals[i].Negated()}));
    } else {
      pairs.emplace_back(v, line_literals[i]);
      model->Add(Implication(line_literals[i], gtl::FindOrDie(encoding, v)));
    }
  }
 
  // Regroup literal with the same value and add for each the clause: If all the
  // tuples containing a value are false, then this value must be false too.
  std::sort(pairs.begin(), pairs.end());
  std::vector<Literal> clause = tuples_with_any;
  for (int i = 0; i < pairs.size();) {
    // We always keep the tuples_with_any at the beginning of the clause.
    clause.resize(tuples_with_any.size());
 
    const IntegerValue value = pairs[i].first;
    for (; i < pairs.size() && pairs[i].first == value; ++i) {
      clause.push_back(pairs[i].second);
    }
 
    // And the "value" literal and load the clause.
    clause.push_back(gtl::FindOrDie(encoding, value).Negated());
    model->Add(ClauseConstraint(clause));
  }
}
 
// Simpler encoding for table constraints with 2 variables.
void AddSizeTwoTable(
    absl::Span<const IntegerVariable> vars,
    const std::vector<std::vector<int64_t>>& tuples,
    const std::vector<absl::flat_hash_set<int64_t>>& values_per_var,
    Model* model) {
  const int n = vars.size();
  CHECK_EQ(n, 2);
  IntegerTrail* const integer_trail = model->GetOrCreate<IntegerTrail>();
 
  std::vector<absl::flat_hash_map<IntegerValue, Literal>> encodings(n);
  for (int i = 0; i < n; ++i) {
    const std::vector<int64_t> reached_values(values_per_var[i].begin(),
                                              values_per_var[i].end());
    integer_trail->UpdateInitialDomain(vars[i],
                                       Domain::FromValues(reached_values));
    if (values_per_var.size() > 1) {
      model->Add(FullyEncodeVariable(vars[i]));
      encodings[i] = GetEncoding(vars[i], model);
    }
  }
 
  // One variable is fixed. Propagation is complete.
  if (values_per_var[0].size() == 1 || values_per_var[1].size() == 1) return;
 
  std::map<LiteralIndex, std::vector<Literal>> left_to_right;
  std::map<LiteralIndex, std::vector<Literal>> right_to_left;
 
  for (const auto& tuple : tuples) {
    const IntegerValue left_value(tuple[0]);
    const IntegerValue right_value(tuple[1]);
    if (!encodings[0].contains(left_value) ||
        !encodings[1].contains(right_value)) {
      continue;
    }
 
    const Literal left = gtl::FindOrDie(encodings[0], left_value);
    const Literal right = gtl::FindOrDie(encodings[1], right_value);
    left_to_right[left.Index()].push_back(right);
    right_to_left[right.Index()].push_back(left);
  }
 
  int num_implications = 0;
  int num_clause_added = 0;
  int num_large_clause_added = 0;
  std::vector<Literal> clause;
  auto add_support_constraint =
      [model, &num_clause_added, &num_large_clause_added, &num_implications,
       &clause](LiteralIndex lit, const std::vector<Literal>& supports,
                int max_support_size) {
        if (supports.size() == max_support_size) return;
        if (supports.size() == 1) {
          model->Add(Implication(Literal(lit), supports.front()));
          num_implications++;
        } else {
          clause.assign(supports.begin(), supports.end());
          clause.push_back(Literal(lit).Negated());
          model->Add(ClauseConstraint(clause));
          num_clause_added++;
          if (supports.size() > max_support_size / 2) {
            num_large_clause_added++;
          }
        }
      };
 
  for (const auto& it : left_to_right) {
    add_support_constraint(it.first, it.second, values_per_var[1].size());
  }
  for (const auto& it : right_to_left) {
    add_support_constraint(it.first, it.second, values_per_var[0].size());
  }
  VLOG(2) << "Table: 2 variables, " << tuples.size() << " tuples encoded using "
          << num_clause_added << " clauses, " << num_large_clause_added
          << " large clauses, " << num_implications << " implications";
}
 
// This method heuristically explores subsets of variables and decide if the
// projection of all tuples nearly fills all the possible combination of
// projected variables domains.
//
// In that case, it creates the complement of the projected tuples and add that
// as a forbidden assignment constraint.
void ExploreSubsetOfVariablesAndAddNegatedTables(
    const std::vector<std::vector<int64_t>>& tuples,
    const std::vector<std::vector<int64_t>>& var_domains,
    absl::Span<const IntegerVariable> vars, Model* model) {
  const int num_vars = var_domains.size();
  for (int start = 0; start < num_vars; ++start) {
    const int limit = start == 0 ? num_vars : std::min(num_vars, start + 3);
    for (int end = start + 1; end < limit; ++end) {
      // TODO(user): If we add negated table for more than one value of
      // end, because the set of variables will be included in each other, we
      // could reduce the number of clauses added. I.e if we excluded
      // (x=2, y=3) there is no need to exclude any of the tuples
      // (x=2, y=3, z=*).
 
