path: root/doc/tutorial/gf-tutorial2_8.txt

Grammatical Framework Tutorial
Author: Aarne Ranta aarne (at) cs.chalmers.se
Last update: %%date(%c)

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[../gf-logo.png]



%--!
=Introduction=

==GF = Grammatical Framework==

The term GF is used for different things:
- a **program** used for working with grammars
- a **programming language** in which grammars can be written
- a **theory** about grammars and languages


This tutorial is primarily about the GF program and 
the GF programming language.
It will guide you
- to use the GF program
- to write GF grammars
- to write programs in which GF grammars are used as components



%--!
==What are GF grammars used for==

A grammar is a definition of a language.
From this definition, different language processing components 
can be derived:
- **parsing**: to analyse the language
- **linearization**: to generate the language
- **translation**: to analyse one language and generate another


A GF grammar can be seen as a declarative program from which these
processing tasks can be automatically derived. In addition, many
other tasks are readily available for GF grammars:
- **morphological analysis**: find out the possible inflection forms of words
- **morphological synthesis**: generate all inflection forms of words
- **random generation**: generate random expressions
- **corpus generation**: generate all expressions
- **treebank generation**: generate a list of trees with multiple linearizations
- **teaching quizzes**: train morphology and translation
- **multilingual authoring**: create a document in many languages simultaneously
- **speech input**: optimize a speech recognition system for your grammar


A typical GF application is based on a **multilingual grammar** involving
translation on a special domain. Existing applications of this idea include
- [Alfa: http://www.cs.chalmers.se/~hallgren/Alfa/Tutorial/GFplugin.html]:
  a natural-language interface to a proof editor
  (languages: English, French, Swedish)
- [KeY http://www.key-project.org/]: 
  a multilingual authoring system for creating software specifications
  (languages: OCL, English, German)
- [TALK http://www.talk-project.org]:
  multilingual and multimodal dialogue systems
  (languages: English, Finnish, French, German, Italian, Spanish, Swedish)
- [WebALT http://webalt.math.helsinki.fi/content/index_eng.html]: 
  a multilingual translator of mathematical exercises
  (languages: Catalan, English, Finnish, French, Spanish, Swedish)
- [Numeral translator http://www.cs.chalmers.se/~bringert/gf/translate/]: 
  number words from 1 to 999,999
  (88 languages)


The specialization of a grammar to a domain makes it possible to
obtain much better translations than in an unlimited machine translation
system. This is due to the well-defined semantics of such domains.
Grammars having this character are called **application grammars**.
They are different from most grammars written by linguists just
because they are multilingual and domain-specific.

However, there is another kind of grammars, which we call **resource grammars**.
These are large, comprehensive grammars that can be used on any domain.
The GF Resource Grammar Library has resource grammars for 10 languages.
These grammars can be used as **libraries** to define application grammars.
In this way, it is possible to write a high-quality grammar without
knowing about linguistics: in general, to write an application grammar
by using the resource library just requires practical knowledge of
the target language. and all theoretical knowledge about its grammar
is given by the libraries.




%--!
==Who is this tutorial for==

This tutorial is mainly for programmers who want to learn to write
application grammars. It will go through GF's programming concepts
without entering too deep into linguistics. Thus it should
be accessible to anyone who has some previous programming experience.

A separate document has been written on how to write resource grammars.
Nevertheless, we will cover some linguistic problems posed by different
languages and how they are solved in GF.


%--!
==The coverage of the tutorial==

The tutorial gives a hands-on introduction to grammar writing.
We start by building a small grammar for the domain of food:
in this grammar, you can say things like
``` 
  this Italian cheese is delicious
```
in English and Italian.

The first English grammar 
[``food.cf`` food.cf]
is written in a context-free
notation (also known as BNF). The BNF format is often a good
starting point for GF grammar development, because it is
simple and widely used. However, the BNF format is not
good for multilingual grammars. While it is possible to
"translate" by just changing the words contained in a 
BNF grammar to words of some other
language, proper translation usually involves more.
For instance, the order of words may have to be changed:
``` 
  Italian cheese ===> formaggio italiano
```
The full GF grammar format is designed to support such
changes, by separating between the **abstract syntax**
(the logical structure) and the **concrete syntax** (the
sequence of words) of expressions. 

There is more than words and word order that makes languages
different. Words can have different forms, and which forms
they have vary from language to language. For instance,
Italian adjectives usually have four forms where English
has just one:
```
  delicious (wine, wines, pizza, pizzas)
  vino delizioso, vini deliziosi, pizza deliziosa, pizze deliziose
```
The **morphology** of a language describes the
forms of its words. While the complete description of morphology
belongs to resource grammars, this tutorial will explain the
programming concepts involved in morphology. This will moreover
make it possible to grow the fragment covered by the food example.
The tutorial will in fact build a miniature resource grammar in order
to give an introduction to linguistically oriented grammar writing.

Thus it is by elaborating the initial ``food.cf`` example that
the tutorial makes a guided tour through all concepts of GF.
While the constructs of the GF language are the main focus,
also the commands of the GF system are introduced as they
are needed. 

To learn how to write GF grammars is not the only goal of
this tutorial. We will also explain the most important 
commands of the GF system. With these commands,
simple applications of grammars, such as translation and
quiz systems, can be built simply by writing scripts for the
system. More complicated applications, such as natural-language
interfaces and dialogue systems, moreover require programming in
some general-purpose language. Thus will briefly explain how
GF grammars are used as components of Haskell, Java, Javascript,
and Prolog grammars. 
%The tutorial concludes with a couple of
%case studies showing how such complete systems can be built.



%--!
==Getting the GF program==

The GF program is open-source free software, which you can download via the
GF Homepage:

[``http://www.cs.chalmers.se/~aarne/GF`` http://www.cs.chalmers.se/~aarne/GF]

There you can download
- binaries for Linux, Mac OS X, and Windows
- source code and documentation
- grammar libraries and examples


If you want to compile GF from source, you need a Haskell compiler.
For the interactive editor, you also need a Java compilers. 
But normally you don't have to compile, and you definitely
don't need to know Haskell or Java to use GF.

We are assuming the availability of a Unix shell. Linux and Mac OS X users
have it automatically, the latter under the name "terminal".
Windows users are recommended to install Cywgin, the free Unix shell for Windows.


%--!
==Running the GF program==

To start the GF program, assuming you have installed it, just type
```
  % gf
```
in the shell. 
You will see GF's welcome message and the prompt ``>``.
The command
```
  > help
```
will give you a list of available commands.

As a common convention in this Tutorial, we will use
- ``%`` as a prompt that marks system commands
- ``>`` as a prompt that marks GF commands


Thus you should not type these prompts, but only the lines that
follow them.



%--!
=The .cf grammar format=

Now you are ready to try out your first grammar.
We start with one that is not written in the GF language, but
in the much more common BNF notation (Backus Naur Form). The GF 
program understands a variant of this notation and translates it
internally to GF's own representation. 

To get started, type (or copy) the following lines into a file named
``food.cf``:
```
Is.        S       ::= Item "is" Quality ;
That.      Item    ::= "that" Kind ;
This.      Item    ::= "this" Kind ;
QKind.     Kind    ::= Quality Kind ;
Cheese.    Kind    ::= "cheese" ;
Fish.      Kind    ::= "fish" ;
Wine.      Kind    ::= "wine" ;
Italian.   Quality ::= "Italian" ;
Boring.    Quality ::= "boring" ;
Delicious. Quality ::= "delicious" ;
Expensive. Quality ::= "expensive" ;
Fresh.     Quality ::= "fresh" ;
Very.      Quality ::= "very" Quality ;
Warm.      Quality ::= "warm" ;
```
For those who know ordinary BNF, the
notation we use includes one extra element: a **label** appearing
as the first element of each rule and terminated by a full stop.

The grammar we wrote defines a set of phrases usable for speaking about food.
It builds **sentences** (``S``) by assigning ``Quality``s to
``Item``s. ``Item``s are build from ``Kind``s by prepending the
word "this" or "that". ``Kind``s are either **atomic**, such as
"cheese" and "wine", or formed by prepending a ``Quality`` to a
``Kind``. A ``Quality`` is either atomic, such as "Italian" and "boring",
or built by another ``Quality`` by prepending "very". Those familiar with
the context-free grammar notation will notice that, for instance, the
following sentence can be built using this grammar:
```
  this delicious Italian wine is very very expensive
```



%--!
==Importing grammars and parsing strings==

The first GF command needed when using a grammar is to **import** it.
The command has a long name, ``import``, and a short name, ``i``.
You can type either
```
  > import food.cf
```
or
```
  > i food.cf
```
to get the same effect.
The effect is that the GF program **compiles** your grammar into an internal
representation, and shows a new prompt when it is ready. It will also show how much
CPU time is consumed:
```
  > i food.cf
  - parsing cf food.cf 12 msec
  16 msec
  >        
```
You can now use GF for **parsing**:
```
  > parse "this cheese is delicious"
  Is (This Cheese) Delicious

  > p "that wine is very very Italian"
  Is (That Wine) (Very (Very Italian))
```
The ``parse`` (= ``p``) command takes a **string**
(in double quotes) and returns an **abstract syntax tree** - the thing
beginning with ``Is``. Trees are built from the rule labels given in the
grammar, and record the ways in which the rules are used to produce the
strings. A tree is, in general, something easier than a string 
for a machine to understand and to process further.

Strings that return a tree when parsed do so in virtue of the grammar
you imported. Try parsing something else, and you fail
```
  > p "hello world"
  Unknown words: hello world
```

**Exercise**. Extend the grammar ``food.cf`` by ten new food kinds and
qualities, and run the parser with new kinds of examples.


**Exercise**. Add a rule that enables questions of the form 
//is this cheese Italian//.



**Exercise**. Add the rule
```
  IsVery. S ::= Item "is" "very" Quality ;
```
and see what happens when parsing ``this wine is very very Italian``.
You have just made the grammar **ambiguous**: it now assigns several
trees to some strings.


**Exercise**. Modify the grammar so that at most one ``Quality`` may
attach to a given ``Kind``. Thus //boring Italian fish// will no longer
be recognized.




%--!
==Generating trees and strings==

You can also use GF for **linearizing**
(``linearize = l``). This is the inverse of
parsing, taking trees into strings:
```
  > linearize Is (That Wine) Warm
  that wine is warm
```
What is the use of this? Typically not that you type in a tree at
the GF prompt. The utility of linearization comes from the fact that
you can obtain a tree from somewhere else. One way to do so is
**random generation** (``generate_random = gr``):
```
  > generate_random
  Is (This (QKind Italian Fish)) Fresh
```
Now you can copy the tree and paste it to the ``linearize command``.
Or, more conveniently, feed random generation into linearization by using
a **pipe**.
```
  > gr | l
  this Italian fish is fresh
```
Pipes in GF work much the same way as Unix pipes: they feed the output
of one command into another command as its input.


%--!
==Visualizing trees==

The gibberish code with parentheses returned by the parser does not
look like trees. Why is it called so? From the abstract mathematical
point of view, trees are a data structure that
represents **nesting**: trees are branching entities, and the branches
are themselves trees. Parentheses give a linear representation of trees,
useful for the computer. But the human eye may prefer to see a visualization;
for this purpose, GF provides the command ``visualizre_tree = vt``, to which
parsing (and any other tree-producing command) can be piped:

``` 
  parse "this delicious cheese is very Italian" | vt
```

[Tree2.png]

This command uses the programs Graphviz and Ghostview, which you
might not have, but which are freely available on the web.



