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tools/openfst-1.6.7/include/fst/replace.h 54.8 KB
8dcb6dfcb   Yannick Estève   first commit
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  // See www.openfst.org for extensive documentation on this weighted
  // finite-state transducer library.
  //
  // Functions and classes for the recursive replacement of FSTs.
  
  #ifndef FST_REPLACE_H_
  #define FST_REPLACE_H_
  
  #include <set>
  #include <string>
  #include <unordered_map>
  #include <utility>
  #include <vector>
  
  #include <fst/log.h>
  
  #include <fst/cache.h>
  #include <fst/expanded-fst.h>
  #include <fst/fst-decl.h>  // For optional argument declarations.
  #include <fst/fst.h>
  #include <fst/matcher.h>
  #include <fst/replace-util.h>
  #include <fst/state-table.h>
  #include <fst/test-properties.h>
  
  namespace fst {
  
  // Replace state tables have the form:
  //
  // template <class Arc, class P>
  // class ReplaceStateTable {
  //  public:
  //   using Label = typename Arc::Label Label;
  //   using StateId = typename Arc::StateId;
  //
  //   using PrefixId = P;
  //   using StateTuple = ReplaceStateTuple<StateId, PrefixId>;
  //   using StackPrefix = ReplaceStackPrefix<Label, StateId>;
  //
  //   // Required constructor.
  //   ReplaceStateTable(
  //       const std::vector<std::pair<Label, const Fst<Arc> *>> &fst_list,
  //       Label root);
  //
  //   // Required copy constructor that does not copy state.
  //   ReplaceStateTable(const ReplaceStateTable<Arc, PrefixId> &table);
  //
  //   // Looks up state ID by tuple, adding it if it doesn't exist.
  //   StateId FindState(const StateTuple &tuple);
  //
  //   // Looks up state tuple by ID.
  //   const StateTuple &Tuple(StateId id) const;
  //
  //   // Lookus up prefix ID by stack prefix, adding it if it doesn't exist.
  //   PrefixId FindPrefixId(const StackPrefix &stack_prefix);
  //
  //  // Looks up stack prefix by ID.
  //  const StackPrefix &GetStackPrefix(PrefixId id) const;
  // };
  
  // Tuple that uniquely defines a state in replace.
  template <class S, class P>
  struct ReplaceStateTuple {
    using StateId = S;
    using PrefixId = P;
  
    ReplaceStateTuple(PrefixId prefix_id = -1, StateId fst_id = kNoStateId,
                      StateId fst_state = kNoStateId)
        : prefix_id(prefix_id), fst_id(fst_id), fst_state(fst_state) {}
  
    PrefixId prefix_id;  // Index in prefix table.
    StateId fst_id;      // Current FST being walked.
    StateId fst_state;   // Current state in FST being walked (not to be
                         // confused with the thse StateId of the combined FST).
  };
  
  // Equality of replace state tuples.
  template <class StateId, class PrefixId>
  inline bool operator==(const ReplaceStateTuple<StateId, PrefixId> &x,
                         const ReplaceStateTuple<StateId, PrefixId> &y) {
    return x.prefix_id == y.prefix_id && x.fst_id == y.fst_id &&
           x.fst_state == y.fst_state;
  }
  
  // Functor returning true for tuples corresponding to states in the root FST.
  template <class StateId, class PrefixId>
  class ReplaceRootSelector {
   public:
    bool operator()(const ReplaceStateTuple<StateId, PrefixId> &tuple) const {
      return tuple.prefix_id == 0;
    }
  };
  
  // Functor for fingerprinting replace state tuples.
  template <class StateId, class PrefixId>
  class ReplaceFingerprint {
   public:
    explicit ReplaceFingerprint(const std::vector<uint64> *size_array)
        : size_array_(size_array) {}
  
    uint64 operator()(const ReplaceStateTuple<StateId, PrefixId> &tuple) const {
      return tuple.prefix_id * size_array_->back() +
             size_array_->at(tuple.fst_id - 1) + tuple.fst_state;
    }
  
   private:
    const std::vector<uint64> *size_array_;
  };
  
  // Useful when the fst_state uniquely define the tuple.
  template <class StateId, class PrefixId>
  class ReplaceFstStateFingerprint {
   public:
    uint64 operator()(const ReplaceStateTuple<StateId, PrefixId> &tuple) const {
      return tuple.fst_state;
    }
  };
  
  // A generic hash function for replace state tuples.
  template <typename S, typename P>
  class ReplaceHash {
   public:
    size_t operator()(const ReplaceStateTuple<S, P>& t) const {
      static constexpr size_t prime0 = 7853;
      static constexpr size_t prime1 = 7867;
      return t.prefix_id + t.fst_id * prime0 + t.fst_state * prime1;
    }
  };
  
  // Container for stack prefix.
  template <class Label, class StateId>
  class ReplaceStackPrefix {
   public:
    struct PrefixTuple {
      PrefixTuple(Label fst_id = kNoLabel, StateId nextstate = kNoStateId)
          : fst_id(fst_id), nextstate(nextstate) {}
  
      Label fst_id;
      StateId nextstate;
    };
  
    ReplaceStackPrefix() {}
  
    ReplaceStackPrefix(const ReplaceStackPrefix &other)
        : prefix_(other.prefix_) {}
  
    void Push(StateId fst_id, StateId nextstate) {
      prefix_.push_back(PrefixTuple(fst_id, nextstate));
    }
  
    void Pop() { prefix_.pop_back(); }
  
    const PrefixTuple &Top() const { return prefix_[prefix_.size() - 1]; }
  
    size_t Depth() const { return prefix_.size(); }
  
   public:
    std::vector<PrefixTuple> prefix_;
  };
  
  // Equality stack prefix classes.
  template <class Label, class StateId>
  inline bool operator==(const ReplaceStackPrefix<Label, StateId> &x,
                         const ReplaceStackPrefix<Label, StateId> &y) {
    if (x.prefix_.size() != y.prefix_.size()) return false;
    for (size_t i = 0; i < x.prefix_.size(); ++i) {
      if (x.prefix_[i].fst_id != y.prefix_[i].fst_id ||
          x.prefix_[i].nextstate != y.prefix_[i].nextstate) {
        return false;
      }
    }
    return true;
  }
  
  // Hash function for stack prefix to prefix id.
  template <class Label, class StateId>
  class ReplaceStackPrefixHash {
   public:
    size_t operator()(const ReplaceStackPrefix<Label, StateId> &prefix) const {
      size_t sum = 0;
      for (const auto &pair : prefix.prefix_) {
        static constexpr size_t prime = 7863;
        sum += pair.fst_id + pair.nextstate * prime;
      }
      return sum;
    }
  };
  
  // Replace state tables.
  