      // Compute the maximum number of such prefix tuples.
      int64_t max_num_prefix_tuples = 1;
      for (int i = start; i <= end; ++i) {
        max_num_prefix_tuples =
            CapProd(max_num_prefix_tuples, var_domains[i].size());
      }
 
      // Abort early.
      if (max_num_prefix_tuples > 2 * tuples.size()) break;
 
      absl::flat_hash_set<absl::Span<const int64_t>> prefixes;
      bool skip = false;
      for (const std::vector<int64_t>& tuple : tuples) {
        prefixes.insert(absl::MakeSpan(&tuple[start], end - start + 1));
        if (prefixes.size() == max_num_prefix_tuples) {
          // Nothing to add with this range [start..end].
          skip = true;
          break;
        }
      }
      if (skip) continue;
      const int num_prefix_tuples = prefixes.size();
 
      std::vector<std::vector<int64_t>> negated_tuples;
 
      int created = 0;
      if (num_prefix_tuples < max_num_prefix_tuples &&
          max_num_prefix_tuples < num_prefix_tuples * 2) {
        std::vector<int64_t> tmp_tuple;
        for (int i = 0; i < max_num_prefix_tuples; ++i) {
          tmp_tuple.clear();
          int index = i;
          for (int j = start; j <= end; ++j) {
            tmp_tuple.push_back(var_domains[j][index % var_domains[j].size()]);
            index /= var_domains[j].size();
          }
          if (!prefixes.contains(tmp_tuple)) {
            negated_tuples.push_back(tmp_tuple);
            created++;
          }
        }
        AddNegatedTableConstraint(vars.subspan(start, end - start + 1),
                                  negated_tuples, model);
        VLOG(2) << "  add negated tables with " << created
                << " tuples on the range [" << start << "," << end << "]";
      }
    }
  }
}
 
}  // namespace
 
// Makes a static decomposition of a table constraint into clauses.
// This uses an auxiliary vector of Literals tuple_literals.
// For every column col, and every value val of that column,
// the decomposition uses clauses corresponding to the equivalence:
// (\/_{row | tuples[row][col] = val} tuple_literals[row]) <=> (vars[col] = val)
void AddTableConstraint(absl::Span<const IntegerVariable> vars,
                        std::vector<std::vector<int64_t>> tuples,
                        Model* model) {
  const int n = vars.size();
  IntegerTrail* integer_trail = model->GetOrCreate<IntegerTrail>();
  const int num_original_tuples = tuples.size();
 
  // Compute the set of possible values for each variable (from the table).
  // Remove invalid tuples along the way.
  std::vector<absl::flat_hash_set<int64_t>> values_per_var(n);
  int index = 0;
  for (int tuple_index = 0; tuple_index < num_original_tuples; ++tuple_index) {
    bool keep = true;
    for (int i = 0; i < n; ++i) {
      const int64_t value = tuples[tuple_index][i];
      if (!values_per_var[i].contains(value) /* cached */ &&
          !integer_trail->InitialVariableDomain(vars[i]).Contains(value)) {
        keep = false;
        break;
      }
    }
    if (keep) {
      std::swap(tuples[tuple_index], tuples[index]);
      for (int i = 0; i < n; ++i) {
        values_per_var[i].insert(tuples[index][i]);
      }
      index++;
    }
  }
  tuples.resize(index);
  const int num_valid_tuples = tuples.size();
 
  if (tuples.empty()) {
    model->GetOrCreate<SatSolver>()->NotifyThatModelIsUnsat();
    return;
  }
 
  if (n == 2) {
    AddSizeTwoTable(vars, tuples, values_per_var, model);
    return;
  }
 
