# Pynini: Grammar compilation in Python


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## Introduction

Pynini (Gorman 2016) is a library for compiling a grammar of strings, regular expressions, and context-dependent rewrite rules into weighted finite-state transducers.

### Design

Pynini supports much of the functionality of Thrax, but whereas Thrax is essentially a compiler for a domain-specific language, Pynini is implemented as a Python extension module. This provides several advantages:

• Pynini users can exploit Python's rich tooling ecosystem, including logging and unit testing frameworks.
• Pynini users can incorporate Thrax primitives like string compilation into executables.

Pynini is based on the OpenFst library, a C++11 template library for weighted finite-state transducers, and particularly its Python extension.

### Outline

This document covers the following areas:

Those who are familiar with FSTs may prefer to jump straight to getting started with Pynini. Or, for a quick start, you may also wish to read our Pynini tutorial for O'Reilly.

## Formal preliminaries

Finite-state transducers (FSTs) are models of computation widely used for speech and language processing:

• Speech recognition: language models, the pronunciation dictionaries, and the lattices produced by the acoustic model are all represented as FSTs; combined, these define the hypothesis space for the recognizer.
• Speech synthesis: FSTs are used for text normalization, the phase of linguistic analysis that converts written text like "meet me at St. Catherine on Bergen St. @ 9:30" to a pronounceable utterance like "meet me at the Saint Catherine on Bergen Street at nine thirty".
• Natural language processing: FSTs are used to represent sequence models such as those used for part of speech tagging, noun chunking, and named entity recognition.
• Information retrieval: finite automata are used to detect dates, times, etc., in web text.

FSTs are less "expressive" than e.g., arbitrary C++ functions or neural networks. However, the formal constraints that limit their expressivity guarantee that FSTs can be efficiently combined, optimized, and searched.

### State machines

Finite automata are simply state machines with a finite number of states. And, a state machine is any devices whose behavior can be modeled using transitions between a number of states. While our finite automata are implemented in software, there are also hardware state machines, like the traditional gumball machine (image credit: Wikimedia Commons):

A working gumball machine can be in one of two states: either its coin slot does, or does not, contain a coin. Initially the gumball machine has an empty coin slot; and turning the knob has no effect on the state of the machine. But once a coin is inserted into the coin slot, turning the knob dispenses a gumball and empties the coin slot. If the knob is turned while the coin slot is vacant, no change of state occurs. However, if the knob is turned while the coin slot is occupied, a gumball is dispensed and the coin slot is cleared. We can represent this schematic description of a gumball machine as a directed graph in which the states are represented by circles and the transitions between states—henceforth, arcs—are represented by arrows between states. Each arc has a pair of labels representing the input consumed, and output emitted, respectively, when that arc is traversed. By convention, the Greek letter ε (epsilon) is used to represent null input and/or output along an arc.

The bold double-circle state labeled 0 simply indicates that that state is "final", whereas the single-circle state 1 is non-final. Here, this encodes the notion that no user would walk away from a gumball machine while there's still a coin in the slot.

### Finite-state acceptors

In what follows, we draw heavily from chapter 1 of Roark & Sproat 2008, and interested readers are advised to consult that resource for further details.

#### Definition

Finite-state acceptors (or FSAs) are the simplest type of FST. Each FSA represents a set of strings (henceforth, a language), similar to a regular expression, as follows. An FSA consists of a five-tuple (Q, S, F, Σ δ) where:

1. Q is a finite set of states,
2. S is the set of start states,
3. F is the set of final states,
4. Σ is the alphabet, and
5. δ : Q × (Σ ∪ ε) → Q is the transition relation.

A path through an FSA (Q, S, F, Σ δ) is simply a sequence of transitions starting with a start state sS, taking only transitions licensed by δ, and terminating at a final state fF. More formally, let a path be a sequence P = p0, p1, …, pn where each element pP is a triple over Q × (Σ ∪ ε) × Q. Each triple consists of a source state, a label in the alphabet (or the null label ε) and a destination state. Then, a path is valid if and only if all of the following conditions hold:

1. The source state for p0 is in S,
2. δ(qi li) = qi + 1 for any pi = (qi, li, qi + 1) ∈ P, and
3. the destination state for pn is in F.

The first condition requires that the path begin at a start state, the second condition requires that all transitions be licensed by δ, and the third condition requires that the final transition be to a final state.

Let the string of a path be the (possibly empty) sequence of labels. For instance, if P is (s, l0, q1, (q1, l1, q2), (q2, l2, q3), then the string of P is simply l0, lsub>1, l2. The language described by an FSA is simply the union of all strings of all its paths.

#### Example

The following FSA represents the Perl-style regular expression ab*cd+e:

Here, the bold circle labeled 0 indicates the start state, and the double circle labeled 4 indicates the final state.

### Finite-state transducers

#### Definition

Finite-state transducers (FSTs) are generalization of FSAs. In the normal case of a two-way transducer, δ is instead a relation from Q × (Σi ∪ ε) × (Σo ∪ ε) → Q where Σiand Σo are the input and output alphabets, respectively. Paths through a FST are defined similarly to the definition given for FSAs above, except that each path corresponds to a set of two strings, an input string over Σi* and an output string over Σo*. Whereas FSAs describe sets of strings, FSTs describe relations between sets of strings.

When the relation described by an FST is such that each input string corresponds to at most one output string, we say that the FST is functional.

#### Example

The following FST represents the relation (a:a)(b:b)*(c:g)(d:f)+(e:e):

### Weighted finite-state transducers

#### Definition

Weighted finite-state transducers (WFSTs) are generalizations of FSTs which use an alternative definition of both F and δ incorporating the notion of weights. FST weights and their operations can be understood by first defining the notion of semiring, which require us to first define the notion monoid.

A monoid is a pair (M, •) where M is a set and • is a binary operation on M such that:

1. • is closed over M: for all a, b in M, abM,
2. there is an identity element eM such that ae = e • = a for all aM, and
3. • is associative; for all a, b, cM, (ab) • c = a • (bc).

A monoid (M, •) is said to be commutative if for all a, bM, ab = ba.

Then, a semiring is a triple (𝕂, ⊕, ⊗) such that:

1. (𝕂, ⊕) is a commutative monoid,
2. (𝕂, ⊗) is a monoid,
3. for all a, b, c ∈ 𝕂, a ⊗ (bc) = (a ⊗ b) ⊕ (ac), and
4. for all a ∈ 𝕂, a0 = 0a = 0 where 0 is the identity element for the monoid (𝕂, ⊕).

In many cases, 𝕂 is the set of real numbers, so a semiring can be denoted simply by specifying the pair of operators (⊕ ⊗). The so-called real semiring is simply (+, ×).

When working with probabilities as weights, we often use the tropical semiring (min, +) and negative log probabilities, taking advantage of the logarithmic identity log(x) + log(y) = log(xy). The tropical semiring (and the associated standard arc type) is the default semiring in Pynini.

At last, we can give the modified definitions for F and delta; for WFSTs. Whereas for unweighted FSTs, F is a set of final states, for WFSTs F is a set of pairs over Q × 𝕂, where the second element is the final weight for that state. And, the transition relation δ for a WFST is from Q × (Σi ∪ ε) × (Σo ∪ ε) × 𝕂 to Q. The definition of paths is parallel to those for unweighted FSTs except that each element in the path is also associated with a weight in 𝕂.

#### Example

WFSTs are a natural representation for conditional probability distributions from strings to strings. For example, consider a text normalization rule which verbalizes 2:00 as two with P = .2 and as two o'clock with P = .8. The following probability (i.e., real-valued) semiring WFST encodes this distribution:

### Conventions used in Pynini

• Pynini represents all acceptors and transducers, weighted or unweighted, as WFSTs. Thus, for example, a weighted finite-state acceptor (WFSA) is represented as a WFST in which input and output labels match in all cases, and an unweighted finite-state transducer is represented by a WFST in which all weights are 1 or 0.
• Pynini only permits one state to be designated the start state.
• Pynini assigns a final weight to all states; a nonfinal state is just one which has a final weight of 0.

## Getting started with Pynini

### Starting Pynini

Install Pynini then simply import the module as follows:

import pynini


### Building FSTs

In Pynini, all FST objects are an instance of a class called pynini.Fst, representing a mutable WFST. The user must specify the arc type at construction time; by default, the standard arc type (and the associated tropical semiring) is are used.

Pynini provides several functions for compiling strings into FSTs; we proceed to review a few of these methods.

#### Constructing acceptors

The acceptor function compiles a (Unicode or byte) string into a deterministic acceptor. The user may specify a final weight using the weight keyword argument; by default, the final weight is 1. The user may also specify the desired arc type using the arc_type keyword argument. The user may also specify how the characters in the string are to be translated into labels of the acceptor. By default (token_type="byte"), each arc in the acceptor corresponds to byte in the input string.

# UTF-8 bytestring, byte arcs.
print pynini.acceptor("Pont l'Evêque")
# Output below...


0 1 P
1 2 o
2 3 n
3 4 t
4 5 <SPACE>
5 6 l
6 7 '
7 8 E
8 9 v
9 10  <0xc3>
10  11  <0xaa>
11  12  q
12  13  u
13  14  e
14


If the user specifies token_type="utf8" then each arc in the FST corresponds to a Unicode codepoint.

# UTF-8 bytestring, Unicode codepoint arcs.
print pynini.acceptor("Pont l'Evêque", token_type="utf8")
# Output below...


0 1 P
1 2 o
2 3 n
3 4 t
4 5 <SPACE>
5 6 l
6 7 '
7 8 E
8 9 v
9 10  <0xea>
10  11  q
11  12  u
12  13  e
13


Sequences of characters enclosed by [ and ] receive special interpretations in byte or utf8 mode. Pynini first attempts to parse any such sequence as an integer using the C standard library function strtoll.

assert pynini.acceptor("b0x61") == pynini.acceptor("baa")  # OK.


If this fails, then Pynini treats the sequence as a sequence of one or more whitespace-delimited "generated" symbols, each of which is given a unique identifier in the resulting FST's symbol table.

x = pynini.acceptor("[It's not much of a cheese shop really]")
y = pynini.acceptor("[It's][not][much][of][a][cheese][shop][really]")
assert x == y   # OK.