%--!
==Some random-generated sentences==

Random generation is a good way to test a grammar; it can also
be fun. So you may want to
generate ten strings with one and the same command:
```
  > gr -number=10 | l
  that wine is boring
  that fresh cheese is fresh
  that cheese is very boring
  this cheese is Italian
  that expensive cheese is expensive
  that fish is fresh
  that wine is very Italian
  this wine is Italian
  this cheese is boring
  this fish is boring
```


%--!
==Systematic generation==

To generate //all// sentence that a grammar
can generate, use the command ``generate_trees = gt``.
```
  > generate_trees | l
  that cheese is very Italian
  that cheese is very boring
  that cheese is very delicious
  that cheese is very expensive
  that cheese is very fresh
  ...
  this wine is expensive
  this wine is fresh
  this wine is warm

```
You get quite a few trees but not all of them: only up to a given
**depth** of trees. To see how you can get more, use the
``help = h`` command,
```
  > help gt
```

**Exercise**. If the command ``gt`` generated all
trees in your grammar, it would never terminate. Why?

**Exercise**. Measure how many trees the grammar gives with depths 4 and 5,
respectively. You use the Unix **word count** command ``wc`` to count lines.
**Hint**. You can pipe the output of a GF command into a Unix command by
using the escape ``?``, as follows:
```
  > generate_trees | ? wc
```





%--!
==More on pipes; tracing==

A pipe of GF commands can have any length, but the "output type"
(either string or tree) of one command must always match the "input type"
of the next command. 

The intermediate results in a pipe can be observed by putting the
**tracing** flag ``-tr`` to each command whose output you
want to see:
```
  > gr -tr | l -tr | p

  Is (This Cheese) Boring
  this cheese is boring
  Is (This Cheese) Boring  
```
This facility is good for test purposes: for instance, you
may want to see if a grammar is **ambiguous**, i.e.
contains strings that can be parsed in more than one way.

**Exercise**. Extend the grammar ``food.cf`` so that it produces ambiguous strings,
and try out the ambiguity test.




%--!
==Writing and reading files==

To save the outputs of GF commands into a file, you can
pipe it to the ``write_file = wf`` command,
```
  > gr -number=10 | l | write_file exx.tmp
```
You can read the file back to GF with the
``read_file = rf`` command,
```
  > read_file exx.tmp | p -lines
```
Notice the flag ``-lines`` given to the parsing
command. This flag tells GF to parse each line of
the file separately. Without the flag, the grammar could
not recognize the string in the file, because it is not
a sentence but a sequence of ten sentences.




%--!
=The .gf grammar format=

To see GF's internal representation of a grammar
that you have imported, you can give the command
``print_grammar = pg``,
```
  > print_grammar
```
The output is quite unreadable at this stage, and you may feel happy that
you did not need to write the grammar in that notation, but that the
GF grammar compiler produced it.

However, we will now start the demonstration 
how GF's own notation gives you
much more expressive power than the ``.cf``
format. We will introduce the ``.gf`` format by presenting
another way of defining the same grammar as in
``food.cf``.
Then we will show how the full GF grammar format enables you
to do things that are not possible in the context-free format.


%--!
==Abstract and concrete syntax==

A GF grammar consists of two main parts:

- **abstract syntax**, defining what syntax trees there are
- **concrete syntax**, defining how trees are linearized into strings 


The context-free format fuses these two things together, but it is always
possible to take them apart. For instance, the sentence formation rule
```
  Is. S ::= Item "is" Quality ;
```
is interpreted as the following pair of GF rules:
```
  fun Is : Item -> Quality -> S ;
  lin Is item quality = {s = item.s ++ "is" ++ quality.s} ;
```
The former rule, with the keyword ``fun``, belongs to the abstract syntax.
It defines the **function**
``Is`` which constructs syntax trees of form
(``Is`` //item// //quality//). 

The latter rule, with the keyword ``lin``, belongs to the concrete syntax.
It defines the **linearization function** for
syntax trees of form (``Is`` //item// //quality//). 


%--!
==Judgement forms==

Rules in a GF grammar are called **judgements**, and the keywords
``fun`` and ``lin`` are used for distinguishing between two
**judgement forms**. Here is a summary of the most important
judgement forms:

  - abstract syntax
  
     | form               | reading                   | 
     | ``cat`` C          | C is a category
     | ``fun`` f ``:`` A  | f is a function of type A 

  - concrete syntax
  
     | form                 | reading        | 
     | ``lincat`` C ``=`` T | category C has linearization type T 
     | ``lin`` f ``=`` t    | function f has linearization t
  


We return to the precise meanings of these judgement forms later.
First we will look at how judgements are grouped into modules, and
show how the food grammar is
expressed by using modules and judgements.


%--!
==Module types==

A GF grammar consists of **modules**, 
into which judgements are grouped. The most important
module forms are

  - ``abstract`` A ``=`` M, abstract syntax A with judgements in
  the module body M.
  - ``concrete`` C ``of`` A ``=`` M, concrete syntax C of the
       abstract syntax A, with judgements in the module body M.


%--!
==Basic types and function types==

The nonterminals of a context-free grammar, i.e. categories,
are called **basic types** in the type system of GF. In addition
to them, there are **function types** such as
```
  Item -> Quality -> S
```
This type is read "a function from iterms and qualities to sentences".
The last type in the arrow-separated sequence is the **value type**
of the function type, the earlier types are its **argument types**.




%--!
==Records and strings==

The linearization type of a category is a **record type**, with
zero of more **fields** of different types. The simplest record
type used for linearization in GF is
```
  {s : Str}
```
which has one field, with **label** ``s`` and type ``Str``.

Examples of records of this type are
```
  {s = "foo"}
  {s = "hello" ++ "world"}
```

Whenever a record ``r`` of type ``{s : Str}`` is given,
``r.s`` is an object of type ``Str``. This is
a special case of the **projection** rule, allowing the extraction
of fields from a record:

- if //r// : ``{`` ... //p// : //T// ... ``}`` then //r.p// : //T//


The type ``Str`` is really the type of **token lists**, but
most of the time one can conveniently think of it as the type of strings,
denoted by string literals in double quotes. 

Notice that
```  "hello world"
is not recommended as an expression of type ``Str``. It denotes
a token with a space in it, and will usually 
not work with the lexical analysis that precedes parsing. A shorthand
exemplified by
```
  ["hello world and people"]  === "hello" ++ "world" ++ "and" ++ "people"
```
can be used for lists of tokens. The expression
```
  []
```
denotes the empty token list.



%--!
==An abstract syntax example==

To express the abstract syntax of ``food.cf`` in
a file ``Food.gf``, we write two kinds of judgements:

- Each category is introduced by a ``cat`` judgement.
- Each rule label is introduced by a ``fun`` judgement,
  with the type formed from the nonterminals of the rule.


```
  abstract Food = {

  cat
    S ; Item ; Kind ; Quality ;

  fun
    Is : Item -> Quality -> S ;
    This, That : Kind -> Item ;
    QKind : Quality -> Kind -> Kind ;
    Wine, Cheese, Fish : Kind ;
    Very : Quality -> Quality ;
    Fresh, Warm, Italian, Expensive, Delicious, Boring : Quality ;
  }
```
Notice the use of shorthands permitting the sharing of
the keyword in subsequent judgements, 
```
  cat S ; Item ;   ===   cat S ; cat Item ; 
```
and of the type in subsequent ``fun`` judgements,
```
  fun Wine, Fish : Kind ;            ===
  fun Wine : Kind ; Fish : Kind ;    ===
  fun Wine : Kind ; fun Fish : Kind ;
```
The order of judgements in a module is free.

**Exercise**. Extend the abstract syntax ``Food`` with ten new
kinds and qualities, and with questions of the form
//is this wine Italian//.
 


%--!
==A concrete syntax example==

Each category introduced in ``Food.gf`` is
given a ``lincat`` rule, and each
function is given a ``lin`` rule. Similar shorthands
apply as in ``abstract`` modules.
```
  concrete FoodEng of Food = {

  lincat
    S, Item, Kind, Quality = {s : Str} ;

  lin
    Is item quality = {s = item.s ++ "is" ++ quality.s} ;
    This kind = {s = "this" ++ kind.s} ;
    That kind = {s = "that" ++ kind.s} ;
    QKind quality kind = {s = quality.s ++ kind.s} ;
    Wine = {s = "wine"} ;
    Cheese = {s = "cheese"} ;
    Fish = {s = "fish"} ;
    Very quality = {s = "very" ++ quality.s} ;
    Fresh = {s = "fresh"} ;
    Warm = {s = "warm"} ;
    Italian = {s = "Italian"} ;
    Expensive = {s = "expensive"} ;
    Delicious = {s = "delicious"} ;
    Boring = {s = "boring"} ;
  }
```

**Exercise**. Extend the concrete syntax ``FoodEng`` so that it
matches the abstract syntax defined in the exercise of the previous
section. What happens if the concrete syntax lacks some of the
new functions?

 


%--!
==Modules and files==

GF uses suffixes to recognize different file formats. The most
important ones are:
- Source files: Module name + ``.gf`` = file name
- Target files: each module is compiled into a ``.gfc`` file.


Import ``FoodEng.gf`` and see what happens:
```
  > i FoodEng.gf
  - compiling Food.gf...   wrote file Food.gfc 16 msec
  - compiling FoodEng.gf...   wrote file FoodEng.gfc 20 msec
```
The GF program does not only read the file 
``FoodEng.gf``, but also all other files that it
depends on - in this case, ``Food.gf``.

For each file that is compiled, a ``.gfc`` file
is generated. The GFC format (="GF Canonical") is the
"machine code" of GF, which is faster to process than
GF source files. When reading a module, GF decides whether
to use an existing ``.gfc`` file or to generate
a new one, by looking at modification times.

**Exercise**.  What happens when you import ``FoodEng.gf`` for 
a second time? Try this in different situations:
- Right after importing it the first time (the modules are kept in
  the memory of GF and need no reloading).
- After issuing the command ``empty`` (``e``), which clears the memory
  of GF.
- After making a small change in ``FoodEng.gf``, be it only an added space.
- After making a change in ``Food.gf``.



%--!
=Multilingual grammars and translation=

The main advantage of separating abstract from concrete syntax is that
one abstract syntax can be equipped with many concrete syntaxes.
A system with this property is called a **multilingual grammar**.

Multilingual grammars can be used for applications such as
translation. Let us build an Italian concrete syntax for
``Food`` and then test the resulting 
multilingual grammar.




%--!
==An Italian concrete syntax==

```
concrete FoodIta of Food = {

  lincat
    S, Item, Kind, Quality = {s : Str} ;

  lin
    Is item quality = {s = item.s ++ "è" ++ quality.s} ;
    This kind = {s = "questo" ++ kind.s} ;
    That kind = {s = "quello" ++ kind.s} ;
    QKind quality kind = {s = kind.s ++ quality.s} ;
    Wine = {s = "vino"} ;
    Cheese = {s = "formaggio"} ;
    Fish = {s = "pesce"} ;
    Very quality = {s = "molto" ++ quality.s} ;
    Fresh = {s = "fresco"} ;
    Warm = {s = "caldo"} ;
    Italian = {s = "italiano"} ;
    Expensive = {s = "caro"} ;
    Delicious = {s = "delizioso"} ;
    Boring = {s = "noioso"} ;

