equivalent.h revision f4c12fce1ee58e670f9c3fce46c40296ba9ee8a2 
1// equivalent.h
2
3// Licensed under the Apache License, Version 2.0 (the "License");
4// you may not use this file except in compliance with the License.
5// You may obtain a copy of the License at
6//
7//     http://www.apache.org/licenses/LICENSE-2.0
8//
9// Unless required by applicable law or agreed to in writing, software
10// distributed under the License is distributed on an "AS IS" BASIS,
11// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
12// See the License for the specific language governing permissions and
13// limitations under the License.
14//
15// Copyright 2005-2010 Google, Inc.
16// Author: wojciech@google.com (Wojciech Skut)
17//
18// \file Functions and classes to determine the equivalence of two
19// FSTs.
20
21#ifndef FST_LIB_EQUIVALENT_H__
22#define FST_LIB_EQUIVALENT_H__
23
24#include <algorithm>
25#include <deque>
26#include <unordered_map>
27using std::tr1::unordered_map;
28using std::tr1::unordered_multimap;
29#include <utility>
30using std::pair; using std::make_pair;
31#include <vector>
32using std::vector;
33
34#include <fst/encode.h>
35#include <fst/push.h>
36#include <fst/union-find.h>
37#include <fst/vector-fst.h>
38
39
40namespace fst {
41
42// Traits-like struct holding utility functions/typedefs/constants for
43// the equivalence algorithm.
44//
45// Encoding device: in order to make the statesets of the two acceptors
46// disjoint, we map Arc::StateId on the type MappedId. The states of
47// the first acceptor are mapped on odd numbers (s -> 2s + 1), and
48// those of the second one on even numbers (s -> 2s + 2). The number 0
49// is reserved for an implicit (non-final) 'dead state' (required for
50// the correct treatment of non-coaccessible states; kNoStateId is
51// mapped to kDeadState for both acceptors). The union-find algorithm
52// operates on the mapped IDs.
53template <class Arc>
54struct EquivalenceUtil {
55  typedef typename Arc::StateId StateId;
56  typedef typename Arc::Weight Weight;
57  typedef StateId MappedId;  // ID for an equivalence class.
58
59  // MappedId for an implicit dead state.
60  static const MappedId kDeadState = 0;
61
62  // MappedId for lookup failure.
63  static const MappedId kInvalidId = -1;
64
65  // Maps state ID to the representative of the corresponding
66  // equivalence class. The parameter 'which_fst' takes the values 1
67  // and 2, identifying the input FST.
68  static MappedId MapState(StateId s, int32 which_fst) {
69    return
70      (kNoStateId == s)
71      ?
72      kDeadState
73      :
74      (static_cast<MappedId>(s) << 1) + which_fst;
75  }
76  // Maps set ID to State ID.
77  static StateId UnMapState(MappedId id) {
78    return static_cast<StateId>((--id) >> 1);
79  }
80  // Convenience function: checks if state with MappedId 's' is final
81  // in acceptor 'fa'.
82  static bool IsFinal(const Fst<Arc> &fa, MappedId s) {
83    return
84      (kDeadState == s) ?
85      false : (fa.Final(UnMapState(s)) != Weight::Zero());
86  }
87  // Convenience function: returns the representative of 'id' in 'sets',
88  // creating a new set if needed.
89  static MappedId FindSet(UnionFind<MappedId> *sets, MappedId id) {
90    MappedId repr = sets->FindSet(id);
91    if (repr != kInvalidId) {
92      return repr;
93    } else {
94      sets->MakeSet(id);
95      return id;
96    }
97  }
98};
99
100template <class Arc> const
101typename EquivalenceUtil<Arc>::MappedId EquivalenceUtil<Arc>::kDeadState;
102
103template <class Arc> const
104typename EquivalenceUtil<Arc>::MappedId EquivalenceUtil<Arc>::kInvalidId;
105
106
107// Equivalence checking algorithm: determines if the two FSTs
108// <code>fst1</code> and <code>fst2</code> are equivalent. The input
109// FSTs must be deterministic input-side epsilon-free acceptors,
110// unweighted or with weights over a left semiring. Two acceptors are
111// considered equivalent if they accept exactly the same set of
112// strings (with the same weights).
113//
114// The algorithm (cf. Aho, Hopcroft and Ullman, "The Design and
115// Analysis of Computer Programs") successively constructs sets of
116// states that can be reached by the same prefixes, starting with a
117// set containing the start states of both acceptors. A disjoint tree
118// forest (the union-find algorithm) is used to represent the sets of
119// states. The algorithm returns 'false' if one of the constructed
120// sets contains both final and non-final states. Returns optional error
121// value (when FLAGS_error_fatal = false).
122//
123// Complexity: quasi-linear, i.e. O(n G(n)), where
124//   n = |S1| + |S2| is the number of states in both acceptors
125//   G(n) is a very slowly growing function that can be approximated
126//        by 4 by all practical purposes.