A bracket character to be interpreted literally must be escaped.

bracket = pynini.acceptor("[")  # OK.


unused = pynini.acceptor("[")  # Not OK: Raises FstStringCompilationError.


Finally, if the user specifies a SymbolTable as the token_type, then the input string is parsed according and the FST labeled according to that table. String parsing failures are logged and raise a FstStringCompilationError exception from within the Python interpreter.

Nearly all FST operations permit a string to be passed in place of an Fst argument; in that case, pynini.acceptor is used (with default arguments) to compile the string into an FST.

#### Constructing transducers

The transducer function creates a transducer from two FSTs, compiling string arguments into FSTs if necessary. The result is a cross-product of the two input FSTs, interpreted as acceptors. Specifically, the transducer is constructed by mapping output arcs in the first FST to epsilon, mapping the input arcs in the second FST to epsilon, then concatenating the two. In the case that both FSTs are strings, the resulting transducer will simply contain the input and output string converted into labels, and the shorter of the two will be padded with epsilons.

As with acceptor, the user may specify a final weight using the weight keyword argument; the final weight is 1. The user also specify the desired arc type using the arc_type keyword argument. The user may also specify how the characters in the input and/or output strings are to be translated into labels of the transducer, if strings are passed in place of FST arguments. By default (input_token_type="byte" and output_token_type="byte"), each arc corresponds to a byte in the input and/or output string.

Whereas transducer is used to construct the cross-product of two strings, the union of many such string pair cross-products is compiled using the string_map and string_file functions. The former takes 2. δ(qi li) = qi + 1 for any pi = (qi, li, qi + 1) ∈ P, and an iterable of pairs of strings (or a string-to-string dictionary) as an argument; the latter reads string pairs from two-column TSV file. Both functions use a compact prefix tree representation which can be used to represent a (multi)map.

As with the above, the user may specify the desired arc type using the arc_type keyword argument. And as with transducer, the user may also specify how the characters in the input and/or output strings are to be translated into labels of the transducer using the appropriate keyword arguments.

### Worked examples

The following examples, taken from Gorman 2016 and Gorman & Sproat 2016, show features of Pynini in action.

#### Finnish vowel harmony

Koskenniemi (1983) provides a number of manually-compiled FSTs modeling Finnish morphophonological patterns. One of these concerns the well-known pattern of Finnish vowel harmony. Many Finnish suffixes have two allomorphs differing only in the backness specification of their vowel. For example, the adessive suffix is usually realized as -llä [lːæː] except when the preceding stem contains one of u, o, and a and there is no intervening y, ö, or ä; in this case, it is -lla [lːɑː]. For example, käde 'hand' has an adessive kädellä, whereas vero 'tax' forms the adessive verolla because the nearest stem vowel is o (Ringen & Heinämäki 1999).

The following gives a Pynini function make_adessive which generates the appropriate form of the noun stem. It first concatenates the stem with an abstract representation of the suffix, and then composes the result with a context-dependent rewrite rule adessive_harmony.

back_vowel = pynini.union("u", "o", "a")
neutral_vowel = pynini.union("i", "e")
front_vowel = pynini.union("y", "ö", "ä")
vowel = pynini.union(back_vowel, neutral_vowel, front_vowel)
archiphoneme = pynini.union("A", "I", "E", "O", "U")
consonant = pynini.union("b", "c", "d", "f", "g", "h", "j", "k", "l", "m", "n",
"p", "q", "r", "s", "t", "v", "w", "x", "z")
sigma_star = pynini.union(vowel, consonant, archiphoneme).closure()
intervener = pynini.union(consonant, neutral_vowel).closure()
back_vowel + intervener, "", sigma_star) *
pynini.cdrewrite(pynini.t("A", "ä"), "", "", sigma_star)
).optimize()



#### Pluralization rules

Consider an application where one is generating text such as, "The current temperature in New York is 25 degrees". Typically one would do this from a template such as the following:

The current temperature in __ is __ degrees


This works fine if one fills in the first blank with the name of a location and the second blank with a number. But what if it’s really cold in New York and the thermometer shows 1° (Fahrenheit)? One does not want this:

The current temperature in New York is 1 degrees


So one needs to have rules that know how to inflect the unit appropriately given the context. The Unicode Consortium has some guidelines for this; they primarily specify how many “plural” forms different languages have and in which circumstances one uses each. From a linguistic point of view the specifications are sometimes nonsensical, but they are widely used and are useful for adding support for new languages.

For handling degrees, we can assume that the default is the plural form, in which case we must singularize the plural form in certain contexts. For the word degrees and many other cases, it’s just a matter of stripping off the final s, but for a word ending in -ches (such as inches) one would want to strip off the es, and for feet and pence the necessary changes are irregular (foot, penny).

One could of course take care of this with a special purpose Python function, but here we consider how simple this is in Pynini. We start by defining singular_map to handle irregular cases, plus the general cases:

singular_map = pynini.union(
pynini.transducer("feet", "foot"),
pynini.transducer("pence", "penny"),
# Any sequence of bytes ending in "ches" strips the "es";
# the last argument -1 is a "weight" that gives this analysis
# a higher priority, if it matches the input.
sigma_star + pynini.transducer("ches", "ch", -1),
# Any sequence of bytes ending in "s" strips the "s".
sigma_star + pynini.transducer("s", ""))


Then as before we can define a context-dependent rewrite rule that performs the singularization just in case the unit is preceded by 1. We define the right context for the application of the rule rc as punctuation, space, or [EOS], a special symbol for the end-of-string.

rc = pynini.union(".", ",", "!", ";", "?", " ", "[EOS]")
singularize = pynini.cdrewrite(singular_map, " 1 ", rc, sigma_star)


Then we can define a convenience function to apply this rule...

def singularize(string):
return pynini.shortestpath(
pynini.compose(string.strip(), singularize)).stringify()


...and use it like so:

>>> singularize("The current temperature in New York is 1 degrees")
"The current temperature in New York is 1 degree"
>>> singularize("That measures 1 feet")
"That measures 1 foot"
>>> singularize("That measures 1.2 feet")
"That measures 1.2 feet"
>>> singularize("That costs just 1 pence")
"That costs just 1 penny"


One can handle other languages by changing the rules appropriately—for example, in Russian, which has http://www.unicode.org/cldr/charts/29/supplemental/language_plural_rules.html[more complex pluralization rules]], one needs several different forms, not just singular and plural—and by changing the conditions under which the various rules apply.

#### T9 disambiguation

T9 (short for "Text on 9 keys"; Grover et al. 1998) is a patented predictive text entry system. In T9, each character in the "plaintext" alphabet is assigned to one of the 9 digit keys (0 is usually reserved to represent space) of the traditional 3x4 touch-tone phone grid. For instance, the message GO HOME is entered as 4604663. Since the resulting "ciphertext" may be highly ambiguous—this sequence could also be read as GO HOOD or as many nonsensical expressions—a hybrid language model/lexicon is used for decoding.

The following snippet implements T9 encoding and decoding in Pynini:

LM = pynini.Fst.read("charlm.fst")
T9_ENCODER = pynini.string_file("t9.tsv").closure()
T9_DECODER = pynini.invert(T9_ENCODER)

def encode_string(plaintext):
return (plaintext * T9_ENCODER).stringify()

def k_best(ciphertext, k):
lattice = (ciphertext * T9_DECODER).project(True) * LM
return pynini.shortestpath(lattice, nshortest=k, unique=True)


The first line reads a language model (LM), represented as a WFSA. The second line reads the encoder table from a TSV file: in this file, each line contains an alphabetic character and the corresponding digit key. By computing the concatenative closure of this map, one obtains an FST T9_ENCODER which can encode arbitrarily-long plaintext strings. The encode_string function applies this FST to arbitrary plaintext strings: application here refers to composition of a string with a transducer followed by output projection. The k_best function first applies a ciphertext string—a bytestring of digits—to the inverse of the encoder FST (T9_DECODER). This creates an intermediate lattice of all possible plaintexts consistent with the T9 ciphertext. This is then scored with—that is, composed with—the character LM. Finally, this function returns the k highest probability plaintexts in the lattice. For the following example, the highest probability plaintext is in fact the correct one:

pt = "THE SINGLE MOST POPULAR CHEESE IN THE WORLD"
ct = encode_string(pt)
print "CIPHERTEXT:", ct
print "DECIPHERED PLAINTEXT:", k_best(ct, 1).stringify()
---+ Output below...


CIPHERTEXT: 8430746453066780767852702433730460843096753
DECIPHERED PLAINTEXT: THE SINGLE MOST POPULAR CHEESE IN THE WORLD


## API reference

This section covers classes (representing FSTs, weights, and the like), their methods (representing various accessors as well as destructive FST algorithms), and functions (representing constructive FST algorithms, weight arithmetic, and the like).

### Objects

#### Arc

An FST arc is essentially a tuple of input label, output label, weight, and the next state ID.

##### Constructor

Arc(ilabel, olabel, weight, nextstate)

This class represents an arc while remaining agnostic about the underlying arc type.  Attributes of the arc can be accessed or mutated, and the arc can be copied.

Attributes:
ilabel: The input label.
olabel: The output label.
weight: The arc weight.
nextstate: The destination state for the arc.


(None.)

#### ArcIterator

An arc iterator is used to iterate through arcs leaving some state in an FST; it is normally constructed by calling the arcs method of an Fst object.

##### Constructor

ArcIterator(ifst, state)

This class is used for iterating over the arcs leaving some state of an FST.
It supports the full C++ API, but most users should just call the arcs
method of an FST object and take advantage of the Pythonic API.


##### Methods

done(self)

Indicates whether the iterator is exhausted or not.

Returns:
True if the iterator is exhausted, False otherwise.


next(self)



flags(self)

Returns the current iterator behavioral flags.

Returns:
The current iterator behavioral flags as an integer.


position(self)

Returns the position of the iterator.

Returns:
The iterator's position, expressed as an integer.


reset(self)

Resets the iterator to the initial position.


seek(self, a)

Advance the iterator to a new position.

Args:
a: The position to seek to.


set_flags(self, flags, mask)

Sets the current iterator behavioral flags.

Args:
flags: The properties to be set.
mask: A mask to be applied to the flags argument before setting them.


value(self)

Returns the current arc.

