Difference between revisions of "Starting a new language with HFST"
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In the next sections we're going to start with the lexicon (<code>lexc</code> file) then progress onto the morphographemics (<code>twol</code> file). |
In the next sections we're going to start with the lexicon (<code>lexc</code> file) then progress onto the morphographemics (<code>twol</code> file). |
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Make sure you have [[Hfst#Compiling_HFST3|HFST3]] compiled. |
Make sure you have [[Hfst#Compiling_HFST3|HFST3]] compiled. You may also want to use [[apertium-init]] to start an minimal language module that will compile (<code>python3 apertium-init.py foo --analyser=hfst</code> where <code>foo</code> is the [[ISO 639-3]] code for the language). |
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==The language== |
==The language== |
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| Accusative || ''-n{y,i}'' || ''-{y,i}'' || Direct object || pagta<u>ny</u> || gaza<u>ny</u> |
| Accusative || ''-n{y,i}'' || ''-{y,i}'' || Direct object || pagta<u>ny</u> || gaza<u>ny</u> |
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|- |
|- |
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| |
| Locative || ''-(n)d{a,e}'' || ''-d{a,e}'' || Time/place || pagta<u>da</u> || gazan<u>da</u> |
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|- |
|- |
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| |
| Ablative || ''-(n)d{a,e}n'' || ''-d{a,e}n'' || Origin || pagta<u>dan</u> || gazan<u>dan</u> |
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|- |
|- |
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|} |
|} |
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| '''Accusative''' || maşgalany || maşgalalary |
| '''Accusative''' || maşgalany || maşgalalary |
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|- |
|- |
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| ''' |
| '''Locative''' || maşgalada || maşgalalarda |
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|- |
|- |
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| ''' |
| '''Ablative''' || maşgaladan || maşgalalardan |
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|- |
|- |
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|} |
|} |
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| '''Accusative''' || esgeri || esgerleri |
| '''Accusative''' || esgeri || esgerleri |
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|- |
|- |
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| ''' |
| '''Locative''' || esgerde || esgerlerde |
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|- |
|- |
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| ''' |
| '''Ablative''' || esgerden || esgerlerden |
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|- |
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|} |
|} |
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==Lexicon== |
==Lexicon== |
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So, after going through the little description above, let's start with the lexicon. The file we're going to make is called <code>apertium- |
So, after going through the little description above, let's start with the lexicon. The file we're going to make is called <code>apertium-tuk.tuk.lexc</code>, and it will contain the lexicon of the transducer. So open up your text editor. |
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===The basics=== |
===The basics=== |
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The symbols <code><</code> and <code>></code> are reserved in <code>lexc</code>, so we need to escape them with <code>%</code> |
The symbols <code><</code> and <code>></code> are reserved in <code>lexc</code>, so we need to escape them with <code>%</code> |
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We also need to define a <code>Root</code> lexicon, which is going to point to a list of stems in the lexicon <code>NounStems</code>. The <code>Root</code> lexicon is analagous to the <code> |
We also need to define a <code>Root</code> lexicon, which is going to point to a list of stems in the lexicon <code>NounStems</code>. The <code>Root</code> lexicon is analagous to the <code><section id="main" type="standard"></code> in [[lttoolbox]]: |
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<pre> |
<pre> |
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<pre> |
<pre> |
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$ hfst-lexc apertium- |
$ hfst-lexc apertium-tuk.tuk.lexc -o tuk.lexc.hfst |
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</pre> |
</pre> |
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(If you do not have <code>hfst-lexc</code> installed, you have a problem -- probably you need to compile hfst with <code>--enable-lexc</code>, but in the meantime you can use <code>hfst-lexc2fst</code> in place of <code>hfst-lexc</code>.) |
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And we can test it both with <code>hfst-fst2strings</code>: |
And we can test it both with <code>hfst-fst2strings</code>: |
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<pre> |
<pre> |
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$ hfst-fst2strings |
$ hfst-fst2strings tuk.lexc.hfst |
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maşgala<n>:maşgala |
maşgala<n>:maşgala |
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esger<n>:esger |
esger<n>:esger |
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</pre> |
</pre> |
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If you bootstrapped a language pair with [[apertium-init]], then you can just run <code>make</code> to compile. |
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===Continuation lexica=== |
===Continuation lexica=== |
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The basic morphotactics of the Turkmen noun is: |
The basic morphotactics of the Turkmen noun is: |