  // A two-level state table for replace. Warning: calls CountStates to compute
  // the number of states of each component FST.
  template <class Arc, class P = ssize_t>
  class VectorHashReplaceStateTable {
   public:
    using Label = typename Arc::Label;
    using StateId = typename Arc::StateId;
  
    using PrefixId = P;
  
    using StateTuple = ReplaceStateTuple<StateId, PrefixId>;
    using StateTable =
        VectorHashStateTable<ReplaceStateTuple<StateId, PrefixId>,
                             ReplaceRootSelector<StateId, PrefixId>,
                             ReplaceFstStateFingerprint<StateId, PrefixId>,
                             ReplaceFingerprint<StateId, PrefixId>>;
    using StackPrefix = ReplaceStackPrefix<Label, StateId>;
    using StackPrefixTable =
        CompactHashBiTable<PrefixId, StackPrefix,
                           ReplaceStackPrefixHash<Label, StateId>>;
  
    VectorHashReplaceStateTable(
        const std::vector<std::pair<Label, const Fst<Arc> *>> &fst_list,
        Label root)
        : root_size_(0) {
      size_array_.push_back(0);
      for (const auto &fst_pair : fst_list) {
        if (fst_pair.first == root) {
          root_size_ = CountStates(*(fst_pair.second));
          size_array_.push_back(size_array_.back());
        } else {
          size_array_.push_back(size_array_.back() +
                                CountStates(*(fst_pair.second)));
        }
      }
      state_table_.reset(
          new StateTable(new ReplaceRootSelector<StateId, PrefixId>,
                         new ReplaceFstStateFingerprint<StateId, PrefixId>,
                         new ReplaceFingerprint<StateId, PrefixId>(&size_array_),
                         root_size_, root_size_ + size_array_.back()));
    }
  
    VectorHashReplaceStateTable(
        const VectorHashReplaceStateTable<Arc, PrefixId> &table)
        : root_size_(table.root_size_),
          size_array_(table.size_array_),
          prefix_table_(table.prefix_table_) {
      state_table_.reset(
          new StateTable(new ReplaceRootSelector<StateId, PrefixId>,
                         new ReplaceFstStateFingerprint<StateId, PrefixId>,
                         new ReplaceFingerprint<StateId, PrefixId>(&size_array_),
                         root_size_, root_size_ + size_array_.back()));
    }
  
    StateId FindState(const StateTuple &tuple) {
      return state_table_->FindState(tuple);
    }
  
    const StateTuple &Tuple(StateId id) const { return state_table_->Tuple(id); }
  
    PrefixId FindPrefixId(const StackPrefix &prefix) {
      return prefix_table_.FindId(prefix);
    }
  
    const StackPrefix& GetStackPrefix(PrefixId id) const {
      return prefix_table_.FindEntry(id);
    }
  
   private:
    StateId root_size_;
    std::vector<uint64> size_array_;
    std::unique_ptr<StateTable> state_table_;
    StackPrefixTable prefix_table_;
  };
  
  // Default replace state table.
  template <class Arc, class P /* = size_t */>
  class DefaultReplaceStateTable
      : public CompactHashStateTable<ReplaceStateTuple<typename Arc::StateId, P>,
                                     ReplaceHash<typename Arc::StateId, P>> {
   public:
    using Label = typename Arc::Label;
    using StateId = typename Arc::StateId;
  
    using PrefixId = P;
    using StateTuple = ReplaceStateTuple<StateId, PrefixId>;
    using StateTable =
        CompactHashStateTable<StateTuple, ReplaceHash<StateId, PrefixId>>;
    using StackPrefix = ReplaceStackPrefix<Label, StateId>;
    using StackPrefixTable =
        CompactHashBiTable<PrefixId, StackPrefix,
                           ReplaceStackPrefixHash<Label, StateId>>;
  
    using StateTable::FindState;
    using StateTable::Tuple;
  
    DefaultReplaceStateTable(
        const std::vector<std::pair<Label, const Fst<Arc> *>> &, Label) {}
  
    DefaultReplaceStateTable(const DefaultReplaceStateTable<Arc, PrefixId> &table)
        : StateTable(), prefix_table_(table.prefix_table_) {}
  
    PrefixId FindPrefixId(const StackPrefix &prefix) {
      return prefix_table_.FindId(prefix);
    }
  
    const StackPrefix &GetStackPrefix(PrefixId id) const {
      return prefix_table_.FindEntry(id);
    }
  
   private:
    StackPrefixTable prefix_table_;
  };
  
  // By default ReplaceFst will copy the input label of the replace arc.
  // The call_label_type and return_label_type options specify how to manage
  // the labels of the call arc and the return arc of the replace FST
  template <class Arc, class StateTable = DefaultReplaceStateTable<Arc>,
            class CacheStore = DefaultCacheStore<Arc>>
  struct ReplaceFstOptions : CacheImplOptions<CacheStore> {
    using Label = typename Arc::Label;
  
    // Index of root rule for expansion.
    Label root;
    // How to label call arc.
    ReplaceLabelType call_label_type = REPLACE_LABEL_INPUT;
    // How to label return arc.
    ReplaceLabelType return_label_type = REPLACE_LABEL_NEITHER;
    // Specifies output label to put on call arc; if kNoLabel, use existing label
    // on call arc. Otherwise, use this field as the output label.
    Label call_output_label = kNoLabel;
    // Specifies label to put on return arc.
    Label return_label = 0;
    // Take ownership of input FSTs?
    bool take_ownership = false;
    // Pointer to optional pre-constructed state table.
    StateTable *state_table = nullptr;
  
    explicit ReplaceFstOptions(const CacheImplOptions<CacheStore> &opts,
                               Label root = kNoLabel)
        : CacheImplOptions<CacheStore>(opts), root(root) {}
  
    explicit ReplaceFstOptions(const CacheOptions &opts, Label root = kNoLabel)
        : CacheImplOptions<CacheStore>(opts), root(root) {}
  
    // FIXME(kbg): There are too many constructors here. Come up with a consistent
    // position for call_output_label (probably the very end) so that it is
    // possible to express all the remaining constructors with a single
    // default-argument constructor. Also move clients off of the "backwards
    // compatibility" constructor, for good.
  
    explicit ReplaceFstOptions(Label root) : root(root) {}
  
    explicit ReplaceFstOptions(Label root, ReplaceLabelType call_label_type,
                               ReplaceLabelType return_label_type,
                               Label return_label)
        : root(root),
          call_label_type(call_label_type),
          return_label_type(return_label_type),
          return_label(return_label) {}
  
    explicit ReplaceFstOptions(Label root, ReplaceLabelType call_label_type,
                               ReplaceLabelType return_label_type,
                               Label call_output_label, Label return_label)
        : root(root),
          call_label_type(call_label_type),
          return_label_type(return_label_type),
          call_output_label(call_output_label),
          return_label(return_label) {}
  
    explicit ReplaceFstOptions(const ReplaceUtilOptions &opts)
        : ReplaceFstOptions(opts.root, opts.call_label_type,
                            opts.return_label_type, opts.return_label) {}
  
    ReplaceFstOptions() : root(kNoLabel) {}
  
    // For backwards compatibility.
    ReplaceFstOptions(int64 root, bool epsilon_replace_arc)
        : root(root),
          call_label_type(epsilon_replace_arc ? REPLACE_LABEL_NEITHER
                                              : REPLACE_LABEL_INPUT),
          call_output_label(epsilon_replace_arc ? 0 : kNoLabel) {}
  };
  
  
  // Forward declaration.
  template <class Arc, class StateTable, class CacheStore>
  class ReplaceFstMatcher;
  
  template <class Arc>
  using FstList = std::vector<std::pair<typename Arc::Label, const Fst<Arc> *>>;
  
  // Returns true if label type on arc results in epsilon input label.
  inline bool EpsilonOnInput(ReplaceLabelType label_type) {
    return label_type == REPLACE_LABEL_NEITHER ||
           label_type == REPLACE_LABEL_OUTPUT;
  }
  
  // Returns true if label type on arc results in epsilon input label.
  inline bool EpsilonOnOutput(ReplaceLabelType label_type) {
    return label_type == REPLACE_LABEL_NEITHER ||
           label_type == REPLACE_LABEL_INPUT;
  }
  