Returns:
The current arc.


#### EncodeMapper

An encode mapper is used to "encode" and "decode" an FST; it is frequently used to apply FST algorithms while viewing the input FST as unweighted, as an acceptor, or both.

##### Constructor

EncodeMapper(arc_type="standard", encode_labels=False, encode_weights=False)

Arc encoder class, wrapping EncodeMapperClass.

This class provides an object which can be used to encode or decode FST arcs.
This is most useful to convert an FST to an unweighted acceptor, on which
some FST operations are more efficient, and then decoding the FST afterwards.

To use an instance of this class to encode or decode a mutable FST, pass it
as the first argument to the FST instance methods encode and decode.

For implementational reasons, it is not currently possible to use an encoder
on disk to construct this class.

Args:
arc_type: A string indicating the arc type.
encode_labels: Should labels be encoded?
encode_weights: Should weights be encoded?


##### Methods

arc_type(self)

Returns a string indicating the arc type.


flags(self)

Returns the encoder's flags.


input_symbols(self)

Returns the encoder's input symbol table, or None if none is present.


output_symbols(self)

Returns the encoder's output symbol table, or None if none is present.


properties(self, mask)

Provides property bits.

Args:

Returns:
A 64-bit bitmask representing the requested properties.


set_input_symbols(self, syms)

Sets the encoder's input symbol table.

Args:
syms: A SymbolTable.


set_output_symbols(self, syms)

Sets the encoder's output symbol table.

Args:
syms: A SymbolTable.


weight_type(self)

Returns a string indicating the weight type.


#### Far

A FAR is an "FST archive": a file used to store multiple FSTs. Each FST in a FAR is stored with a string key.

The Far object behaves similarly to a Python file handle: it can be opened for reading, or for writing. When opened for reading, it behaves similarly to an iterator. FSTs can be written or read using Python's [] syntax, as well as via instance methods. A Far can also be used as part of a PEP-343 context guard.

##### Constructor

Far(filename, mode="r", arc_type="standard", far_type="default")

Pynini FAR ("Fst ARchive") object.

This class is used to either read FSTs from or write FSTs to a FAR. When
opening a FAR for writing, the user may also specify the desired arc type
and FAR type.

Args:
filename: A string indicating the filename.
mode: FAR IO mode; one of: "r" (open for reading), "w" (open for writing).
arc_type: Desired arc type; ignored if the FAR is opened for reading.
far_type: Desired FAR type; ignored if the FAR is opened for reading.


##### Methods

arc_type(self)

Returns a string indicating the arc type.


closed(self)

Indicates whether the FAR is closed for IO.


far_type(self)

Returns a string indicating the FAR type.


mode(self)

Returns a char indicating the FAR's current mode.


name(self)

Returns the FAR's filename.


###### Methods when opened for reading (mode="r")

done(self)

Indicates whether the iterator is exhausted or not.

Returns:
True if the iterator is exhausted, False otherwise.


find(self, key)

Sets the current position to the first entry greater than or equal to the key (a
string) and indicates whether or not a match was found.

Args:
key: A string key.

Returns:
True if the key was found, False otherwise.


get_fst(self)

Returns the FST at the current position.

Returns:
A copy of the FST at the current position.


get_key(self)

Returns the string key at the current position.

Returns:
The string key at the current position.


next(self)



reset(self)

Resets the iterator to the initial position.


###### Methods when opened for writing (mode="w")

add(self, key, fst)

Adds an FST to the FAR (when open for writing).

This methods adds an FST to the FAR which can be retrieved with the
specified string key.

This method is provided for compatibility with the C++ API only; most users
should use the Pythonic API.

Args:
key: The string used to key the input FST.
fst: The FST to write to the FAR.

Raises:
FstOpError: Cannot invoke method in current mode.
FstOpError: Incompatible or invalid arc type.


close(self)

Closes the FAR and flushes to disk (when open for writing).

Raises:
FstOpError: Cannot invoke method in current mode.


#### Fst

This class represents a mutable, expanded FST. In addition to its constructor, it can be created using the string compilation function acceptor, the cross-product construction functions transducer, string_file, and string_map, and by various constructive FST algorithms.

##### Constructors

Fst(arc_type="standard")

This class wraps a mutable FST and exposes all destructive methods.

Args:
arc_type: An optional string indicating the arc type for the FST.


Fst.read(filename)

Reads an FST from a file.

Args:
filename: The string location of the input file.

Returns:
An FST.

Raises:


Fst.from_pywrapfst(ifst)

Constructs a Pynini FST from a pywrapfst._Fst.

This class method converts an FST from the pywrapfst module (pywrapfst._Fst
or its subclass pywrapfst._MutableFst) to a Pynini.Fst. This is essentially
a downcasting operation which grants the object additional instance methods,
including an enhanced closure, paths, and stringify. The input FST is
not invalidated, but mutation of the input or output object while the other
is still in scope will trigger a deep copy.

Args:
ifst: Input FST of type pywrapfst._Fst.

Returns:
An FST.


##### Methods

Note that many methods here are "destructive" in that they mutate the underlying FST. Wherever possible, such methods return self to allow method chaining.

add_arc(self, state, arc)

Adds a new arc to the FST and return self.

Args:
state: The integer index of the source state.

Returns:
self.

Raises:
FstIndexError: State index out of range.
FstOpdexError: Incompatible or invalid weight type.


add_state(self)

Adds a new state to the FST and returns the state ID.

Returns:
The integer index of the new state.


arc_type(self)

Returns a string indicating the arc type.


arcs(self, state)

Returns an iterator over arcs leaving the specified state.

Args:
state: The source state ID.

Returns:
An ArcIterator.


arcsort(self, sort_type="ilabel")

Sorts arcs leaving each state of the FST.

This operation destructively sorts arcs leaving each state using either
input or output labels.

Args:
sort_type: Either "ilabel" (sort arcs according to input labels) or
"olabel" (sort arcs according to output labels).

Returns:
self.

Raises:
FstArgError: Unknown sort type.


closure(self, lower)
closure(self, lower, upper)

Computes concatenative closure.

This operation destructively converts the FST to its concatenative closure.
If A transduces string x to y with weight w, then the zero-argument form
A.closure() mutates A so that it transduces between empty strings with
weight 1, transduces string x to y with weight w, transduces xx to yy with
weight w \otimes w, string xxx to yyy with weight w \otimes w \otimes w
(and so on).

When called with two optional positive integer arguments, these act as
lower and upper bounds, respectively, for the number of cycles through the
original FST that the mutated FST permits. Therefore, =A.closure(0, 1)=
mutates A so that it permits 0 or 1 cycles; i.e., the mutated A transduces
between empty strings or transduces x to y.

When called with one optional positive integer argument, this argument
acts as the lower bound, with the upper bound implicitly set to infinity.
Therefore, A.closure(1) performs a mutation roughly equivalent to
A.closure() except that the former does not transduce between empty
strings.

The following are the equivalents for the closure-style syntax used in
Perl-style regular expressions:

Regexp:             This method:            Copy shortcuts:

/x?/                x.closure(0, 1)         x.ques
/x*/                x.closure()             x.star
/x+/                x.closure(1)            x.plus
/x{N}/              x.closure(N, N)
/x{M,N}/            x.closure(M, N)
/x{N,}/             x.closure(N)
/x{,N}/             x.closure(0, N)

Args:
lower: lower bound.
upper: upper bound.

Returns:
self.


concat(self, ifst)

Computes the concatenation (product) of two FSTs.

This operation destructively concatenates the FST with a second FST. If A
transduces string x to y with weight a and B transduces string w to v with
weight b, then their concatenation transduces string xw to yv with weight
a \otimes b.

Args:
ifst: The second input Fst.

Returns:
self.

Raises:
FstOpError: Operation failed.
FstSymbolTableMergeError: Unable to resolve symbol table conflict
without relabeling.


connect(self)

Removes unsuccessful paths.

This operation destructively trims the FST, removing states and arcs that
are not part of any successful path.

Returns:
self.


copy(self)

Makes a copy of the FST.


decode(self, encoder)

Decodes encoded labels and/or weights.

This operation reverses the encoding performed by encode.

Args:
encoder: An EncodeMapper object used to encode the FST.

Returns:
self.


delete_arcs(self, state, n=0)

Deletes arcs leaving a particular state.

Args:
state: The integer index of a state.
n: An optional argument indicating how many arcs to be deleted. If this
argument is omitted or passed as zero, all arcs from this state are
deleted.

Returns:
self.

Raises:
FstIndexError: State index out of range.


delete_states(self, states=None)

Deletes states.

Args:
states: An optional iterable of integer indices of the states to be
deleted. If this argument is omitted, all states are deleted.

Returns:
self.

Raises:
FstIndexError: State index out of range.


draw(self, filename, isymbols=None, osymbols=None, ssymbols=None,
acceptor=False, title="", width=8.5, height=11, portrait=False,
vertical=False, ranksep=0.4, nodesep=0.25, fontsize=14,
precision=5, float_format="g", show_weight_one=False):

Writes out the FST in Graphviz text format.

This method writes out the FST in the dot graph description language. The
graph can be rendered using the dot executable provided by Graphviz.

Args:
filename: The string location of the output dot/Graphviz file.
isymbols: An optional symbol table used to label input symbols.
osymbols: An optional symbol table used to label output symbols.
ssymbols: An optional symbol table used to label states.
acceptor: Should the figure be rendered in acceptor format if possible?
title: An optional string indicating the figure title.
width: The figure width, in inches.
height: The figure height, in inches.
portrait: Should the figure be rendered in portrait rather than
landscape?
vertical: Should the figure be rendered bottom-to-top rather than
left-to-right?
ranksep: The minimum separation separation between ranks, in inches.
nodesep: The minimum separation between nodes, in inches.
fontsize: Font size, in points.
precision: Numeric precision for floats, in number of chars.
float_format: One of: 'e', 'f' or 'g'.
show_weight_one: Should weights equivalent to semiring One be printed?


encode(self, encoder)

Encodes labels and/or weights.

This operation allows for the representation of a weighted transducer as a
weighted acceptor, an unweighted transducer, or an unweighted acceptor by
considering the pair (input label, output label), the pair (input label,
weight), or the triple (input label, output label, weight) as a single
label. Applying this operation mutates the EncodeMapper argument, which
can then be used to decode.