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:{{sc|stem}} {{sc|plural?}} {{sc|possessive?}} {{sc|case}} |
:{{sc|stem}} {{sc|plural?}} {{sc|possessive?}} {{sc|case}} {{sc|copula?}} |
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Where <code>?</code> denotes optionality. We're just working with number and case here, so let's describe the inflection, first we can start with number. In the section of the file <code>LEXICON Ninfl</code>, add the following line: |
Where <code>?</code> denotes optionality. We're just working with number and case here, so let's describe the inflection, first we can start with number. In the section of the file <code>LEXICON Ninfl</code>, add the following line: |
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| <code>:</code> || The symbol <code>:</code> delimits the left and right sides (or surface side, and lexical side) |
| <code>:</code> || The symbol <code>:</code> delimits the left and right sides (or surface side, and lexical side) |
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|- |
|- |
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| <code> |
| <code>%>l%{A%}r</code> || This is the surface form, which is split into: |
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|- |
|- |
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| <code>%></code> || The morpheme boundary delimiter (we'll talk about this later, but you put it in between morphemes where changes might happen. |
| <code>%></code> || The morpheme boundary delimiter (we'll talk about this later, but you put it in between morphemes where changes might happen. |
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(You need to have them though) |
(You need to have them though) |
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For comparison, in lttoolbox (using · for morpheme boundary and <s n="A"/> for the {A}) for , this would look something like: |
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<pre> |
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<e><p><l>·l<s n="A"/>r</l><r><s n="n"/><s n="pl"/></e> |
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</pre> |
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So, we've added the first of our inflections, the plural. We need to do two things before we can test it. First we need to add <code>%{A%}</code> to the <code>Multichar_Symbols</code> section of the file, so scroll to the top and add it, you should get something like: |
So, we've added the first of our inflections, the plural. We need to do two things before we can test it. First we need to add <code>%{A%}</code> to the <code>Multichar_Symbols</code> section of the file, so scroll to the top and add it, you should get something like: |
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<pre> |
<pre> |
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$ hfst-lexc |
$ hfst-lexc apertium-tuk.tuk.lexc -o tuk.lexc.hfst |
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</pre> |
</pre> |
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<pre> |
<pre> |
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$ hfst-fst2strings |
$ hfst-fst2strings tuk.lexc.hfst |
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maşgala<n><pl>:maşgala>l{A}r |
maşgala<n><pl>:maşgala>l{A}r |
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maşgala<n>:maşgala |
maşgala<n>:maşgala |
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What we basically want to say is "if the stem contains front vowels, then we want the front vowel alternation, if it contains back vowels then we want the back vowel alternation". And at the same time, remove the morpheme boundary. So let's give it a shot. |
What we basically want to say is "if the stem contains front vowels, then we want the front vowel alternation, if it contains back vowels then we want the back vowel alternation". And at the same time, remove the morpheme boundary. So let's give it a shot. |
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We're going to make a new file <code>apertium- |
We're going to make a new file <code>apertium-tuk.tuk.twol</code>. |
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First we need to define the alphabet: |
First we need to define the alphabet: |
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Here, the left side is the morphotactic form, and the right side is the surface form. Doing <code>%>:0</code> changes <code>%></code> into <code>0</code>, which is the same as deleting it. The <code>0</code> symbol is not output. |
Here, the left side is the morphotactic form, and the right side is the surface form. Doing <code>%>:0</code> changes <code>%></code> into <code>0</code>, which is the same as deleting it. The <code>0</code> symbol is not output. |
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So, the final alphabet section will look like this: |
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<pre> |
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Alphabet |
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A B Ç D E Ä F G H I J Ž K L M N Ň O Ö P R S Ş T U Ü W Y Ý Z |
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a b ç d e ä f g h i j ž k l m n ň o ö p r s ş t u ü w y ý z |
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%{A%}:a %>:0 ; |
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</pre> |
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Next we need to define some "sets" to work with, these are basically for giving mnemonics to features, like "front vowel" and "back vowel" which we want to refer to later in the rules: |
Next we need to define some "sets" to work with, these are basically for giving mnemonics to features, like "front vowel" and "back vowel" which we want to refer to later in the rules: |
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</pre> |
</pre> |
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The rule is basically saying: " |