}
```

**Exercise**. Write a concrete syntax of ``Food`` for some other language.
You will probably end up with grammatically incorrect output - but don't
worry about this yet.

**Exercise**. If you have written ``Food`` for German, Swedish, or some
other language, test with random or exhaustive generation what constructs
come out incorrect, and prepare a list of those ones that cannot be helped
with the currently available fragment of GF.


%--!
==Using a multilingual grammar==

Import the two grammars in the same GF session.
```
  > i FoodEng.gf
  > i FoodIta.gf
```
Try generation now:
```
  > gr | l
  quello formaggio molto noioso è italiano

  > gr | l -lang=FoodEng
  this fish is warm
```
Translate by using a pipe:
```
  > p -lang=FoodEng "this cheese is very delicious" | l -lang=FoodIta
  questo formaggio è molto delizioso
```
Generate a **multilingual treebank**, i.e. a set of trees with their
translations in different languages:
```
  > gr -number=2 | tree_bank

  Is (That Cheese) (Very Boring)
  quello formaggio è molto noioso
  that cheese is very boring

  Is (That Cheese) Fresh
  quello formaggio è fresco
  that cheese is fresh
```
The ``lang`` flag tells GF which concrete syntax to use in parsing and
linearization. By default, the flag is set to the last-imported grammar.
To see what grammars are in scope and which is the main one, use the command
``print_options = po``:
```
  > print_options
  main abstract :     Food
  main concrete :     FoodIta
  actual concretes :  FoodIta FoodEng
```
You can change the main grammar by the command ``change_main = cm``:
```
  > change_main FoodEng
  main abstract :     Food
  main concrete :     FoodEng
  actual concretes :  FoodIta FoodEng
```


%--!
==Translation session==

If translation is what you want to do with a set of grammars, a convenient
way to do it is to open a ``translation_session = ts``. In this session,
you can translate between all the languages that are in scope.
A dot ``.`` terminates the translation session.
```
  > ts

  trans> that very warm cheese is boring
  quello formaggio molto caldo è noioso
  that very warm cheese is boring

  trans> questo vino molto italiano è molto delizioso
  questo vino molto italiano è molto delizioso
  this very Italian wine is very delicious

  trans> .
  >
```



%--!
==Translation quiz==

This is a simple language exercise that can be automatically
generated from a multilingual grammar. The system generates a set of
random sentences, displays them in one language, and checks the user's
answer given in another language. The command ``translation_quiz = tq``
makes this in a subshell of GF.
```
  > translation_quiz FoodEng FoodIta

  Welcome to GF Translation Quiz.
  The quiz is over when you have done at least 10 examples
  with at least 75 % success.
  You can interrupt the quiz by entering a line consisting of a dot ('.').

  this fish is warm
  questo pesce è caldo
  > Yes.
  Score 1/1

  this cheese is Italian
  questo formaggio è noioso
  > No, not questo formaggio è noioso, but
  questo formaggio è italiano

  Score 1/2
  this fish is expensive
```
You can also generate a list of translation exercises and save it in a
file for later use, by the command ``translation_list = tl``
```
  > translation_list -number=25 FoodEng FoodIta | write_file transl.txt
```
The ``number`` flag gives the number of sentences generated.




%--!
=Grammar architecture=

==Extending a grammar==

The module system of GF makes it possible to **extend** a
grammar in different ways. The syntax of extension is
shown by the following example. We extend ``Food`` by
adding a category of questions and two new functions.
```
  abstract Morefood = Food ** {
    cat
      Question ;
    fun
      QIs : Item -> Quality -> Question ;
      Pizza : Kind ;
      
  }
```
Parallel to the abstract syntax, extensions can
be built for concrete syntaxes:
```
  concrete MorefoodEng of Morefood = FoodEng ** {
    lincat
      Question = {s : Str} ;
    lin
      QIs item quality = {s = "is" ++ item.s ++ quality.s} ;
      Pizza = {s = "pizza"} ;
  }
```
The effect of extension is that all of the contents of the extended
and extending module are put together. We also say that the new 
module **inherits** the contents of the old module.



%--!
==Multiple inheritance==

Specialized vocabularies can be represented as small grammars that
only do "one thing" each. For instance, the following are grammars
for fruit and mushrooms
```
  abstract Fruit = {
    cat Fruit ;
    fun Apple, Peach : Fruit ;
  }

  abstract Mushroom = {
    cat Mushroom ;
    fun Cep, Agaric : Mushroom ;
  }
```
They can afterwards be combined into bigger grammars by using
**multiple inheritance**, i.e. extension of several grammars at the
same time:
```
  abstract Foodmarket = Food, Fruit, Mushroom ** {
    fun 
      FruitKind    : Fruit    -> Kind ;
      MushroomKind : Mushroom -> Kind ;
    }
```
At this point, you would perhaps like to go back to
``Food`` and take apart ``Wine`` to build a special
``Drink`` module.


%--!
==Visualizing module structure==

When you have created all the abstract syntaxes and
one set of concrete syntaxes needed for ``Foodmarket``,
your grammar consists of eight GF modules. To see how their
dependences look like, you can use the command 
``visualize_graph = vg``,
```
  > visualize_graph
```
and the graph will pop up in a separate window. 

The graph uses

- oval boxes for abstract modules
- square boxes  for concrete modules
- black-headed arrows for inheritance
- white-headed arrows for the concrete-of-abstract relation


[Foodmarket.png]


Just as the ``visualize_tree = vt`` command, the open source tools
Ghostview and Graphviz are needed.


%--!
==System commands==

To document your grammar, you may want to print the
graph into a file, e.g. a ``.png`` file that
can be included in an HTML document. You can do this
by first printing the graph into a file ``.dot`` and then
processing this file with the ``dot`` program (from the Graphviz package).
```
  > pm -printer=graph | wf Foodmarket.dot
  > ! dot -Tpng Foodmarket.dot > Foodmarket.png
```
The latter command is a Unix command, issued from GF by using the
shell escape symbol ``!``. The resulting graph was shown in the previous section.

The command ``print_multi = pm`` is used for printing the current multilingual
grammar in various formats, of which the format ``-printer=graph`` just
shows the module dependencies. Use ``help`` to see what other formats
are available:
```
  > help pm
  > help -printer
  > help help
```
Another form of system commands are those usable in GF pipes. The escape symbol
is then ``?``.
```
  > generate_trees | ? wc
```



%--!
=Resource modules=


==The golden rule of functional programming==

In comparison to the ``.cf`` format, the ``.gf`` format looks rather
verbose, and demands lots more characters to be written. You have probably
done this by the copy-paste-modify method, which is a common way to
avoid repeating work.

However, there is a more elegant way to avoid repeating work than the copy-and-paste
method. The **golden rule of functional programming** says that
- whenever you find yourself programming by copy-and-paste, write a function instead.


A function separates the shared parts of different computations from the
changing parts, its **arguments**, or **parameters**. 
In functional programming languages, such as
[Haskell http://www.haskell.org], it is possible to share much more 
code with functions than in imperative languages such as C and Java.


==Operation definitions==

GF is a functional programming language, not only in the sense that
the abstract syntax is a system of functions (``fun``), but also because
functional programming can be used to define concrete syntax. This is
done by using a new form of judgement, with the keyword ``oper`` (for
**operation**), distinct from ``fun`` for the sake of clarity.
Here is a simple example of an operation:
```
  oper ss : Str -> {s : Str} = \x -> {s = x} ;
```
The operation can be **applied** to an argument, and GF will
**compute** the application into a value. For instance,
```
  ss "boy"  ===>  {s = "boy"}
```
(We use the symbol ``===>`` to indicate how an expression is 
computed into a value; this symbol is not a part of GF)

Thus an ``oper`` judgement includes the name of the defined operation,
its type, and an expression defining it. As for the syntax of the defining
expression, notice the **lambda abstraction** form ``\x -> t`` of
the function.



%--!
==The ``resource`` module type==

Operator definitions can be included in a concrete syntax. 
But they are not really tied to a particular set of linearization rules.
They should rather be seen as **resources**
usable in many concrete syntaxes.

The ``resource`` module type can be used to package
``oper`` definitions into reusable resources. Here is
an example, with a handful of operations to manipulate
strings and records.
```
  resource StringOper = {
    oper
      SS : Type = {s : Str} ;
      ss : Str -> SS = \x -> {s = x} ;
      cc : SS -> SS -> SS = \x,y -> ss (x.s ++ y.s) ;
      prefix : Str -> SS -> SS = \p,x -> ss (p ++ x.s) ;
  }
```
Resource modules can extend other resource modules, in the
same way as modules of other types can extend modules of the
same type. Thus it is possible to build resource hierarchies.



%--!
==Opening a resource==

Any number of ``resource`` modules can be
**opened** in a ``concrete`` syntax, which
makes definitions contained
in the resource usable in the concrete syntax. Here is
an example, where the resource ``StringOper`` is
opened in a new version of ``FoodEng``.
```
  concrete Food2Eng of Food = open StringOper in {

  lincat
    S, Item, Kind, Quality = SS ;

  lin
    Is item quality = cc item (prefix "is" quality) ;
    This k = prefix "this" k ;
    That k = prefix "that" k ;
    QKind k q = cc k q ;
    Wine = ss "wine" ;
    Cheese = ss "cheese" ;
    Fish = ss "fish" ;
    Very = prefix "very" ;
    Fresh = ss "fresh" ;
    Warm = ss "warm" ;
    Italian = ss "Italian" ;
    Expensive = ss "expensive" ;
    Delicious = ss "delicious" ;
    Boring = ss "boring" ;

