# Grammatical Framework Tutorial

Aarne Ranta
Version 3.1.2, November 2008

- Overview
  - Outline
  - Slides
- Lesson 1: Getting Started with GF
  - What GF is
  - GF grammars and language processing tasks
  - Getting the GF system
  - Running the GF system
  - A "Hello World" grammar
    - The program: abstract syntax and concrete syntaxes
    - Using grammars in the GF system
    - Exercises on the Hello World grammar
  - Using grammars from outside GF
  - GF scripts
  - What else can be done with the grammar
  - Embedded grammar applications
- Lesson 2: Designing a grammar for complex phrases
  - The abstract syntax Food
  - The concrete syntax FoodEng
    - Exercises on the Food grammar
  - Commands for testing grammars
    - Generating trees and strings
    - Exercises on generation
    - More on pipes: tracing
    - Writing and reading files
    - Visualizing trees
    - System commands
  - An Italian concrete syntax
    - Exercises on multilinguality
  - Free variation
  - More application of multilingual grammars
    - Multilingual treebanks
    - Translation quiz
  - Context-free grammars and GF
    - The "cf" grammar format
    - Restrictions of context-free grammars
  - Modules and files
  - Using operations and resource modules
    - Operation definitions
    - The ``resource`` module type
    - Opening a resource
    - Partial application
    - Testing resource modules
  - Grammar architecture
    - Extending a grammar
    - Multiple inheritance
- Lesson 3: Grammars with parameters
  - The problem: words have to be inflected
  - Parameters and tables
  - Inflection tables and paradigms
    - Exercises on morphology
  - Using parameters in concrete syntax
    - Agreement
    - Determiners
    - Parametric vs. inherent features
  - An English concrete syntax for Foods with parameters
  - More on inflection paradigms
    - Worst-case functions
    - Smart paradigms
    - Exercises on regular patterns
    - Function types with variables
    - Separating operation types and definitions
    - Overloading of operations
    - Morphological analysis and morphology quiz
  - The Italian Foods grammar
    - Exercises on using parameters
  - Discontinuous constituents
  - Strings at compile time vs. run time
    - Supplementary constructs for concrete syntax
- Lesson 4: Using the resource grammar library
  - The coverage of the library
  - The structure of the library
    - Lexical vs. phrasal rules
    - Lexical categories
    - Lexical rules
    - Resource lexicon
    - Phrasal categories
    - Syntactic combinations
    - Example syntactic combination
  - The resource API
    - A miniature resource API: categories
    - A miniature resource API: rules
    - A miniature resource API: structural words
    - A miniature resource API: paradigms
    - A miniature resource API: more paradigms
    - Exercises
  - Example: English
    - English example: linearization types and combination rules
    - English example: lexical rules
    - English example: exercises
  - Functor implementation of multilingual grammars
    - New language by copy and paste
    - Functors: functions on the module level
    - Code for the Foods functor
    - Code for the LexFoods interface
    - Code for a German instance of the lexicon
    - Code for a German functor instantiation
    - Adding languages to a functor implementation
    - Example: adding Finnish
    - A design pattern
    - Functors: exercises
  - Restricted inheritance
    - A problem with functors
    - Restricted inheritance: include or exclude
    - The functor problem solved
  - Grammar reuse
    - Library exercises
  - Tenses
- Lesson 5: Refining semantics in abstract syntax
  - Dependent types
    - A dependent type system
    - Examples of devices and actions
    - Linearization and parsing with dependent types
    - Solving metavariables
  - Polymorphism
    - Dependent types: exercises
  - Proof objects
    - Proof-carrying documents
  - Restricted polymorphism
    - Example: classes for switching and dimming
  - Variable bindings
    - Higher-order abstract syntax
    - Higher-order abstract syntax: linearization
    - Eta expansion
    - Parsing variable bindings
    - Exercises on variable bindings
  - Semantic definitions
    - Computing a tree
    - Definitional equality
    - Judgement forms for constructors
    - Exercises on semantic definitions
  - Lesson 6: Grammars of formal languages
    - Arithmetic expressions
    - Concrete syntax: a simple approach
  - Lexing and unlexing
    - Most common lexers and unlexers
  - Precedence and fixity
    - Precedence as a parameter
    - Fixities
    - Exercises on precedence
  - Code generation as linearization
    - Programs with variables
    - Exercises on code generation
- Lesson 7: Embedded grammars
  - Functionalities of an embedded grammar format
  - The portable grammar format
    - Haskell: the EmbedAPI module
    - First application: a translator
    - Producing GFCC for the translator
    - A translator loop
    - A question-answer system
    - Abstract syntax of the query system
    - Exporting GF datatypes to Haskell
    - The question-answer function
    - Converting between Haskell and GF trees
    - Putting it all together: the transfer definition
    - Putting it all together: the Main module
    - Putting it all together: the Makefile
  - Web server applications
  - JavaScript applications
    - Compiling to JavaScript
    - Using the JavaScript grammar
  - Language models for speech recognition
    - More speech recognition grammar formats

# Overview

This is a hands-on introduction to grammar writing in GF.

Main ingredients of GF:

- linguistics
- functional programming

Prerequisites:

- some previous experience from some programming language
- the basics of using computers, e.g. the use of text editors and the management of files.
- knowledge of Unix commands is useful but not necessary
- knowledge of many natural languages may add fun to experience

## Outline

Lesson 1: a multilingual "Hello World" grammar. English, Finnish, Italian.