Args:
encoder: An EncodeMapper object to be used as the encoder.

Returns:
self.


final(self, state)

Returns the final weight of a state.

Args:
state: The integer index of a state.

Returns:
The final Weight of that state.

Raises:
FstIndexError: State index out of range.


fst_type(self)

Returns a string indicating the FST type.


input_symbols(self)

Returns the FST's input symbol table, or None if none is present.


invert(self)

Inverts the FST's transduction.

This operation destructively inverts the FST's transduction by exchanging
input and output labels.

Returns:
self.


minimize(self, delta=0.0009765625, allow_nondet=False)

Minimizes the FST.

This operation destructively performs the minimization of deterministic
weighted automata and transducers. If the input FST A is an acceptor, this
operation produces the minimal acceptor B equivalent to A, i.e. the
acceptor with a minimal number of states that is equivalent to A. If the
input FST A is a transducer, this operation internally builds an equivalent
transducer with a minimal number of states. However, this minimality is
obtained by allowing transition having strings of symbols as output labels,
this known in the litterature as a real-time transducer. Such transducers
are not directly supported by the library. This function will convert such
transducer by expanding each string-labeled transition into a sequence of
transitions. This will results in the creation of new states, hence losing
the minimality property.

Args:
delta: Comparison/quantization delta.
allow_nondet: Attempt minimization of non-deterministic FST?

Returns:
self.


mutable_arcs(self, state)

Returns a mutable iterator over arcs leaving the specified state.

Args:
state: The source state ID.

Returns:
A MutableArcIterator.


mutable_input_symbols(self)

Returns the FST's (mutable) input symbol table, or None if none is present.


mutable_output_symbols(self)

Returns the FST's (mutable) output symbol table, or None if none is present.


num_arcs(self, state)

Returns the number of arcs leaving a state.

Args:
state: The integer index of a state.

Returns:
The number of arcs leaving that state.

Raises:
FstIndexError: State index out of range.


num_input_epsilons(self, state)

Returns the number of arcs with epsilon input labels leaving a state.

Args:
state: The integer index of a state.

Returns:
The number of epsilon-input-labeled arcs leaving that state.

Raises:
FstIndexError: State index out of range.


num_output_epsilons(self, state)

Returns the number of arcs with epsilon output labels leaving a state.

Args:
state: The integer index of a state.

Returns:
The number of epsilon-output-labeled arcs leaving that state.

Raises:
FstIndexError: State index out of range.


num_states(self)

Returns the number of states.


optimize(self, compute_props=False)

Performs a generic optimization of the FST.

This operation destructively optimizes the FST using epsilon-removal,
arc-sum mapping, determinization, and minimization (where possible). The
algorithm is as follows:

* If the FST is not (known to be) epsilon-free, perform epsilon-removal.
* Combine identically labeled multi-arcs and sum their weights.
* If the FST does not have idempotent weights, halt.
* If the FST is not (known to be) deterministic:
- If the FST is a (known) acceptor:
* If the FST is not (known to be) unweighted and/or acyclic, encode
weights.
- Otherwise, encode labels and, if the FST is not (known to be)
unweighted, encode weights.
- Determinize the FST.
* Minimize the FST.
* Decode the FST and combine identically-labeled multi-arcs and sum their
weights, if the FST was previously encoded.

By default, FST properties are not computed if they are not already set.

This optimization may result in a reduction of size (due to epsilon-removal,
arc-sum mapping, and minimization) and possibly faster composition, but
determinization (a prerequisite of minimization) may result in an
exponential blowup in size in the worst case. Judicious use of optimization
is a bit of a black art.

Args:
compute_props: Should unknown FST properties be computed to help choose
appropriate optimizations?

Returns:
self.


output_symbols(self)

Returns the FST's output symbol table, or None if none is present.


paths(self, token_type="byte)

Creates iterator over all string paths in an acyclic FST.

This method returns an iterator over all paths (represented as pairs of
strings and an associated path weight) through an acyclic FST. This
operation is only feasible when the FST is acyclic. Depending on the
requested token type, the arc labels along the input and output sides of a
path are interpreted as UTF-8-encoded Unicode strings, raw bytes, or a
concatenation of string labels from a symbol table.

Args:
input_token_type: A string indicating how the input strings are to be
constructed from arc labels---one of: "byte" (interprets arc labels
as raw bytes), "utf8" (interprets arc labels as Unicode code points),
or "symbol" (interprets arc labels using the input symbol table).
output_token_type: A string indicating how the output strings are to be
constructed from arc labels---one of: "byte" (interprets arc labels
as raw bytes), "utf8" (interprets arc labels as Unicode code points),
or "symbol" (interprets arc labels using the input symbol table).
rm_epsilon: Should epsilons be removed?

Raises:
FstArgError: Unknown token type.
FstArgError: FST is not acyclic.


project(self, project_output=False)

Converts the FST to an acceptor using input or output labels.

This operation destructively projects an FST onto its domain or range by
either copying each arc's input label to its output label (the default) or
vice versa.

Args:
project_output: Should the output labels be projected?

Returns:
self.


properties(self, mask)

Provides property bits.

Args:

Returns:
A 64-bit bitmask representing the requested properties.


prune(self, delta=0.0009765625, nstate=NO_STATE_ID, weight=None)

Removes paths with weights below a certain threshold.

This operation deletes states and arcs in the input FST that do not belong
to a successful path whose weight is no more (w.r.t the natural semiring
order) than the threshold t otimes-times the weight of the shortest path in
the input FST. Weights must be commutative and have the path property.

Args:
delta: Comparison/quantization delta.
nstate: State number threshold.
weight: A Weight or weight string indicating the desired weight threshold
below which paths are pruned; if omitted, no paths are pruned.

Returns:
self.


push(self, delta=0.0009765625, remove_total_weight=False, to_final=False)

Pushes weights towards the initial or final states.

This operation destructively produces an equivalent transducer by pushing
the weights towards the initial state or toward the final states. When
pushing weights towards the initial state, the sum of the weight of the
outgoing transitions and final weight at any non-initial state is equal to
one in the resulting machine. When pushing weights towards the final states,
the sum of the weight of the incoming transitions at any state is equal to
one. Weights need to be left distributive when pushing towards the initial
state and right distributive when pushing towards the final states.

Args:
delta: Comparison/quantization delta.
remove_total_weight: If pushing weights, should the total weight be
removed?
to_final: Push towards final states?

Returns:
self.


relabel_pairs(self, ipairs=None, opairs=None)

Replaces input and/or output labels using pairs of labels.

This operation destructively relabels the input and/or output labels of the
FST using pairs of the form (old_ID, new_ID); omitted indices are
identity-mapped.

Args:
ipairs: An iterable containing (older index, newer index) integer pairs.
opairs: An iterable containing (older index, newer index) integer pairs.

Returns:
self.

Raises:
FstArgError: No relabeling pairs specified.


relabel_tables(self, old_isymbols=None, new_isymbols=None,
unknown_isymbol="", attach_new_isymbols=True,
old_osymbols=None, new_osymbols=None,
unknown_osymbol="", attach_new_osymbols=True)

Replaces input and/or output labels using SymbolTables.

This operation destructively relabels the input and/or output labels of the
FST using user-specified symbol tables; omitted symbols are identity-mapped.

Args:
old_isymbols: The old SymbolTable for input labels, defaulting to the
FST's input symbol table.
new_isymbols: A SymbolTable used to relabel the input labels
unknown_isymbol: Input symbol to use to relabel OOVs (if empty,
OOVs raise an exception)
attach_new_isymbols: Should new_isymbols be made the FST's input symbol
table?
old_osymbols: The old SymbolTable for output labels, defaulting to the
FST's output symbol table.
new_osymbols: A SymbolTable used to relabel the output labels.
unknown_osymbol: Outnput symbol to use to relabel OOVs (if empty,
OOVs raise an exception)
attach_new_isymbols: Should new_osymbols be made the FST's output symbol
table?

Returns:
self.

Raises:
FstArgError: No SymbolTable specified.


reserve_arcs(self, state, n)

Reserve n arcs at a particular state (best effort).

Args:
state: The integer index of a state.
n: The number of arcs to reserve.

Returns:
self.

Raises:
FstIndexError: State index out of range.


reserve_states(self, n)

Reserve n states (best effort).

Args:
n: The number of states to reserve.

Returns:
self.


reweight(self, potentials, to_final=False)

Reweights an FST using an iterable of potentials.

This operation destructively reweights an FST according to the potentials
and in the direction specified by the user. An arc of weight w, with an
origin state of potential p and destination state of potential q, is
reweighted by p^{-1} otimes (w otimes q) when reweighting towards the
initial state, and by (p otimes w) otimes q^{-1} when reweighting towards
the final states. The weights must be left distributive when reweighting
towards the initial state and right distributive when reweighting towards
the final states (e.g., TropicalWeight and LogWeight).

Args:
potentials: An iterable of Weight or weight strings.
to_final: Push towards final states?

Returns:
self.


rmepsilon(self, connect=True, delta=0.0009765625, nstate=NO_STATE_ID,
weight=None)

Removes epsilon transitions.

This operation destructively removes epsilon transitions, i.e., those where
both input and output labels are epsilon) from an FST.

Args:
connect: Should output be trimmed?
delta: Comparison/quantization delta.
nstate: State number threshold.
weight: A Weight or weight string indicating the desired weight threshold
below which paths are pruned; if omitted, no paths are pruned.

Returns:
self.


set_final(self, state, weight)

Sets the final weight for a state.

Args:
state: The integer index of a state.
weight: A Weight or weight string indicating the desired final weight; if
omitted, it is set to semiring One.

Raises:
FstIndexError: State index out of range.
FstOpError: Incompatible or invalid weight.


set_input_symbols(self, syms)

Sets the input symbol table.

Passing None as a value will delete the input symbol table.

Args:
syms: A SymbolTable.

Returns:
self.


set_output_symbols(self, syms)

Sets the output symbol table.

Passing None as a value will delete the output symbol table.

Args:
syms: A SymbolTable.

Returns:
self.


set_properties(self, props, mask)

Sets the properties bits.