  }
```
**Exercise**. Use the same string operations to write ``FoodIta``
more concisely.



%--!
==Partial application==

GF, like Haskell, permits **partial application** of
functions. An example of this is the rule
```
  lin This k = prefix "this" k ;
```
which can be written more concisely
```
  lin This = prefix "this" ;
```
The first form is perhaps more intuitive to write
but, once you get used to partial application, you will appreciate its
conciseness and elegance. The logic of partial application 
is known as **currying**, with a reference to Haskell B. Curry. 
The idea is that any //n//-place function can be defined as a 1-place
function whose value is an //n-//1 -place function. Thus
```
  oper prefix : Str -> SS -> SS ;
```
can be used as a 1-place function that takes a ``Str`` into a
function ``SS -> SS``. The expected linearization of ``This`` is exactly
a function of such a type, operating on an argument of type ``Kind``
whose linearization is of type ``SS``. Thus we can define the
linearization directly as ``prefix "this"``.

**Exercise**. Define an operation ``infix`` analogous to ``prefix``,
such that it allows you to write
```
  lin Is = infix "is" ;
```


%--!
==Testing resource modules==

To test a ``resource`` module independently, you must import it
with the flag ``-retain``, which tells GF to retain ``oper`` definitions
in the memory; the usual behaviour is that ``oper`` definitions
are just applied to compile linearization rules 
(this is called **inlining**) and then thrown away.
```
  > i -retain StringOper.gf
```
The command ``compute_concrete = cc`` computes any expression
formed by operations and other GF constructs. For example,
```
  > compute_concrete prefix "in" (ss "addition")
  {
    s : Str = "in" ++ "addition"
  }
```



%--!
==Division of labour==

Using operations defined in resource modules is a
way to avoid repetitive code.
In addition, it enables a new kind of modularity
and division of labour in grammar writing: grammarians familiar with
the linguistic details of a language can make their knowledge
available through resource grammar modules, whose users only need
to pick the right operations and not to know their implementation
details. 

In the following sections, we will go through some
such linguistic details. The programming constructs needed when
doing this are useful for all GF programmers, even if they don't
hand-code the linguistics of their applications  but get them
from libraries. It is also useful to know something about the
linguistic concepts of inflection, agreement, and parts of speech.




%--!
=Morphology=

Suppose we want to say, with the vocabulary included in
``Food.gf``, things like
```
  all Italian wines are delicious
```
The new grammatical facility we need are the plural forms
of nouns and verbs (//wines, are//), as opposed to their
singular forms.

The introduction of plural forms requires two things:
- the **inflection** of nouns and verbs in singular and plural
- the **agreement** of the verb to subject: 
  the verb must have the same number as the subject


Different languages have different rules of inflection and agreement.
For instance, Italian has also agreement in gender (masculine vs. feminine).
We want to express such special features of languages in the
concrete syntax while ignoring them in the abstract syntax.

To be able to do all this, we need one new judgement form
and many new expression forms.
We also need to generalize linearization types
from strings to more complex types.

**Exercise**. Make a list of the possible forms that nouns,
adjectives, and verbs can have in some languages that you know.


%--!
==Parameters and tables==

We define the **parameter type** of number in Englisn by
using a new form of judgement:
```
  param Number = Sg | Pl ;
```
To express that ``Kind`` expressions in English have a linearization
depending on number, we replace the linearization type ``{s : Str}``
with a type where the ``s`` field is a **table** depending on number:
```
  lincat Kind = {s : Number => Str} ;
```
The **table type** ``Number => Str`` is in many respects similar to
a function type (``Number -> Str``). The main difference is that the
argument type of a table type must always be a parameter type. This means
that the argument-value pairs can be listed in a finite table. The following
example shows such a table:
```
  lin Cheese = {s = table {
    Sg => "cheese" ;
    Pl => "cheeses"
    }
  } ;
```
The table consists of **branches**, where a **pattern** on the
left of the arrow ``=>`` is assigned a **value** on the right.

The application of a table to a parameter is done by the **selection**
operator ``!``. For instance,
```
   table {Sg => "cheese" ; Pl => "cheeses"} ! Pl
```
is a selection that computes into the value ``"cheeses"``.
This computation is performed by **pattern matching**: return
the value from the first branch whose pattern matches the
selection argument. Thus
```
   table {Sg => "cheese" ; Pl => "cheeses"} ! Pl 
   ===> "cheeses"
```

**Exercise**. In a previous exercise, we make a list of the possible 
forms that nouns, adjectives, and verbs can have in some languages that 
you know. Now take some of the results and implement them by
using parameter type definitions and tables. Write them into a ``resource``
module, which you can test by using the command ``compute_concrete``.



%--!
==Inflection tables and paradigms==

All English common nouns are inflected in number, most of them in the
same way: the plural form is obtained from the singular by adding the
ending //s//. This rule is an example of 
a **paradigm** - a formula telling how the inflection
forms of a word are formed.

From the GF point of view, a paradigm is a function that takes a **lemma** -
also known as a **dictionary form** - and returns an inflection
table of desired type. Paradigms are not functions in the sense of the
``fun`` judgements of abstract syntax (which operate on trees and not
on strings), but operations defined in ``oper`` judgements.
The following operation defines the regular noun paradigm of English:
```
  oper regNoun : Str -> {s : Number => Str} = \x -> {
    s = table {
      Sg => x ;
      Pl => x + "s"
      }
    } ;
```
The **gluing** operator ``+`` tells that
the string held in the variable ``x`` and the ending ``"s"`` 
are written together to form one **token**. Thus, for instance,
```
  (regNoun "cheese").s ! Pl  ---> "cheese" + "s"  --->  "cheeses"
```

**Exercise**. Identify cases in which the ``regNoun`` paradigm does not
apply in English, and implement some alternative paradigms.

**Exercise**. Implement a paradigm for regular verbs in English.

**Exercise**. Implement some regular paradigms for other languages you have
considered in earlier exercises.


%--!
==Worst-case functions and data abstraction==

Some English nouns, such as ``mouse``, are so irregular that
it makes no sense to see them as instances of a paradigm. Even
then, it is useful to perform **data abstraction** from the
definition of the type ``Noun``, and introduce a constructor
operation, a **worst-case function** for nouns:
```
  oper mkNoun : Str -> Str -> Noun = \x,y -> {
    s = table {
      Sg => x ;
      Pl => y
      }
    } ;
```
Thus we can define
```
  lin Mouse = mkNoun "mouse" "mice" ;
```
and
```
  oper regNoun : Str -> Noun = \x -> 
    mkNoun x (x + "s") ;
```
instead of writing the inflection tables explicitly.

The grammar engineering advantage of worst-case functions is that
the author of the resource module may change the definitions of
``Noun`` and ``mkNoun``, and still retain the
interface (i.e. the system of type signatures) that makes it
correct to use these functions in concrete modules. In programming
terms, ``Noun`` is then treated as an **abstract datatype**.



%--!
==A system of paradigms using Prelude operations==

In addition to the completely regular noun paradigm ``regNoun``, 
some other frequent noun paradigms deserve to be
defined, for instance,
```
  sNoun : Str -> Noun = \kiss  -> mkNoun kiss (kiss + "es") ;
```
What about nouns like //fly//, with the plural //flies//? The already
available solution is to use the longest common prefix
//fl// (also known as the **technical stem**) as argument, and define
```
  yNoun : Str -> Noun = \fl -> mkNoun (fl + "y") (fl + "ies") ;
```
But this paradigm would be very unintuitive to use, because the technical stem
is not an existing form of the word. A better solution is to use
the lemma and a string operator ``init``, which returns the initial segment (i.e.
all characters but the last) of a string:
```
  yNoun : Str -> Noun = \fly -> mkNoun fly (init fly + "ies") ;  
```
The operation ``init`` belongs to a set of operations in the
resource module ``Prelude``, which therefore has to be
``open``ed so that ``init`` can be used. Its dual is ``last``:
```
  > cc init "curry"
  "curr"

  > cc last "curry"
  "y"
```
As generalizations of the library functions ``init`` and ``last``, GF has
two predefined funtions:
``Predef.dp``, which "drops" suffixes of any length, 
and ``Predef.tk``, which "takes" a prefix
just omitting a number of characters from the end. For instance,
```
  > cc Predef.tk 3 "worried"
  "worr"
  > cc Predef.dp 3 "worried"
  "ied"
```
The prefix ``Predef`` is given to a handful of functions that could
not be defined internally in GF. They are available in all modules
without explicit ``open`` of the module ``Predef``.





%--!
==Pattern matching==

We have so far built all expressions of the ``table`` form 
from branches whose patterns are constants introduced in
``param`` definitions, as well as constant strings. 
But there are more expressive patterns. Here is a summary of the possible forms:
- a variable pattern (identifier other than constant parameter) matches anything
- the wild card ``_`` matches anything
- a string literal pattern, e.g. ``"s"``, matches the same string 
- a disjunctive pattern ``P | ... | Q`` matches anything that
     one of the disjuncts matches


Pattern matching is performed in the order in which the branches
appear in the table: the branch of the first matching pattern is followed.

As syntactic sugar, one-branch tables can be written concisely,
```
  \\P,...,Q => t  ===  table {P => ... table {Q => t} ...}
```
Finally, the ``case`` expressions common in functional
programming languages are syntactic sugar for table selections:
```
  case e of {...} ===  table {...} ! e
```


%--!
==An intelligent noun paradigm using pattern matching==

It may be hard for the user of a resource morphology to pick the right
inflection paradigm. A way to help this is to define a more intelligent
paradigm, which chooses the ending by first analysing the lemma.
The following variant for English regular nouns puts together all the
previously shown paradigms, and chooses one of them on the basis of
the final letter of the lemma (found by the prelude operator ``last``).
```
  regNoun : Str -> Noun = \s -> case last s of {
    "s" | "z" => mkNoun s (s + "es") ;
    "y"       => mkNoun s (init s + "ies") ;
    _         => mkNoun s (s + "s")
    } ;
```
This definition displays many GF expression forms not shown befores;
these forms are explained in the next section.

The paradigms ``regNoun`` does not give the correct forms for
all nouns. For instance, //mouse - mice// and
//fish - fish// must be given by using ``mkNoun``.
Also the word //boy// would be inflected incorrectly; to prevent
this, either use ``mkNoun`` or modify 
``regNoun`` so that the ``"y"`` case does not
apply if the second-last character is a vowel.

**Exercise**. Extend the ``regNoun`` paradigm so that it takes care
of all variations there are in English. Test it with the nouns
//ax//, //bamboo//, //boy//, //bush//, //hero//, //match//. 
**Hint**. The library functions ``Predef.dp`` and ``Predef.tk``
are useful in this task.

**Exercise**. The same rules that form plural nouns in English also
apply in the formation of third-person singular verbs. 
Write a regular verb paradigm that uses this idea, but first
rewrite ``regNoun`` so that the analysis needed to build //s//-forms
is factored out as a separate ``oper``, which is shared with
``regVerb``.




%--!
==Morphological resource modules==

A common idiom is to
gather the ``oper`` and ``param`` definitions
needed for inflecting words in
a language into a morphology module. Here is a simple
example, [``MorphoEng`` resource/MorphoEng.gf].
```
  --# -path=.:prelude

  resource MorphoEng = open Prelude in {

    param
      Number = Sg | Pl ;

    oper
      Noun, Verb : Type = {s : Number => Str} ;

      mkNoun : Str -> Str -> Noun = \x,y -> {
        s = table {
          Sg => x ;
          Pl => y
          }
        } ;

      regNoun : Str -> Noun = \s -> case last s of {
        "s" | "z" => mkNoun s (s + "es") ;
        "y"       => mkNoun s (init s + "ies") ;
        _         => mkNoun s (s + "s")
        } ;

      mkVerb : Str -> Str -> Verb = \x,y -> mkNoun y x ;

      regVerb : Str -> Verb = \s -> case last s of {
        "s" | "z" => mkVerb s (s + "es") ;
        "y"       => mkVerb s (init s + "ies") ;
        "o"       => mkVerb s (s + "es") ;
        _         => mkVerb s (s + "s")
        } ;
  }
```
The first line gives as a hint to the compiler the
**search path** needed to find all the other modules that the
module depends on. The directory ``prelude`` is a subdirectory of
``GF/lib``; to be able to refer to it in this simple way, you can
set the environment variable ``GF_LIB_PATH`` to point to this
directory.



=Using parameters in concrete syntax=

We can now enrich the concrete syntax definitions to
comprise morphology. This will involve a more radical
variation between languages (e.g. English and Italian)
then just the use of different words. In general,
parameters and linearization types are different in
different languages - but this does not prevent the 
use of a common abstract syntax.


%--!
==Parametric vs. inherent features, agreement==

The rule of subject-verb agreement in English says that the verb
phrase must be inflected in the number of the subject. This
means that a noun phrase (functioning as a subject), inherently
//has// a number, which it passes to the verb. The verb does not
//have// a number, but must be able to //receive// whatever number the
subject has. This distinction is nicely represented by the
different linearization types of **noun phrases** and **verb phrases**:
```
  lincat NP = {s : Str ; n : Number} ;
  lincat VP = {s : Number => Str} ;
```
We say that the number of ``NP`` is an **inherent feature**,
whereas the number of  ``NP`` is a **variable feature** (or a
**parametric feature**).

The agreement rule itself is expressed in the linearization rule of
the predication function:
```
  lin PredVP np vp = {s = np.s ++ vp.s ! np.n} ;
```
The following section will present
``FoodsEng``, assuming the abstract syntax ``Foods`` 
that is similar to ``Food`` but also has the
plural determiners ``These`` and ``Those``.
The reader is invited to inspect the way in which agreement works in
the formation of sentences.


%--!
==English concrete syntax with parameters==

The grammar uses both 
[``Prelude`` ../../lib/prelude/Prelude.gf] and
[``MorphoEng`` resource/MorphoEng].
We will later see how to make the grammar even
more high-level by using a resource grammar library
and parametrized modules.
```
--# -path=.:resource:prelude