Lesson 2: a larger grammar for the domain of food. English and Italian.

Lesson 3: parameters - morphology and agreement.

Lesson 4: using the resource grammar library.

Lesson 5: semantics - dependent types, variable bindings, and semantic definitions.

Lesson 6: implementing formal languages.

Lesson 7: embedded grammar applications.

## Slides

You can chop this tutorial into a set of slides by the command

        htmls gf-tutorial.html

where the program `htmls` is distributed with GF (see below), in

`GF/src/tools/Htmls.hs`

The slides will appear as a set of files beginning with `01-gf-tutorial.htmls`.

Internal links will not work in the slide format, except for those in the upper left corner of each slide, and the links behind the "Contents" link.

# Lesson 1: Getting Started with GF

Goals:

- install and run GF
- write the first GF grammar: a "Hello World" grammar in three languages
- use GF for translation and multilingual generation

## What GF is

We use the term GF for three different things:

- a system (computer program) used for working with grammars
- a programming language in which grammars can be written
- a theory about grammars and languages

The GF system is an implementation of the GF programming language, which in turn is built on the ideas of the GF theory.

The focus of this tutorial is on using the GF programming language.

At the same time, we learn the way of thinking in the GF theory.

We make the grammars run on a computer by using the GF system.

## GF grammars and language processing tasks

A GF program is called a grammar.

A grammar defines a language.

From this definition, language processing components can be derived:

- parsing: to analyse the language
- linearization: to generate the language
- translation: to analyse one language and generate another

In general, a GF grammar is multilingual:

- many languages in one grammar
- translations between them

## Getting the GF system

Open-source free software, downloaded via the GF Homepage:

`digitalgrammars.com/gf`

There you find

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

Many examples in this tutorial are online.

Normally you don't have to compile GF yourself.

But, if you do want to compile GF from source, you need the Haskell compiler GHC.

We assume a Unix shell: Bash in Linux, "terminal" in Mac OS X, or Cygwin in Windows. But you can do most things even without Cygwin in Windows.

## Running the GF system

Type `gf` in the Unix (or Cygwin) shell:

        % gf

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, 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 characters that follow them.

## A "Hello World" grammar

Like most programming language tutorials, we start with a program that prints "Hello World" on the terminal.

Extra features:

- Multilinguality: the message is printed in many languages.
- Reversibility: in addition to printing, you can parse the message and translate it to other languages.

### The program: abstract syntax and concrete syntaxes

A GF program, in general, is a multilingual grammar. Its main parts are

- an abstract syntax
- one or more concrete syntaxes

The abstract syntax defines what meanings can be expressed in the grammar

- Greetings, where we greet a Recipient, which can be World or Mum or Friends

GF code for the abstract syntax:

        -- a "Hello World" grammar
        abstract Hello = {

          flags startcat = Greeting ;

          cat Greeting ; Recipient ;

          fun
            Hello : Recipient -> Greeting ;
            World, Mum, Friends : Recipient ;
        }

The code has the following parts:

- a comment (optional), saying what the module is doing
- a module header indicating that it is an abstract syntax module named `Hello`
- a module body in braces, consisting of
  - a startcat flag declaration stating that `Greeting` is the default start category for parsing and generation
  - category declarations introducing two categories, i.e. types of meanings
  - function declarations introducing three meaning-building functions

English concrete syntax (mapping from meanings to strings):

        concrete HelloEng of Hello = {

          lincat Greeting, Recipient = {s : Str} ;

          lin
            Hello recip = {s = "hello" ++ recip.s} ;
            World = {s = "world"} ;
            Mum = {s = "mum"} ;
            Friends = {s = "friends"} ;
        }

The major parts of this code are:

- a module header indicating that it is a concrete syntax of the abstract syntax `Hello`, itself named `HelloEng`
- a module body in curly brackets, consisting of
  - linearization type definitions stating that `Greeting` and `Recipient` are records with a string `s`
  - linearization definitions telling what records are assigned to each of the meanings defined in the abstract syntax

Notice the concatenation `++` and the record projection `.`.

Finnish and an Italian concrete syntaxes:

        concrete HelloFin of Hello = {
          lincat Greeting, Recipient = {s : Str} ;
          lin
            Hello recip = {s = "terve" ++ recip.s} ;
            World = {s = "maailma"} ;
            Mum = {s = "äiti"} ;
            Friends = {s = "ystävät"} ;
        }

        concrete HelloIta of Hello = {
          lincat Greeting, Recipient = {s : Str} ;
          lin
            Hello recip = {s = "ciao" ++ recip.s} ;
            World = {s = "mondo"} ;
            Mum = {s = "mamma"} ;
            Friends = {s = "amici"} ;
        }

### Using grammars in the GF system

In order to compile the grammar in GF, we create four files, one for each module, named Modulename`.gf`:

        Hello.gf  HelloEng.gf  HelloFin.gf  HelloIta.gf

The first GF command: import a grammar.