The rule is basically saying: "A {A} is e if the previous letters are anything except back vowels, then there is a morpheme boundary, then there are no back vowels, and at some point there is a front vowel" |
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Next up, to compile the rule and test it: |
Next up, to compile the rule and test it: |
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<pre> |
<pre> |
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$ hfst-twolc -R -i apertium- |
$ hfst-twolc -R -i apertium-tuk.tuk.twol -o tuk.twol.hfst |
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Reading input from |
Reading input from tuk.twol. |
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Writing output to |
Writing output to tuk.twol.hfst. |
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Reading alphabet. |
Reading alphabet. |
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Reading sets. |
Reading sets. |
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<pre> |
<pre> |
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$ hfst-compose-intersect -1 |
$ hfst-compose-intersect -1 tuk.lexc.hfst -2 tuk.twol.hfst -o tuk.autogen.hfst |
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</pre> |
</pre> |
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<pre> |
<pre> |
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$ hfst-fst2strings |
$ hfst-fst2strings tuk.autogen.hfst |
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maşgala<n>:maşgala |
maşgala<n>:maşgala |
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maşgala<n><pl>:maşgalalar |
maşgala<n><pl>:maşgalalar |
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<pre> |
<pre> |
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$ hfst-fst2fst -O -i |
$ hfst-fst2fst -O -i tuk.autogen.hfst -o tuk.autogen.hfst.ol |
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</pre> |
</pre> |
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<pre> |
<pre> |
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$ echo "^maşgala<n><pl>$" | hfst-proc -g |
$ echo "^maşgala<n><pl>$" | hfst-proc -g tuk.autogen.hfst.ol |
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maşgalalar |
maşgalalar |
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</pre> |
</pre> |
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<pre> |
<pre> |
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$ echo "^esger<n><pl>$" | hfst-proc -g |
$ echo "^esger<n><pl>$" | hfst-proc -g tuk.autogen.hfst.ol |
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esgerler |
esgerler |
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</pre> |
</pre> |
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<pre> |
<pre> |
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$ hfst-invert -i |
$ hfst-invert -i tuk.autogen.hfst -o tuk.automorf.hfst |
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$ hfst-fst2strings |
$ hfst-fst2strings tuk.automorf.hfst |
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maşgala:maşgala<n> |
maşgala:maşgala<n> |
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maşgalalar:maşgala<n><pl> |
maşgalalar:maşgala<n><pl> |
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<pre> |
<pre> |
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$ hfst-fst2fst -O -i |
$ hfst-fst2fst -O -i tuk.automorf.hfst -o tuk.automorf.hfst.ol |
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</pre> |
</pre> |
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<pre> |
<pre> |
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$ echo "maşgalalar" | hfst-proc |
$ echo "maşgalalar" | hfst-proc tuk.automorf.hfst.ol |
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^maşgalalar/maşgala<n><pl>$ |
^maşgalalar/maşgala<n><pl>$ |
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$ echo "esgerler" | hfst-proc |
$ echo "esgerler" | hfst-proc tuk.automorf.hfst.ol |
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^esgerler/esger<n><pl>$ |
^esgerler/esger<n><pl>$ |
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</pre> |
</pre> |
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<references/> |
<references/> |
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== |
==See also== |
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* [[Starting a new language with lttoolbox]] |
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[[Category:HFST]] |
[[Category:HFST]] |
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[[Category:Writing dictionaries]] |
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[[Category:Quickstart]] |
Latest revision as of 09:31, 29 October 2015
- For information on how to install HFST, see HFST
This page is going to describe how to start a new language with HFST. There are some great references out there to the lexc and twol formalisms, for example the FSMBook, but a lot of them deal with the proprietary Xerox implementations, not the free HFST implementation.
While the actual formalisms are more or less identical, the commands used to compile them are not necessarily the same. HFST has a much more Unix-compatible philosophy. So we're going to take advantage of this. As most Indo-European languages, and isolating languages can be dealt with fairly easily in lttoolbox, we're going to deal with a language that is not from this family, and one that has more complex morphology that isn't easily dealt with in lttoolbox.
Preliminaries[edit]
A morphological transducer in HFST has two principle files, one is a lexc
file. This defines how morphemes in the language are joined together, morphotactics. The other file can be a twol
(two-level rules) or xfst
(sequential rewrite rules) file. These describe what changes happen when these morphemes are joined together, morphographemics (or morphophonology). For example,
- Morphotactics:
wolf<n><pl>
→wolf + s
- Morphographemics:
wolf + s
→wolves
Here we're going to deal with twol
, the two-level rules. If you're interested in xfst
, there is a nice tutorial on the Foma site.