Args:
props: The properties to be set.
mask: A mask to be applied to the props argument before setting the
FST's properties.

Returns:
self.


set_start(self, state)

Sets a state to be the initial state.

Args:
state: The integer index of a state.

Returns:
self.

Raises:
FstIndexError: State index out of range.


start(self)

Returns the start state.


states(self)

Returns an iterator over all states in the FST.

Returns:
A StateIterator object for the FST.


stringify(self, token_type="byte")

Creates a string from a string FST.

This method returns the string recognized by the FST as a Python byte or
Unicode string. This is only well-defined when the FST is an acceptor and a
"string" FST (meaning that the start state is numbered 0, and there is
exactly one transition from each state i to each state i + 1, there are no
other transitions, and the last state is final). Depending on the requested
token type, the arc labels are interpreted as a UTF-8-encoded Unicode
string, raw bytes, or as a concatenation of string labels from the output
symbol table.

The underlying routine reads only the output labels, so if the FST is
not an acceptor, it will be treated as the output projection of the FST.

Args:
token_type: A string indicating how the string is to be constructed from
arc labels---one of: "byte" (interprets arc labels as raw bytes),
"utf8" (interprets arc labels as Unicode code points), or a
SymbolTable.
rm_epsilon: Should epsilons be removed?

Returns:
The string corresponding to (an output projection) of the FST.

Raises:
FstArgError: FST is not a string.
FstArgError: Unknown token type.


text(self, isymbols=None, osymbols=None, ssymbols=None, acceptor=False,
show_weight_one=False, missing_sym="")

Produces a human-readable string representation of the FST.

This method generates a human-readable string representation of the FST.
The caller may optionally specify SymbolTables used to label input labels,
output labels, or state labels, respectively.

Args:
isymbols: An optional symbol table used to label input symbols.
osymbols: An optional symbol table used to label output symbols.
ssymbols: An optional symbol table used to label states.
acceptor: Should the FST be rendered in acceptor format if possible?
show_weight_one: Should weights equivalent to semiring One be printed?
missing_symbol: The string to be printed when symbol table lookup fails.

Returns:
A formatted string representing the machine.


Note: use print f as an alias for print f.text().

topsort(self)

Sorts transitions by state IDs.

This operation destructively topologically sorts the FST, if it is acyclic;
otherwise it remains unchanged. Once sorted, all transitions are from lower
state IDs to higher state IDs

Returns:
self.


union(self, ifst)

Computes the union (sum) of two FSTs.

This operation destructively computes the union (sum) of two FSTs. If A
transduces string x to y with weight a and B transduces string w to v with
weight b, then their union transduces x to y with weight a and w to v with
weight b.

Args:
ifst: The second input FST.

Returns:
self.

Raises:
FstOpError: Operation failed.
FstSymbolTableMergeError: Unable to resolve symbol table conflict
without relabeling.


verify(self)

Verifies that an FST's contents are sane.

Returns:
True if the contents are sane, False otherwise.


Note: use this in combination with assertions in tests or library code.

weight_type(self)

Provides the FST's weight type.

Returns:
A string representing the weight type.


write(self, filename)

Serializes FST to a file.

This method writes the FST to a file in a binary format.

Args:
filename: The string location of the output file.

Raises:
FstIOError: Write failed.


write_to_string(self)

Serializes FST to a string.

Returns:
A string.


#### MPdtParentheses

The MPDT parentheses class stores triples consisting of "push" and "pop" parentheses arc labels and an associated stack label for a multi-pushdown transducer.

##### Constructor

MPdtParentheses()

Multi-pushdown transducer parentheses class.

This class wraps a vector of pairs of arc labels in which the first label is
interpreted as a "push" stack operation and the second represents the
corresponding "pop" operation, and an equally sized vector which assigns each
pair to a stack. The library currently only permits two stacks (numbered 1 and
2) to be used.

A MPDT is expressed as an (Fst, MPdtParentheses) pair for the purposes of all
supported MPDT operations.


##### Methods

add_triple(self, push, pop, assignment)

Adds a triple of (left parenthesis, right parenthesis, stack assignment)
triples to the object.

Args:
push: An arc label to be interpreted as a "push" operation.
pop: An arc label to be interpreted as a "pop" operation.
assignment: An arc label indicating what stack the parentheses pair
is assigned to.


copy(self)

Makes a copy of this MPdtParentheses object.

Returns:
A deep copy of the MPdtParentheses object.


MPdtParentheses.read(filename)

Reads parentheses/assignment triples from a text file.

This class method creates a new MPdtParentheses object from a pairs of
integer labels in a text file.

Args:
filename: The string location of the input file.

Returns:
A new MPdtParentheses instance.

Raises:


write(self, filename)

Writes parentheses triples to a text file.

This method writes the MPdtParentheses object to a text file.

Args:
filename: The string location of the output file.

Raises:
FstIOError: Write failed.


#### MutableArcIterator

A mutable arc iterator is used to iterate through arcs leaving some state in the FST; it is normally constructed by calling the mutable_arcs method of an Fst object.

This iterator is invalidated if the underlying FST is mutated by any other means than the set_value method.

##### Constructor

MutableArcIterator(ifst, state)

This class is used for iterating over the arcs leaving some state of an FST,
also permitting mutation of the current arc.


##### Methods

done(self)

Indicates whether the iterator is exhausted or not.


flags(self)

Returns the current iterator behavioral flags.

Returns:
The current iterator behavioral flags as an integer.


next(self)



position(self)

Returns the position of the iterator.

Returns:
The iterator's position, expressed as an integer.


reset(self)

Resets the iterator to the initial position.


seek(self, a)

Advance the iterator to a new position.

Args:
a: The position to seek to.


set_flags(self, flags, mask)

Sets the current iterator behavioral flags.

Args:
flags: The properties to be set.
mask: A mask to be applied to the flags argument before setting them.


set_value(self, arc)

Replace the current arc with a new arc.

Args:
arc: The arc to replace the current arc with.


value(self)

Returns the current arc.


#### PdtParentheses

The PDT parentheses class stores pairs of "push" and "pop" parentheses arc labels for a pushdown transducer.

##### Constructor

PdtParentheses()

Pushdown transducer parentheses class.

This class wraps a vector of pairs of arc labels in which the first label is
interpreted as a "push" stack operation and the second represents the
corresponding "pop" operation. When efficiency is desired, the push and pop
indices should be contiguous.

A PDT is expressed as an (Fst, PdtParentheses) pair for the purposes of all
supported PDT operations.


##### Methods

add_pair(self, push, pop)

Adds a pair of parentheses to the set.

Args:
push: An arc label to be interpreted as a "push" operation.
pop: An arc label to be interpreted as a "pop" operation.


copy(self)

Makes a copy of this PdtParentheses object.

Returns:
A deep copy of the PdtParentheses object.


PdtParentheses.read(filename)

Reads parentheses pairs from a text file.

This class method creates a new PdtParentheses object from a pairs of
integer labels in a text file.

Args:
filename: The string location of the input file.

Returns:
A new PdtParentheses instance.

Raises:


write(self, filename)

Writes parentheses triples to a text file.

This method writes the MPdtParentheses object to a text file.

Args:
filename: The string location of the output file.

Raises:
FstIOError: Write failed.


#### StateIterator

An state iterator is used to iterate through state IDs in an FST; it is normally constructed by calling the states method of an Fst object.

##### Constructor

StateIterator(ifst)

This class is used for iterating over the states in an FST.


##### Methods

done(self)

Indicates whether the iterator is exhausted or not.

Returns:
True if the iterator is exhausted, False otherwise.


next(self)



reset(self)

Resets the iterator to the initial position.


value(self)

Returns the current state index.


#### StringPaths

The string paths object is used to iterate through paths of an acyclic FST, in the forms of input and output label sequences, input and output strings, and path weights.

##### Constructor

StringPaths(fst, token_type="byte", isymbols=None, osymbols=None)

Iterator for string paths in acyclic FST.

This class provides an iterator over all paths (represented as pairs of
strings and an associated path weight) through an acyclic FST. This
operation is only feasible when the FST is acyclic. Depending on the
requested token type, the arc labels along the input and output sides of a
path are interpreted as UTF-8-encoded Unicode strings, raw bytes, or a
concatenation of string labels from a symbol table. This class is normally
created by invoking the paths method of Fst.

Args:
input_token_type: A string indicating how the input strings are to be
constructed from arc labels---one of: "byte" (interprets arc labels
as raw bytes), "utf8" (interprets arc labels as Unicode code points),
or a SymbolTable.
output_token_type: A string indicating how the output strings are to be
constructed from arc labels---one of: "byte" (interprets arc labels
as raw bytes), "utf8" (interprets arc labels as Unicode code points),
or a SymbolTable.
rm_epsilon: Should epsilons be removed?

Raises:
FstArgError: Unknown token type.
FstArgError: FST is not acyclic.


##### Methods

done(self)

Indicates whether the iterator is exhausted or not.


error(self)

Indicates whether the StringPaths has encountered an error.

Returns:
True if the StringPaths is in an errorful state, False otherwise.


ilabels(self)

Returns the input labels for the current path.

Returns:
A list of input labels for the current path.


istring(self)

Returns the current path's input string.

This method is provided for compatibility with the C++ API only; most users
should use the Pythonic API.

Returns:
The path's input string.


next(self)



iter_istrings(self)

Generates all input strings in the FST.

This method returns a generator over all input strings in the path. The
caller is responsible for resetting the iterator if desired.

Yields:
All input strings.


iter_ostrings(self)

Generates all output strings in the FST.

This method returns a generator over all output strings in the path. The
caller is responsible for resetting the iterator if desired.

Yields:
All output strings.


iter_weights(self)

Generates all path weights in the FST.

This method returns a generator over all path weights. The caller is
responsible for resetting the iterator if desired.

Yields:
All weights.


olabels(self)

Returns the output labels for the current path.

Returns:
A list of output labels for the current path.


ostring(self)

Returns the current path's output string.

Returns:
The path's output string.


weight(self)

Returns the current path's total weight.

Returns:
The path's Weight.


#### SymbolTable

A symbol table stores a bidirectional mapping between arc labels and "symbols" (strings). They can be attached to FSTs or encoders, or iterated through using a SymbolTableIterator.