  // Returns true if for either the call or return arc ilabel != olabel.
  template <class Label>
  bool ReplaceTransducer(ReplaceLabelType call_label_type,
                         ReplaceLabelType return_label_type,
                         Label call_output_label) {
    return call_label_type == REPLACE_LABEL_INPUT ||
           call_label_type == REPLACE_LABEL_OUTPUT ||
           (call_label_type == REPLACE_LABEL_BOTH &&
            call_output_label != kNoLabel) ||
           return_label_type == REPLACE_LABEL_INPUT ||
           return_label_type == REPLACE_LABEL_OUTPUT;
  }
  
  template <class Arc>
  uint64 ReplaceFstProperties(typename Arc::Label root_label,
                              const FstList<Arc> &fst_list,
                              ReplaceLabelType call_label_type,
                              ReplaceLabelType return_label_type,
                              typename Arc::Label call_output_label,
                              bool *sorted_and_non_empty) {
    using Label = typename Arc::Label;
    std::vector<uint64> inprops;
    bool all_ilabel_sorted = true;
    bool all_olabel_sorted = true;
    bool all_non_empty = true;
    // All nonterminals are negative?
    bool all_negative = true;
    // All nonterminals are positive and form a dense range containing 1?
    bool dense_range = true;
    Label root_fst_idx = 0;
    for (Label i = 0; i < fst_list.size(); ++i) {
      const auto label = fst_list[i].first;
      if (label >= 0) all_negative = false;
      if (label > fst_list.size() || label <= 0) dense_range = false;
      if (label == root_label) root_fst_idx = i;
      const auto *fst = fst_list[i].second;
      if (fst->Start() == kNoStateId) all_non_empty = false;
      if (!fst->Properties(kILabelSorted, false)) all_ilabel_sorted = false;
      if (!fst->Properties(kOLabelSorted, false)) all_olabel_sorted = false;
      inprops.push_back(fst->Properties(kCopyProperties, false));
    }
    const auto props = ReplaceProperties(
        inprops, root_fst_idx, EpsilonOnInput(call_label_type),
        EpsilonOnInput(return_label_type), EpsilonOnOutput(call_label_type),
        EpsilonOnOutput(return_label_type),
        ReplaceTransducer(call_label_type, return_label_type, call_output_label),
        all_non_empty, all_ilabel_sorted, all_olabel_sorted,
        all_negative || dense_range);
    const bool sorted = props & (kILabelSorted | kOLabelSorted);
    *sorted_and_non_empty = all_non_empty && sorted;
    return props;
  }
  
  namespace internal {
  
  // The replace implementation class supports a dynamic expansion of a recursive
  // transition network represented as label/FST pairs with dynamic replacable
  // arcs.
  template <class Arc, class StateTable, class CacheStore>
  class ReplaceFstImpl
      : public CacheBaseImpl<typename CacheStore::State, CacheStore> {
   public:
    using Label = typename Arc::Label;
    using StateId = typename Arc::StateId;
    using Weight = typename Arc::Weight;
  
    using State = typename CacheStore::State;
    using CacheImpl = CacheBaseImpl<State, CacheStore>;
    using PrefixId = typename StateTable::PrefixId;
    using StateTuple = ReplaceStateTuple<StateId, PrefixId>;
    using StackPrefix = ReplaceStackPrefix<Label, StateId>;
    using NonTerminalHash = std::unordered_map<Label, Label>;
  
    using FstImpl<Arc>::SetType;
    using FstImpl<Arc>::SetProperties;
    using FstImpl<Arc>::WriteHeader;
    using FstImpl<Arc>::SetInputSymbols;
    using FstImpl<Arc>::SetOutputSymbols;
    using FstImpl<Arc>::InputSymbols;
    using FstImpl<Arc>::OutputSymbols;
  
    using CacheImpl::PushArc;
    using CacheImpl::HasArcs;
    using CacheImpl::HasFinal;
    using CacheImpl::HasStart;
    using CacheImpl::SetArcs;
    using CacheImpl::SetFinal;
    using CacheImpl::SetStart;
  
    friend class ReplaceFstMatcher<Arc, StateTable, CacheStore>;
  
    ReplaceFstImpl(const FstList<Arc> &fst_list,
                   const ReplaceFstOptions<Arc, StateTable, CacheStore> &opts)
        : CacheImpl(opts),
          call_label_type_(opts.call_label_type),
          return_label_type_(opts.return_label_type),
          call_output_label_(opts.call_output_label),
          return_label_(opts.return_label),
          state_table_(opts.state_table ? opts.state_table
                                        : new StateTable(fst_list, opts.root)) {
      SetType("replace");
      // If the label is epsilon, then all replace label options are equivalent,
      // so we set the label types to NEITHER for simplicity.
      if (call_output_label_ == 0) call_label_type_ = REPLACE_LABEL_NEITHER;
      if (return_label_ == 0) return_label_type_ = REPLACE_LABEL_NEITHER;
      if (!fst_list.empty()) {
        SetInputSymbols(fst_list[0].second->InputSymbols());
        SetOutputSymbols(fst_list[0].second->OutputSymbols());
      }
      fst_array_.push_back(nullptr);
      for (Label i = 0; i < fst_list.size(); ++i) {
        const auto label = fst_list[i].first;
        const auto *fst = fst_list[i].second;
        nonterminal_hash_[label] = fst_array_.size();
        nonterminal_set_.insert(label);
        fst_array_.emplace_back(opts.take_ownership ? fst : fst->Copy());
        if (i) {
          if (!CompatSymbols(InputSymbols(), fst->InputSymbols())) {
            FSTERROR() << "ReplaceFstImpl: Input symbols of FST " << i
                       << " do not match input symbols of base FST (0th FST)";
            SetProperties(kError, kError);
          }
          if (!CompatSymbols(OutputSymbols(), fst->OutputSymbols())) {
            FSTERROR() << "ReplaceFstImpl: Output symbols of FST " << i
                       << " do not match output symbols of base FST (0th FST)";
            SetProperties(kError, kError);
          }
        }
      }
      const auto nonterminal = nonterminal_hash_[opts.root];
      if ((nonterminal == 0) && (fst_array_.size() > 1)) {
        FSTERROR() << "ReplaceFstImpl: No FST corresponding to root label "
                   << opts.root << " in the input tuple vector";
        SetProperties(kError, kError);
      }
      root_ = (nonterminal > 0) ? nonterminal : 1;
      bool all_non_empty_and_sorted = false;
      SetProperties(ReplaceFstProperties(opts.root, fst_list, call_label_type_,
                                         return_label_type_, call_output_label_,
                                         &all_non_empty_and_sorted));
      // Enables optional caching as long as sorted and all non-empty.
      always_cache_ = !all_non_empty_and_sorted;
      VLOG(2) << "ReplaceFstImpl::ReplaceFstImpl: always_cache = "
              << (always_cache_ ? "true" : "false");
    }
  
    ReplaceFstImpl(const ReplaceFstImpl &impl)
        : CacheImpl(impl),
          call_label_type_(impl.call_label_type_),
          return_label_type_(impl.return_label_type_),
          call_output_label_(impl.call_output_label_),
          return_label_(impl.return_label_),
          always_cache_(impl.always_cache_),
          state_table_(new StateTable(*(impl.state_table_))),
          nonterminal_set_(impl.nonterminal_set_),
          nonterminal_hash_(impl.nonterminal_hash_),
          root_(impl.root_) {
      SetType("replace");
      SetProperties(impl.Properties(), kCopyProperties);
      SetInputSymbols(impl.InputSymbols());
      SetOutputSymbols(impl.OutputSymbols());
      fst_array_.reserve(impl.fst_array_.size());
      fst_array_.emplace_back(nullptr);
      for (Label i = 1; i < impl.fst_array_.size(); ++i) {
        fst_array_.emplace_back(impl.fst_array_[i]->Copy(true));
      }
    }
  
    // Computes the dependency graph of the replace class and returns
    // true if the dependencies are cyclic. Cyclic dependencies will result
    // in an un-expandable FST.
    bool CyclicDependencies() const {
      const ReplaceUtilOptions opts(root_);
      ReplaceUtil<Arc> replace_util(fst_array_, nonterminal_hash_, opts);
      return replace_util.CyclicDependencies();
    }
  
