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OpenFst Quick Tour

Finding and Using the Library

The OpenFst Library is a C++ template library. To use it from C++, include lib/fstlib-inl.h and link to lib/libfst.{a,so}. You may instead use just those include files for the classes and functions that you will need. All classes and functions are in the fst namespace; the examples below assume you are within that namespace for brevity.

As an alternative interface, there are shell-level commands in bin that operate on file representations of FSTs.

Example FST

The following picture depicts a finite state transducer:

symbolicfst.jpg

The initial state is label 0. There can only be one initial state. The final state is 2 with final weight of 3.5. Any state with non-infinite final weight is a final state. There is an arc (or transition) from state 0 to 1 with input label a, output label x, and weight 0.5. This FST transduces, for instance, the string ac to xz with weight 6.5 (the sum of the arc and final weights). Note we have assumed the library default Weight type for this description.

Creating FSTs

FSTs can be created with constructors and mutators from C++ or from text files at the shell-level. We will show how to create the above example FST both ways.

Creating FSTs Using Constructors and Mutators From C++

The following code will create our example FST within C++:

// A vector FST is a general mutable FST
StdVectorFst fst;

// Adds state 0 to the initially empty FST and make it the start state.
fst.AddState();   // 1st state will be state 0 (returned by AddState)
fst.SetStart(0);  // arg is state ID

// Adds two arcs exiting state 0.
// Arc constructor args: ilabel, olabel, weight, dest state ID.
fst.AddArc(0, StdArc(1, 1, 0.5, 1));  // 1st arg is src state ID
fst.AddArc(0, StdArc(2, 2, 1.5, 1)); 

// Adds state 1 and its arc.
fst.AddState();
fst.AddArc(1, StdArc(3, 3, 2.5, 2));

// Adds state 2 and set its final weight.
fst.AddState();
fst.SetFinal(2, 3.5);  // 1st arg is state ID, 2nd arg weight

We can save this FST to a file with:

fst.Write("binary.fst");

Creating FSTs Using Text Files from the Shell

FSTs can be specified using a text file in the AT&T FSM format Supporting this format permits interoperability with the AT&T FSM binary tools, which can be downloaded for non-commercial purposes.

We can create the text FST file for our example as follows:

# arc format: src dest ilabel olabel [weight]
# final state format: state [weight]
# lines may occur in any order except initial state must be first line
# unspecified weights default to 0.0 (for the library-default Weight type)
cat >text.fst <<EOF
0 1 a x .5
0 1 b y 1.5
1 2 c z 2.5
2 3.5
EOF

The internal representation of an arc label is an integer. We must provide the mapping from symbols to integers explicitly with a symbol table file, also in AT&T format:

$ cat >isyms.txt <<EOF
<eps> 0
a 1
b 2
c 3
EOF

$ cat >osyms.txt <<EOF
<eps> 0
x 1
y 2
z 3
EOF

You may use any string for a label; you may use any non-negative integer for a label ID. The zero label ID is reserved for the epsilon label, which is the empty string. We have included 0 in our table, even though it is not used in our example. Since subsequent FST operations might add epsilons, it is good practice to include a symbol for it.

This text FST must be converted into a binary FST file before it can be used by the OpenFst library.

# Creates binary Fst from text file.
# The symbolic labels will be converted into integers using the symbol table files
$ fstcompile -isymbols=isyms.txt -osymbols=osyms.txt text.fst binary.fst

# As above but the symbol tables are stored with the FST.
$ fstcompile -isymbols=isyms.txt -osymbols=osyms.txt -keep_isymbols -keep_osymbols text.fst binary.fst

If the labels are represented as non-negative integers in the text FST, then the symbol table files can be omitted. In any case, the internal representation of the FST is:

numericfst.jpg

Once a binary FST is created, it can be used with the other shell-level programs. It can be loaded inside C++ with:

Fst *fst = StdFst::Read("binary.fst");

Accessing FSTs

FSTs can be examined from C++ accessors or from shell-level commands that read the binary files.