##### Constructor

SymbolTable(name="<unspecified>")

Mutable SymbolTable class.

This class wraps the library SymbolTable and exposes both const (i.e.,
access) and non-const (i.e., mutation) methods of wrapped object.


##### Methods

add_symbol(self, symbol, key=NO_SYMBOL)

Adds a symbol to the table and returns the index.

This method adds a symbol to the table. The caller can optionally
specify a non-negative integer index for the key.

Args:
symbol: A symbol string.
key: An index for the symbol; if not specified, the next index will be
used.

Returns:
The integer key of the new symbol.


add_table(self, syms)

Adds another SymbolTable to this table.

This method merges another symbol table into the current table. All key
values will be offset by the current available key.

Args:
syms: A SymbolTable to be merged with the current table.


available_key(self)

Returns an integer indicating the next available key index in the table.


checksum(self)

Returns a string indicating the label-agnostic MD5 checksum for the table.


copy(self)

Returns a mutable copy of the SymbolTable.


find(self, key)

Given a symbol or index, finds the other one.

This method returns the index associated with a symbol key, or the symbol
associated with a index key.

Args:
key: Either a string or an index.

Returns:
If key is a string, the associated index; if key is an integer, the
associated symbol.

Raises:


get_nth_key(self, pos)

Retrieves the integer index of the n-th key in the table.

Args:
pos: The n-th key to retrieve.

Returns:
The integer index of the n-th key.

Raises:


labeled_checksum(self)

Returns a string indicating the label-dependent MD5 checksum for the table.


member(self, key)

Given a symbol or index, returns whether it is found in the table.

This method returns a boolean indicating whether the given symbol or index
is present in the table. If one intends to perform subsequent lookup, it is
better to simply call the find method, catching the KeyError.

Args:
key: Either a string or an index.

Returns:
Whether or not the key is present (as a string or a index) in the table.


name(self)

Returns the symbol table's name.


name(self)

Returns the symbol table's name.


SymbolTable.read(filename)

Reads symbol table from binary file.

This class method creates a new SymbolTable from a symbol table binary file.

Args:
filename: The string location of the input binary file.

Returns:
A new SymbolTable instance.


SymbolTable.read_text(filename)

Reads symbol table from text file.

This class method creates a new SymbolTable from a symbol table text file.

Args:
filename: The string location of the input text file.
allow_negative_labels: Should negative labels be allowed? (Not
recommended; may cause conflicts).

Returns:
A new SymbolTable instance.


write(self, filename)

Serializes symbol table to a file.

This methods writes the SymbolTable to a file in binary format.

Args:
filename: The string location of the output file.

Raises:
FstIOError: Write failed.


write_text(self, filename)

Writes symbol table to text file.

This method writes the SymbolTable to a file in human-readable format.

Args:
filename: The string location of the output file.

Raises:
FstIOError: Write failed.


#### SymbolTableIterator

A symbol table iterator is used to iterate through label/string pairs stored in a symbol table.

##### Constructor

SymbolTableIterator(syms)

This class is used for iterating over a symbol table.


##### Methods

done(self)

Indicates whether the iterator is exhausted or not.

Returns:
True if the iterator is exhausted, False otherwise.


next(self)



reset(self)

Resets the iterator to the initial position.


symbol(self)

Returns the current symbol string.

This method returns the current symbol string at this point in the table.

Returns:
A symbol string.


value(self)

Returns the current integer index of the symbol.

Returns:
An integer index.


#### Weight

The weight class is used to represent weights which can be attached to arcs or final states in an FST. They also support semiring-dependent arithmetic operations via external functions.

##### Constructors

Weight(weight_type, weight_string)

FST weight class.

This class represents an FST weight. When passed as an argument to an FST
operation, it should have the weight type of the input FST(s) to said
operation.

Args:
weight_type: A string indicating the weight type.
weight_string: A string indicating the underlying weight.

Raises:


Weight.NoWeight(weight_type)

Constructs a non-member weight in the semiring.


Weight.One(weight_type)

Constructs semiring One.


Weight.Zero(weight_type)

Constructs semiring zero.


##### Methods

copy(self)

Returns a copy of the Weight.


type(self)

Returns a string indicating the weight type.


#### Fst algorithm functions

FSTs support the following constructive FST algorithm functions, which take one or more Fst arguments and (usually) return a new Fst.

arcmap(ifst, delta=0.0009765625, map_type="identity", weight=None)

Constructively applies a transform to all arcs and final states.

This operation transforms each arc and final state in the input FST using
one of the following:

* identity: maps to self.
* input_epsilon: replaces all input labels with epsilon.
* invert: reciprocates all non-Zero weights.
* output_epsilon: replaces all output labels with epsilon.
* plus: adds a constant to all weights.
* quantize: quantizes weights.
* rmweight: replaces all non-Zero weights with 1.
* superfinal: redirects final states to a new superfinal state.
* times: right-multiplies a constant to all weights.
* to_log: converts weights to the log semiring.
* to_log64: converts weights to the log64 semiring.
* to_standard: converts weights to the tropical ("standard") semiring.

Args:
ifst: The input FST.
delta: Comparison/quantization delta (ignored unless map_type is
quantize).
map_type: A string matching a known mapping operation (see above).
weight: A Weight or weight string passed to the arc-mapper; ignored unless
map_type is plus (in which case it defaults to semiring Zero) or
times (in which case it defaults to semiring One).

Returns:
An FST.

Raises:
FstArgError: Unknown map type.


cdrewrite(tau, lambda, rho, sigma_star, direction="ltr", mode="obl")

Generates a transducer expressing a context-dependent rewrite rule.

This operation compiles a transducer representing a context-dependent
rewrite rule of the form

phi -> psi / lambda __ rho

over a finite vocabulary. To apply the resulting transducer, simply compose
it with an input string or lattice.

Args:
tau: A (weighted) transducer representing phi -> psi.
lambda: An unweighted acceptor representing the left context.
rho: An unweighted acceptor representing the right context.
sigma_star: A cyclic, unweighted acceptor representing the closure over the
alphabet.
direction: A string specifying the direction of rule application; one of:
"ltr" (left-to-right application), "rtl" (right-to-left application),
or "sim" (simultaneous application).
mode: A string specifying the mode of rule application; one of: "obl"
(obligatory application), "opt" (optional application).

Returns:
An rewrite rule FST.

Raises:
FstArgError: Unknown cdrewrite direction type.
FstArgError: Unknown cdrewrite mode type.
FstOpError: Operation failed.


compose(ifst1, ifst2, compose_filter="auto", connect=True)

Constructively composes two FSTs.

This operation computes the composition of two FSTs. If A transduces string
x to y with weight a and B transduces y to z with weight b, then their
composition transduces string x to z with weight a otimes b. The output
labels of the first transducer or the input labels of the second transducer
must be sorted (or otherwise support appropriate matchers).

Args:
ifst1: The first input FST.
ifst2: The second input FST.
compose_filter: A string matching a known composition filter; one of:
"alt_sequence", "auto", "match", "null", "sequence", "trivial".
connect: Should output be trimmed?

Returns:
A composed FST.


containment(ifst, sigma_star)

Creates a containment of a transducer with respect to a universal language.

This operation constructs a transducer consisting of the input FST optionally
preceded and/or followed by any string from some alphabet.

Args:
ifst: The input FST.
sigma_star: A cyclic, unweighted acceptor representing the closure over the
alphabet.

Returns:
An FST.

Raises:
FstOpError: Operation failed.
FstSymbolTableMergeError: Unable to resolve symbol table conflict without
relabeling.


convert(ifst, fst_type=None)

Constructively converts an FST to a new internal representation.

Args:
ifst: The input FST.
fst_type: A string indicating the FST type to convert to, or None if
no conversion is desired.

Returns:
An equivalent Fst converted to the desired FST type.


determinize(ifst, delta=0.0009765625, det_type="functional",
nstate=NO_STATE_ID, subsequential_label=0, weight=None,
incremental_subsequential_label=False)

Constructively determinizes a weighted FST.

This operations creates an equivalent FST that has the property that no
state has two transitions with the same input label. For this algorithm,
epsilon transitions are treated as regular symbols (cf. rmepsilon).

Args:
ifst: The input FST.
delta: Comparison/quantization delta.
det_type: Type of determinization; one of: "functional" (input transducer is
functional), "nonfunctional" (input transducer is not functional) and
disambiguate" (input transducer is not functional but only keep the min
of ambiguous outputs).
nstate: State number threshold.
subsequential_label: Input label of arc corresponding to residual final
output when producing a subsequential transducer.
weight: A Weight or weight string indicating the desired weight threshold
below which paths are pruned; if omitted, no paths are pruned.
increment_subsequential_label: Increment subsequential when creating
several arcs for the residual final output at a given state.

Returns:
An equivalent deterministic FST.

Raises:
FstArgError: Unknown determinization type.


disambiguate(ifst, delta=0.0009765625, nstate=NO_STATE_ID,
subsequential_label=0, weight=None):

Constructively disambiguates a weighted transducer.

This operation disambiguates a weighted transducer. The result will be an
equivalent FST that has the property that no two successful paths have the
same input labeling. For this algorithm, epsilon transitions are treated as
regular symbols (cf. rmepsilon).

Args:
ifst: The input FST.
delta: Comparison/quantization delta.
nstate: State number threshold.
subsequential_label: Input label of arc corresponding to residual final
output when producing a subsequential transducer.
weight: A Weight or weight string indicating the desired weight threshold
below which paths are pruned; if omitted, no paths are pruned.

Returns:
An equivalent disambiguated FST.


difference(ifst1, ifst2, compose_filter="auto", connect=True)

Constructively computes the difference of two FSTs.

This operation computes the difference between two FSAs. Only strings that are
in the first automaton but not in second are retained in the result. The first
argument must be an acceptor; the second argument must be an unweighted,
epsilon-free, deterministic acceptor. The output labels of the first
transducer or the input labels of the second transducer must be sorted (or
otherwise support appropriate matchers).

Args:
ifst1: The first input FST.
ifst2: The second input FST.
compose_filter: A string matching a known composition filter; one of:
"alt_sequence", "auto", "match", "null", "sequence", "trivial".
connect: Should the output FST be trimmed?