    StateId Start() {
      if (!HasStart()) {
        if (fst_array_.size() == 1) {
          SetStart(kNoStateId);
          return kNoStateId;
        } else {
          const auto fst_start = fst_array_[root_]->Start();
          if (fst_start == kNoStateId) return kNoStateId;
          const auto prefix = GetPrefixId(StackPrefix());
          const auto start =
              state_table_->FindState(StateTuple(prefix, root_, fst_start));
          SetStart(start);
          return start;
        }
      } else {
        return CacheImpl::Start();
      }
    }
  
    Weight Final(StateId s) {
      if (HasFinal(s)) return CacheImpl::Final(s);
      const auto &tuple = state_table_->Tuple(s);
      auto weight = Weight::Zero();
      if (tuple.prefix_id == 0) {
        const auto fst_state = tuple.fst_state;
        weight = fst_array_[tuple.fst_id]->Final(fst_state);
      }
      if (always_cache_ || HasArcs(s)) SetFinal(s, weight);
      return weight;
    }
  
    size_t NumArcs(StateId s) {
      if (HasArcs(s)) {
        return CacheImpl::NumArcs(s);
      } else if (always_cache_) {  // If always caching, expands and caches state.
        Expand(s);
        return CacheImpl::NumArcs(s);
      } else {  // Otherwise computes the number of arcs without expanding.
        const auto tuple = state_table_->Tuple(s);
        if (tuple.fst_state == kNoStateId) return 0;
        auto num_arcs = fst_array_[tuple.fst_id]->NumArcs(tuple.fst_state);
        if (ComputeFinalArc(tuple, nullptr)) ++num_arcs;
        return num_arcs;
      }
    }
  
    // Returns whether a given label is a non-terminal.
    bool IsNonTerminal(Label label) const {
      if (label < *nonterminal_set_.begin() ||
          label > *nonterminal_set_.rbegin()) {
        return false;
      } else {
        return nonterminal_hash_.count(label);
      }
      // TODO(allauzen): be smarter and take advantage of all_dense or
      // all_negative. Also use this in ComputeArc. This would require changes to
      // Replace so that recursing into an empty FST lead to a non co-accessible
      // state instead of deleting the arc as done currently. The current use
      // correct, since labels are sorted if all_non_empty is true.
    }
  
    size_t NumInputEpsilons(StateId s) {
      if (HasArcs(s)) {
        return CacheImpl::NumInputEpsilons(s);
      } else if (always_cache_ || !Properties(kILabelSorted)) {
        // If always caching or if the number of input epsilons is too expensive
        // to compute without caching (i.e., not ilabel-sorted), then expands and
        // caches state.
        Expand(s);
        return CacheImpl::NumInputEpsilons(s);
      } else {
        // Otherwise, computes the number of input epsilons without caching.
        const auto tuple = state_table_->Tuple(s);
        if (tuple.fst_state == kNoStateId) return 0;
        size_t num = 0;
        if (!EpsilonOnInput(call_label_type_)) {
          // If EpsilonOnInput(c) is false, all input epsilon arcs
          // are also input epsilons arcs in the underlying machine.
          num = fst_array_[tuple.fst_id]->NumInputEpsilons(tuple.fst_state);
        } else {
          // Otherwise, one need to consider that all non-terminal arcs
          // in the underlying machine also become input epsilon arc.
          ArcIterator<Fst<Arc>> aiter(*fst_array_[tuple.fst_id], tuple.fst_state);
          for (; !aiter.Done() && ((aiter.Value().ilabel == 0) ||
                                   IsNonTerminal(aiter.Value().olabel));
               aiter.Next()) {
            ++num;
          }
        }
        if (EpsilonOnInput(return_label_type_) &&
            ComputeFinalArc(tuple, nullptr)) {
          ++num;
        }
        return num;
      }
    }
  
    size_t NumOutputEpsilons(StateId s) {
      if (HasArcs(s)) {
        return CacheImpl::NumOutputEpsilons(s);
      } else if (always_cache_ || !Properties(kOLabelSorted)) {
        // If always caching or if the number of output epsilons is too expensive
        // to compute without caching (i.e., not olabel-sorted), then expands and
        // caches state.
        Expand(s);
        return CacheImpl::NumOutputEpsilons(s);
      } else {
        // Otherwise, computes the number of output epsilons without caching.
        const auto tuple = state_table_->Tuple(s);
        if (tuple.fst_state == kNoStateId) return 0;
        size_t num = 0;
        if (!EpsilonOnOutput(call_label_type_)) {
          // If EpsilonOnOutput(c) is false, all output epsilon arcs are also
          // output epsilons arcs in the underlying machine.
          num = fst_array_[tuple.fst_id]->NumOutputEpsilons(tuple.fst_state);
        } else {
          // Otherwise, one need to consider that all non-terminal arcs in the
          // underlying machine also become output epsilon arc.
          ArcIterator<Fst<Arc>> aiter(*fst_array_[tuple.fst_id], tuple.fst_state);
          for (; !aiter.Done() && ((aiter.Value().olabel == 0) ||
                                   IsNonTerminal(aiter.Value().olabel));
               aiter.Next()) {
            ++num;
          }
        }
        if (EpsilonOnOutput(return_label_type_) &&
            ComputeFinalArc(tuple, nullptr)) {
          ++num;
        }
        return num;
      }
    }
  
    uint64 Properties() const override { return Properties(kFstProperties); }
  
    // Sets error if found, and returns other FST impl properties.
    uint64 Properties(uint64 mask) const override {
      if (mask & kError) {
        for (Label i = 1; i < fst_array_.size(); ++i) {
          if (fst_array_[i]->Properties(kError, false)) {
            SetProperties(kError, kError);
          }
        }
      }
      return FstImpl<Arc>::Properties(mask);
    }
  
    // Returns the base arc iterator, and if arcs have not been computed yet,
    // extends and recurses for new arcs.
    void InitArcIterator(StateId s, ArcIteratorData<Arc> *data) {
      if (!HasArcs(s)) Expand(s);
      CacheImpl::InitArcIterator(s, data);
      // TODO(allauzen): Set behaviour of generic iterator.
      // Warning: ArcIterator<ReplaceFst<A>>::InitCache() relies on current
      // behaviour.
    }
  
    // Extends current state (walk arcs one level deep).
    void Expand(StateId s) {
      const auto tuple = state_table_->Tuple(s);
      if (tuple.fst_state == kNoStateId) {  // Local FST is empty.
        SetArcs(s);
        return;
      }
      ArcIterator<Fst<Arc>> aiter(*fst_array_[tuple.fst_id], tuple.fst_state);
      Arc arc;
      // Creates a final arc when needed.
      if (ComputeFinalArc(tuple, &arc)) PushArc(s, arc);
      // Expands all arcs leaving the state.
      for (; !aiter.Done(); aiter.Next()) {
        if (ComputeArc(tuple, aiter.Value(), &arc)) PushArc(s, arc);
      }
      SetArcs(s);
    }
  
    void Expand(StateId s, const StateTuple &tuple,
                const ArcIteratorData<Arc> &data) {
      if (tuple.fst_state == kNoStateId) {  // Local FST is empty.
        SetArcs(s);
        return;
      }
      ArcIterator<Fst<Arc>> aiter(data);
      Arc arc;
      // Creates a final arc when needed.
      if (ComputeFinalArc(tuple, &arc)) AddArc(s, arc);
      // Expands all arcs leaving the state.
      for (; !aiter.Done(); aiter.Next()) {
        if (ComputeArc(tuple, aiter.Value(), &arc)) AddArc(s, arc);
      }
      SetArcs(s);
    }
  