        > import HelloEng.gf

All commands also have short names; here:

        > i HelloEng.gf

The GF system will compile your grammar into an internal representation and show the CPU time was consumed, followed by a new prompt:

        > i HelloEng.gf
        - compiling Hello.gf...   wrote file Hello.gfo 8 msec
        - compiling HelloEng.gf...   wrote file HelloEng.gfo 12 msec

        12 msec
        >

You can use GF for parsing (`parse` = `p`)

        > parse "hello world"
        Hello World

Parsing takes a string into an abstract syntax tree.

The notation for trees is that of function application:

        function argument1 ... argumentn

Parentheses are only needed for grouping.

Parsing something that is not in grammar will fail:

        > parse "hello dad"
        Unknown words: dad

        > parse "world hello"
        no tree found

You can also use GF for linearization (`linearize = l`). It takes trees into strings:

        > linearize Hello World
        hello world

Translation: pipe linearization to parsing:

        > import HelloEng.gf
        > import HelloIta.gf

        > parse -lang=HelloEng "hello mum" | linearize -lang=HelloIta
        ciao mamma

Default of the language flag (`-lang`): the last-imported concrete syntax.

Multilingual generation:

        > parse -lang=HelloEng "hello friends" | linearize
        terve ystävät
        ciao amici
        hello friends

Linearization is by default to all available languages.

### Exercises on the Hello World grammar

1. Test the parsing and translation examples shown above, as well as some other examples, in different combinations of languages.

2. Extend the grammar `Hello.gf` and some of the concrete syntaxes by five new recipients and one new greeting form.

3. Add a concrete syntax for some other languages you might know.

4. Add a pair of greetings that are expressed in one and the same way in one language and in two different ways in another. For instance, good morning and good afternoon in English are both expressed as buongiorno in Italian. Test what happens when you translate buongiorno to English in GF.

5. Inject errors in the `Hello` grammars, for example, leave out some line, omit a variable in a `lin` rule, or change the name in one occurrence of a variable. Inspect the error messages generated by GF.

## Using grammars from outside GF

You can use the `gf` program in a Unix pipe.

- echo a GF command
- pipe it into GF with grammar names as arguments

        % echo "l Hello World" | gf HelloEng.gf HelloFin.gf HelloIta.gf

You can also write a script, a file containing the lines

        import HelloEng.gf
        import HelloFin.gf
        import HelloIta.gf
        linearize Hello World

## GF scripts

If we name this script `hello.gfs`, we can do

        $ gf --run <hello.gfs

        ciao mondo
        terve maailma
        hello world

The option `--run` removes prompts, CPU time, and other messages.

See Lesson 7, for stand-alone programs that don't need the GF system to run.

Exercise. (For Unix hackers.) Write a GF application that reads an English string from the standard input and writes an Italian translation to the output.

## What else can be done with the grammar

Some more functions that will be covered:

- 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 their linearizations
- teaching quizzes: train morphology and translation
- multilingual authoring: create a document in many languages simultaneously
- speech input: optimize a speech recognition system for a grammar

## Embedded grammar applications

Application programs, using techniques from Lesson 7:

- compile grammars to new formats, such as speech recognition grammars
- embed grammars in Java and Haskell programs
- build applications using compilation and embedding:
  - voice commands
  - spoken language translators
  - dialogue systems
  - user interfaces
  - localization: render the messages printed by a program in different languages

# Lesson 2: Designing a grammar for complex phrases

Goals:

- build a larger grammar: phrases about food in English and Italian
- learn to write reusable library functions ("operations")
- learn the basics of GF's module system

## The abstract syntax Food

Phrases usable for speaking about food:

- the start category is `Phrase`
- a `Phrase` can be built by assigning a `Quality` to an `Item` (e.g. this cheese is Italian)
- an`Item` is build from a `Kind` by prefixing this or that (e.g. this wine)
- a `Kind` is either atomic (e.g. cheese), or formed qualifying a given `Kind` with a `Quality` (e.g. Italian cheese)
- a `Quality` is either atomic (e.g. Italian, or built by modifying a given `Quality` with the word very (e.g. very warm)

Abstract syntax:

        abstract Food = {

          flags startcat = Phrase ;

          cat
            Phrase ; Item ; Kind ; Quality ;

          fun
            Is : Item -> Quality -> Phrase ;
            This, That : Kind -> Item ;
            QKind : Quality -> Kind -> Kind ;
            Wine, Cheese, Fish : Kind ;
            Very : Quality -> Quality ;
            Fresh, Warm, Italian, Expensive, Delicious, Boring : Quality ;
        }

Example `Phrase`

        Is (This (QKind Delicious (QKind Italian Wine))) (Very (Very Expensive))
        this delicious Italian wine is very very expensive

## The concrete syntax FoodEng

        concrete FoodEng of Food = {

          lincat
            Phrase, 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"} ;
        }

Test the grammar for parsing:

        > import FoodEng.gf
        > parse "this delicious wine is very very Italian"
        Is (This (QKind Delicious Wine)) (Very (Very Italian))

Parse in other categories setting the `cat` flag:

        p -cat=Kind "very Italian wine"
        QKind (Very Italian) Wine

### Exercises on the Food grammar

1. Extend the `Food` grammar by ten new food kinds and qualities, and run the parser with new kinds of examples.

2. Add a rule that enables question phrases of the form is this cheese Italian.

3. Enable the optional prefixing of phrases with the words "excuse me but". Do this in such a way that the prefix can occur at most once.