Returns:
An FST representing the difference of the two FSTs.


epsnormalize(ifst, eps_norm_output=False)

Constructively epsilon-normalizes an FST.

This operation creates an equivalent FST that is epsilon-normalized. An
acceptor is epsilon-normalized if it is epsilon-removed (cf. rmepsilon).
A transducer is input epsilon-normalized if, in addition, along any path, all
arcs with epsilon input labels follow all arcs with non-epsilon input labels.
Output epsilon-normalized is defined similarly. The input FST must be
functional.

Args:
ifst: The input FST.
eps_norm_output: Should the FST be output epsilon-normalized?

Returns:
An equivalent epsilon-normalized FST.


equal(ifst1, ifst2, delta=0.0009765625)

Are two FSTs equal?

This function tests whether two FSTs have the same states with the same
numbering and the same transitions with the same labels and weights in the
same order.

Args:
ifst1: The first input FST.
ifst2: The second input FST.
delta: Comparison/quantization delta.

Returns:
True if the FSTs satisfy the above condition, else False.


equivalent(ifst1, ifst2, delta=0.0009765625)

Are the two acceptors equivalent?

This operation tests whether two epsilon-free deterministic weighted
acceptors are equivalent, that is if they accept the same strings with the
same weights.

Args:
ifst1: The first input FST.
ifst2: The second input FST.
delta: Comparison/quantization delta.

Returns:
True if the FSTs satisfy the above condition, else False.

Raises:
FstOpError: Equivalence test encountered error.


intersect(ifst1, ifst2, compose_filter="auto", connect=True)

Constructively intersects two FSTs.

This operation computes the intersection (Hadamard product) of two FSTs.
Only strings that are in both automata are retained in the result. The two
arguments must be acceptors. One of the arguments must be label-sorted (or
otherwise support appropriate matchers).

Args:
ifst1: The first input FST.
ifst2: The second input FST.
compose_filter: A string matching a known composition filter; one of:
"alt_sequence", "auto", "match", "null", "sequence", "trivial".
connect: Should output be trimmed?

Returns:
An intersected FST.


leniently_compose(ifst1, ifst2, compose_filter="auto", connect=True)

Constructively leniently-composes two FSTs.

This operation computes the lenient composition of two FSTs. The lenient
composition of two FSTs the priority union of their composition and the
left-hand side argument, where priority union is simply union in which the
left-hand side argument's relations have "priority" over the right-hand side
argument's relations.

Args:
ifst: The input FST.
sigma_star: A cyclic, unweighted acceptor representing the closure over the
alphabet.
compose_filter: A string matching a known composition filter; one of:
"alt_sequence", "auto", "match", "null", "sequence", "trivial".
connect: Should output be trimmed?

Returns:
A leniently-composed FST.

Raises:
FstOpError: Operation failed.
FstSymbolTableMergeError: Unable to resolve symbol table conflict without
relabeling.


prune(ifst, delta=0.0009765625, nstate=NO_STATE_ID, weight=None)

Constructively removes paths with weights below a certain threshold.

This operation deletes states and arcs in the input FST that do not belong
to a successful path whose weight is no more (w.r.t the natural semiring
order) than the threshold t otimes-times the weight of the shortest path in
the input FST. Weights must be commutative and have the path property.

Args:
ifst: The input FST.
delta: Comparison/quantization delta.
nstate: State number threshold.
weight: A Weight or weight string indicating the desired weight threshold
below which paths are pruned; if omitted, no paths are pruned.

Returns:
A pruned FST.


push(ifst, delta=0.0009765625, push_weights=False, push_labels=False,
remove_common_affix=False, remove_total_weight=False, to_final=False)

Constructively pushes weights/labels towards initial or final states.

This operation produces an equivalent transducer by pushing the weights
and/or the labels towards the initial state or toward the final states.

When pushing weights towards the initial state, the sum of the weight of the
outgoing transitions and final weight at any non-initial state is equal to 1
in the resulting machine. When pushing weights towards the final states, the
sum of the weight of the incoming transitions at any state is equal to 1.
Weights need to be left distributive when pushing towards the initial state
and right distributive when pushing towards the final states.

Pushing labels towards the initial state consists in minimizing at every
state the length of the longest common prefix of the output labels of the
outgoing paths. Pushing labels towards the final states consists in
minimizing at every state the length of the longest common suffix of the
output labels of the incoming paths.

Args:
ifst: The input FST.
delta: Comparison/quantization delta.
push_weights: Should weights be pushed?
push_labels: Should labels be pushed?
remove_common_affix: If pushing labels, should common prefix/suffix be
removed?
remove_total_weight: If pushing weights, should total weight be removed?
to_final: Push towards final states?

Returns:
An equivalent pushed FST.


randgen(ifst, npath=1, seed=0, select="uniform", max_length=2147483647,
weight=False, remove_total_weight=False)

Randomly generate successful paths in an FST.

This operation randomly generates a set of successful paths in the input FST.
This relies on a mechanism for selecting arcs, specified using the select
argument. The default selector, "uniform", randomly selects a transition
using a uniform distribution. The "log_prob" selector randomly selects a
transition w.r.t. the weights treated as negative log probabilities after
normalizing for the total weight leaving the state. In all cases, finality is
treated as a transition to a super-final state.

Args:
ifst: The input FST.
npath: The number of random paths to generate.
seed: An optional seed value for random path generation; if zero, the
current time and process ID is used.
select: A string matching a known random arc selection type; one of:
"uniform", "log_prob", "fast_log_prob".
max_length: The maximum length of each random path.
weighted: Should the output be weighted by path count?
remove_total_weight: Should the total weight be removed (ignored when
weighted is False)?

Returns:
An FST containing one or more random paths.


reverse(ifst, require_superinitial=True)

Constructively reverses an FST's transduction.

This operation reverses an FST. If A transduces string x to y with weight a,
then the reverse of A transduces the reverse of x to the reverse of y with
weight a.Reverse(). (Typically, a = a.Reverse() and Arc = RevArc, e.g.,
TropicalWeight and LogWeight.) In general, e.g., when the weights only form a
left or right semiring, the output arc type must match the input arc type.

Args:
ifst: The input FST.
require_superinitial: Should a superinitial state be created?

Returns:
A reversed FST.


rmepsilon(ifst, connect=True, delta=0.0009765625, nstate=NO_STATE_ID,
queue_type="auto", reverse=False, weight=None)

Constructively removes epsilon transitions from an FST.

This operation removes epsilon transitions (those where both input and output
labels are epsilon) from an FST.

Args:
ifst: The input FST.
connect: Should output be trimmed?
delta: Comparison/quantization delta.
nstate: State number threshold.
queue_type: A string matching a known queue type; one of: "auto", "fifo",
"lifo", "shortest", "state", "top".
reverse: Should epsilon transitions be removed in reverse order?
weight: A string indicating the desired weight threshold; paths with
weights below this threshold will be pruned.

Returns:
An equivalent FST with no epsilon transitions.


shortestdistance(ifst, delta=0.0009765625, nstate=NO_STATE_ID,
queue_type="auto", reverse=False)

Compute the shortest distance from the initial or final state.

This operation computes the shortest distance from the initial state (when
reverse is False) or from every state to the final state (when reverse is
True). The shortest distance from p to q is the otimes-sum of the weights of
all the paths between p and q. The weights must be right (if reverse is
False) or left (if reverse is True) distributive, and k-closed (i.e., 1
otimes x otimes x^2 otimes ... otimes x^{k + 1} = 1 otimes x otimes x^2
otimes ... otimes x^k; e.g., TropicalWeight).

Args:
ifst: The input FST.
delta: Comparison/quantization delta.
nstate: State number threshold (ignored if reverse is True).
queue_type: A string matching a known queue type; one of: "auto", "fifo",
"lifo", "shortest", "state", "top" (ignored if reverse is
True).
reverse: Should the reverse distance (from each state to the final state)
be computed?

Returns:
A list of Weight objects representing the shortest distance for each state.


shortestpath(ifst, delta=0.0009765625, nshortest=1, nstate=NO_STATE_ID,
queue_type="auto", unique=False, weight=None)

Construct an FST containing the shortest path(s) in the input FST.

This operation produces an FST containing the n-shortest paths in the input
FST. The n-shortest paths are the n-lowest weight paths w.r.t. the natural
semiring order. The single path that can be read from the ith of at most n
transitions leaving the initial state of the resulting FST is the ith
shortest path. The weights need to be right distributive and have the path
property. They also need to be left distributive as well for n-shortest with
n > 1 (e.g., TropicalWeight).

Args:
ifst: The input FST.
delta: Comparison/quantization delta.
nshortest: The number of paths to return.
nstate: State number threshold.
queue_type: A string matching a known queue type; one of: "auto", "fifo",
"lifo", "shortest", "state", "top".
unique: Should the resulting FST only contain distinct paths? (Requires
the input FST to be an acceptor; epsilons are treated as if they are
regular symbols.)
weight: A Weight or weight string indicating the desired weight threshold
below which paths are pruned; if omitted, no paths are pruned.

Returns:
An FST containing the n-shortest paths.


state_map(ifst, map_type)

Constructively applies a transform to all states.

This operation transforms each state according to the requested map type.
Note that currently, only one state-mapping operation is supported.

Args:
ifst: The input FST.
map_type: A string matching a known mapping operation; one of: "arc_sum"
(sum weights of identically-labeled multi-arcs), "arc_unique" (deletes
non-unique identically-labeled multi-arcs).

Returns:
An FST with states remapped.

Raises:
FstArgError: Unknown map type.


synchronize(ifst)

Constructively synchronizes an FST.

This operation synchronizes a transducer. The result will be an equivalent
FST that has the property that during the traversal of a path, the delay is
either zero or strictly increasing, where the delay is the difference between
the number of non-epsilon output labels and input labels along the path. For
the algorithm to terminate, the input transducer must have bounded delay,
i.e., the delay of every cycle must be zero.

Args:
ifst: The input FST.

Returns:
An equivalent synchronized FST.


The following destructive methods of Fst also can be called as constructive functions; like the above, these take one or more Fst arguments and return a new Fst.