Accessing FSTs from C++

Here is the standard representation of an arc:

struct StdArc {
 typedef int Label;
 typedef TropicalWeight Weight;  // see "FST Weights" below
 typedef int StateId; 
 
 Label ilabel;
 Label olabel;
 Weight weight;
 StateId nextstate;
};

Here are some example accesses of an FST:

typedef StdArc::StateId StateId;

# Gets the initial state; if kNoState => empty FST.
StateId initial_state = fst.Start();

# Iterates over the FSTs states.
for (StateIterator<StdFst> siter(fst); !siter.Done(); siter.Next()) 
  StateId state_id = siter.Value();

# Iterates over state i's arcs.
for (ArcIterator<StdFst> aiter(fst, i); !aiter.Done(); aiter.Next())
  const StdArc &arc = aiter.Value();

There are various conventions that must be observed when accessing FSTs.

Printing and Summarizing FSTs from the Shell

The following command will print out an FST in AT&T text format:

# Print FST using symbol table files.
$ fstprint -isymbols=isyms.txt -osymbols=osyms.txt binary.fst text.fst

If the symbol table files are omitted, the FST will be printed with numeric labels unless the symbol tables are stored with the FST (e.g., with fstcompile -keep_isymbols -keep_osymbols).

Summary information about an FST can be obtained with:

$ fstinfo binary.fst
fst type                        vector
arc type                        standard
input symbol table              isyms.txt
output symbol table             osyms.txt
# of states                     3
# of arcs                       3
initial state                   0
# of final states               1
# of input/output epsilons      0
# of input epsilons             0
# of output epsilons            0
# of accessible states          3
# of coaccessible states        3
# of connected states           3
# of strongly conn components   3
expanded                        y
mutable                         y
acceptor                        y
input deterministic             y
output deterministic            y
input/output epsilons           n
input epsilons                  n
output epsilons                 n
input label sorted              y
output label sorted             y
weighted                        y
cyclic                          n
cyclic at initial state         n
top sorted                      y
accessible                      y
coaccessible                    y

FST Operations

Calling FST Operations

The FST operations can be invoked either at the C++ level or from shell-level commands.

Calling FST Operations from C++

To invoke FST operations from C++, the FST class hierarchy must first be introduced:

The FST interface hierarchy consists of the following abstract class templates:

  • Fst<Arc> : supports access operations described above
  • ExpandedFst<Arc> : an Fst that additionally supports NumStates()
  • MutableFst<Arc> : an ExpandedFst that supports the various mutating operations like AddStates() and SetStart().

Specific, non-abstract FSTs include these class templates:

  • VectorFst<Arc>: a general-purpose mutable FST
  • ConstFst<Arc>: a general-purpose expanded, immutable FST
  • ComposeFst<Arc>: an unexpanded, delayed composition of two FSTs

These classes are templated on the arc to allow customization. The class StdFst is a typedef for Fst<StdArc>. Similar typedefs exist for all the above templates.

For the state and arc iterators, you will get the greatest efficiency if you specify the most specific FST class as the iterator template argument (e.g., ArcIterator<StdVectorFst> rather than ArcIterator<StdFst> for a known StdVectorFst).

The C++ FST operations come in three general forms:

  • Destructive: When an operation, like Connect, modifies its input, it has the form:
void Connect(MutableFst<Arc> *fst);
  • Constructive: When an operation, like Reverse, creates a new expanded Fst, it has the form:
void Reverse(const Fst<Arc> &infst, MutableFst<Arc> *outfst);
  • Delayed: When an operation, like ComposeFst, creates a lazy-evaluated Fst, it is a new unexpanded Fst class of the form:
ComposeFst<Arc>(const Fst<Arc> &fst1, const Fst<Arc> &fst2); 

Delayed Fsts have constant time-class constructors. When components of delayed Fsts are accessed through the Fst interface, the automaton is built dynamically, just enough to respond to the accesses requested. It is important that the object access conventions are observed for correct operation.

Several operations, like Union, come in more than one of the above forms.