127//
128template <class Arc>
129bool Equivalent(const Fst<Arc> &fst1,
130                const Fst<Arc> &fst2,
131                double delta = kDelta, bool *error = 0) {
132  typedef typename Arc::Weight Weight;
133  if (error) *error = false;
134
135  // Check that the symbol table are compatible
136  if (!CompatSymbols(fst1.InputSymbols(), fst2.InputSymbols()) ||
137      !CompatSymbols(fst1.OutputSymbols(), fst2.OutputSymbols())) {
138    FSTERROR() << "Equivalent: input/output symbol tables of 1st argument "
139               << "do not match input/output symbol tables of 2nd argument";
140    if (error) *error = true;
141    return false;
142  }
143  // Check properties first:
144  uint64 props = kNoEpsilons | kIDeterministic | kAcceptor;
145  if (fst1.Properties(props, true) != props) {
146    FSTERROR() << "Equivalent: first argument not an"
147               << " epsilon-free deterministic acceptor";
148    if (error) *error = true;
149    return false;
150  }
151  if (fst2.Properties(props, true) != props) {
152    FSTERROR() << "Equivalent: second argument not an"
153               << " epsilon-free deterministic acceptor";
154    if (error) *error = true;
155    return false;
156  }
157
158  if ((fst1.Properties(kUnweighted , true) != kUnweighted)
159      || (fst2.Properties(kUnweighted , true) != kUnweighted)) {
160    VectorFst<Arc> efst1(fst1);
161    VectorFst<Arc> efst2(fst2);
162    Push(&efst1, REWEIGHT_TO_INITIAL, delta);
163    Push(&efst2, REWEIGHT_TO_INITIAL, delta);
164    ArcMap(&efst1, QuantizeMapper<Arc>(delta));
165    ArcMap(&efst2, QuantizeMapper<Arc>(delta));
166    EncodeMapper<Arc> mapper(kEncodeWeights|kEncodeLabels, ENCODE);
167    ArcMap(&efst1, &mapper);
168    ArcMap(&efst2, &mapper);
169    return Equivalent(efst1, efst2);
170  }
171
172  // Convenience typedefs:
173  typedef typename Arc::StateId StateId;
174  typedef EquivalenceUtil<Arc> Util;
175  typedef typename Util::MappedId MappedId;
176  enum { FST1 = 1, FST2 = 2 };  // Required by Util::MapState(...)
177
178  MappedId s1 = Util::MapState(fst1.Start(), FST1);
179  MappedId s2 = Util::MapState(fst2.Start(), FST2);
180
181  // The union-find structure.
182  UnionFind<MappedId> eq_classes(1000, Util::kInvalidId);
183
184  // Initialize the union-find structure.
185  eq_classes.MakeSet(s1);
186  eq_classes.MakeSet(s2);
187
188  // Data structure for the (partial) acceptor transition function of
189  // fst1 and fst2: input labels mapped to pairs of MappedId's
190  // representing destination states of the corresponding arcs in fst1
191  // and fst2, respectively.
192  typedef
193    unordered_map<typename Arc::Label, pair<MappedId, MappedId> >
194    Label2StatePairMap;
195
196  Label2StatePairMap arc_pairs;
197
198  // Pairs of MappedId's to be processed, organized in a queue.
199  deque<pair<MappedId, MappedId> > q;
200
201  bool ret = true;
202  // Early return if the start states differ w.r.t. being final.
203  if (Util::IsFinal(fst1, s1) != Util::IsFinal(fst2, s2)) {
204    ret = false;
205  }
206
207  // Main loop: explores the two acceptors in a breadth-first manner,
208  // updating the equivalence relation on the statesets. Loop
209  // invariant: each block of states contains either final states only
210  // or non-final states only.
211  for (q.push_back(make_pair(s1, s2)); ret && !q.empty(); q.pop_front()) {
212    s1 = q.front().first;
213    s2 = q.front().second;
214
215    // Representatives of the equivalence classes of s1/s2.
216    MappedId rep1 = Util::FindSet(&eq_classes, s1);
217    MappedId rep2 = Util::FindSet(&eq_classes, s2);
218
219    if (rep1 != rep2) {
220      eq_classes.Union(rep1, rep2);
221      arc_pairs.clear();
222
223      // Copy outgoing arcs starting at s1 into the hashtable.
224      if (Util::kDeadState != s1) {
225        ArcIterator<Fst<Arc> > arc_iter(fst1, Util::UnMapState(s1));
226        for (; !arc_iter.Done(); arc_iter.Next()) {
227          const Arc &arc = arc_iter.Value();
228          if (arc.weight != Weight::Zero()) {  // Zero-weight arcs
229                                                   // are treated as
230                                                   // non-exisitent.
231            arc_pairs[arc.ilabel].first = Util::MapState(arc.nextstate, FST1);
232          }
233        }
234      }
235      // Copy outgoing arcs starting at s2 into the hashtable.
236      if (Util::kDeadState != s2) {
237        ArcIterator<Fst<Arc> > arc_iter(fst2, Util::UnMapState(s2));
238        for (; !arc_iter.Done(); arc_iter.Next()) {
239          const Arc &arc = arc_iter.Value();
240          if (arc.weight != Weight::Zero()) {  // Zero-weight arcs
241                                                   // are treated as
242                                                   // non-existent.
243            arc_pairs[arc.ilabel].second = Util::MapState(arc.nextstate, FST2);
244          }
245        }
246      }
247      // Iterate through the hashtable and process pairs of target
248      // states.
249      for (typename Label2StatePairMap::const_iterator
250             arc_iter = arc_pairs.begin();
251           arc_iter != arc_pairs.end();
252           ++arc_iter) {
253        const pair<MappedId, MappedId> &p = arc_iter->second;
254        if (Util::IsFinal(fst1, p.first) != Util::IsFinal(fst2, p.second)) {
255          // Detected inconsistency: return false.
256          ret = false;
257          break;
258        }
259        q.push_back(p);
260      }
261    }
262  }
263
264  if (fst1.Properties(kError, false) || fst2.Properties(kError, false)) {
265    if (error) *error = true;
266    return false;
267  }
268
269  return ret;
270}
271
272}  // namespace fst
273
274#endif  // FST_LIB_EQUIVALENT_H__
275