Beesley 2000

Introduction to the xfst Interface

ReviewIntroduction to MorphologyRelations and Transducers

Introduction to xfst

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Basic Formal-Language Review

• What is a Symbol?

• What is an Alphabet?

• What is a string (=word)?

• What is a Language?

• What basic operations can be performed on Sets?

• What basic operations can be performed on Languages?

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Formal Languages and Natural Languages

• Any set of strings is a formal language

L1 = { “a”, “aa”, “aaa”, “aaaa”, “aaaaa”, …}

L2 = { “zzmy”, “niwhiuhew”, “sjehuiwheu” }

L3 = { “dog”, “cat”, “elephant” }

• The systems that we write will “accept” or “map” words in a formal language.

• In practical natural-language processing, we try to make these formal languages as close as possible to a natural language, e.g. Swahili. I.e. we try to model a natural language, as perfectly as possible, in our grammars.

• We write our grammars using xfst and lexc.

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Concatenation can form “Real” Words

work talk walk

Root Language

ing ed s

Suffix Language

working worked works talking talked talks walking walked walks

The concatenation of the Suffix language after the Root language.

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Concatenation can also form Bad Words

try plot wiggle

Root Language

ing ed s

Suffix Language

Raw Concatenation Result/Level/Language:

*trys *tryed trying plots *ploted *ploting wiggles *wiggleed *wiggleing

tries tried trying plots plotted plotting wiggles wiggled wiggling

Desired Final Result/Level/Language

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InuktitutParis+mut+nngau+juma+niraq+lauq+si+ma+nngit+jungaPari mu nngau juma nira lauq si ma nngit tunga

Paris ‘Paris’mut terminalis-casenngau direction-tojuma wantniraq declare thatlauq pastsi perfectivema resulting statenngit negativejunga 1P pres. indic

“I never said that I wanted to go to Paris”

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Concatenative-Agglutinative (Aymara)

Lexical: uta+ma+na-ka+p+xa+raki-i+waSurface: uta ma n ka p xa rak i wauta = house (noun stem)+ma = 2nd person possessive (your)+na = in (case suffix)-ka = locative (also verbalizes)+p = plural+xa = perfect aspect+raki = also-i = 3rd person present tense+wa = topic (primary emphasis)

“also they are in your house”

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Morphology

• In most languages, morphemes are just concatenations of symbols from the alphabet of the language.

• In most languages, words are just concatenations of morphemes.

• But raw concatenation often gives us abstract, morphophonemic, not-yet-correct words.

• There are alternations between the raw concatenations and the desired final words.

• There are two challenges in natural-language Morphology:• Morphotactics: describe word-formation

• Alternation: describe mappings between raw concatenations and final forms

• Both can be modeled and computed using finite-state methods

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Transducers

Recall that finite-state transducers can “map” from one string of symbols to a different string of symbols.

c a n t a r +Verb +PInd +1P +Sg

c a n t є є є o є є

We can also use transducers to map between abstract, not-yet-correct forms (usually built by simple concatenation) and correct forms.

w i g g l e i n g

w i g g l є i n g

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Regular Relations

• A Regular Language is a set of strings, e.g. { “cat”, “fly”, “big” }.

• An ordered pair of strings, notated <“upper”, “lower”>, relates two strings, e.g. <“wiggleing”, “wiggling”>.

• A Regular Relation is a set of ordered pairs of strings, e.g.

{ <“cat+N”, “cat”> , <“fly+N”, “fly”> , <“fly+V”, “fly”>, <“big+A”, “big”> }

Or

{ <“cat”, “cats”> , <“zebra”, “zebras”> , <“deer”, “deer”>, <“ox”, “oxen”>, <“child”, “children”> }

• The set of upper-side strings in a relation is a Regular Language.

• The set of lower-side strings in a relation is a Regular Language.

• A Regular Relation is a “mapping” between two Regular Languages. Each string in one of the languages is “related” to one or more strings of the other language.

• A Regular Relation can be encoded in a Finite-State Transducer (FST).

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Relations, Analysis and Generation

Given a transducer (relation), and a string, we can see the mappings of the relation via Analysis and Generation:

c a n t a r +Verb +PInd +1P +Sg

c a n t o

Upper-side string: c a n t a r +Verb +PInd +1P +Sg

Lower-side string: c a n t o

Apply the transducer in a downward direction to the upper-side string to perform Generation.

