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Pynini (Gorman 2016, Gorman & Sproat 2021) is a library for compiling a grammar of strings, regular expressions, and context-dependent rewrite rules into weighted finite-state transducers.
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 is based on the OpenFst library, a C++17 template library for weighted finite-state transducers, and particularly its Python extension.
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.
Finite-state transducers (FSTs) are models of computation widely used for speech and language processing:
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.
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.
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.
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:
A path through an FSA (Q, S, F, Σ δ) is simply a sequence of transitions starting with a start state s ∈ S, taking only transitions licensed by δ, and terminating at a final state f ∈ F. More formally, let a path be a sequence P = p_{0}, p_{1}, …, p_{n} where each element p ∈ P 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:
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, l_{0}, q_{1}, (q_{1}, l_{1}, q_{2}), (q_{2}, l_{2}, q_{3}), then the string of P is simply l_{0}, l_{1}, l_{2}. The language described by an FSA is simply the union of all strings of all its paths.
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 (FSTs) are generalization of FSAs. In the normal case of a two-way transducer, δ is instead a relation from Q × (Σ_{i} ∪ ε) × (Σ_{o} ∪ ε) → Q where Σ_{i}and Σ_{o} 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 Σ_{i}^{*} and an output string over Σ_{o}^{*}. 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.
The following FST represents the relation (a:a)(b:b)*(c:g)(d:f)+(e:e)
:
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:
A monoid (M, •) is said to be commutative if for all a, b ∈ M, a • b = b • a.
Then, a semiring is a triple (𝕂, ⊕, ⊗) such that:
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 × (Σ_{i} ∪ ε) × (Σ_{o} ∪ 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.
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:
Install Pynini then simply import the module as follows:
import pynini
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.
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. Finally, the user may specify whether or not the symbol table used should be attached to the FST using the attach_symbols
keyword argument, which defaults to True
.
# UTF-8 bytestring, byte arcs. print(pynini.accep("Pont l'Evêque")) # Output below... 0 1 80 80 1 2 111 111 2 3 110 110 3 4 116 116 4 5 32 32 5 6 108 108 6 7 39 39 7 8 69 69 8 9 118 118 9 10 195 195 10 11 170 170 11 12 113 113 12 13 117 117 13 14 101 101 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.accep("Pont l'Evêque", token_type="utf8")) # Output below... 0 1 80 80 1 2 111 111 2 3 110 110 3 4 116 116 4 5 32 32 5 6 108 108 6 7 39 39 7 8 69 69 8 9 118 118 9 10 195 195 10 11 170 170 11 12 113 113 12 13 117 117 13 14 101 101 14
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.accept("b[0x61][97]") == pynini.accep("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.accep("[It's not much of a cheese shop really]") y = pynini.accep("[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.accep("\[") # OK. unused = pynini.accep("[") # 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.
The cross
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. Finally, the user may specify whether or not symbol tables should be attached to the FST.
Whereas cross
is used to construct the cross-product of two strings or automata, the union of many string pair cross-products is compiled using the string_map
and string_file
functions. The former takes iterables of strings as an argument; the latter reads string pairs from one-to-three 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.
The following examples, taken from Gorman 2016 and Gorman & Sproat 2016, show features of Pynini in action.
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() adessive = "llA" intervener = pynini.union(consonant, neutral_vowel).closure() adessive_harmony = (pynini.cdrewrite(pynini.cross("A", "a"), back_vowel + intervener, "", sigma_star) @ pynini.cdrewrite(pynini.cross("A", "ä"), "", "", sigma_star) ).optimize() def make_adessive(stem): ur = stem + adessive sr = ur @ adessive_harmony return sr.string()
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).string() def k_best(ciphertext, k): lattice = ciphertext @ T9_DECODER lattice.project("output") lattice @= 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).string()) ---+ Output below... CIPHERTEXT: 8430746453066780767852702433730460843096753 DECIPHERED PLAINTEXT: THE SINGLE MOST POPULAR CHEESE IN THE WORLD
The APIs are documented in-module. To learn about a function or method, use Python's built-in help
function (or in Jupyter, use the ?
prefix operator); e.g., import pynini; help(pynini)
.
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. 2021. Finite-State Text Processing. Morgan & Claypool.
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 University Press.