concrete FoodsEng of Foods = open Prelude, MorphoEng in {

  lincat
    S, Quality = SS ; 
    Kind = {s : Number => Str} ; 
    Item = {s : Str ; n : Number} ; 

  lin
    Is item quality = ss (item.s ++ (mkVerb "are" "is").s ! item.n ++ quality.s) ;
    This  = det Sg "this" ;
    That  = det Sg "that" ;
    These = det Pl "these" ;
    Those = det Pl "those" ;
    QKind quality kind = {s = \\n => quality.s ++ kind.s ! n} ;
    Wine = regNoun "wine" ;
    Cheese = regNoun "cheese" ;
    Fish = mkNoun "fish" "fish" ;
    Very = prefixSS "very" ;
    Fresh = ss "fresh" ;
    Warm = ss "warm" ;
    Italian = ss "Italian" ;
    Expensive = ss "expensive" ;
    Delicious = ss "delicious" ;
    Boring = ss "boring" ;

  oper
    det : Number -> Str -> Noun -> {s : Str ; n : Number} = \n,d,cn -> {
      s = d ++ cn.s ! n ;
      n = n
      } ;
}
```



%--!
==Hierarchic parameter types==

The reader familiar with a functional programming language such as
[Haskell http://www.haskell.org] must have noticed the similarity
between parameter types in GF and **algebraic datatypes** (``data`` definitions
in Haskell). The GF parameter types are actually a special case of algebraic
datatypes: the main restriction is that in GF, these types must be finite.
(It is this restriction that makes it possible to invert linearization rules into
parsing methods.)

However, finite is not the same thing as enumerated. Even in GF, parameter
constructors can take arguments, provided these arguments are from other
parameter types - only recursion is forbidden. Such parameter types impose a
hierarchic order among parameters. They are often needed to define
the linguistically most accurate parameter systems.

To give an example, Swedish adjectives
are inflected in number (singular or plural) and
gender (uter or neuter). These parameters would suggest 2*2=4 different
forms. However, the gender distinction is done only in the singular. Therefore,
it would be inaccurate to define adjective paradigms using the type
``Gender => Number => Str``. The following hierarchic definition
yields an accurate system of three adjectival forms.
```
  param AdjForm = ASg Gender | APl ;
  param Gender  = Utr | Neutr ;
```
Here is an example of pattern matching, the paradigm of regular adjectives.
```
  oper regAdj : Str -> AdjForm => Str = \fin -> table {
    ASg Utr   => fin ;
    ASg Neutr => fin + "t" ;
    APl       => fin + "a" ;
    }
```
A constructor can be used as a pattern that has patterns as arguments. For instance,
the adjectival paradigm in which the two singular forms are the same, 
can be defined
```
  oper plattAdj : Str -> AdjForm => Str = \platt -> table {
    ASg _ => platt ;
    APl   => platt + "a" ;
    }
```


%--!
==Morphological analysis and morphology quiz==

Even though morphology is in GF
mostly used as an auxiliary for syntax, it
can also be useful on its own right. The command ``morpho_analyse = ma``
can be used to read a text and return for each word the analyses that
it has in the current concrete syntax.
```
  > rf bible.txt | morpho_analyse
```
In the same way as translation exercises, morphological exercises can
be generated, by the command ``morpho_quiz = mq``. Usually,
the category is set to be something else than ``S``. For instance,
```
  > cd GF/lib/resource-1.0/
  > i french/IrregFre.gf
  > morpho_quiz -cat=V

  Welcome to GF Morphology Quiz.
  ...

  réapparaître : VFin VCondit  Pl  P2
  réapparaitriez
  > No, not réapparaitriez, but
  réapparaîtriez
  Score 0/1
```
Finally, a list of morphological exercises can be generated
off-line and saved in a
file for later use, by the command ``morpho_list = ml``
```
  > morpho_list -number=25 -cat=V | wf exx.txt
```
The ``number`` flag gives the number of exercises generated.



%--!
==Discontinuous constituents==

A linearization type may contain more strings than one. 
An example of where this is useful are English particle
verbs, such as //switch off//. The linearization of
a sentence may place the object between the verb and the particle:
//he switched it off//.

The following judgement defines transitive verbs as
**discontinuous constituents**, i.e. as having a linearization
type with two strings and not just one. 
```
  lincat TV = {s : Number => Str ; part : Str} ;
```
This linearization rule
shows how the constituents are separated by the object in complementization.
```
  lin PredTV tv obj = {s = \\n => tv.s ! n ++ obj.s ++ tv.part} ;
```
There is no restriction in the number of discontinuous constituents
(or other fields) a  ``lincat`` may contain. The only condition is that
the fields must be of finite types, i.e. built from records, tables,
parameters, and ``Str``, and not functions. 

A mathematical result
about parsing in GF says that the worst-case complexity of parsing
increases with the number of discontinuous constituents. This is
potentially a reason to avoid discontinuous constituents.
Moreover, the parsing and linearization commands only give accurate
results for categories whose linearization type has a unique ``Str`` 
valued field labelled ``s``. Therefore, discontinuous constituents
are not a good idea in top-level categories accessed by the users
of a grammar application.


%--!
==Free variation==

Sometimes there are many alternative ways to define a concrete syntax.
For instance, the verb negation in English can be expressed both by
//does not// and //doesn't//. In linguistic terms, these expressions
are in **free variation**. The ``variants`` construct of GF can
be used to give a list of strings in free variation. For example,
```
  NegVerb verb = {s = variants {["does not"] ; "doesn't} ++ verb.s ! Pl} ;
```
An empty variant list
```
  variants {}
```
can be used e.g. if a word lacks a certain form.

In general, ``variants`` should be used cautiously. It is not
recommended for modules aimed to be libraries, because the
user of the library has no way to choose among the variants.



==Overloading of operations==

Large libraries, such as the GF Resource Grammar Library, may define 
hundreds of names, which can be unpractical
for both the library writer and the user. The writer has to invent longer 
and longer names which are not always intuitive,
and the user has to learn or at least be able to find all these names. 
A solution to this problem, adopted by languages such as C++, is **overloading**:
the same name can be used for several functions. When such a name is used, the
compiler performs **overload resolution** to find out which of the possible functions
is meant. The resolution is based on the types of the functions: all functions that
have the same name must have different types.

In C++, functions with the same name can be scattered everywhere in the program.
In GF, they must be grouped together in ``overload`` groups. Here is an example
of an overload group, defining four ways to define nouns in Italian:
```
  oper mkN = overload {
    mkN : Str -> N                  = -- regular nouns
    mkN : Str -> Gender -> N        = -- regular nouns with unexpected gender
    mkN : Str -> Str -> N           = -- irregular nouns
    mkN : Str -> Str -> Gender -> N = -- irregular nouns with unexpected gender
  }
```
All of the following uses of ``mkN`` are easy to resolve:
```
  lin Pizza = mkN "pizza" ;         -- Str -> N
  lin Hand  = mkN "mano" Fem ;      -- Str -> Gender -> N
  lin Man   = mkN "uomo" "uomini" ; -- Str -> Str -> N
```






%--!
=More constructs for concrete syntax=

In this chapter, we go through constructs that are not necessary in simple grammars
or when the concrete syntax relies on libraries, but very useful when writing advanced
concrete syntax implementations, such as resource grammar libraries.


%--!
==Local definitions==

Local definitions ("``let`` expressions") are used in functional
programming for two reasons: to structure the code into smaller
expressions, and to avoid repeated computation of one and
the same expression. Here is an example, from 
[``MorphoIta`` resource/MorphoIta.gf]:
```
  oper regNoun : Str -> Noun = \vino -> 
        let 
          vin = init vino ;
          o   = last vino
        in
        case o of {
          "a"       => mkNoun Fem  vino (vin + "e") ;
          "o" | "e" => mkNoun Masc vino (vin + "i") ;
          _         => mkNoun Masc vino vino         
          } ;
```


==Record extension and subtyping==

Record types and records can be **extended** with new fields. For instance,
in German it is natural to see transitive verbs as verbs with a case.
The symbol ``**`` is used for both constructs.
```
  lincat TV = Verb ** {c : Case} ;