    // If acpp is null, only returns true if a final arcp is required, but does
    // not actually compute it.
    bool ComputeFinalArc(const StateTuple &tuple, Arc *arcp,
                         uint32 flags = kArcValueFlags) {
      const auto fst_state = tuple.fst_state;
      if (fst_state == kNoStateId) return false;
      // If state is final, pops the stack.
      if (fst_array_[tuple.fst_id]->Final(fst_state) != Weight::Zero() &&
          tuple.prefix_id) {
        if (arcp) {
          arcp->ilabel = (EpsilonOnInput(return_label_type_)) ? 0 : return_label_;
          arcp->olabel =
              (EpsilonOnOutput(return_label_type_)) ? 0 : return_label_;
          if (flags & kArcNextStateValue) {
            const auto &stack = state_table_->GetStackPrefix(tuple.prefix_id);
            const auto prefix_id = PopPrefix(stack);
            const auto &top = stack.Top();
            arcp->nextstate = state_table_->FindState(
                StateTuple(prefix_id, top.fst_id, top.nextstate));
          }
          if (flags & kArcWeightValue) {
            arcp->weight = fst_array_[tuple.fst_id]->Final(fst_state);
          }
        }
        return true;
      } else {
        return false;
      }
    }
  
    // Computes an arc in the FST corresponding to one in the underlying machine.
    // Returns false if the underlying arc corresponds to no arc in the resulting
    // FST.
    bool ComputeArc(const StateTuple &tuple, const Arc &arc, Arc *arcp,
                    uint32 flags = kArcValueFlags) {
      if (!EpsilonOnInput(call_label_type_) &&
          (flags == (flags & (kArcILabelValue | kArcWeightValue)))) {
        *arcp = arc;
        return true;
      }
      if (arc.olabel == 0 || arc.olabel < *nonterminal_set_.begin() ||
          arc.olabel > *nonterminal_set_.rbegin()) {  // Expands local FST.
        const auto nextstate =
            flags & kArcNextStateValue
                ? state_table_->FindState(
                      StateTuple(tuple.prefix_id, tuple.fst_id, arc.nextstate))
                : kNoStateId;
        *arcp = Arc(arc.ilabel, arc.olabel, arc.weight, nextstate);
      } else {
        // Checks for non-terminal.
        const auto it = nonterminal_hash_.find(arc.olabel);
        if (it != nonterminal_hash_.end()) {  // Recurses into non-terminal.
          const auto nonterminal = it->second;
          const auto nt_prefix =
              PushPrefix(state_table_->GetStackPrefix(tuple.prefix_id),
                         tuple.fst_id, arc.nextstate);
          // If the start state is valid, replace; othewise, the arc is implicitly
          // deleted.
          const auto nt_start = fst_array_[nonterminal]->Start();
          if (nt_start != kNoStateId) {
            const auto nt_nextstate = flags & kArcNextStateValue
                                          ? state_table_->FindState(StateTuple(
                                                nt_prefix, nonterminal, nt_start))
                                          : kNoStateId;
            const auto ilabel =
                (EpsilonOnInput(call_label_type_)) ? 0 : arc.ilabel;
            const auto olabel =
                (EpsilonOnOutput(call_label_type_))
                    ? 0
                    : ((call_output_label_ == kNoLabel) ? arc.olabel
                                                        : call_output_label_);
            *arcp = Arc(ilabel, olabel, arc.weight, nt_nextstate);
          } else {
            return false;
          }
        } else {
          const auto nextstate =
              flags & kArcNextStateValue
                  ? state_table_->FindState(
                        StateTuple(tuple.prefix_id, tuple.fst_id, arc.nextstate))
                  : kNoStateId;
          *arcp = Arc(arc.ilabel, arc.olabel, arc.weight, nextstate);
        }
      }
      return true;
    }
  
    // Returns the arc iterator flags supported by this FST.
    uint32 ArcIteratorFlags() const {
      uint32 flags = kArcValueFlags;
      if (!always_cache_) flags |= kArcNoCache;
      return flags;
    }
  
    StateTable *GetStateTable() const { return state_table_.get(); }
  
    const Fst<Arc> *GetFst(Label fst_id) const {
      return fst_array_[fst_id].get();
    }
  
    Label GetFstId(Label nonterminal) const {
      const auto it = nonterminal_hash_.find(nonterminal);
      if (it == nonterminal_hash_.end()) {
        FSTERROR() << "ReplaceFstImpl::GetFstId: Nonterminal not found: "
                   << nonterminal;
      }
      return it->second;
    }
  
    // Returns true if label type on call arc results in epsilon input label.
    bool EpsilonOnCallInput() { return EpsilonOnInput(call_label_type_); }
  
   private:
    // The unique index into stack prefix table.
    PrefixId GetPrefixId(const StackPrefix &prefix) {
      return state_table_->FindPrefixId(prefix);
    }
  
    // The prefix ID after a stack pop.
    PrefixId PopPrefix(StackPrefix prefix) {
      prefix.Pop();
      return GetPrefixId(prefix);
    }
  
    // The prefix ID after a stack push.
    PrefixId PushPrefix(StackPrefix prefix, Label fst_id, StateId nextstate) {
      prefix.Push(fst_id, nextstate);
      return GetPrefixId(prefix);
    }
  
    // Runtime options
    ReplaceLabelType call_label_type_;    // How to label call arc.
    ReplaceLabelType return_label_type_;  // How to label return arc.
    int64 call_output_label_;  // Specifies output label to put on call arc
    int64 return_label_;       // Specifies label to put on return arc.
    bool always_cache_;        // Disable optional caching of arc iterator?
  
    // State table.
    std::unique_ptr<StateTable> state_table_;
  
    // Replace components.
    std::set<Label> nonterminal_set_;
    NonTerminalHash nonterminal_hash_;
    std::vector<std::unique_ptr<const Fst<Arc>>> fst_array_;
    Label root_;
  };
  
  }  // namespace internal
  
  //
  // ReplaceFst supports dynamic replacement of arcs in one FST with another FST.
  // This replacement is recursive. ReplaceFst can be used to support a variety of
  // delayed constructions such as recursive
  // transition networks, union, or closure. It is constructed with an array of
  // FST(s). One FST represents the root (or topology) machine. The root FST
  // refers to other FSTs by recursively replacing arcs labeled as non-terminals
  // with the matching non-terminal FST. Currently the ReplaceFst uses the output
  // symbols of the arcs to determine whether the arc is a non-terminal arc or
  // not. A non-terminal can be any label that is not a non-zero terminal label in
  // the output alphabet.
  //
  // Note that the constructor uses a vector of pairs. These correspond to the
  // tuple of non-terminal Label and corresponding FST. For example to implement
  // the closure operation we need 2 FSTs. The first root FST is a single
  // self-loop arc on the start state.
  //
  // The ReplaceFst class supports an optionally caching arc iterator.
  //
  // The ReplaceFst needs to be built such that it is known to be ilabel- or
  // olabel-sorted (see usage below).
  //
  // Observe that Matcher<Fst<A>> will use the optionally caching arc iterator
  // when available (the FST is ilabel-sorted and matching on the input, or the
  // FST is olabel -orted and matching on the output).  In order to obtain the
  // most efficient behaviour, it is recommended to set call_label_type to
  // REPLACE_LABEL_INPUT or REPLACE_LABEL_BOTH and return_label_type to
  // REPLACE_LABEL_OUTPUT or REPLACE_LABEL_NEITHER. This means that the call arc
  // does not have epsilon on the input side and the return arc has epsilon on the
  // input side) and matching on the input side.
  //
  // This class attaches interface to implementation and handles reference
  // counting, delegating most methods to ImplToFst.
  template <class A, class T /* = DefaultReplaceStateTable<A> */,
            class CacheStore /* = DefaultCacheStore<A> */>
  class ReplaceFst
      : public ImplToFst<internal::ReplaceFstImpl<A, T, CacheStore>> {
   public:
    using Arc = A;
    using Label = typename Arc::Label;
    using StateId = typename Arc::StateId;
    using Weight = typename Arc::Weight;
  
    using StateTable = T;
    using Store = CacheStore;
    using State = typename CacheStore::State;
    using Impl = internal::ReplaceFstImpl<Arc, StateTable, CacheStore>;
    using CacheImpl = internal::CacheBaseImpl<State, CacheStore>;
  
    using ImplToFst<Impl>::Properties;
  
    friend class ArcIterator<ReplaceFst<Arc, StateTable, CacheStore>>;
    friend class StateIterator<ReplaceFst<Arc, StateTable, CacheStore>>;
    friend class ReplaceFstMatcher<Arc, StateTable, CacheStore>;
  