• arcsort
• closure
• concat
• connect
• decode and encode
• invert
• minimize
• optimize
• project
• relabel_pairs
• relabel_tables
• reweight
• topsort

#### Fst binary operators

The following FST functions have binary operator aliases:

• compose: *
• concat: +
• union: |

#### Weight arithmetic functions

Weights of the same semiring also support the following arithmetic functions, which return a new Weight.

plus(lhs, rhs)

Computes the sum of two Weights in the same semiring.

This function computes lhs oplus rhs, raising an exception if lhs and rhs
are not in the same semiring.

Args:
lhs: left-hand side Weight.
rhs: right-hand side Weight.

Returns:
A Weight object.

Raises:


times(lhs, rhs)

Computes the product of two Weights in the same semiring.

This function computes lhs otimes rhs, raising an exception if lhs and rhs
are not in the same semiring.

Args:
lhs: left-hand side Weight.
rhs: right-hand side Weight.

Returns:
A Weight object.

Raises:


power(lhs, rhs)

Computes the iterated product of a weight.

Args:
w: The weight.
n: The power.

Returns:
A Weight object.

Raises:


divide(lhs, rhs)

Computes the quotient of two Weights in the same semiring.

This function computes lhs oslash rhs, raising an exception if lhs and rhs
are not in the same semiring. As there is no way to specify whether to use
left vs. right division, this assumes a commutative semiring in which these
are equivalent operations.

Args:
lhs: left-hand side Weight.
rhs: right-hand side Weight.

Returns:
A Weight object.

Raises:


### Exceptions

The following exceptions (all errors) are raised by Pynini:

• FstError: abstract base class for all exceptions raised by Pynini.
• FstArgError: signals an error parsing input arguments (often in matching string arguments to underlying enum values).
• FstBadWeightError: signals an error in constructing a weight.
• FstDeletedConstructorError: signals an attempt to use a constructor which is syntactically-required, but not implemented; should not be encountered by end users.
• FstIndexError: signals an attempt to access an out-of-range state ID.
• FstIOError: signals an error in reading or writing from disk.
• FstOpError: signals an error returned by an FST algorithm; usually, this is accompanied by a low-level logging statement.
• FstStringCompilationError: signals an error compiling a string into an FST.
• FstSymbolTableMergeError: signals that a symbol table relabeling is required but disabled (using the --fst_relabel_symbol_conflicts command-line flag).

### Getting the hang of iterators

The Fst class provides four types of iterators (plus SymbolTableIterator), which are usually constructed by invoking the appropriate method of Fst. Some examples are given below.

There are two ways to use the various iterators. First, they can be used similarly to Python built-in iterators over file handles or containers by placing them in a =for=-loop, but they also expose a lower-level interface used as follows:

while not iterator.done():
# Do something with =iterator= in its current state.


Attempting to use an iterator which is "done" may result in a fatal error. To return an iterator to the initial state, one can use the reset method.

The method states creates a StateIterator. When iterated over, it yields state ID integers in strictly increasing order. For example, the following function computes the number of arcs and states of an FST (a simple measure of FST size).

def size(f):
"""Computes the number of arcs and states in the FST."""
return sum(1 + f.num_arcs(state) for state in f.states())


The method arcs returns an ArcIterator. When iterated over, it yields all arcs leaving some state of the FST. For example, the following function computes the in-degree (number of incoming states) for a given state ID.

def indegree(f, state):
"""Computes in-degree of some state in an FST."""
n = 0
for s in f.states():
for arc in f.arcs(s):
n += (arc.nextstate == state)
return n


The method mutable_arcs is similar to arcs, except that the MutableArcIterator it returns has a special method set_value which replaces the arc at the iterator's current position. For example, the following function changes input labels matching a certain label to epsilon (cf. the relabel_pairs method).

def map_ilabel_to_epsilon(f, ilabel):
"""Maps any input label matching =ilabel= to epsilon (0)."""
for state in f.states():
aiter = f.mutable_arcs(state)
while not aiter.done():
arc = aiter.value()
if arc.ilabel == ilabel:
arc.ilabel = 0
aiter.set_value(arc)
aiter.next()
return f


The method paths returns a StringPaths iterator. This provides several ways to access the paths of an acyclic FST. For each path, one can access the associated input or output labels (lists of integers) via the ilabels and olabels methods, the input and output strings via the istring and ostring methods, and the path weight via the weight method. One can also access all input strings, all output strings, and all path weights using the iter_istrings, iter_ostrings, and iter_weights methods, which return generators; after invoking any one of these methods, the caller must reset the iterator to reuse it.

### FST properties

Each Fst has an associated set of stored properties which assert facts about the FST. These can be queried by using the properties method. Some properties are binary (either true or false), whereas others are ternary (true, false, or unknown). The first argument to properties is a bitmask for the desired property/properties, and the second argument indicates whether unknown properties are to be computed.

Querying known properties is a constant time operation.

# Optimizes the FST if it is *not known* to be deterministic.
if f.properties(pynini.I_DETERMINISTIC, False) != pynini.I_DETERMINISTIC:
f.optimize()


Computing unknown properties is linear in the size of the FST in the worst case.

# Optimizes the FST if it is not already deterministic and epsilon-free.
desired_props = pynini.I_DETERMINISTIC | pynini.NO_EPSILONS
if f.properties(desired_props, True) != desired_props:
f.optimize()


An FST's properties can be set using the set_properties method.

### Creating FSTs manually

Fst (which happens to be the constructor for the Fst class) creates an FST with no states or arcs. Before this FST can be used, one must first add a state, set it as initial, and set it or some other state as final.

f = pynini.Fst()
assert not f.verify()  # OK.
f.set_start(state)
f.set_final(state)
assert f.verify()      # OK.


The epsilon_machine function eliminates the need for this tedious book-keeping; it returns a one-state, no-arc machine.

A much more useful initialization is provided by epsilon_machine, which creates a one-state, no-arc acceptor which matches the empty string.

f = pynini.epsilon_machine()
assert f.num_states() == 0                                       # OK.
assert f.num_arcs(e.start()) == 0                                # OK.
assert f.final(e.start()) == pynini.Weight.One(f.weight_type())  # OK.


In general, attempting to refer to a state before it has been added to the FST will raise an FstIndexError.

f = pynini.epsilon_machine()
f.set_start(1)  # Not OK: Raises FstIndexError.


f = pynini.epsilon_machine()
f.set_final(1)  # Not OK: Raises FstIndexError.


f = pynini.epsilon_machine()
f.add_arc(1, Arc(65, 65, 0, f.start()))  # Not OK: Raises FstIndexError.


The one exception to this generalization is that one may add an arc to a destination state not yet added to the FST, though the FST will not verify until that state is added.

f = pynini.epsilon_machine()
assert not f.verify()  # OK, though arc has non-existent destination state ID.
state = f.add_state()  # OK, now that state exists.
f.set_final(state)
assert f.verify()      # OK.


### Pushdown transducers and multi-pushdown transducers

Pushdown transducers (PDTs) and multi-pushdown transducers (MPDTs) are generalizations of (W)FSTs.

PDTs represent the class of weighted context-free grammars, and are often used to encode a set of phrase-structure grammar rules. A PDT can be thought of as an WFST in which the machine's memory is augmented by a stack. In Pynini a PDT is encoded as an WFST in which some transitions represent "push" or "pop" operations (represented as open and close parentheses) rather than input and/or output symbols. A path through the PDT is then valid if and only if it is a path through the WFST and the open and close parenthese balance along that path.

MPDTs are a further generalization of PDTs in which there is more than one stack. While these are computationally quite unconstrained, in speech and language applications they are primarily used for modeling full reduplication.

In Pynini, there is no one class representing a PDT or MPDT; rather they are represented as a pair of an Fst and a table of parentheses labels. The PdtParentheses object contains pairs of open/close parentheses labels, and the MPdtParentheses object also contains an integer representing the identity of the stack each parentheses pair is assigned to.

PDTs and MPDT support composition with an FST using the pdt_compose and mpdt_compose functions, respectively. These functions take Fst arguments, a parentheses table (PdtParentheses or MPdtParentheses, respectively), and a specification of whether the right or the left argument should be interpreted as a (M)PDT. Both functions return the FST component of an (M)PDT; the parentheses table can be reused.

(f, fparens) = ipdt
opdt = (pynini.pdt_compose(f, g, fparens), fparens)


The pdt_expand and mpdt_expand functions can be used to expand a PDT or MPDT into an FST (assuming they have a finite expansion). They take an Fst and parentheses table and return an Fst. The boolean keep_parentheses argument species whether or not expansion should keep parentheses arcs; by default, they are removed during expansion.

(f, fparens) = ipdt
g = pynini.pdt_expand(f, fparens)


The pdt_reverse and mpdt_reverse functions can be used to reverse a PDT or MPDT. Both take as arguments the two components of an (M)PDT. The pdt_reverse function return the FST component of a PDT, whereas the mpdt_function also returns a modified MPdtParentheses object.

(f, fparen) = impdt
(g, gparens) = pynini.mpdt_reverse(f, fparens)


Finally, PDTs (but not MPDTs) provide a shortest-path operation using pdt_shortestpath. Note that the result is an FST rather than a PDT.

(f, fparens) = ipdt
sp = pynini.pdt_shortestpath(f, fparens)


## References

Gorman, Kyle. 2016. Pynini: A Python library for weighted finite-state grammar compilation. In Proceedings of the ACL Workshop on Statistical NLP and Weighted Automata, 75-80.

Gorman, Kyle, and Richard Sproat. 2016. How to get superior text processing in Python with Pynini. O'Reilly Ideas blog, accessed 11/16/16.

Grover, Dale L., Martin T. King, and Clifford A. Kushler. 1998. Reduced keyboard disambiguating computer. US Patent 5,818,437.

Koskenniemi, Kimmo. 1983. Two-level morphology: A general computational model for word-form recognition and production. Doctoral dissertation, University of Helsinki.

Ringen, Catherine O. and Orvokki Heinämäki. 1999. Variation in Finnish vowel harmony: An OT account. Natural Language and Linguistic Theory 17(2): 303-337.

Roark, Brian, and Richard Sproat. 2007. Computational approaches to syntax and morphology. Oxford: Oxford University Press.

Topic revision: r8 - 2017-09-05 - KyleGorman

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