  lin Follow = regVerb "folgen" ** {c = Dative} ; 
```
To extend a record type or a record with a field whose label it
already has is a type error.

A record type //T// is a **subtype** of another one //R//, if //T// has
all the fields of //R// and possibly other fields. For instance,
an extension of a record type is always a subtype of it.

If //T// is a subtype of //R//, an object of //T// can be used whenever
an object of //R// is required. For instance, a transitive verb can
be used whenever a verb is required.

**Contravariance** means that a function taking an //R// as argument
can also be applied to any object of a subtype //T//.



==Tuples and product types==

Product types and tuples are syntactic sugar for record types and records:
```
  T1 * ... * Tn   ===   {p1 : T1 ; ... ; pn : Tn}
  <t1, ...,  tn>  ===   {p1 = T1 ; ... ; pn = Tn}
```
Thus the labels ``p1, p2,...`` are hard-coded.


==Record and tuple patterns==

Record types of parameter types are also parameter types.
A typical example is a record of agreement features, e.g. French
```
  oper Agr : PType = {g : Gender ; n : Number ; p : Person} ;
```
Notice the term ``PType`` rather than just ``Type`` referring to
parameter types. Every ``PType`` is also a ``Type``, but not vice-versa.

Pattern matching is done in the expected way, but it can moreover
utilize partial records: the branch
```
  {g = Fem} => t
```
in a table of type ``Agr => T`` means the same as
```
  {g = Fem ; n = _ ; p = _} => t
```
Tuple patterns are translated to record patterns in the
same way as tuples to records; partial patterns make it
possible to write, slightly surprisingly,
```
  case <g,n,p> of {
    <Fem> => t
    ...
    }
```


%--!
==Regular expression patterns==

To define string operations computed at compile time, such
as in morphology, it is handy to use regular expression patterns:
 - //p// ``+`` //q// : token consisting of //p// followed by //q//
 - //p// ``*``       : token //p// repeated 0 or more times
                                     (max the length of the string to be matched)
 - ``-`` //p//       : matches anything that //p// does not match
 - //x// ``@`` //p// : bind to //x// what //p// matches
 - //p// ``|`` //q// : matches what either //p// or //q// matches


The last three apply to all types of patterns, the first two only to token strings.
As an example, we give a rule for the formation of English word forms
ending with an //s// and used in the formation of both plural nouns and
third-person present-tense verbs.
```
  add_s : Str -> Str = \w -> case w of {
    _ + "oo"                           => w + "s" ;   -- bamboo
    _ + ("s" | "z" | "x" | "sh" | "o") => w + "es" ;  -- bus, hero
    _ + ("a" | "o" | "u" | "e") + "y"  => w + "s" ;   -- boy
    x + "y"                            => x + "ies" ; -- fly
    _                                  => w + "s"     -- car
    } ;
```
Here is another example, the plural formation in Swedish 2nd declension.
The second branch uses a variable binding with ``@`` to cover the cases where an
unstressed pre-final vowel //e// disappears in the plural
(//nyckel-nycklar, seger-segrar, bil-bilar//):
```
  plural2 : Str -> Str = \w -> case w of {
    pojk + "e"                       => pojk + "ar" ;
    nyck + "e" + l@("l" | "r" | "n") => nyck + l + "ar" ;
    bil                              => bil + "ar"
    } ;
```


Semantics: variables are always bound to the **first match**, which is the first
in the sequence of binding lists ``Match p v`` defined as follows. In the definition,
``p`` is a pattern and ``v`` is a value. The semantics is given in Haskell notation.
```
  Match (p1|p2) v = Match p1 ++ U Match p2 v
  Match (p1+p2) s = [Match p1 s1 ++ Match p2 s2 | 
                       i <- [0..length s], (s1,s2) = splitAt i s]
  Match p*      s = [[]] if Match "" s ++ Match p s ++ Match (p+p) s ++... /= []
  Match -p      v = [[]] if Match p v = []
  Match c       v = [[]] if c == v  -- for constant and literal patterns c
  Match x       v = [[(x,v)]]       -- for variable patterns x
  Match x@p     v = [[(x,v)]] + M   if M = Match p v /= []
  Match p       v = [] otherwise    -- failure
```
Examples:
- ``x + "e" + y`` matches ``"peter"`` with ``x = "p", y = "ter"``
- ``x + "er"*`` matches ``"burgerer"`` with ``x = "burg"





%--!
==Prefix-dependent choices==

Sometimes a token has different forms depending on the token
that follows. An example is the English indefinite article,
which is //an// if a vowel follows, //a// otherwise.
Which form is chosen can only be decided at run time, i.e.
when a string is actually build. GF has a special construct for
such tokens, the ``pre`` construct exemplified in
```
  oper artIndef : Str = 
    pre {"a" ; "an" / strs {"a" ; "e" ; "i" ; "o"}} ;
```
Thus
```
  artIndef ++ "cheese"  --->  "a" ++ "cheese"
  artIndef ++ "apple"   --->  "an" ++ "apple"
```
This very example does not work in all situations: the prefix
//u// has no general rules, and some problematic words are
//euphemism, one-eyed, n-gram//. It is possible to write
```
  oper artIndef : Str = 
    pre {"a" ; 
         "a"  / strs {"eu" ; "one"} ;
         "an" / strs {"a" ; "e" ; "i" ; "o" ; "n-"}
        } ;
```


==Predefined types==

GF has the following predefined categories in abstract syntax:
```
  cat Int ;     -- integers, e.g. 0, 5, 743145151019
  cat Float ;   -- floats,   e.g. 0.0, 3.1415926
  cat String ;  -- strings,  e.g. "", "foo", "123"
```
The objects of each of these categories are **literals**
as indicated in the comments above. No ``fun`` definition
can have a predefined category as its value type, but
they can be used as arguments. For example:
```
  fun StreetAddress : Int -> String -> Address ;
  lin StreetAddress number street = {s = number.s ++ street.s} ;

  -- e.g. (StreetAddress 10 "Downing Street") : Address
```
FIXME: The linearization type is ``{s : Str}`` for all these categories.



%--!
=Using the resource grammar library=

%!include: ../intro-resource.txt


%--!
=Overview of the resource grammar library=

%!include: ../overview-resource.txt



=More concepts of abstract syntax=

This section is about the use of the type theory part of GF for 
including more semantics in grammars. Some of the subsections present
ideas that have not yet been used in real-world applications, and whose
tool support outside the GF program is not complete.


==GF as a logical framework==

In this section, we will show how 
to encode advanced semantic concepts in an abstract syntax.
We use concepts inherited from **type theory**. Type theory
is the basis of many systems known as **logical frameworks**, which are
used for representing mathematical theorems and their proofs on a computer.
In fact, GF has a logical framework as its proper part:
this part is the abstract syntax.

In a logical framework, the formalization of a mathematical theory
is a set of type and function declarations. The following is an example
of such a theory, represented as an ``abstract`` module in GF.
```
abstract Arithm = {
  cat
    Prop ;                        -- proposition
    Nat ;                         -- natural number
  fun
    Zero : Nat ;                  -- 0
    Succ : Nat -> Nat ;           -- successor of x
    Even : Nat -> Prop ;          -- x is even
    And  : Prop -> Prop -> Prop ; -- A and B
    } 
```
A concrete syntax is given below, as an example of using the resource grammar
library.



==Dependent types==

**Dependent types** are a characteristic feature of GF,
inherited from the
**constructive type theory** of Martin-Löf and
distinguishing GF from most other grammar formalisms and
functional programming languages.
The initial main motivation for developing GF was, indeed,
to have a grammar formalism with dependent types.
As can be inferred from the fact that we introduce them only now,
after having written lots of grammars without them,
dependent types are no longer the only motivation for GF.
But they are still important and interesting.


Dependent types can be used for stating stronger
**conditions of well-formedness** than non-dependent types.
A simple example is postal addresses. Ignoring the other details,
let us take a look at addresses consisting of
a street, a city, and a country. 
```
abstract Address = {
  cat 
    Address ; Country ; City ; Street ;

  fun
    mkAddress : Country -> City -> Street -> Address ;

    UK, France : Country ;
    Paris, London, Grenoble : City ;
    OxfordSt, ShaftesburyAve, BdRaspail, RueBlondel, AvAlsaceLorraine : Street ;
  }
```
The linearization rules are straightforward,
```
  lin
    mkAddress country city street = 
      ss (street.s ++ "," ++ city.s ++ "," ++ country.s) ;
    UK = ss ("U.K.") ;
    France = ss ("France") ;
    Paris = ss ("Paris") ;
    London = ss ("London") ;
    Grenoble = ss ("Grenoble") ;
    OxfordSt = ss ("Oxford" ++ "Street") ;
    ShaftesburyAve = ss ("Shaftesbury" ++ "Avenue") ;
    BdRaspail = ss ("boulevard" ++ "Raspail") ;
    RueBlondel = ss ("rue" ++ "Blondel") ;
    AvAlsaceLorraine = ss ("avenue" ++ "Alsace-Lorraine") ;
```
Notice that, in ``mkAddress``, we have
reversed the order of the constituents. The type of ``mkAddress``
in the abstract syntax takes its arguments in a "logical" order,
with increasing precision. (This order is sometimes even used in the 
concrete syntax of addresses, e.g. in Russia).

Both existing and non-existing addresses are recognized by this
grammar. The non-existing ones in the following randomly generated
list have afterwards been marked by *:
```
  > gr -cat=Address -number=7 | l

  * Oxford Street , Paris , France
  * Shaftesbury Avenue , Grenoble , U.K.
  boulevard Raspail , Paris , France
  * rue Blondel , Grenoble , U.K.
  * Shaftesbury Avenue , Grenoble , France
  * Oxford Street , London , France
  * Shaftesbury Avenue , Grenoble , France
```
Dependent types provide a way to guarantee that addresses are
well-formed. What we do is to include **contexts** in
``cat`` judgements:
```
  cat 
    Address ; 
    Country ; 
    City Country ; 
    Street (x : Country)(City x) ;
```
The first two judgements are as before, but the third one makes
``City`` dependent on ``Country``: there are no longer just cities,
but cities of the U.K. and cities of France. The fourth judgement
makes ``Street`` dependent on ``City``; but since 
``City`` is itself dependent on ``Country``, we must
include them both in the context, moreover guaranteeing that
the city is one of the given country. Since the context itself
is built by using a dependent type, we have to use variables
to indicate the dependencies. The judgement we used for ``City``
is actually shorthand for
```
  cat City (x : Country)
```
which is only possible if the subsequent context does not depend on ``x``.

The ``fun`` judgements of the grammar are modified accordingly:
```
  fun
    mkAddress : (x : Country) -> (y : City x) -> Street x y -> Address ;
  
    UK : Country ;
    France : Country ;
    Paris : City France ; 
    London : City UK ; 
    Grenoble : City France ;
    OxfordSt : Street UK London ; 
    ShaftesburyAve : Street UK London ;
    BdRaspail : Street France Paris ; 
    RueBlondel : Street France Paris ; 
    AvAlsaceLorraine : Street France Grenoble ;
```
Since the type of ``mkAddress`` now has dependencies among
its argument types, we have to use variables just like we used in
the context of ``Street`` above. What we claimed to be the
"logical" order of the arguments is now forced by the type system
of GF: a variable must be declared (=bound) before it can be
referenced (=used).

The effect of dependent types is that the *-marked addresses above are
no longer well-formed. What the GF parser actually does is that it
initially accepts them (by using a context-free parsing algorithm)
and then rejects them (by type checking). The random generator does not produce
illegal addresses (this could be useful in bulk mailing!).
The linearization algorithm does not care about type dependencies;
actually, since the //categories// (ignoring their arguments)
are the same in both abstract syntaxes,
we use the same concrete syntax
for both of them.

**Remark**. Function types //without//
variables are actually a shorthand notation: writing
```
  fun PredV1 : NP -> V1 -> S
```
is shorthand for 
```
  fun PredV1 : (x : NP) -> (y : V1) -> S
```
or any other naming of the variables. Actually the use of variables
sometimes shortens the code, since we can write e.g.
```
  oper triple : (x,y,z : Str) -> Str = ...
```
If a bound variable is not used, it can here, as elsewhere in GF, be replaced by
a wildcard:
```
  oper triple : (_,_,_ : Str) -> Str = ...
```



==Dependent types in concrete syntax==

The **functional fragment** of GF
terms and types comprises function types, applications, lambda
abstracts, constants, and variables. This fragment is similar in
abstract and concrete syntax. In particular,
dependent types are also available in concrete syntax.
We have not made use of them yet,
but we will now look at one example of how they
can be used.

Those readers who are familiar with functional programming languages
like ML and Haskell, may already have missed **polymorphic**
functions. For instance, Haskell programmers have access to
the functions
```
  const :: a -> b -> a
  const c _ = c

  flip :: (a -> b -> c) -> b -> a -> c
  flip f y x = f x y
```
which can be used for any given types ``a``,``b``, and ``c``.

The GF counterpart of polymorphic functions are **monomorphic**
functions with explicit **type variables**. Thus the above
definitions can be written
```
  oper const :(a,b : Type) -> a -> b -> a =
    \_,_,c,_ -> c ;