Apply the transducer in an upward direction to the lower-side string to perform Analysis.

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Transducers encode Finite-State Relations

• Let a Relation X include the ordered string pairs

{ <“cantar+Verb+PInd+1P+Sg”, “canto”>,

<“canto+Noun+Masc+Sg”, “canto”> }

• What is the upper-side Language of this Relation?

• What is the lower-side Language of this Relation?

• How can such a relation be encoded?

• What do you get when you analyze the string “canto”?

• What do you get when you generate from the string “cantar+Verb+PInd+1P+Sg”?

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Rules and Infinite Relations

One or both of the Languages related by a Relation can be infinite, e.g. the relation that relates lower-case words to their upper-case versions:

{ <“a”, “A”>, <“aa”, “AA”>, <“dog”, “DOG”>, … }

a:A

b:B

c:C

d:DEtc, (assume arcs for all other symbols in the alphabet)

Apply this network in a downward direction to the input string “cad”. What is the output?

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Alternation Rules

• We will write finite-state rules to describe alternations between abstract morphophonemic words and well-formed surface words.

• These rules compile into finite-state transducers (relations) that can be used to compute these mappings.

• Typically the upper language of a rule FST is the Universal Language, the set of all possible strings.

• Typically the lower language is like the upper language, except for the alternations controlled by the rule.

• Strings that don’t match the rule are mapped unchanged.

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Rule Application = Composition

• Composition is an operation that merges two transducers “vertically”. Let X be a transducer that contains the single ordered pair < “dog”, “chien”>. Let Y be a transducer that contains the single ordered pair <“chien”, “Hund”>. The composition of X over Y, notated X .o. Y, is the relation that contains the ordered pair <“dog”, “Hund”> .

• Composition merges any two transducers. If the shared middle level has a non-empty intersection, then the result will be a non-empty relation.

• Rule application is done via composition.

• Composition is a difficult topic that we will return to many times. Read pp 28-34 and do exercise 1.10.3 on page 37.

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Review: Basic Concepts

Language = a set of strings/wordsRegular Language = a set of string/words that can be

generated using concatenation, union, iteration and similar operations

Simple Finite-State Automaton (“Acceptor”)= a finite-state machine that accepts/recognizes a regular language

Regular Relation = a mapping between two regular languages

Finite-State Transducer (FST) = a two-level finite-state automaton that maps between two regular languages (performs look-up and generation)

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Regular Expressions

• A compact formula for describing a regular language or regular relation.

• The regular-expression language is a metalanguage.

• Think of regular expressions as the “programming language” of xfst

• Each implementation of regular expressions is slightly different (Python, Perl, emacs, …)

• We will have to learn the Xerox flavor of regular expressions as used in xfst.

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Regular Expressions Denoting a Language

Regular Expression

Regular Language

Finite-State Automaton

(“acceptor”)

describes compiles into

accepts/recognizes

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Regular Expression Denoting a Relation

Regular Expression

Regular Relation

Finite-State Transducer

describes compiles into

maps

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Introduction to xfst

xfst is an interface giving access to the finite-state operations (algorithms such as union, concatenation, iteration, intersection).

xfst includes a powerful and efficient regular-expression compiler.

xfst includes the lookup operation (‘apply up’) and the generation operation (‘apply down’) so that we can test our networks. For small examples, we can also print out all the words in the language using the command ‘print words’.

We have to learn the Xerox regular-expression metalanguage.

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Xerox Regular-Expression Operators I

a a simple symbol c a t a CONCATENATION of three symbols[ c a t ] grouping brackets? denotes any single symbol%+Noun or “+Noun”%+Verb or “+Verb”%+Adj or “+Adj” single symbols with multicharacter print

names(aka “multicharacter symbols”)

cat Beware: this will be compiled by xfst as a single multicharacter symbol

{cat} explosion brackets: equivalent to [ c a t ]

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Xerox Regular Expression Operators II

[]0 two ways to denote the empty (zero-length) string

Now, where A and B are arbitrarily complex regular expressions:[A] bracketing; equivalent to AA | B union(A) optional; equivalent to [ A | 0 ]A & B intersectionA B concatenation (N.B. the space between A and B)A - B subtraction

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Xerox Regular-Expression Operators III

A* Kleene star; zero or more iterationsA+ Kleene plus; one or more iterations?* The Universal Language~A The complement of language A;

equivalent to [ ?* - A]~[?*] The empty language (i.e. it contains no

strings at all, not even the zero-length string)%+ the literal plus-sign symbol%* the literal asterisk symbol

and similarly for %?, %(, %), %~, etc.