    ReplaceFst(const std::vector<std::pair<Label, const Fst<Arc> *>> &fst_array,
               Label root)
        : ImplToFst<Impl>(std::make_shared<Impl>(
              fst_array, ReplaceFstOptions<Arc, StateTable, CacheStore>(root))) {}
  
    ReplaceFst(const std::vector<std::pair<Label, const Fst<Arc> *>> &fst_array,
               const ReplaceFstOptions<Arc, StateTable, CacheStore> &opts)
        : ImplToFst<Impl>(std::make_shared<Impl>(fst_array, opts)) {}
  
    // See Fst<>::Copy() for doc.
    ReplaceFst(const ReplaceFst<Arc, StateTable, CacheStore> &fst,
               bool safe = false)
        : ImplToFst<Impl>(fst, safe) {}
  
    // Get a copy of this ReplaceFst. See Fst<>::Copy() for further doc.
    ReplaceFst<Arc, StateTable, CacheStore> *Copy(
        bool safe = false) const override {
      return new ReplaceFst<Arc, StateTable, CacheStore>(*this, safe);
    }
  
    inline void InitStateIterator(StateIteratorData<Arc> *data) const override;
  
    void InitArcIterator(StateId s, ArcIteratorData<Arc> *data) const override {
      GetMutableImpl()->InitArcIterator(s, data);
    }
  
    MatcherBase<Arc> *InitMatcher(MatchType match_type) const override {
      if ((GetImpl()->ArcIteratorFlags() & kArcNoCache) &&
          ((match_type == MATCH_INPUT && Properties(kILabelSorted, false)) ||
           (match_type == MATCH_OUTPUT && Properties(kOLabelSorted, false)))) {
        return new ReplaceFstMatcher<Arc, StateTable, CacheStore>
            (this, match_type);
      } else {
        VLOG(2) << "Not using replace matcher";
        return nullptr;
      }
    }
  
    bool CyclicDependencies() const { return GetImpl()->CyclicDependencies(); }
  
    const StateTable &GetStateTable() const {
      return *GetImpl()->GetStateTable();
    }
  
    const Fst<Arc> &GetFst(Label nonterminal) const {
      return *GetImpl()->GetFst(GetImpl()->GetFstId(nonterminal));
    }
  
   private:
    using ImplToFst<Impl>::GetImpl;
    using ImplToFst<Impl>::GetMutableImpl;
  
    ReplaceFst &operator=(const ReplaceFst &) = delete;
  };
  
  // Specialization for ReplaceFst.
  template <class Arc, class StateTable, class CacheStore>
  class StateIterator<ReplaceFst<Arc, StateTable, CacheStore>>
      : public CacheStateIterator<ReplaceFst<Arc, StateTable, CacheStore>> {
   public:
    explicit StateIterator(const ReplaceFst<Arc, StateTable, CacheStore> &fst)
        : CacheStateIterator<ReplaceFst<Arc, StateTable, CacheStore>>(
              fst, fst.GetMutableImpl()) {}
  };
  
  // Specialization for ReplaceFst, implementing optional caching. It is be used
  // as follows:
  //
  //   ReplaceFst<A> replace;
  //   ArcIterator<ReplaceFst<A>> aiter(replace, s);
  //   // Note: ArcIterator< Fst<A>> is always a caching arc iterator.
  //   aiter.SetFlags(kArcNoCache, kArcNoCache);
  //   // Uses the arc iterator, no arc will be cached, no state will be expanded.
  //   // Arc flags can be used to decide which component of the arc need to be
  //   computed.
  //   aiter.SetFlags(kArcILabelValue, kArcValueFlags);
  //   // Wants the ilabel for this arc.
  //   aiter.Value();  // Does not compute the destination state.
  //   aiter.Next();
  //   aiter.SetFlags(kArcNextStateValue, kArcNextStateValue);
  //   // Wants the ilabel and next state for this arc.
  //   aiter.Value();  // Does compute the destination state and inserts it
  //                   // in the replace state table.
  //   // No additional arcs have been cached at this point.
  template <class Arc, class StateTable, class CacheStore>
  class ArcIterator<ReplaceFst<Arc, StateTable, CacheStore>> {
   public:
    using StateId = typename Arc::StateId;
  
    using StateTuple = typename StateTable::StateTuple;
  
    ArcIterator(const ReplaceFst<Arc, StateTable, CacheStore> &fst, StateId s)
        : fst_(fst),
          s_(s),
          pos_(0),
          offset_(0),
          flags_(kArcValueFlags),
          arcs_(nullptr),
          data_flags_(0),
          final_flags_(0) {
      cache_data_.ref_count = nullptr;
      local_data_.ref_count = nullptr;
      // If FST does not support optional caching, forces caching.
      if (!(fst_.GetImpl()->ArcIteratorFlags() & kArcNoCache) &&
          !(fst_.GetImpl()->HasArcs(s_))) {
        fst_.GetMutableImpl()->Expand(s_);
      }
      // If state is already cached, use cached arcs array.
      if (fst_.GetImpl()->HasArcs(s_)) {
        (fst_.GetImpl())
            ->internal::template CacheBaseImpl<
                typename CacheStore::State,
                CacheStore>::InitArcIterator(s_, &cache_data_);
        num_arcs_ = cache_data_.narcs;
        arcs_ = cache_data_.arcs;      // arcs_ is a pointer to the cached arcs.
        data_flags_ = kArcValueFlags;  // All the arc member values are valid.
      } else {  // Otherwise delay decision until Value() is called.
        tuple_ = fst_.GetImpl()->GetStateTable()->Tuple(s_);
        if (tuple_.fst_state == kNoStateId) {
          num_arcs_ = 0;
        } else {
          // The decision to cache or not to cache has been defered until Value()
          // or
          // SetFlags() is called. However, the arc iterator is set up now to be
          // ready for non-caching in order to keep the Value() method simple and
          // efficient.
          const auto *rfst = fst_.GetImpl()->GetFst(tuple_.fst_id);
          rfst->InitArcIterator(tuple_.fst_state, &local_data_);
          // arcs_ is a pointer to the arcs in the underlying machine.
          arcs_ = local_data_.arcs;
          // Computes the final arc (but not its destination state) if a final arc
          // is required.
          bool has_final_arc = fst_.GetMutableImpl()->ComputeFinalArc(
              tuple_, &final_arc_, kArcValueFlags & ~kArcNextStateValue);
          // Sets the arc value flags that hold for final_arc_.
          final_flags_ = kArcValueFlags & ~kArcNextStateValue;
          // Computes the number of arcs.
          num_arcs_ = local_data_.narcs;
          if (has_final_arc) ++num_arcs_;
          // Sets the offset between the underlying arc positions and the
          // positions
          // in the arc iterator.
          offset_ = num_arcs_ - local_data_.narcs;
          // Defers the decision to cache or not until Value() or SetFlags() is
          // called.
          data_flags_ = 0;
        }
      }
    }
  