In the next sections we're going to start with the lexicon (lexc
file) then progress onto the morphographemics (twol
file).
Make sure you have HFST3 compiled. You may also want to use apertium-init to start an minimal language module that will compile (python3 apertium-init.py foo --analyser=hfst
where foo
is the ISO 639-3 code for the language).
The language[edit]
The language we're going to model today — well, start to model — is Turkmen, a Turkic language spoken in Turkmenistan. The language pair we will supposedly be working on is Turkish--Turkmen. We're going to try and model the basic inflection (number, case) of the category of nouns. The basic inflection for Turkmen nouns is: Six cases, two numbers, and possessive. Suffixes can have different forms depending on if they are attached to a vowel ending stem, or a consonant ending stem.
Vowel harmony[edit]
Simplifying a lot,[1] we can say that stems in Turkmen can be one of two types, back-vowel stems, or front-vowel stems. Back-vowel stems, such as mugallym "teacher" only have back vowels, and front-vowel stems, such as kädi "pumpkin" have only front vowels. The back vowels in Turkmen are: a, y, o, and u. The front vowels are: ä, e, i, ö, and ü.
So, when adding a suffix to a stem, we need to know what vowels are in the stem in order to choose the right vowel to put in the suffix.
Number[edit]
Number in Turkmen can either be undefined (where there is no suffix) or plural, where the suffix is -lar or -ler. The first is used with back vowels, and the second with front vowels.
Case[edit]
We use a more compact representation below to show the suffixes for case. In between { and } are vowel alternations in the suffixes, and in between ( and ) are epentheses.
Case | Suffix | Usage | Example | ||
---|---|---|---|---|---|
V | C | V | C | ||
Nominative | Indicates the subject of the sentence | pagta | gazan | ||
Genitive | -n{y,i,u,ü}ň | -{y,i,u,ü}ň | Indicates possession | pagtanyň | gazanyň |
Dative | -{a,ä} , -n{a,e} | -{a,e} | Indirect object (directed action) | pagta | gazana |
Accusative | -n{y,i} | -{y,i} | Direct object | pagtany | gazany |
Locative | -(n)d{a,e} | -d{a,e} | Time/place | pagtada | gazanda |
Ablative | -(n)d{a,e}n | -d{a,e}n | Origin | pagtadan | gazandan |
Full paradigm[edit]
Note: This does not include the possessive.
maşgala "family" | ||
---|---|---|
Case | Singular | Plural |
Nominative | maşgala | maşgalalar |
Genitive | maşgalanyň | maşgalalaryň |
Dative | maşgala | maşgalalara |
Accusative | maşgalany | maşgalalary |
Locative | maşgalada | maşgalalarda |
Ablative | maşgaladan | maşgalalardan |
esger "soldier" | ||
---|---|---|
Case | Singular | Plural |
Nominative | esger | esgerler |
Genitive | esgeriň | esgerleriň |
Dative | esgere | esgerlere |
Accusative | esgeri | esgerleri |
Locative | esgerde | esgerlerde |
Ablative | esgerden | esgerlerden |
Lexicon[edit]
So, after going through the little description above, let's start with the lexicon. The file we're going to make is called apertium-tuk.tuk.lexc
, and it will contain the lexicon of the transducer. So open up your text editor.
The basics[edit]
The first thing we need to define are the tags that we want to produce. In lttoolbox, this is done through the <sdefs>
section of the .dix
file.
Multichar_Symbols %<n%> ! Noun %<nom%> ! Nominative %<pl%> ! Plural
The symbols <
and >
are reserved in lexc
, so we need to escape them with %
We also need to define a Root
lexicon, which is going to point to a list of stems in the lexicon NounStems
. The Root
lexicon is analagous to the <section id="main" type="standard">
in lttoolbox:
LEXICON Root NounStems ;
Now let's add our two words:
LEXICON NounStems maşgala Ninfl ; ! "family" esger Ninfl ; ! "soldier"
First we put the stem, then we put the paradigm (or continuation class) that it belongs to, in this case Ninfl
, and finally, in a comment (the comment symbol is !
) we put the translation.