  // It is easier to compute this before compression, as compression will merge
  // tuples.
  int num_prefix_tuples = 0;
  {
    absl::flat_hash_set<absl::Span<const int64_t>> prefixes;
    for (const std::vector<int64_t>& tuple : tuples) {
      prefixes.insert(absl::MakeSpan(tuple.data(), n - 1));
    }
    num_prefix_tuples = prefixes.size();
  }
 
  std::vector<std::vector<int64_t>> var_domains(n);
  for (int j = 0; j < n; ++j) {
    var_domains[j].assign(values_per_var[j].begin(), values_per_var[j].end());
    std::sort(var_domains[j].begin(), var_domains[j].end());
  }
  CHECK_GT(vars.size(), 2);
  ExploreSubsetOfVariablesAndAddNegatedTables(tuples, var_domains, vars, model);
 
  // The variable domains have been computed. Fully encode variables.
  // Note that in some corner cases (like duplicate vars), as we call
  // UpdateInitialDomain(), the domain of other variable could become more
  // restricted that values_per_var. For now, we do not try to reach a fixed
  // point here.
  std::vector<absl::flat_hash_map<IntegerValue, Literal>> encodings(n);
  for (int i = 0; i < n; ++i) {
    const std::vector<int64_t> reached_values(values_per_var[i].begin(),
                                              values_per_var[i].end());
    integer_trail->UpdateInitialDomain(vars[i],
                                       Domain::FromValues(reached_values));
    if (values_per_var.size() > 1) {
      model->Add(FullyEncodeVariable(vars[i]));
      encodings[i] = GetEncoding(vars[i], model);
    }
  }
 
  // Compress tuples.
  const int64_t any_value = std::numeric_limits<int64_t>::min();
  std::vector<int64_t> domain_sizes;
  for (int i = 0; i < n; ++i) {
    domain_sizes.push_back(values_per_var[i].size());
  }
  CompressTuples(domain_sizes, any_value, &tuples);
  const int num_compressed_tuples = tuples.size();
 
  // Detect if prefix tuples are all different.
  const bool prefixes_are_all_different = num_prefix_tuples == num_valid_tuples;
  if (VLOG_IS_ON(2)) {
    // Compute the maximum number of prefix tuples.
    int64_t max_num_prefix_tuples = 1;
    for (int i = 0; i + 1 < n; ++i) {
      max_num_prefix_tuples =
          CapProd(max_num_prefix_tuples, values_per_var[i].size());
    }
 
    std::string message = absl::StrCat(
        "Table: ", n, " variables, original tuples = ", num_original_tuples);
    if (num_valid_tuples != num_original_tuples) {
      absl::StrAppend(&message, ", valid tuples = ", num_valid_tuples);
    }
    if (prefixes_are_all_different) {
      if (num_prefix_tuples < max_num_prefix_tuples) {
        absl::StrAppend(&message, ", partial prefix = ", num_prefix_tuples, "/",
                        max_num_prefix_tuples);
      } else {
        absl::StrAppend(&message, ", full prefix = true");
      }
    } else {
      absl::StrAppend(&message, ", num prefix tuples = ", num_prefix_tuples);
    }
    if (num_compressed_tuples != num_valid_tuples) {
      absl::StrAppend(&message,
                      ", compressed tuples = ", num_compressed_tuples);
    }
    VLOG(2) << message;
  }
 
  if (tuples.size() == 1) {
    // Nothing more to do.
    return;
  }
 
  // Create one Boolean variable per tuple to indicate if it can still be
  // selected or not. Note that we don't enforce exactly one tuple to be
  // selected because these variables are just used by this constraint, so
  // only the information "can't be selected" is important.
  //
  // TODO(user): If a value in one column is unique, we don't need to
  // create a new BooleanVariable corresponding to this line since we can use
  // the one corresponding to this value in that column.
  //
  // Note that if there is just one tuple, there is no need to create such
  // variables since they are not used.
  std::vector<Literal> tuple_literals;
  tuple_literals.reserve(tuples.size());
  if (tuples.size() == 2) {
    tuple_literals.emplace_back(model->Add(NewBooleanVariable()), true);
    tuple_literals.emplace_back(tuple_literals[0].Negated());
  } else if (tuples.size() > 2) {
    for (int i = 0; i < tuples.size(); ++i) {
      tuple_literals.emplace_back(model->Add(NewBooleanVariable()), true);
    }
    model->Add(ClauseConstraint(tuple_literals));
  }
 
  std::vector<Literal> active_tuple_literals;
  std::vector<IntegerValue> active_values;
  std::vector<Literal> any_tuple_literals;
  for (int i = 0; i < n; ++i) {
    if (values_per_var[i].size() == 1) continue;
 
    active_tuple_literals.clear();
    active_values.clear();
    any_tuple_literals.clear();
    for (int j = 0; j < tuple_literals.size(); ++j) {
      const int64_t v = tuples[j][i];
 