    ~ArcIterator() {
      if (cache_data_.ref_count) --(*cache_data_.ref_count);
      if (local_data_.ref_count) --(*local_data_.ref_count);
    }
  
    void ExpandAndCache() const  {
      // TODO(allauzen): revisit this.
      // fst_.GetImpl()->Expand(s_, tuple_, local_data_);
      // (fst_.GetImpl())->CacheImpl<A>*>::InitArcIterator(s_,
      //                                               &cache_data_);
      //
      fst_.InitArcIterator(s_, &cache_data_);  // Expand and cache state.
      arcs_ = cache_data_.arcs;      // arcs_ is a pointer to the cached arcs.
      data_flags_ = kArcValueFlags;  // All the arc member values are valid.
      offset_ = 0;                   // No offset.
    }
  
    void Init() {
      if (flags_ & kArcNoCache) {  // If caching is disabled
        // arcs_ is a pointer to the arcs in the underlying machine.
        arcs_ = local_data_.arcs;
        // Sets the arcs value flags that hold for arcs_.
        data_flags_ = kArcWeightValue;
        if (!fst_.GetMutableImpl()->EpsilonOnCallInput()) {
          data_flags_ |= kArcILabelValue;
        }
        // Sets the offset between the underlying arc positions and the positions
        // in the arc iterator.
        offset_ = num_arcs_ - local_data_.narcs;
      } else {
        ExpandAndCache();
      }
    }
  
    bool Done() const { return pos_ >= num_arcs_; }
  
    const Arc &Value() const {
      // If data_flags_ is 0, non-caching was not requested.
      if (!data_flags_) {
        // TODO(allauzen): Revisit this.
        if (flags_ & kArcNoCache) {
          // Should never happen.
          FSTERROR() << "ReplaceFst: Inconsistent arc iterator flags";
        }
        ExpandAndCache();
      }
      if (pos_ - offset_ >= 0) {  // The requested arc is not the final arc.
        const auto &arc = arcs_[pos_ - offset_];
        if ((data_flags_ & flags_) == (flags_ & kArcValueFlags)) {
          // If the value flags match the recquired value flags then returns the
          // arc.
          return arc;
        } else {
          // Otherwise, compute the corresponding arc on-the-fly.
          fst_.GetMutableImpl()->ComputeArc(tuple_, arc, &arc_,
                                            flags_ & kArcValueFlags);
          return arc_;
        }
      } else {  // The requested arc is the final arc.
        if ((final_flags_ & flags_) != (flags_ & kArcValueFlags)) {
          // If the arc value flags that hold for the final arc do not match the
          // requested value flags, then
          // final_arc_ needs to be updated.
          fst_.GetMutableImpl()->ComputeFinalArc(tuple_, &final_arc_,
                                                 flags_ & kArcValueFlags);
          final_flags_ = flags_ & kArcValueFlags;
        }
        return final_arc_;
      }
    }
  
    void Next() { ++pos_; }
  
    size_t Position() const { return pos_; }
  
    void Reset() { pos_ = 0; }
  
    void Seek(size_t pos) { pos_ = pos; }
  
    uint32 Flags() const { return flags_; }
  
    void SetFlags(uint32 flags, uint32 mask) {
      // Updates the flags taking into account what flags are supported
      // by the FST.
      flags_ &= ~mask;
      flags_ |= (flags & fst_.GetImpl()->ArcIteratorFlags());
      // If non-caching is not requested (and caching has not already been
      // performed), then flush data_flags_ to request caching during the next
      // call to Value().
      if (!(flags_ & kArcNoCache) && data_flags_ != kArcValueFlags) {
        if (!fst_.GetImpl()->HasArcs(s_)) data_flags_ = 0;
      }
      // If data_flags_ has been flushed but non-caching is requested before
      // calling Value(), then set up the iterator for non-caching.
      if ((flags & kArcNoCache) && (!data_flags_)) Init();
    }
  
   private:
    const ReplaceFst<Arc, StateTable, CacheStore> &fst_;  // Reference to the FST.
    StateId s_;                                           // State in the FST.
    mutable StateTuple tuple_;  // Tuple corresponding to state_.
  
    ssize_t pos_;             // Current position.
    mutable ssize_t offset_;  // Offset between position in iterator and in arcs_.
    ssize_t num_arcs_;        // Number of arcs at state_.
    uint32 flags_;            // Behavorial flags for the arc iterator
    mutable Arc arc_;         // Memory to temporarily store computed arcs.
  
    mutable ArcIteratorData<Arc> cache_data_;  // Arc iterator data in cache.
    mutable ArcIteratorData<Arc> local_data_;  // Arc iterator data in local FST.
  
    mutable const Arc *arcs_;     // Array of arcs.
    mutable uint32 data_flags_;   // Arc value flags valid for data in arcs_.
    mutable Arc final_arc_;       // Final arc (when required).
    mutable uint32 final_flags_;  // Arc value flags valid for final_arc_.
  
    ArcIterator(const ArcIterator &) = delete;
    ArcIterator &operator=(const ArcIterator &) = delete;
  };
  
  template <class Arc, class StateTable, class CacheStore>
  class ReplaceFstMatcher : public MatcherBase<Arc> {
   public:
    using Label = typename Arc::Label;
    using StateId = typename Arc::StateId;
    using Weight = typename Arc::Weight;
  
    using FST = ReplaceFst<Arc, StateTable, CacheStore>;
    using LocalMatcher = MultiEpsMatcher<Matcher<Fst<Arc>>>;
  
    using StateTuple = typename StateTable::StateTuple;
  
    // This makes a copy of the FST.
    ReplaceFstMatcher(const ReplaceFst<Arc, StateTable, CacheStore> &fst,
                      MatchType match_type)
        : owned_fst_(fst.Copy()),
          fst_(*owned_fst_),
          impl_(fst_.GetMutableImpl()),
          s_(fst::kNoStateId),
          match_type_(match_type),
          current_loop_(false),
          final_arc_(false),
          loop_(kNoLabel, 0, Weight::One(), kNoStateId) {
      if (match_type_ == fst::MATCH_OUTPUT) {
        std::swap(loop_.ilabel, loop_.olabel);
      }
      InitMatchers();
    }
  
    // This doesn't copy the FST.
    ReplaceFstMatcher(const ReplaceFst<Arc, StateTable, CacheStore> *fst,
                      MatchType match_type)
        : fst_(*fst),
          impl_(fst_.GetMutableImpl()),
          s_(fst::kNoStateId),
          match_type_(match_type),
          current_loop_(false),
          final_arc_(false),
          loop_(kNoLabel, 0, Weight::One(), kNoStateId) {
      if (match_type_ == fst::MATCH_OUTPUT) {
        std::swap(loop_.ilabel, loop_.olabel);
      }
      InitMatchers();
    }
  
    // This makes a copy of the FST.
    ReplaceFstMatcher(
        const ReplaceFstMatcher<Arc, StateTable, CacheStore> &matcher,
        bool safe = false)
        : owned_fst_(matcher.fst_.Copy(safe)),
          fst_(*owned_fst_),
          impl_(fst_.GetMutableImpl()),
          s_(fst::kNoStateId),
          match_type_(matcher.match_type_),
          current_loop_(false),
          final_arc_(false),
          loop_(fst::kNoLabel, 0, Weight::One(), fst::kNoStateId) {
      if (match_type_ == fst::MATCH_OUTPUT) {
        std::swap(loop_.ilabel, loop_.olabel);
      }
      InitMatchers();
    }
  