  oper flip : (a,b,c : Type) -> (a -> b ->c) -> b -> a -> c =
    \_,_,_,f,x,y -> f y x ;
```
When the operations are used, the type checker requires
them to be equipped with all their arguments; this may be a nuisance
for a Haskell or ML programmer. 



==Expressing selectional restrictions==

This section introduces a way of using dependent types to
formalize a notion known as **selectional restrictions**
in linguistics. We first present a mathematical model
of the notion, and then integrate it in a linguistically
motivated syntax.

In linguistics, a 
grammar is usually thought of as being about **syntactic well-formedness**
in a rather liberal sense: an expression can be well-formed without
being meaningful, in other words, without being
**semantically well-formed**.
For instance, the sentence
```
  the number 2 is equilateral
```
is syntactically well-formed but semantically ill-formed.
It is well-formed because it combines a well-formed
noun phrase ("the number 2") with a well-formed
verb phrase ("is equilateral") and satisfies the agreement 
rule saying that the verb phrase is inflected in the
number of the noun phrase:
```
  fun PredVP : NP -> VP -> S ;
  lin PredVP np v = {s = np.s ++ vp.s ! np.n} ;
```
It is ill-formed because the predicate "is equilateral"
is only defined for triangles, not for numbers.

In a straightforward type-theoretical formalization of
mathematics, domains of mathematical objects
are defined as types. In GF, we could write
```
  cat Nat ;
  cat Triangle ;
  cat Prop ;
```
for the types of natural numbers, triangles, and propositions,
respectively.
Noun phrases are typed as objects of basic types other than
``Prop``, whereas verb phrases are functions from basic types
to ``Prop``. For instance,
```
  fun two : Nat ;
  fun Even : Nat -> Prop ;
  fun Equilateral : Triangle -> Prop ;
```
With these judgements, and the linearization rules
```
  lin two = ss ["the number 2"] ;
  lin Even x = ss (x.s ++ ["is even"]) ;
  lin Equilateral x = ss (x.s ++ ["is equilateral"]) ;
```
we can form the proposition ``Even two``
```
  the number 2 is even
```
but no proposition linearized to
```
  the number 2 is equilateral
```
since ``Equilateral two`` is not a well-formed type-theoretical object.
It is not even accepted by the context-free parser.

When formalizing mathematics, e.g. in the purpose of
computer-assisted theorem proving, we are certainly interested
in semantic well-formedness: we want to be sure that a proposition makes
sense before we make the effort of proving it. The straightforward typing
of nouns and predicates shown above is the way in which this
is guaranteed in various proof systems based on type theory.
(Notice that it is still possible to form //false// propositions,
e.g. "the number 3 is even".
False and meaningless are different things.)

As shown by the linearization rules for ``two``, ``Even``,
etc, it //is// possible to use straightforward mathematical typings
as the abstract syntax of a grammar. However, this syntax is not very
expressive linguistically: for instance, there is no distinction between
adjectives and verbs. It is hard to give rules for structures like
adjectival modification ("even number") and conjunction of predicates
("even or odd"). 

By using dependent types, it is simple to combine a linguistically
motivated system of categories with mathematically motivated
type restrictions. What we need is a category of domains of
individual objects,
```
  cat Dom
```
and dependencies of other categories on this:
```
  cat 
    S ;            -- sentence
    V1 Dom ;       -- one-place verb with specific subject type
    V2 Dom Dom ;   -- two-place verb with specific subject and object types
    A1 Dom ;       -- one-place adjective
    A2 Dom Dom ;   -- two-place adjective
    NP Dom ;       -- noun phrase for an object of specific type
    Conj ;         -- conjunction
    Det ;          -- determiner
```
Having thus parametrized categories on domains, we have to reformulate
the rules of predication, etc, accordingly. This is straightforward:
```
  fun
    PredV1  : (A   : Dom) -> NP A -> V1 A -> S ;
    ComplV2 : (A,B : Dom) -> V2 A B -> NP B -> V1 A ;
    DetCN   :                Det -> (A : Dom) -> NP A ;
    ModA1   :                (A : Dom) -> A1 A -> Dom ;
    ConjS   :                Conj -> S -> S -> S ;
    ConjV1  : (A   : Dom) -> Conj -> V1 A -> V1 A -> V1 A ;
```
In linearization rules,
we use wildcards for the domain arguments,
because they don't affect linearization:
```
  lin
    PredV1 _ np vp = ss (np.s ++ vp.s) ;
    ComplV2 _ _ v2 np = ss (v2.s ++ np.s) ;
    DetCN det cn = ss (det.s ++ cn.s) ;
    ModA1 cn a1 = ss (a1.s ++ cn.s) ;
    ConjS conj s1 s2 = ss (s1.s ++ conj.s ++ s2.s) ;
    ConjV1 _ conj v1 v2 = ss (v1.s ++ conj.s ++ v2.s) ;
```
The domain arguments thus get suppressed in linearization.
Parsing initially returns metavariables for them,
but type checking can usually restore them
by inference from those arguments that are not suppressed.

One traditional linguistic example of domain restrictions
(= selectional restrictions) is the contrast between the two sentences
```
  John plays golf
  golf plays John
```
To explain the contrast, we introduce the functions
```
  human : Dom ; 
  game : Dom ;
  play : V2 human game ;
  John : NP human ;
  Golf : NP game ;
```
Both sentences still pass the context-free parser,
returning trees with lots of metavariables of type ``Dom``:
```
  PredV1 ?0 John (ComplV2 ?1 ?2 play Golf)
  PredV1 ?0 Golf (ComplV2 ?1 ?2 play John)
```
But only the former sentence passes the type checker, which moreover
infers the domain arguments:
```
  PredV1 human John (ComplV2 human game play Golf)
```
To try this out in GF, use ``pt = put_term`` with the **tree transformation**
that solves the metavariables by type checking:
```
  > p -tr "John plays golf" | pt -transform=solve
  > p -tr "golf plays John" | pt -transform=solve
```
In the latter case, no solutions are found.

A known problem with selectional restrictions is that they can be more
or less liberal. For instance,
```
  John loves Mary
  John loves golf
```
should both make sense, even though ``Mary`` and ``golf``
are of different types. A natural solution in this case is to
formalize ``love`` as a **polymorphic** verb, which takes
a human as its first argument but an object of any type as its second
argument:
```
  fun love : (A : Dom) -> V2 human A ;
  lin love _ = ss "loves" ;
```
In other words, it is possible for a human to love anything.

A problem related to polymorphism is **subtyping**: what
is meaningful for a ``human`` is also meaningful for
a ``man`` and a ``woman``, but not the other way round.
One solution to this is **coercions**: functions that
"lift" objects from subtypes to supertypes.


==Case study: selectional restrictions and statistical language models TODO==


==Proof objects==

Perhaps the most well-known idea in constructive type theory is
the **Curry-Howard isomorphism**, also known as the
**propositions as types principle**. Its earliest formulations
were attempts to give semantics to the logical systems of
propositional and predicate calculus. In this section, we will consider
a more elementary example, showing how the notion of proof is useful
outside mathematics, as well.

We first define the category of unary (also known as Peano-style)
natural numbers:
```
  cat Nat ; 
  fun Zero : Nat ;
  fun Succ : Nat -> Nat ;
```
The **successor function** ``Succ`` generates an infinite
sequence of natural numbers, beginning from ``Zero``.

We then define what it means for a number //x// to be //less than//
a number //y//. Our definition is based on two axioms:
- ``Zero`` is less than ``Succ`` //y// for any //y//.
- If //x// is less than //y//, then``Succ`` //x// is less than ``Succ`` //y//.


The most straightforward way of expressing these axioms in type theory
is as typing judgements that introduce objects of a type ``Less`` //x y //:
```
  cat Less Nat Nat ; 
  fun lessZ : (y : Nat) -> Less Zero (Succ y) ;
  fun lessS : (x,y : Nat) -> Less x y -> Less (Succ x) (Succ y) ;
```
Objects formed by ``lessZ`` and ``lessS`` are
called **proof objects**: they establish the truth of certain
mathematical propositions.
For instance, the fact that 2 is less that
4 has the proof object
```
  lessS (Succ Zero) (Succ (Succ (Succ Zero)))
        (lessS Zero (Succ (Succ Zero)) (lessZ (Succ Zero)))
```
whose type is
```
  Less (Succ (Succ Zero)) (Succ (Succ (Succ (Succ Zero))))
```
which is the formalization of the proposition that 2 is less than 4.

GF grammars can be used to provide a **semantic control** of
well-formedness of expressions. We have already seen examples of this:
the grammar of well-formed addresses and the grammar with
selectional restrictions above. By introducing proof objects
we have now added a very powerful technique of expressing semantic conditions.

A simple example of the use of proof objects is the definition of
well-formed //time spans//: a time span is expected to be from an earlier to
a later time:
```
  from 3 to 8
```
is thus well-formed, whereas
```
  from 8 to 3
```
is not. The following rules for spans impose this condition
by using the ``Less`` predicate:
```
  cat Span ;
  fun span : (m,n : Nat) -> Less m n -> Span ;
```
A possible practical application of this idea is **proof-carrying documents**:
to be semantically well-formed, the abstract syntax of a document must contain a proof
of some property, although the proof is not shown in the concrete document.
Think, for instance, of small documents describing flight connections:

//To fly from Gothenburg to Prague, first take LH3043 to Frankfurt, then OK0537 to Prague.//

The well-formedness of this text is partly expressible by dependent typing: 
```
  cat
    City ;
    Flight City City ;
  fun
    Gothenburg, Frankfurt, Prague : City ;
    LH3043 : Flight Gothenburg Frankfurt ;
    OK0537 : Flight Frankfurt Prague ;
```
This rules out texts saying //take OK0537 from Gothenburg to Prague//. However, there is a
further condition saying that it must be possible to change from LH3043 to OK0537 in Frankfurt.
This can be modelled as a proof object of a suitable type, which is required by the constructor
that connects flights.
```
  cat
    IsPossible (x,y,z : City)(Flight x y)(Flight y z) ;
  fun
    Connect : (x,y,z : City) -> 
      (u : Flight x y) -> (v : Flight y z) -> 
        IsPossible x y z u v -> Flight x z ;
```



==Variable bindings==

Mathematical notation and programming languages have lots of
expressions that **bind** variables. For instance,
a universally quantifier proposition
```
  (All x)B(x)
```
consists of the **binding** ``(All x)`` of the variable ``x``,
and the **body** ``B(x)``, where the variable ``x`` can have
**bound occurrences**. 

Variable bindings appear in informal mathematical language as well, for
instance,
```
  for all x, x is equal to x