## Commands for testing grammars

### Generating trees and strings

Random generation (`generate_random = gr`): build build a random tree in accordance with an abstract syntax:

        > generate_random
        Is (This (QKind Italian Fish)) Fresh

By using a pipe, random generation can be fed into linearization:

        > generate_random | linearize
        this Italian fish is fresh

Use the `number` flag to generate several trees:

        > gr -number=4 | l
        that wine is boring
        that fresh cheese is fresh
        that cheese is very boring
        this cheese is Italian

To generate all phrases that a grammar can produce, use `generate_trees = gt`.

        > generate_trees | l
        that cheese is very Italian
        that cheese is very boring
        that cheese is very delicious
        ...
        this wine is fresh
        this wine is warm

The default depth is 3; the depth can be set by using the `depth` flag:

        > generate_trees -depth=2 | l

What options a command has can be seen by the `help = h` command:

        > help gr
        > help gt

### Exercises on generation

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

2. Measure how many trees the grammar gives with depths 4 and 5, respectively. Hint. You can use the Unix word count command `wc` to count lines.

### More on pipes: tracing

Put the tracing option `-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

Useful for test purposes: the pipe above can show if a grammar is ambiguous, i.e. contains strings that can be parsed in more than one way.

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

### Writing and reading files

To save the outputs into a file, pipe it to the `write_file = wf` command,

        > gr -number=10 | linearize | write_file -file=exx.tmp

To read a file to GF, use the `read_file = rf` command,

        > read_file -file=exx.tmp -lines | parse

The flag `-lines` tells GF to read each line of the file separately.

Files with examples can be used for regression testing of grammars - the most systematic way to do this is by treebanks; see here.

### Visualizing trees

Parentheses give a linear representation of trees, useful for the computer.

Human eye may prefer to see a visualization: `visualize_tree = vt`:

        > parse "this delicious cheese is very Italian" | visualize_tree

The tree is generated in postscript (`.ps`) file. The `-view` option is used for telling what command to use to view the file. Its default is `"gv"`, which works on most Linux installations. On a Mac, one would probably write

        > parse "this delicious cheese is very Italian" | visualize_tree -view="open"

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

You can save the temporary file `_grph.dot`, which the command `vt` produces.

Then you can process this file with the `dot` program (from the Graphviz package).

        % dot -Tpng _grph.dot > mytree.png

### System commands

You can give a system command without leaving GF: `!` followed by a Unix command,

        > ! dot -Tpng grphtmp.dot > mytree.png
        > ! open mytree.png

A system command may also receive its argument from a GF pipes. It then has the name `sp` = `system_pipe`:

        > generate_trees -depth=4 | sp -command="wc -l"

This command example returns the number of generated trees.

Exercise. Measure how many trees the grammar `FoodEng` gives with depths 4 and 5, respectively. Use the Unix word count command `wc` to count lines, and a system pipe from a GF command into a Unix command.

## An Italian concrete syntax

Just (?) replace English words with their dictionary equivalents:

        concrete FoodIta of Food = {

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

          lin
            Is item quality = {s = item.s ++ "è" ++ quality.s} ;
            This kind = {s = "questo" ++ kind.s} ;
            That kind = {s = "quel" ++ 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"} ;
        }

Not just replacing words:

The order of a quality and the kind it modifies is changed in

          QKind quality kind = {s = kind.s ++ quality.s} ;

Thus Italian says `vino italiano` for `Italian wine`.

(Some Italian adjectives are put before the noun. This distinction can be controlled by parameters, which are introduced in Lesson 3.)

### Exercises on multilinguality

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

2. 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. You can return to your list after having worked out Lesson 3.

## Free variation

Semantically indistinguishable ways of expressing a thing.

The variants construct of GF expresses free variation. For example,

        lin Delicious = {s = "delicious" | "exquisit" | "tasty"} ;

By default, the `linearize` command shows only the first variant from such lists; to see them all, use the option `-all`:

        > p "this exquisit wine is delicious" | l -all
        this delicious wine is delicious
        this delicious wine is exquisit
        ...

An equivalent notation for variants is

        lin Delicious = {s = variants {"delicious" ; "exquisit" ; "tasty"}} ;

This notation also allows the limiting case: an empty variant list,

        variants {}

It can be used e.g. if a word lacks a certain inflection form.

Free variation works for all types in concrete syntax; all terms in a variant list must be of the same type.

## More application of multilingual grammars

### Multilingual treebanks

Multilingual treebank: a set of trees with their linearizations in different languages:

        > gr -number=2 | l -treebank

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

        Is (That Cheese) Fresh
        quel formaggio è fresco
        that cheese is fresh

### Translation quiz

`translation_quiz = tq`: generate random sentences, display them in one language, and check the user's answer given in another language.

        > translation_quiz -from=FoodEng -to=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

## Context-free grammars and GF

### The "cf" grammar format

The grammar `FoodEng` can be written in a BNF format as follows:

        Is.        Phrase  ::= 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" ;

GF can convert BNF grammars into GF. BNF files are recognized by the file name suffix `.cf` (for context-free):

        > import food.cf

The compiler creates separate abstract and concrete modules internally.

### Restrictions of context-free grammars

Separating concrete and abstract syntax allows three deviations from context-free grammar:

- permutation: changing the order of constituents
- suppression: omitting constituents
- reduplication: repeating constituents

Exercise. Define the non-context-free copy language `{x x | x <- (a|b)*}` in GF.