Calling FST Operations from the Shell

The shell-level FST operations typically read one or more input binary FST files, call internally the corresponding C++ operation and then write an output binary FST file. Specifically, they have the form:

  • Unary Operations:
fstunaryop in.fst out.fst
  • Binary Operations:
fstbinaryop in1.fst in2.fst out.fst

Example Use: FST Application

One of the most useful finite-state operations is composition, which produces the relational composition of two transductions. It can be used, for example, to apply a transduction to some input:

FST Application from C++

// Reads in an input FST.
StdFst *input = StdFst::Read("input.fst");

// Reads in the transduction model.
StdFst *model = StdFst::Read("model.fst");

// The FSTs must be sorted along the dimensions they will be joined.
// In fact, only one needs to be so sorted.
// This could have instead been done for "model.fst" when it was created.
ArcSort(input, StdOLabelCompare());
ArcSort(model, StdILabelCompare());

// Container for composition result.
StdVectorFst result;

// Create the composed FST
Compose(*input, *model, &result);

// Just keeps the output labels
Project(&result,  PROJECT_OUTPUT);

FST Application from the Shell

# The FSTs must be sorted along the dimensions they will be joined.
# In fact, only one needs to be so sorted.
# This could have instead been done for "model.fst" when it was created.
$ arcsort --sort_type=olabel input.fst input_sorted.fst
$ arcsort --sort_type=ilabel model.fst model_sorted.fst

# Creates the composed FST
$ fstcompose input_sorted.fst model_sorted.fst comp.fst

# Just keeps the output label
$ fstproject -project_output comp.fst result.fst

Available FST Operations

Click on operation name for additional information.

Operation Usage Description
ArcSort ArcSort(&A, compare); sorts arcs using compare function object
  ArcSortFst<Arc, Compare>(A, compare);  
  fstarcsort [--sort_type=$type] a.fst out.fst  
Closure Closure(&A, type); A* = <eps> U A U AA U AAA ...
  ClosureFst<Arc>(A, type);  
  fstclosure a.fst out.fst  
Compose Compose(A, B, &C); composition of binary relations A and B
  ComposeFst<Arc>(A, B);  
  fstcompose a.fst b.fst out.fst  
Concat Concat(&A, B); contains the strings in A followed by B
  ConcatFst<Arc>(A,B);  
  fstconcat a.fst b.fst out.fst  
Connect Connect(&A); removes states and arcs not on a path from the start to a final state
  fstconnect a.fst out.fst  
Determinize Determinize(A, &B); creates equiv. FST with no state with two arcs with the same input label
  DeterminizeFst<Arc>(A);  
  fstdeterminize a.fst out.fst  
Difference Difference(A, B, &C); contains strings in A but not B; B unweighted
  DifferenceFst<Arc>(A, B);  
  fstdifference a.fsa b.dfa out.fsa  
EpsNormalize EpsNormalize(A, &B, type); creates equiv. FST with any input (output) epsilons at path ends
  fstepsnormalize [--eps_norm_output]  
Equivalent Equivalent(A, B) determines if acceptors A and B accept the same strings with the same weights
  fstequivalent a.dfa b.dfa  
Intersect Intersect(A, B, &C); contains strings both in A and B
IntersectFst<Arc>(A, B);  
  fstintersect a.fsa b.fsa out.fsa  
Invert Invert(&A); inverse binary relation; exchanges input and output labels
  InvertFst<Arc>(A);  
  fstinvert a.fst out.fst  
Map Map(&A, mapper); transforms arcs in an FST
  Map(A, &B, mapper);  
  MapFst<Arc, Mapper>(A, mapper);  
  fstmap [--delta=$d] [--map=$type] [--weight=$w] in.fst out.fst  
Minimize Minimize(&A); transforms to equiv. deterministic FSA with fewest states and arcs
  Minimize(&A, &B); transforms to equiv. deterministic FST with fewest states and arcs
  fstminimize a.fst out1.fst [out2.fst]  
Project Project(&A, type); creates acceptor of just the input or output strings
  ProjectFst<Arc>(A, type);  
  fstproject [--project_output] a.fst out.fst  
Prune Prune(&A, threshold); remove paths outside a threshold of best path
  fstprune [--weight=$w] a.fst out.fst  
Push Push<Arc, Type>(&A, flags); create equiv. FST pushing weights and/or output labels toward initial \or final states
  fstpush [--push_labels] [--push_weights] [--to_final] a.fst out.fst  
Relabel Relabel(&A, isyms, osyms); changes input and output label IDs
  RelabelFst<Arc>(A, isyms, osyms);  
  fstrelabel [--relabel_isymbols=$isyms] [--relabel_osymbols=$osyms] in.fst out.fst  
Replace Replace(fst_label_pairs, &B, root_label); replaces non-terminals with FSTs analogous to an RTN
ReplaceFst<Arc>(fst_label_pairs, root_label);  
  fstreplace root.fst rootlabel [rule1.fst rule1.label ...] out.fst  
Reweight Reweight(&A, potentials, type); creates equiv. FST changing arc weights based on potentials
  fstreweight [--to_final] in.fst potentials.txt out.fst  
Reverse Reverse(A, &B); contains the reversed strings in A
  fstreverse a.fst out.fst  
RmEpsilon RmEpsilon(&A); creates equiv. FST with no input/output epsilons
  RmEpsilonFst<Arc>(A);  
  fstrmepsilon a.fst out.fst  
ShortestDistance ShortestDistance(A, &distance); shortest distance from initial state to each state
  ShortestDistance(A, &distance, true); shortest distance from each state to final states
  fstshortestdistance [--reverse] in.fst [distance.txt]  
ShortestPath ShortestPath(A, &B, nshortest=1); N-best paths
  fstshortestpath [--nshortest=$n] a.fst out.fst  
Synchronize SynchronizeFst<Arc>(A); synchronizes an Fst
  fstsynchronize a.fst out.fst  
TopSort TopSort(&A); sorts an acyclic FST so that all transitions are from lower to higher state IDs
  fsttopsort a.fst out.fst  
Union Union(&A, B); contains strings in A or B
  UnionFst<Arc>(A, B);  
  fstunion a.fst b.fst out.fst  