  the function that for any numbers x and y returns the maximum of x+y
  and x*y
```
In type theory, variable-binding expression forms can be formalized
as functions that take functions as arguments. The universal
quantifier is defined
```
  fun All : (Ind -> Prop) -> Prop
```
where ``Ind`` is the type of individuals and ``Prop``,
the type of propositions. If we have, for instance, the equality predicate
```
  fun Eq : Ind -> Ind -> Prop
```
we may form the tree
```
  All (\x -> Eq x x)
```
which corresponds to the ordinary notation
```
  (All x)(x = x).
```


An abstract syntax where trees have functions as arguments, as in
the two examples above, has turned out to be precisely the right
thing for the semantics and computer implementation of
variable-binding expressions. The advantage lies in the fact that
only one variable-binding expression form is needed, the lambda abstract
``\x -> b``, and all other bindings can be reduced to it.
This makes it easier to implement mathematical theories and reason
about them, since variable binding is tricky to implement and
to reason about. The idea of using functions as arguments of
syntactic constructors is known as **higher-order abstract syntax**.

The question now arises: how to define linearization rules
for variable-binding expressions?
Let us first consider universal quantification,
```
  fun All : (Ind -> Prop) -> Prop
```
We write
```
  lin All B = {s = "(" ++ "All" ++ B.$0 ++ ")" ++ B.s}
```
to obtain the form shown above. 
This linearization rule brings in a new GF concept - the ``$0``
field of ``B`` containing a bound variable symbol.
The general rule is that, if an argument type of a function is
itself a function type ``A -> C``, the linearization type of
this argument is the linearization type of ``C``
together with a new field ``$0 : Str``. In the linearization rule
for ``All``, the argument ``B`` thus has the linearization
type
```
  {$0 : Str ; s : Str},
```
since the linearization type of ``Prop`` is
```
  {s : Str}
```
In other words, the linearization of a function
consists of a linearization of the body together with a
field for a linearization of the bound variable.
Those familiar with type theory or lambda calculus
should notice that GF requires trees to be in
**eta-expanded** form in order to be linearizable:
any function of type
```
  A -> B
```
always has a syntax tree of the form
```
  \x -> b
```
where ``b : B`` under the assumption ``x : A``.
It is in this form that an expression can be analysed
as having a bound variable and a body. 


Given the linearization rule
```
  lin Eq a b = {s = "(" ++ a.s ++ "=" ++ b.s ++ ")"}
```
the linearization of
```
  \x -> Eq x x
```
is the record
```
  {$0 = "x", s = ["( x = x )"]}
```
Thus we can compute the linearization of the formula,
```
  All (\x -> Eq x x)  --> {s = "[( All x ) ( x = x )]"}.
```

How did we get the //linearization// of the variable ``x``
into the string ``"x"``? GF grammars have no rules for
this: it is just hard-wired in GF that variable symbols are
linearized into the same strings that represent them in
the print-out of the abstract syntax. 


To be able to //parse// variable symbols, however, GF needs to know what
to look for (instead of e.g. trying to parse //any//
string as a variable). What strings are parsed as variable symbols 
is defined in the lexical analysis part of GF parsing 
```
  > p -cat=Prop -lexer=codevars "(All x)(x = x)"
  All (\x -> Eq x x)
```
(see more details on lexers below). If several variables are bound in the 
same argument, the labels are ``$0, $1, $2``, etc. 



==Semantic definitions==

We have seen that,
just like functional programming languages, GF has declarations
of functions, telling what the type of a function is.
But we have not yet shown how to **compute**
these functions: all we can do is provide them with arguments
and linearize the resulting terms.
Since our main interest is the well-formedness of expressions,
this has not yet bothered 
us very much. As we will see, however, computation does play a role
even in the well-formedness of expressions when dependent types are
present.

GF has a form of judgement for **semantic definitions**,
recognized by the key word ``def``. At its simplest, it is just
the definition of one constant, e.g.
```
  def one = Succ Zero ;
```
We can also define a function with arguments,
```
  def Neg A = Impl A Abs ;
```
which is still a special case of the most general notion of
definition, that of a group of **pattern equations**:
```
  def 
    sum x Zero = x ;
    sum x (Succ y) = Succ (Sum x y) ;
```
To compute a term is, as in functional programming languages,
simply to follow a chain of reductions until no definition
can be applied. For instance, we compute
```
  Sum one one -->
  Sum (Succ Zero) (Succ Zero) -->
  Succ (sum (Succ Zero) Zero) -->
  Succ (Succ Zero)
```
Computation in GF is performed with the ``pt`` command and the
``compute`` transformation, e.g.
```
  > p -tr "1 + 1" | pt -transform=compute -tr | l
  sum one one
  Succ (Succ Zero)
  s(s(0))
```

The ``def`` definitions of a grammar induce a notion of
**definitional equality** among trees: two trees are
definitionally equal if they compute into the same tree.
Thus, trivially, all trees in a chain of computation
(such as the one above)
are definitionally equal to each other. So are the trees
```
  sum Zero (Succ one)
  Succ one
  sum (sum Zero Zero) (sum (Succ Zero) one)
```
and infinitely many other trees.

A fact that has to be emphasized about ``def`` definitions is that
they are //not// performed as a first step of linearization.
We say that **linearization is intensional**, which means that
the definitional equality of two trees does not imply that
they have the same linearizations. For instance, each of the seven terms
shown above has a different linearizations in arithmetic notation:
```
  1 + 1
  s(0) + s(0)
  s(s(0) + 0)
  s(s(0))
  0 + s(0)
  s(1)
  0 + 0 + s(0) + 1
```
This notion of intensionality is
no more exotic than the intensionality of any **pretty-printing**
function of a programming language (function that shows
the expressions of the language as strings). It is vital for
pretty-printing to be intensional in this sense - if we want,
for instance, to trace a chain of computation by pretty-printing each
intermediate step, what we want to see is a sequence of different
expression, which are definitionally equal.

What is more exotic is that GF has two ways of referring to the
abstract syntax objects. In the concrete syntax, the reference is intensional.
In the abstract syntax, the reference is extensional, since
**type checking is extensional**. The reason is that,
in the type theory with dependent types, types may depend on terms.
Two types depending on terms that are definitionally equal are
equal types. For instance,
```
  Proof (Odd one)
  Proof (Odd (Succ Zero))
```
are equal types. Hence, any tree that type checks as a proof that
1 is odd also type checks as a proof that the successor of 0 is odd.
(Recall, in this connection, that the
arguments a category depends on never play any role
in the linearization of trees of that category,
nor in the definition of the linearization type.)

In addition to computation, definitions impose a
**paraphrase** relation on expressions:
two strings are paraphrases if they
are linearizations of trees that are
definitionally equal.
Paraphrases are sometimes interesting for
translation: the **direct translation**
of a string, which is the linearization of the same tree
in the targer language, may be inadequate because it is e.g.
unidiomatic or ambiguous. In such a case, 
the translation algorithm may be made to consider
translation by a paraphrase.

To stress express the distinction between
**constructors** (=**canonical** functions)
and other functions, GF has a judgement form
``data`` to tell that certain functions are canonical, e.g.
```
  data Nat = Succ | Zero ;
```
Unlike in Haskell, but similarly to ALF (where constructor functions
are marked with a flag ``C``), 
new constructors can be added to
a type with new ``data`` judgements. The type signatures of constructors
are given separately, in ordinary ``fun`` judgements.
One can also write directly
```
  data Succ : Nat -> Nat ;
```
which is equivalent to the two judgements
```
  fun Succ : Nat -> Nat ;
  data Nat = Succ ;
```


==Case study: representing anaphoric reference TODO==


=Transfer modules TODO=

Transfer means noncompositional tree-transforming operations.
The command ``apply_transfer = at`` is typically used in a pipe:
```
  > p "John walks and John runs" | apply_transfer aggregate | l
  John walks and runs
```
See the
[sources ../../transfer/examples/aggregation] of this example.

See the
[transfer language documentation  ../transfer.html]
for more information.


=Practical issues TODO=


==Lexers and unlexers==

Lexers and unlexers can be chosen from
a list of predefined ones, using the flags``-lexer`` and `` -unlexer`` either
in the grammar file or on the GF command line.

Given by ``help -lexer``, ``help -unlexer``:
```
    The default is words.
    -lexer=words         tokens are separated by spaces or newlines
    -lexer=literals      like words, but GF integer and string literals recognized
    -lexer=vars          like words, but "x","x_...","$...$" as vars, "?..." as meta
    -lexer=chars         each character is a token
    -lexer=code          use Haskell's lex
    -lexer=codevars      like code, but treat unknown words as variables, ?? as meta
    -lexer=text          with conventions on punctuation and capital letters
    -lexer=codelit       like code, but treat unknown words as string literals
    -lexer=textlit       like text, but treat unknown words as string literals
    -lexer=codeC         use a C-like lexer
    -lexer=ignore        like literals, but ignore unknown words
    -lexer=subseqs       like ignore, but then try all subsequences from longest

    The default is unwords.
    -unlexer=unwords     space-separated token list (like unwords)
    -unlexer=text        format as text: punctuation, capitals, paragraph <p>
    -unlexer=code        format as code (spacing, indentation)
    -unlexer=textlit     like text, but remove string literal quotes
    -unlexer=codelit     like code, but remove string literal quotes
    -unlexer=concat      remove all spaces
    -unlexer=bind        like identity, but bind at "&+"
```


==Efficiency of grammars==

Issues:

- the choice of datastructures in ``lincat``s
- the value of the ``optimize`` flag 
- parsing efficiency: ``-fcfg`` vs. others


==Speech input and output==

The``speak_aloud = sa`` command sends a string to the speech
synthesizer 
[Flite http://www.speech.cs.cmu.edu/flite/doc/].
It is typically used via a pipe:
```  generate_random | linearize | speak_aloud
The result is only satisfactory for English.

The ``speech_input = si`` command receives a string from a
speech recognizer that requires the installation of
[ATK http://mi.eng.cam.ac.uk/~sjy/software.htm].
It is typically used to pipe input to a parser:
```  speech_input -tr | parse
The method words only for grammars of English.

Both Flite and ATK are freely available through the links
above, but they are not distributed together with GF.


==Multilingual syntax editor==

The 
[Editor User Manual http://www.cs.chalmers.se/~aarne/GF2.0/doc/javaGUImanual/javaGUImanual.htm]
describes the use of the editor, which works for any multilingual GF grammar.

Here is a snapshot of the editor:

[../quick-editor.png]
 
The grammars of the snapshot are from the
[Letter grammar package http://www.cs.chalmers.se/~aarne/GF/examples/letter].



==Interactive Development Environment (IDE)==

Forthcoming.


==Communicating with GF==

Other processes can communicate with the GF command interpreter,
and also with the GF syntax editor. Useful flags when invoking GF are
- ``-batch`` suppresses the promps and structures the communication with XML tags.
- ``-s`` suppresses non-output non-error messages and XML tags.
-- ``-nocpu`` suppresses CPU time indication.

Thus the most silent way to invoke GF is
```
  gf -batch -s -nocpu
```



==Embedded grammars in Haskell, Java, and Prolog==

GF grammars can be used as parts of programs written in the
following languages. The links give more documentation.

- [Java http://www.cs.chalmers.se/~bringert/gf/gf-java.html]
- [Haskell http://www.cs.chalmers.se/~aarne/GF/src/GF/Embed/EmbedAPI.hs]
- [Prolog http://www.cs.chalmers.se/~peb/software.html]


==Alternative input and output grammar formats==

A summary is given in the following chart of GF grammar compiler phases:
[../gf-compiler.png]


=Larger case studies TODO=

==Interfacing formal and natural languages==

[Formal and Informal Software Specifications http://www.cs.chalmers.se/~krijo/thesis/thesisA4.pdf],
PhD Thesis by
[Kristofer Johannisson http://www.cs.chalmers.se/~krijo], is an extensive example of this.
The system is based on a multilingual grammar relating the formal language OCL with
English and German.

A simpler example will be explained here.


==A multimodal dialogue system==

See TALK project deliverables, [TALK homepage http://www.talk-project.org]




% Appendices

=Appendix A: The GF Command Help=

%!include: ../gf-help.txt


=Appendix B: GF Quick Reference=

%!include: ../gf-reference.txt


=Appendix C: Resource Grammar Synopsis=

%!include: ../resource-synopsis.txt