## Modules and files

GF uses suffixes to recognize different file formats:

- Source files: Modulename`.gf`
- Target files: Modulename`.gfo`

Importing generates target from source:

        > i FoodEng.gf
        - compiling Food.gf...   wrote file Food.gfo 16 msec
        - compiling FoodEng.gf...   wrote file FoodEng.gfo 20 msec

The `.gfo` format (="GF Object") is precompiled GF, which is faster to load than source GF (`.gf`).

When reading a module, GF decides whether to use an existing `.gfo` 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`.

## Using operations and resource modules

### Operation definitions

The golden rule of functional programmin:

Whenever you find yourself programming by copy-and-paste, write a function instead.

Functions in concrete syntax are defined using the keyword `oper` (for operation), distinct from `fun` for the sake of clarity.

Example:

        oper ss : Str -> {s : Str} = \x -> {s = x} ;

The operation can be applied to an argument, and GF will compute the value:

        ss "boy" ===> {s = "boy"}

The symbol `===>` will be used for computation.

Notice the lambda abstraction form

- `\`x `->` t

This is read:

- function with variable x and function body t

For lambda abstraction with multiple arguments, we have the shorthand

        \x,y -> t   ===  \x -> \y -> t

Linearization rules actually use syntactic sugar for abstraction:

        lin f x = t   ===  lin f = \x -> t

### The ``resource`` module type

The `resource` module type is used to package `oper` definitions into reusable resources.

        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) ;
        }

### Opening a resource

Any number of `resource` modules can be opened in a `concrete` syntax.

        concrete FoodEng 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" ;
        }

### Partial application

The rule

        lin This k = prefix "this" k ;

can be written more concisely

        lin This = prefix "this" ;

Part of the art in functional programming: decide the order of arguments in a function, so that partial application can be used as much as possible.

For instance, `prefix` is typically applied to linearization variables with constant strings. Hence we put the `Str` argument before the `SS` argument.

Exercise. Define an operation `infix` analogous to `prefix`, such that it allows you to write

        lin Is = infix "is" ;

### Testing resource modules

Import with the flag `-retain`,

        > import -retain StringOper.gf

Compute the value with `compute_concrete = cc`,

        > compute_concrete prefix "in" (ss "addition")
        {s : Str = "in" ++ "addition"}

## Grammar architecture

### Extending a grammar

A new module can extend an old one:

        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: all of the contents of the extended and extending modules are put together.

In other words: the new module inherits the contents of the old module.

Simultaneous extension and opening:

        concrete MorefoodIta of Morefood = FoodIta ** open StringOper in {
          lincat
            Question = SS ;
          lin
            QIs item quality = ss (item.s ++ "è" ++ quality.s) ;
            Pizza = ss "pizza" ;
        }

Resource modules can extend other resource modules - thus it is possible to build resource hierarchies.

### Multiple inheritance

Extend several grammars at the same time:

        abstract Foodmarket = Food, Fruit, Mushroom ** {
          fun
            FruitKind    : Fruit    -> Kind ;
            MushroomKind : Mushroom -> Kind ;
          }

where

        abstract Fruit = {
          cat Fruit ;
          fun Apple, Peach : Fruit ;
        }

        abstract Mushroom = {
          cat Mushroom ;
          fun Cep, Agaric : Mushroom ;
        }

Exercise. Refactor `Food` by taking apart `Wine` into a special `Drink` module.

# Lesson 3: Grammars with parameters

Goals:

- implement sophisticated linguistic structures:
  - morphology: the inflection of words
  - agreement: rules for selecting word forms in syntactic combinations

- Cover all GF constructs for concrete syntax

It is possible to skip this chapter and go directly to the next, since the use of the GF Resource Grammar library makes it unnecessary to use parameters: they could be left to library implementors.

## The problem: words have to be inflected

Plural forms are needed in things like

these Italian wines are delicious
This 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 types of inflection and agreement.

- Italian has also gender (masculine vs. feminine).

In a multilingual grammar, we want to ignore such distinctions in abstract syntax.

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 English by a new form of judgement:

        param Number = Sg | Pl ;

This judgement defines the parameter type `Number` by listing its two constructors, `Sg` and `Pl` (singular and plural).

We give `Kind` a linearization type that has a table depending on number:

        lincat Kind = {s : Number => Str} ;

The table type `Number => Str` is similar a function type (`Number -> Str`).

Difference: the argument must be a parameter type. Then the argument-value pairs can be listed in a finite table.

Here is a table:

        lin Cheese = {
          s = table {
            Sg => "cheese" ;
            Pl => "cheeses"
          }
        } ;

The table has branches, with a pattern on the left of the arrow `=>` and a value on the right.

The application of a table is done by the selection operator `!`.

It which is computed by pattern matching: return the value from the first branch whose pattern matches the argument. For instance,

         table {Sg => "cheese" ; Pl => "cheeses"} ! Pl
         ===> "cheeses"

Case expressions are syntactic sugar:

        case e of {...} ===  table {...} ! e

Since they are familiar to Haskell and ML programmers, they can come out handy when writing GF programs.

Constructors can take arguments from other parameter types.

Example: forms of English verbs (except be):

        param VerbForm = VPresent Number | VPast | VPastPart | VPresPart ;

Fact expressed: only present tense has number variation.