      if (v == any_value) {
        any_tuple_literals.push_back(tuple_literals[j]);
      } else {
        active_tuple_literals.push_back(tuple_literals[j]);
        active_values.push_back(IntegerValue(v));
      }
    }
 
    if (!active_tuple_literals.empty()) {
      ProcessOneColumn(active_tuple_literals, active_values, encodings[i],
                       any_tuple_literals, model);
    }
  }
 
  if (prefixes_are_all_different) {
    // The first n-1 columns are all different, this encodes the implication
    // table (tuple of size n - 1) implies value. We can add an optional
    // propagation that should lead to better explanation.
    // For each tuple, we add a clause prefix => last value.
    std::vector<Literal> clause;
    for (int j = 0; j < tuples.size(); ++j) {
      clause.clear();
      bool tuple_is_valid = true;
      for (int i = 0; i + 1 < n; ++i) {
        // Ignore fixed variables.
        if (values_per_var[i].size() == 1) continue;
 
        const int64_t v = tuples[j][i];
        // Ignored 'any' created during compression.
        if (v == any_value) continue;
 
        const IntegerValue value(v);
        if (!encodings[i].contains(value)) {
          tuple_is_valid = false;
          break;
        }
        clause.push_back(gtl::FindOrDie(encodings[i], value).Negated());
      }
      if (!tuple_is_valid) continue;
 
      // Add the target of the implication.
      const IntegerValue target_value = IntegerValue(tuples[j][n - 1]);
      if (!encodings[n - 1].contains(target_value)) continue;
      const Literal target_literal =
          gtl::FindOrDie(encodings[n - 1], target_value);
      clause.push_back(target_literal);
      model->Add(ClauseConstraint(clause));
    }
  }
}
 
void AddNegatedTableConstraint(absl::Span<const IntegerVariable> vars,
                               std::vector<std::vector<int64_t>> tuples,
                               Model* model) {
  const int n = vars.size();
  auto* integer_trail = model->GetOrCreate<IntegerTrail>();
  auto* integer_encoder = model->GetOrCreate<IntegerEncoder>();
 
  // Remove unreachable tuples.
  int index = 0;
  while (index < tuples.size()) {
    bool remove = false;
    for (int i = 0; i < n; ++i) {
      if (!integer_trail->InitialVariableDomain(vars[i]).Contains(
              tuples[index][i])) {
        remove = true;
        break;
      }
    }
    if (remove) {
      tuples[index] = tuples.back();
      tuples.pop_back();
    } else {
      index++;
    }
  }
 
  if (tuples.empty()) {
    return;
  }
 
  // Compress tuples.
  const int64_t any_value = std::numeric_limits<int64_t>::min();
  std::vector<int64_t> domain_sizes;
  for (int i = 0; i < n; ++i) {
    domain_sizes.push_back(
        integer_trail->InitialVariableDomain(vars[i]).Size());
  }
  CompressTuples(domain_sizes, any_value, &tuples);
 
  // Collect all relevant var == value literal.
  std::vector<absl::flat_hash_map<int64_t, Literal>> mapping(n);
  for (int i = 0; i < n; ++i) {
    for (const auto pair : integer_encoder->PartialDomainEncoding(vars[i])) {
      mapping[i][pair.value.value()] = pair.literal;
    }
  }
 
  // For each tuple, forbid the variables values to be this tuple.
  std::vector<Literal> clause;
  for (const std::vector<int64_t>& tuple : tuples) {
    bool add_tuple = true;
    clause.clear();
    for (int i = 0; i < n; ++i) {
      const int64_t value = tuple[i];
      if (value == any_value) continue;
 
      // If a literal associated to var == value exist, use it, otherwise
      // just use (and eventually create) the two literals var >= value + 1
      // and var <= value - 1.
      if (mapping[i].contains(value)) {
        clause.push_back(gtl::FindOrDie(mapping[i], value).Negated());
      } else {
        const int64_t lb = model->Get(LowerBound(vars[i]));
        const int64_t ub = model->Get(UpperBound(vars[i]));
 