FST Weights

An arc weight in an FST gives the cost of taking that transition. The OpenFst library supports multiple types of weights -- in fact, any C++ class that meets various properties can be used as the Weight type specified in the Arc template parameter of an Fst. A few weight types are predefined in the library that will normally meet your needs. Among a weight's properties, it must have associated binary functions Plus(x, y) and Times(x, y) and static member functions Zero() and One(). See FST Weights for constraints on these operations and other properties of weights. Plus is used to combine the weight of two identically labeled alternative paths, while Times is used to combine weights along a path or when matching paths in composition or intersection. A state is final if and only its final weight is not equal to Zero. A transition with weight One is, in essense, "free". A transition with weight Zero is not allowed (since Zero weight paths present technical problems with some algorithms).

The following are useful weight types:

Name Set Plus Times Zero One
Boolean { 0, 1 } or and 0 1
Real [0, inf] + * 0 1
Log [-inf, inf] -log(e^-x + e^-y) + inf 0
Tropical [-inf, inf] min + inf 0

The boolean weight type is used for the familiar unweighted automata (but see tropical below). The real weight type is appropriate when the transition weights represent probabilities. The log weight type is appropriate when the transition weights represent negative log probabilities (more numerically stable than the isomorphic, under log(), real weight type). The tropical weight type is appropriate for shortest path operations and is identical to the log except it uses min for the Plus operation.

The OpenFst library predefines TropicalWeight [bad link?] and LogWeight [bad link?] as well as the corresponding StdArc and LogArc. These weight classes represent their weight in a single precision float that is a constructor argument. That float can be directly accessed with member function Value(). For unweighted automata, it is convenient and customary in this library to use TropicalWeight restricted to Zero and One. StdArc is the default arc type for the FST binaries.

From the shell-level, the FST arc type can be specified to fstcompile with the --arc_type flag; StdArc is the default.

-- MichaelRiley - 25 May 2007

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