Example table: the forms of the verb drink:

        table {
          VPresent Sg => "drinks" ;
          VPresent Pl => "drink" ;
          VPast       => "drank" ;
          VPastPart   => "drunk" ;
          VPresPart   => "drinking"
          }

Exercise. In an earlier exercise (previous section), you made 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

A morphological paradigm is a formula telling how a class of words is inflected.

From the GF point of view, a paradigm is a function that takes a lemma (also known as a dictionary form, or a citation form) and returns an inflection table.

The following operation defines the regular noun paradigm of English:

        oper regNoun : Str -> {s : Number => Str} = \dog -> {
          s = table {
            Sg => dog ;
            Pl => dog + "s"
            }
          } ;

The gluing operator `+` glues strings to one token:

        (regNoun "cheese").s ! Pl  ===> "cheese" + "s"  ===>  "cheeses"

A more complex example: regular verbs,

        oper regVerb : Str -> {s : VerbForm => Str} = \talk -> {
          s = table {
            VPresent Sg => talk + "s" ;
            VPresent Pl => talk ;
            VPresPart   => talk + "ing" ;
            _           => talk + "ed"
            }
          } ;

The catch-all case for the past tense and the past participle uses a wild card pattern `_`.

### Exercises on morphology

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

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

## Using parameters in concrete syntax

Purpose: a more radical variation between languages than just the use of different words and word orders.

We add to the grammar `Food` two rules for forming plural items:

        fun These, Those : Kind -> Item ;

We also add a noun which in Italian has the feminine case:

        fun Pizza : Kind ;

This will force us to deal with gender-

### Agreement

In English, the phrase-forming rule

        fun Is : Item -> Quality -> Phrase ;

is affected by the number because of subject-verb agreement: the verb of a sentence must be inflected in the number of the subject,

        Is (This Pizza) Warm   ===>  "this pizza is warm"
        Is (These Pizza) Warm  ===>  "these pizzas are warm"

It is the copula (the verb be) that is affected:

        oper copula : Number -> Str = \n ->
          case n of {
            Sg => "is" ;
            Pl => "are"
            } ;

The subject `Item` must have such a number to provide to the copula:

        lincat Item = {s : Str ; n : Number} ;

Now we can write

        lin Is item qual = {s = item.s ++ copula item.n ++ qual.s} ;

### Determiners

How does an `Item` subject receive its number? The rules

        fun This, These : Kind -> Item ;

add determiners, either this or these, which require different this pizza vs. these pizzas.

Thus `Kind` must have both singular and plural forms:

        lincat Kind = {s : Number => Str} ;

We can write

        lin This kind = {
          s = "this" ++ kind.s ! Sg ;
          n = Sg
        } ;

        lin These kind = {
          s = "these" ++ kind.s ! Pl ;
          n = Pl
        } ;

To avoid copy-and-paste, we can factor out the pattern of determination,

        oper det :
          Str -> Number -> {s : Number => Str} -> {s : Str ; n : Number} =
            \det,n,kind -> {
            s = det ++ kind.s ! n ;
            n = n
          } ;

Now we can write

        lin This  = det Sg "this" ;
        lin These = det Pl "these" ;

In a more lexicalized grammar, determiners would be a category:

        lincat Det = {s : Str ; n : Number} ;
        fun Det : Det -> Kind -> Item ;
        lin Det det kind = {
            s = det.s ++ kind.s ! det.n ;
            n = det.n
          } ;

### Parametric vs. inherent features

`Kind`s have number as a parametric feature: both singular and plural can be formed,

        lincat Kind = {s : Number => Str} ;

`Item`s have number as an inherent feature: they are inherently either singular or plural,

        lincat Item = {s : Str ; n : Number} ;

Italian `Kind` will have parametric number and inherent gender:

        lincat Kind = {s : Number => Str ; g : Gender} ;

Questions to ask when designing parameters:

- existence: what forms are possible to build by morphological and other means?
- need: what features are expected via agreement or government?

Dictionaries give good advice:

uomo, pl. uomini, n.m. "man"
tells that uomo is a masculine noun with the plural form uomini. Hence, parametric number and an inherent gender.

For words, inherent features are usually given as lexical information.

For combinations, they are inherited from some part of the construction (typically the one called the head). Italian modification:

        lin QKind qual kind =
          let gen = kind.g in {
            s = table {n => kind.s ! n ++ qual.s ! gen ! n} ;
            g = gen
            } ;

Notice

- local definition (`let` expression)
- variable pattern `n`

## An English concrete syntax for Foods with parameters

We use some string operations from the library `Prelude` are used.

         concrete FoodsEng of Foods = open Prelude in {

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

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

        param
          Number = Sg | Pl ;

        oper
          det : Number -> Str -> {s : Number => Str} -> {s : Str ; n : Number} =
            \n,d,cn -> {
              s = d ++ cn.s ! n ;
              n = n
            } ;
          noun : Str -> Str -> {s : Number => Str} =
            \man,men -> {s = table {
              Sg => man ;
              Pl => men
              }
            } ;
          regNoun : Str -> {s : Number => Str} =
            \car -> noun car (car + "s") ;
          copula : Number -> Str =
            \n -> case n of {
              Sg => "is" ;
              Pl => "are"
              } ;
        }

## More on inflection paradigms

Let us extend the English noun paradigms so that we can deal with all nouns, not just the regular ones. The goal is to provide a morphology module that makes it easy to add words to a lexicon.