        // TODO(user): test the full initial domain instead of just checking
        // the bounds. That shouldn't change too much since the literals added
        // below will be trivially true or false though.
        if (value < lb || value > ub) {
          add_tuple = false;
          break;
        }
        if (value > lb) {
          clause.push_back(integer_encoder->GetOrCreateAssociatedLiteral(
              IntegerLiteral::LowerOrEqual(vars[i], IntegerValue(value - 1))));
        }
        if (value < ub) {
          clause.push_back(integer_encoder->GetOrCreateAssociatedLiteral(
              IntegerLiteral::GreaterOrEqual(vars[i],
                                             IntegerValue(value + 1))));
        }
      }
    }
    if (add_tuple) model->Add(ClauseConstraint(clause));
  }
}
 
std::function<void(Model*)> LiteralTableConstraint(
    const std::vector<std::vector<Literal>>& literal_tuples,
    const std::vector<Literal>& line_literals) {
  return [=](Model* model) {
    CHECK_EQ(literal_tuples.size(), line_literals.size());
    const int num_tuples = line_literals.size();
    if (num_tuples == 0) return;
    const int tuple_size = literal_tuples[0].size();
    if (tuple_size == 0) return;
    for (int i = 1; i < num_tuples; ++i) {
      CHECK_EQ(tuple_size, literal_tuples[i].size());
    }
 
    absl::flat_hash_map<LiteralIndex, std::vector<LiteralIndex>>
        line_literals_per_literal;
    for (int i = 0; i < num_tuples; ++i) {
      const LiteralIndex selected_index = line_literals[i].Index();
      for (const Literal l : literal_tuples[i]) {
        line_literals_per_literal[l.Index()].push_back(selected_index);
      }
    }
 
    // line_literals[i] == true => literal_tuples[i][j] == true.
    // literal_tuples[i][j] == false => line_literals[i] == false.
    for (int i = 0; i < num_tuples; ++i) {
      const Literal line_is_selected = line_literals[i];
      for (const Literal lit : literal_tuples[i]) {
        model->Add(Implication(line_is_selected, lit));
      }
    }
 
    // Exactly one selected literal is true.
    model->Add(ExactlyOneConstraint(line_literals));
 
    // If all selected literals of the lines containing a literal are false,
    // then the literal is false.
    for (const auto& p : line_literals_per_literal) {
      std::vector<Literal> clause;
      for (const auto& index : p.second) {
        clause.push_back(Literal(index));
      }
      clause.push_back(Literal(p.first).Negated());
      model->Add(ClauseConstraint(clause));
    }
  };
}
 
std::function<void(Model*)> TransitionConstraint(
    const std::vector<IntegerVariable>& vars,
    const std::vector<std::vector<int64_t>>& automaton, int64_t initial_state,
    const std::vector<int64_t>& final_states) {
  return [=](Model* model) {
    IntegerTrail* integer_trail = model->GetOrCreate<IntegerTrail>();
    const int n = vars.size();
    CHECK_GT(n, 0) << "No variables in TransitionConstraint().";
 
    // Test precondition.
    {
      std::set<std::pair<int64_t, int64_t>> unique_transition_checker;
      for (const std::vector<int64_t>& transition : automaton) {
        CHECK_EQ(transition.size(), 3);
        const std::pair<int64_t, int64_t> p{transition[0], transition[1]};
        CHECK(!gtl::ContainsKey(unique_transition_checker, p))
            << "Duplicate outgoing transitions with value " << transition[1]
            << " from state " << transition[0] << ".";
        unique_transition_checker.insert(p);
      }
    }
 
    // Construct a table with the possible values of each vars.
    std::vector<absl::flat_hash_set<int64_t>> possible_values(n);
    for (int time = 0; time < n; ++time) {
      const auto domain = integer_trail->InitialVariableDomain(vars[time]);
      for (const std::vector<int64_t>& transition : automaton) {
        // TODO(user): quadratic algo, improve!
        if (domain.Contains(transition[1])) {
          possible_values[time].insert(transition[1]);
        }
      }
    }
 
    // Compute the set of reachable state at each time point.
    std::vector<std::set<int64_t>> reachable_states(n + 1);
    reachable_states[0].insert(initial_state);
    reachable_states[n] = {final_states.begin(), final_states.end()};
 
    // Forward.
    //
    // TODO(user): filter using the domain of vars[time] that may not contain
    // all the possible transitions.
    for (int time = 0; time + 1 < n; ++time) {
      for (const std::vector<int64_t>& transition : automaton) {
        if (!gtl::ContainsKey(reachable_states[time], transition[0])) continue;
        if (!gtl::ContainsKey(possible_values[time], transition[1])) continue;
        reachable_states[time + 1].insert(transition[2]);
      }
    }
 