    // Creates a local matcher for each component FST in the RTN. LocalMatcher is
    // a multi-epsilon wrapper matcher. MultiEpsilonMatcher is used to match each
    // non-terminal arc, since these non-terminal
    // turn into epsilons on recursion.
    void InitMatchers() {
      const auto &fst_array = impl_->fst_array_;
      matcher_.resize(fst_array.size());
      for (Label i = 0; i < fst_array.size(); ++i) {
        if (fst_array[i]) {
          matcher_[i].reset(
              new LocalMatcher(*fst_array[i], match_type_, kMultiEpsList));
          auto it = impl_->nonterminal_set_.begin();
          for (; it != impl_->nonterminal_set_.end(); ++it) {
            matcher_[i]->AddMultiEpsLabel(*it);
          }
        }
      }
    }
  
    ReplaceFstMatcher<Arc, StateTable, CacheStore> *Copy(
        bool safe = false) const override {
      return new ReplaceFstMatcher<Arc, StateTable, CacheStore>(*this, safe);
    }
  
    MatchType Type(bool test) const override {
      if (match_type_ == MATCH_NONE) return match_type_;
      const auto true_prop =
          match_type_ == MATCH_INPUT ? kILabelSorted : kOLabelSorted;
      const auto false_prop =
          match_type_ == MATCH_INPUT ? kNotILabelSorted : kNotOLabelSorted;
      const auto props = fst_.Properties(true_prop | false_prop, test);
      if (props & true_prop) {
        return match_type_;
      } else if (props & false_prop) {
        return MATCH_NONE;
      } else {
        return MATCH_UNKNOWN;
      }
    }
  
    const Fst<Arc> &GetFst() const override { return fst_; }
  
    uint64 Properties(uint64 props) const override { return props; }
  
    // Sets the state from which our matching happens.
    void SetState(StateId s) final {
      if (s_ == s) return;
      s_ = s;
      tuple_ = impl_->GetStateTable()->Tuple(s_);
      if (tuple_.fst_state == kNoStateId) {
        done_ = true;
        return;
      }
      // Gets current matcher, used for non-epsilon matching.
      current_matcher_ = matcher_[tuple_.fst_id].get();
      current_matcher_->SetState(tuple_.fst_state);
      loop_.nextstate = s_;
      final_arc_ = false;
    }
  
    // Searches for label from previous set state. If label == 0, first
    // hallucinate an epsilon loop; otherwise use the underlying matcher to
    // search for the label or epsilons. Note since the ReplaceFst recursion
    // on non-terminal arcs causes epsilon transitions to be created we use
    // MultiEpsilonMatcher to search for possible matches of non-terminals. If the
    // component FST
    // reaches a final state we also need to add the exiting final arc.
    bool Find(Label label) final {
      bool found = false;
      label_ = label;
      if (label_ == 0 || label_ == kNoLabel) {
        // Computes loop directly, avoiding Replace::ComputeArc.
        if (label_ == 0) {
          current_loop_ = true;
          found = true;
        }
        // Searches for matching multi-epsilons.
        final_arc_ = impl_->ComputeFinalArc(tuple_, nullptr);
        found = current_matcher_->Find(kNoLabel) || final_arc_ || found;
      } else {
        // Searches on a sub machine directly using sub machine matcher.
        found = current_matcher_->Find(label_);
      }
      return found;
    }
  
    bool Done() const final {
      return !current_loop_ && !final_arc_ && current_matcher_->Done();
    }
  
    const Arc &Value() const final {
      if (current_loop_) return loop_;
      if (final_arc_) {
        impl_->ComputeFinalArc(tuple_, &arc_);
        return arc_;
      }
      const auto &component_arc = current_matcher_->Value();
      impl_->ComputeArc(tuple_, component_arc, &arc_);
      return arc_;
    }
  
    void Next() final {
      if (current_loop_) {
        current_loop_ = false;
        return;
      }
      if (final_arc_) {
        final_arc_ = false;
        return;
      }
      current_matcher_->Next();
    }
  
    ssize_t Priority(StateId s) final { return fst_.NumArcs(s); }
  
   private:
    std::unique_ptr<const ReplaceFst<Arc, StateTable, CacheStore>> owned_fst_;
    const ReplaceFst<Arc, StateTable, CacheStore> &fst_;
    internal::ReplaceFstImpl<Arc, StateTable, CacheStore> *impl_;
    LocalMatcher *current_matcher_;
    std::vector<std::unique_ptr<LocalMatcher>> matcher_;
    StateId s_;             // Current state.
    Label label_;           // Current label.
    MatchType match_type_;  // Supplied by caller.
    mutable bool done_;
    mutable bool current_loop_;  // Current arc is the implicit loop.
    mutable bool final_arc_;     // Current arc for exiting recursion.
    mutable StateTuple tuple_;   // Tuple corresponding to state_.
    mutable Arc arc_;
    Arc loop_;
  
    ReplaceFstMatcher &operator=(const ReplaceFstMatcher &) = delete;
  };
  
  template <class Arc, class StateTable, class CacheStore>
  inline void ReplaceFst<Arc, StateTable, CacheStore>::InitStateIterator(
      StateIteratorData<Arc> *data) const {
    data->base =
        new StateIterator<ReplaceFst<Arc, StateTable, CacheStore>>(*this);
  }
  
  using StdReplaceFst = ReplaceFst<StdArc>;
  
  // Recursively replaces arcs in the root FSTs with other FSTs.
  // This version writes the result of replacement to an output MutableFst.
  //
  // Replace supports replacement of arcs in one Fst with another FST. This
  // replacement is recursive. Replace takes an array of FST(s). One FST
  // represents the root (or topology) machine. The root FST refers to other FSTs
  // by recursively replacing arcs labeled as non-terminals with the matching
  // non-terminal FST. Currently Replace uses the output symbols of the arcs to
  // determine whether the arc is a non-terminal arc or not. A non-terminal can be
  // any label that is not a non-zero terminal label in the output alphabet.
  //
  // Note that input argument is a vector of pairs. These correspond to the tuple
  // of non-terminal Label and corresponding FST.
  template <class Arc>
  void Replace(const std::vector<std::pair<typename Arc::Label, const Fst<Arc> *>>
                   &ifst_array,
               MutableFst<Arc> *ofst,
               ReplaceFstOptions<Arc> opts = ReplaceFstOptions<Arc>()) {
    opts.gc = true;
    opts.gc_limit = 0;  // Caches only the last state for fastest copy.
    *ofst = ReplaceFst<Arc>(ifst_array, opts);
  }
  
  template <class Arc>
  void Replace(const std::vector<std::pair<typename Arc::Label, const Fst<Arc> *>>
                   &ifst_array,
               MutableFst<Arc> *ofst, const ReplaceUtilOptions &opts) {
    Replace(ifst_array, ofst, ReplaceFstOptions<Arc>(opts));
  }
  
  // For backwards compatibility.
  template <class Arc>
  void Replace(const std::vector<std::pair<typename Arc::Label, const Fst<Arc> *>>
                   &ifst_array,
               MutableFst<Arc> *ofst, typename Arc::Label root,
               bool epsilon_on_replace) {
    Replace(ifst_array, ofst, ReplaceFstOptions<Arc>(root, epsilon_on_replace));
  }
  
  template <class Arc>
  void Replace(const std::vector<std::pair<typename Arc::Label, const Fst<Arc> *>>
                   &ifst_array,
               MutableFst<Arc> *ofst, typename Arc::Label root) {
    Replace(ifst_array, ofst, ReplaceFstOptions<Arc>(root));
  }
  
  }  // namespace fst
  
  #endif  // FST_REPLACE_H_