### Worst-case functions

We perform data abstraction from the type of nouns by writing a a worst-case function:

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

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

        oper regNoun : Str -> Noun = \x -> mkNoun x (x + "s") ;

Then we can define

        lincat N = Noun ;
        lin Mouse = mkNoun "mouse" "mice" ;
        lin House = regNoun "house" ;

where the underlying types are not seen.

We are free to change the undelying definitions, e.g. add case (nominative or genitive) to noun inflection:

        param Case = Nom | Gen ;

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

Now we have to redefine the worst-case function

        oper mkNoun : Str -> Str -> Noun = \x,y -> {
          s = table {
            Sg => table {
              Nom => x ;
              Gen => x + "'s"
              } ;
            Pl => table {
              Nom => y ;
              Gen => y + case last y of {
                "s" => "'" ;
                _   => "'s"
              }
            }
          } ;

But up from this level, we can retain the old definitions

        lin Mouse = mkNoun "mouse" "mice" ;
        oper regNoun : Str -> Noun = \x -> mkNoun x (x + "s") ;

In the last definition of `mkNoun`, we used a case expression on the last character of the plural, as well as the `Prelude` operation

        last : Str -> Str ;

returning the string consisting of the last character.

The case expression uses pattern matching over strings, which is supported in GF, alongside with pattern matching over parameters.

### Smart paradigms

The regular dog-dogs paradigm has predictable variations:

- nouns ending with an y: fly-flies, except if a vowel precedes the y: boy-boys
- nouns ending with s, ch, and a number of other endings: bus-buses, leech-leeches

We could provide alternative paradigms:

        noun_y : Str -> Noun = \fly -> mkNoun fly (init fly + "ies") ;
        noun_s : Str -> Noun = \bus -> mkNoun bus (bus + "es") ;

(The Prelude function `init` drops the last character of a token.)

Drawbacks:

- it can be difficult to select the correct paradigm
- it can be difficult to remember the names of the different paradigms

Better solution: a smart paradigm:

        regNoun : Str -> Noun = \w ->
          let
            ws : Str = case w of {
              _ + ("a" | "e" | "i" | "o") + "o" => w + "s" ;  -- bamboo
              _ + ("s" | "x" | "sh" | "o")      => w + "es" ; -- bus, hero
              _ + "z"                           => w + "zes" ;-- quiz
              _ + ("a" | "e" | "o" | "u") + "y" => w + "s" ;  -- boy
              x + "y"                           => x + "ies" ;-- fly
              _                                 => w + "s"    -- car
              }
          in
          mkNoun w ws

GF has regular expression patterns:

- disjunctive patterns P `|` Q
- concatenation patterns P `+` Q

The patterns are ordered in such a way that, for instance, the suffix `"oo"` prevents bamboo from matching the suffix `"o"`.

### Exercises on regular patterns

1. 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`.

2. Extend the verb paradigms to cover all verb forms in English, with special care taken of variations with the suffix ed (e.g. try-tried, use-used).

3. Implement the German Umlaut operation on word stems. The operation changes the vowel of the stressed stem syllable as follows: a to ä, au to äu, o to ö, and u to ü. You can assume that the operation only takes syllables as arguments. Test the operation to see whether it correctly changes Arzt to Ärzt, Baum to Bäum, Topf to Töpf, and Kuh to Küh.

### Function types with variables

In Lesson 5, dependent function types need a notation that binds a variable to the argument type, as in

        switchOff : (k : Kind) -> Action k

Function types without variables are actually a shorthand:

        PredVP : NP -> VP -> S

means

        PredVP : (x : NP) -> (y : VP) -> S

or any other naming of the variables.

Sometimes variables shorten the code, since they can share a type:

        octuple : (x,y,z,u,v,w,s,t : Str) -> Str

If a bound variable is not used, it can be replaced by a wildcard:

        octuple : (_,_,_,_,_,_,_,_ : Str) -> Str

A good practice is to indicate the number of arguments:

        octuple : (x1,_,_,_,_,_,_,x8 : Str) -> Str

For inflection paradigms, it is handy to use heuristic variable names, looking like the expected forms:

        mkNoun : (mouse,mice : Str) -> Noun

### Separating operation types and definitions

In librarues, it is useful to group type signatures separately from definitions. It is possible to divide an `oper` judgement,

        oper regNoun : Str -> Noun ;
        oper regNoun s = mkNoun s (s + "s") ;

and put the parts in different places.

With the `interface` and `instance` module types (see here): the parts can even be put to different files.

### Overloading of operations

Overloading: different functions can be given the same name, as e.g. in C++.

The compiler performs overload resolution, which works as long as the functions have different types.

In GF, the functions must be grouped together in `overload` groups.

Example: different ways to define nouns in English:

        oper mkN : overload {
          mkN : (dog : Str) -> Noun ;         -- regular nouns
          mkN : (mouse,mice : Str) -> Noun ;  -- irregular nouns
        }

Cf. dictionaries: if the word is regular, just one form is needed. If it is irregular, more forms are given.

The definition can be given separately, or at the same time, as the types:

        oper mkN = overload {
          mkN : (dog : Str) -> Noun = regNoun ;
          mkN : (mouse,mice : Str) -> Noun = mkNoun ;
        }

Exercise. Design a system of English verb paradigms presented by an overload group.