    // Backward.
    for (int time = n - 1; time > 0; --time) {
      std::set<int64_t> new_set;
      for (const std::vector<int64_t>& transition : automaton) {
        if (!gtl::ContainsKey(reachable_states[time], transition[0])) continue;
        if (!gtl::ContainsKey(possible_values[time], transition[1])) continue;
        if (!gtl::ContainsKey(reachable_states[time + 1], transition[2]))
          continue;
        new_set.insert(transition[0]);
      }
      reachable_states[time].swap(new_set);
    }
 
    // We will model at each time step the current automaton state using Boolean
    // variables. We will have n+1 time step. At time zero, we start in the
    // initial state, and at time n we should be in one of the final states. We
    // don't need to create Booleans at at time when there is just one possible
    // state (like at time zero).
    absl::flat_hash_map<IntegerValue, Literal> encoding;
    absl::flat_hash_map<IntegerValue, Literal> in_encoding;
    absl::flat_hash_map<IntegerValue, Literal> out_encoding;
    for (int time = 0; time < n; ++time) {
      // All these vector have the same size. We will use them to enforce a
      // local table constraint representing one step of the automaton at the
      // given time.
      std::vector<Literal> tuple_literals;
      std::vector<IntegerValue> in_states;
      std::vector<IntegerValue> transition_values;
      std::vector<IntegerValue> out_states;
      for (const std::vector<int64_t>& transition : automaton) {
        if (!gtl::ContainsKey(reachable_states[time], transition[0])) continue;
        if (!gtl::ContainsKey(possible_values[time], transition[1])) continue;
        if (!gtl::ContainsKey(reachable_states[time + 1], transition[2]))
          continue;
 
        // TODO(user): if this transition correspond to just one in-state or
        // one-out state or one variable value, we could reuse the corresponding
        // Boolean variable instead of creating a new one!
        tuple_literals.push_back(
            Literal(model->Add(NewBooleanVariable()), true));
        in_states.push_back(IntegerValue(transition[0]));
 
        transition_values.push_back(IntegerValue(transition[1]));
 
        // On the last step we don't need to distinguish the output states, so
        // we use zero.
        out_states.push_back(time + 1 == n ? IntegerValue(0)
                                           : IntegerValue(transition[2]));
      }
 
      // Fully instantiate vars[time].
      // Tricky: because we started adding constraints that can propagate, the
      // possible values returned by encoding might not contains all the value
      // computed in transition_values.
      {
        std::vector<IntegerValue> s = transition_values;
        gtl::STLSortAndRemoveDuplicates(&s);
 
        encoding.clear();
        if (s.size() > 1) {
          std::vector<int64_t> values;
          values.reserve(s.size());
          for (IntegerValue v : s) values.push_back(v.value());
          integer_trail->UpdateInitialDomain(vars[time],
                                             Domain::FromValues(values));
          model->Add(FullyEncodeVariable(vars[time]));
          encoding = GetEncoding(vars[time], model);
        } else {
          // Fix vars[time] to its unique possible value.
          CHECK_EQ(s.size(), 1);
          const int64_t unique_value = s.begin()->value();
          model->Add(LowerOrEqual(vars[time], unique_value));
          model->Add(GreaterOrEqual(vars[time], unique_value));
        }
      }
 
      // For each possible out states, create one Boolean variable.
      {
        std::vector<IntegerValue> s = out_states;
        gtl::STLSortAndRemoveDuplicates(&s);
 
        out_encoding.clear();
        if (s.size() == 2) {
          const BooleanVariable var = model->Add(NewBooleanVariable());
          out_encoding[s.front()] = Literal(var, true);
          out_encoding[s.back()] = Literal(var, false);
        } else if (s.size() > 1) {
          for (const IntegerValue state : s) {
            const Literal l = Literal(model->Add(NewBooleanVariable()), true);
            out_encoding[state] = l;
          }
        }
      }
 
      // Now we link everything together.
      //
      // Note that we do not need the ExactlyOneConstraint(tuple_literals)
      // because it is already implicitely encoded since we have exactly one
      // transition value.
      if (!in_encoding.empty()) {
        ProcessOneColumn(tuple_literals, in_states, in_encoding, {}, model);
      }
      if (!encoding.empty()) {
        ProcessOneColumn(tuple_literals, transition_values, encoding, {},
                         model);
      }
      if (!out_encoding.empty()) {
        ProcessOneColumn(tuple_literals, out_states, out_encoding, {}, model);
      }
      in_encoding = out_encoding;
    }
  };
}
 
}  // namespace sat
}  // namespace operations_research