And define the most basic of inflection, that is, tagging the bare stem with <n>
to indicate a noun:
LEXICON Ninfl %<n%>: # ;
This LEXICON
should go before the NounStems
lexicon. The #
symbol is the end-of-word boundary. It is very important to have this, as it tells the transducer where to stop.
Compiling[edit]
So, now we've got our basic lexicon, let's compile it and test it. We compile with hfst-lexc
:
$ hfst-lexc apertium-tuk.tuk.lexc -o tuk.lexc.hfst
(If you do not have hfst-lexc
installed, you have a problem -- probably you need to compile hfst with --enable-lexc
, but in the meantime you can use hfst-lexc2fst
in place of hfst-lexc
.)
And we can test it both with hfst-fst2strings
:
$ hfst-fst2strings tuk.lexc.hfst maşgala<n>:maşgala esger<n>:esger
If you bootstrapped a language pair with apertium-init, then you can just run make
to compile.
Continuation lexica[edit]
So, we've managed to describe that maşgala and esger are nouns, but what about the inflection. This is where continuation lexica come in. These are like paradigms in lttoolbox.
The basic morphotactics of the Turkmen noun is:
- stem plural? possessive? case copula?
Where ?
denotes optionality. We're just working with number and case here, so let's describe the inflection, first we can start with number. In the section of the file LEXICON Ninfl
, add the following line:
%<n%>%<pl%>:%>l%{A%}r # ;
Phew, that looks pretty complicated!! Well, perhaps, but each part has it's reason, let's describe them:
Part | Description |
---|---|
%<n%>%<pl%> |
The part on the left side defines the analysis, in this case noun, plural. Note, this is in contrast to lttoolbox, where the analysis is usually on the right side. |
: |
The symbol : delimits the left and right sides (or surface side, and lexical side)
|
%>l%{A%}r |
This is the surface form, which is split into: |
%> |
The morpheme boundary delimiter (we'll talk about this later, but you put it in between morphemes where changes might happen. |
l%{A%}r |
The surface morpheme, in this case -lar or -ler |
%{A%} |
An "archivowel"... a placeholder for a vowel that can be either a or e |
# |
The end of word boundary |
; |
End of line |
Part of the reason it looks complicated is all of the %
symbols. If we remove them it looks far more readable:
<n><pl>:>l{A}r # ;
(You need to have them though)
For comparison, in lttoolbox (using · for morpheme boundary and for the {A}) for , this would look something like:
<e><p><l>·l<s n="A"/>r</l><r><s n="n"/><s n="pl"/></e>
So, we've added the first of our inflections, the plural. We need to do two things before we can test it. First we need to add %{A%}
to the Multichar_Symbols
section of the file, so scroll to the top and add it, you should get something like:
Multichar_Symbols %<n%> ! Noun %<nom%> ! Nominative %<pl%> ! Plural %{A%} ! Archivowel 'a' or 'e'
Now save the file. The next thing we need to do is compile again:
$ hfst-lexc apertium-tuk.tuk.lexc -o tuk.lexc.hfst
And then we can test:
$ hfst-fst2strings tuk.lexc.hfst maşgala<n><pl>:maşgala>l{A}r maşgala<n>:maşgala esger<n><pl>:esger>l{A}r esger<n>:esger
Ok, so this is cool, but it also kind of sucks, these aren't real surface forms. We'll never see esger>l{A}r in any text. The surface form we're looking for is esgerler. So how do we get that ?
Enter twol
[edit]
The idea of twol
is to take the surface forms produced by lexc and apply rules to them to change them into real surface forms. So, this is where we change -l{A}r into -lar or -ler.
What we basically want to say is "if the stem contains front vowels, then we want the front vowel alternation, if it contains back vowels then we want the back vowel alternation". And at the same time, remove the morpheme boundary. So let's give it a shot.
We're going to make a new file apertium-tuk.tuk.twol
.
First we need to define the alphabet:
Alphabet A B Ç D E Ä F G H I J Ž K L M N Ň O Ö P R S Ş T U Ü W Y Ý Z a b ç d e ä f g h i j ž k l m n ň o ö p r s ş t u ü w y ý z %{A%}:a ;
You don't have to define the upper and lower case on separate lines, but it can help make it clearer.