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Denoting Relations

A .x. B the “cross-product”; relates every string in A to every string in B, and vice versa; e.g.

[ g o .x. w e n t ] relates “go” and “went”

a:b shorthand for [ a .x. b ]%+Pl:s shorthand for [ %+Pl .x. s ]%+Past:{ed} shorthand for [ %+Past .x. e d ]%+Prog:{ing} shorthand for [ %+Prog .x. i n g ]

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Useful Abbreviations

$A denotes the language of all strings that contain A; equivalent to [ ?* A ?* ], e.g.

$b denotes the language of all strings that contain a ‘b’ anywhere

A/B denotes the language of all strings in A, ignoring any strings from B, e.g.

a*/b contains “a”, “aa”, “aaa”, … “ba”, “ab”, “aba”, ...\A any single symbol, minus strings in A; i.e. [ ? - A ],

e.g.\b denotes any single symbol, except a ‘b’

Beware: NOT to be confused with~A the complement of A; i.e. [ ?* - A ]

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Basic xfst interface commands

UnixPrompt% xfstxfst> helpxfst> help union netxfst> exitxfst> read regex [ d o g | c a t ] ;xfst> read regex < myfile.regexxfst> apply up dogxfst> apply down dogxfst> pop stackxfst> clear stackxfst> save stack myfile.fsm

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xfst saves networks in a LIFO stack

xfst> read regex [ d o g | c a t ] ;orxfst> read regex < myfile.regexcauses the compiled network to be “pushed” onto the stack. When you type xfst> pop stackthe top network is popped off the stack and discarded. When you typexfst> apply up dogthe top network on the stack is applied in an upward direction (lookup) on the

string “dog”, and the related string or strings are printed. When you typexfst> clear stackthe entire stack is popped and left empty. When you type xfst> save stack myfile.fsmthe contents of the stack are written in binary (compiled) form to the

indicated file.

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Setting Variables

xfst> define Myvarpops the top network off of the stack and saves it as the value of

Myvar, which can be used in subsequent regular expressionsxfst> define Myvar2 [ d o g | c a t ] ;assigns a value to Myvar2 without modifying the stack. It is

equivalent to the two commandsxfst> read regex [ d o g | c a t ] ;xfst> define Myvar2

xfst> undefine Myvarundefines Myvar and recycles the memory

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Using Variables in Regular Expressions

xfst> define var1 [ b i r d | f r o g | d o g ] ;xfst> define var2 [ d o g | c a t ] ;

You can now use var1 and var2 in subsequent regular expressions:

xfst> define var3 var1 | var2 ;xfst> define var4 var1 var2 ;xfst> define var5 var1 & var2 ;xfst> define var6 var1 - var2 ;

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Performing network operations on the stack

xfst> read regex [ d o g | c a t ] ;xfst> read regex [ m o u s e | r a t ] ;xfst> read regex [ d e e r | s q u i r r e l ] ;xfst> union net

‘union net’ will pop its arguments off of the stack one at a time, perform the union operation, and push the result back onto the stack, leaving just one network on the stack. Enter the command ‘words’ to see the resulting language.

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The xfst Stack

• Assume that two networks have already been pushed onto the stack.

• If we then invoke a stack-based operation like ‘union net’, the xfst algorithm pops its first argument from the top of the stack, then the second argument:

NetA | NetB

NetA

NetB

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Remember that the stack is last-in, first-out (LIFO)

Ordered operations like ‘minus net’ and ‘compose net’ are often difficult to get right. E.g. Assume that we want to compute A - B on the stack. Try this

xfst> define A [ d o g | c a t | m o u s e | r a t ] ;xfst> define B [ d o g | m o u s e | e l e p h a n t ] ;Now push the arguments onto the LIFO stack in the right order and

invoke ‘minus net’. If you have a defined variable X, you can push its value onto the stack using

xfst> push Xorxfst> read regex X ;

What is your answer? Type ‘words’ to see the language of the resulting network.