We also want to define at this point, that whatever happens, we want to remove the morpheme boundaries %>
from the surface forms, so add the following line just below the last line of lower case letters, and before the ;
:
%>:0
Here, the left side is the morphotactic form, and the right side is the surface form. Doing %>:0
changes %>
into 0
, which is the same as deleting it. The 0
symbol is not output.
So, the final alphabet section will look like this:
Alphabet A B Ç D E Ä F G H I J Ž K L M N Ň O Ö P R S Ş T U Ü W Y Ý Z a b ç d e ä f g h i j ž k l m n ň o ö p r s ş t u ü w y ý z %{A%}:a %>:0 ;
Next we need to define some "sets" to work with, these are basically for giving mnemonics to features, like "front vowel" and "back vowel" which we want to refer to later in the rules:
Sets Consonant = B Ç D F G H J Ž K L M N Ň P R S Ş T W Z b ç d f g h j ž k l m n ň p r s ş t w z ; Vowel = A E Ä I O Ö U Ü Y Ý a e ä i o ö u ü y ý ; FrontVowel = Ä E I Ö Ü ä e i ö ü ; BackVowel = A Y O U a y o u ; NonBack = Consonant FrontVowel %> ; NonFront = Consonant BackVowel %> ;
So now we've got everything set up, to add the rule, there is a new section, Rules
:
Rules "Front harmony in suffixes" %{A%}:e <=> FrontVowel: NonBack:* %>: NonBack:* _ ;
The rule is basically saying: "A {A} is e if the previous letters are anything except back vowels, then there is a morpheme boundary, then there are no back vowels, and at some point there is a front vowel"
Next up, to compile the rule and test it:
$ hfst-twolc -R -i apertium-tuk.tuk.twol -o tuk.twol.hfst Reading input from tuk.twol. Writing output to tuk.twol.hfst. Reading alphabet. Reading sets. Reading rules and compiling their contexts and centers. Compiling and storing rules. Compiling rules. Storing rules.
With the power of intersecting composition![edit]
In order to get the final transducer, what we need to do is combine the morphotactic model (lexc
) with the morphographemic model (twol
). There is a way of doing this called "intersecting composition" which is fairly efficient. There is also a tool in HFST called hfst-compose-intersect
which is what we'll be using.
$ hfst-compose-intersect -1 tuk.lexc.hfst -2 tuk.twol.hfst -o tuk.autogen.hfst
Now we can test the final transducer:
$ hfst-fst2strings tuk.autogen.hfst maşgala<n>:maşgala maşgala<n><pl>:maşgalalar esger<n>:esger esger<n><pl>:esgerler
Great!! We have the desired forms.
Analysis and generation[edit]
The transducer we made above was for generation, but we can't yet use it with hfst-proc
because of the format. If we want to use it with hfst-proc
, all we need to do is change the format, with the following command:
$ hfst-fst2fst -O -i tuk.autogen.hfst -o tuk.autogen.hfst.ol
Now we should be able to generate both of our plurals:
$ echo "^maşgala<n><pl>$" | hfst-proc -g tuk.autogen.hfst.ol maşgalalar
and
$ echo "^esger<n><pl>$" | hfst-proc -g tuk.autogen.hfst.ol esgerler
But what if we want to analyse some words ? Well, then we need to invert the transducer. This is changing the left side to the right side, and the right side to the left side, let's do it in two stages so we can see the results:
$ hfst-invert -i tuk.autogen.hfst -o tuk.automorf.hfst $ hfst-fst2strings tuk.automorf.hfst maşgala:maşgala<n> maşgalalar:maşgala<n><pl> esger:esger<n> esgerler:esger<n><pl>
As we can see, now the left side is the surface form, and the right side the analysis. Now just to convert the analyser to optimised lookup format:
$ hfst-fst2fst -O -i tuk.automorf.hfst -o tuk.automorf.hfst.ol
And do some analysis:
$ echo "maşgalalar" | hfst-proc tuk.automorf.hfst.ol ^maşgalalar/maşgala<n><pl>$ $ echo "esgerler" | hfst-proc tuk.automorf.hfst.ol ^esgerler/esger<n><pl>$
Troubleshooting[edit]
Here is a brief troubleshooting checklist for when you do something, but it isn't working:
- Are all your multicharacter symbols defined ? Including archivowels/consonants. If you think you added them, triple check. This goes for problems in
twol
as well as inlexc
.
Notes[edit]
- ↑ This is actually supercomplicated, but for this didactic example, it'll do