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The xfst Stack and Ordered Operations

• To perform NetA – NetB, the B net must be pushed onto the stack first, then the A net, so that they can be popped off in the reverse order.

• When performing operations on the stack, try to visualize the stack itself.

NetA - NetB

NetA

NetB

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A little concatenation example

xfst> define Root [ w a l k | t a l k | w o r k ] ;xfst> define Prefix [ 0 | r e ] ;xfst> define Suffix [ 0 | s | e d | i n g ] ;xfst> read regex Prefix Root Suffix ;xfst> wordsxfst> apply up walking

Try to get the same result by starting with the same three definitions and then pushing them on the stack, invoking ‘concatenate net’ to perform the concatenation. Remember that concatenation is an ordered operation.

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xfst file types

Regex files: contain only a regular expression, terminated with a semicolon and newline.

xfst> read regex [ d o g | c a t ] ;xfst> read regex < myfile.regex

Binary files: contain an already compiled network or networks, e.g.xfst> save stack myfile.fsmxfst> load stack myfile.fsm

Script files: contain a list of xfst commands; run them with ‘source’xfst> source myfile.script

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The Simplest Replace Rules

Replace rules are a very powerful extension to the regular-expression metalanguage. Here is the simplest kind needed for the kaNpat and Portuguese-pronunciation exercises. The arrow -> is typed as a hyphen followed by a right angle-bracket. The || operator consists of two vertical bars typed together. The _ is the underscore.

Rule Schema: upper -> lowerupper -> lower || leftcontext _ rightcontext

E.g.xfst> read regex s -> z || [ a | e | i | o | u ] _ [ a | e | i | o

| u ] ;xfst> apply down casaWhat is this rule intended to do? What comes out?

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kaNpat example

Assume a language that joins morpheme kaN (with an underspecified nasal N) and morpheme pat into the underlying or morphophonemic form kaNpat. This language then has “alternation” rules that dictate that N, when followed by p, gets realized as m. And p, when preceded by m, gets realized as m. The derivation looks like

Underlying input: kaNpatRule1: N -> m || _ pOutput of Rule1: kampatRule2: p -> m || m _Output of Rule2: kammat

The composition operation (.o.) reduces the derivational cascade of transducer networks into a single transducer network.

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Your first cascade of rules

xfst> define Rule1 N -> m || _ p ;xfst> define Rule2 p -> m || m _ ;xfst> read regex Rule1 .o. Rule2 ;xfst> apply down kaNpat

What is the output?Now restart (with ‘clear stack’), define the two Rules as shown

above, push them on the stack in the right order, and perform the composition on the stack using ‘compose net’. What is your result? (Remember that the networks must be pushed in the right order.)

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Rule Abbreviations

Multiple left-hand sides, separated by commas:b -> p, d -> t, g -> k || _ .#.

Multiple right-hand sides, separated by commas:e -> i || _ (s) .#. , .#. p _ r

Use .#. to refer to either the very beginning or the very end of a word.

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Typing Accented Letters in xemacs

The COMPOSE key is to the right of the space bar.

COMPOSE a ” yields äCOMPOSE a ’ áCOMPOSE a ` àCOMPOSE a ^ âCOMPOSE a ~ ãCOMPOSE c , ç

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A Trick for Testing Multiple Words

The exercise will ask you to write a cascade of rules that map orthographical strings to something more like a phonemic notation.

1. Type the test words into a file, e.g. “wordlist”

2. Compile your rules, compose them, and put the result on The Stack

3. Test all the words using the following syntax:

xfst[1]: apply down < wordlist

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Assignment

• Read Chapter 2 (The Systematic Introduction) when you can.

• For hands-on practice right now start reading Chapter 3, doing the examples as you go along.

• Do the kaNpat exercise in section 3.5.3 and the Southern Brazilian Portuguese exercise in 3.5.4 (p. 134).

• Progress to Bambona (p. 140) and Monish (p. 146) if you can.