### Morphological analysis and morphology quiz

The command `morpho_analyse = ma` can be used to read a text and return for each word its analyses (in the current grammar):

        > read_file bible.txt | morpho_analyse

The command `morpho_quiz = mq` generates inflection exercises.

        % gf -path=alltenses:prelude $GF_LIB_PATH/alltenses/IrregFre.gfc

        > morpho_quiz -cat=V

        Welcome to GF Morphology Quiz.
        ...

        réapparaître : VFin VCondit  Pl  P2
        réapparaitriez
        > No, not réapparaitriez, but
        réapparaîtriez
        Score 0/1

To create a list for later use, use the command `morpho_list = ml`

        > morpho_list -number=25 -cat=V | write_file exx.txt

## The Italian Foods grammar

Parameters include not only number but also gender.

      concrete FoodsIta of Foods = open Prelude in {

        param
          Number = Sg | Pl ;
          Gender = Masc | Fem ;

Qualities are inflected for gender and number, whereas kinds have a parametric number and an inherent gender. Items have an inherent number and gender.

        lincat
          Phr = SS ;
          Quality = {s : Gender => Number => Str} ;
          Kind = {s : Number => Str ; g : Gender} ;
          Item = {s : Str ; g : Gender ; n : Number} ;

A Quality is an adjective, with one form for each gender-number combination.

        oper
          adjective : (_,_,_,_ : Str) -> {s : Gender => Number => Str} =
            \nero,nera,neri,nere -> {
              s = table {
                Masc => table {
                  Sg => nero ;
                  Pl => neri
                  } ;
                Fem => table {
                  Sg => nera ;
                  Pl => nere
                  }
                }
            } ;

Regular adjectives work by adding endings to the stem.

          regAdj : Str -> {s : Gender => Number => Str} = \nero ->
            let ner = init nero
            in adjective nero (ner + "a") (ner + "i") (ner + "e") ;

For noun inflection, we are happy to give the two forms and the gender explicitly:

          noun : Str -> Str -> Gender -> {s : Number => Str ; g : Gender} =
            \vino,vini,g -> {
              s = table {
                Sg => vino ;
                Pl => vini
                } ;
              g = g
            } ;

We need only number variation for the copula.

          copula : Number -> Str =
            \n -> case n of {
              Sg => "è" ;
              Pl => "sono"
              } ;

Determination is more complex than in English, because of gender:

          det : Number -> Str -> Str -> {s : Number => Str ; g : Gender} ->
              {s : Str ; g : Gender ; n : Number} =
            \n,m,f,cn -> {
              s = case cn.g of {Masc => m ; Fem => f} ++ cn.s ! n ;
              g = cn.g ;
              n = n
            } ;

The complete set of linearization rules:

        lin
          Is item quality =
            ss (item.s ++ copula item.n ++ quality.s ! item.g ! item.n) ;
          This  = det Sg "questo" "questa" ;
          That  = det Sg "quel"   "quella" ;
          These = det Pl "questi" "queste" ;
          Those = det Pl "quei"   "quelle" ;
          QKind quality kind = {
            s = \\n => kind.s ! n ++ quality.s ! kind.g ! n ;
            g = kind.g
            } ;
          Wine = noun "vino" "vini" Masc ;
          Cheese = noun "formaggio" "formaggi" Masc ;
          Fish = noun "pesce" "pesci" Masc ;
          Pizza = noun "pizza" "pizze" Fem ;
          Very qual = {s = \\g,n => "molto" ++ qual.s ! g ! n} ;
          Fresh = adjective "fresco" "fresca" "freschi" "fresche" ;
          Warm = regAdj "caldo" ;
          Italian = regAdj "italiano" ;
          Expensive = regAdj "caro" ;
          Delicious = regAdj "delizioso" ;
          Boring = regAdj "noioso" ;
        }

### Exercises on using parameters

1. Experiment with multilingual generation and translation in the `Foods` grammars.

2. Add items, qualities, and determiners to the grammar, and try to get their inflection and inherent features right.

3. Write a concrete syntax of `Food` for a language of your choice, now aiming for complete grammatical correctness by the use of parameters.

4. Measure the size of the context-free grammar corresponding to `FoodsIta`. You can do this by printing the grammar in the context-free format (`print_grammar -printer=bnf`) and counting the lines.

## Discontinuous constituents

A linearization record may contain more strings than one, and those strings can be put apart in linearization.

Example: English particle verbs, (switch off). The object can appear between:

he switched it off

The verb switch off is called a discontinuous constituents.

We can define transitive verbs and their combinations as follows:

        lincat TV = {s : Number => Str ; part : Str} ;

        fun AppTV : Item -> TV -> Item -> Phrase ;

        lin AppTV subj tv obj =
          {s = subj.s ++ tv.s ! subj.n ++ obj.s ++ tv.part} ;

Exercise. Define the language `a^n b^n c^n` in GF, i.e. any number of a's followed by the same number of b's and the same number of c's. This language is not context-free, but can be defined in GF by using discontinuous constituents.

## Strings at compile time vs. run time

Tokens are created in the following ways:

- quoted string: `"foo"`
- gluing : `t + s`
- predefined operations `init, tail, tk, dp`
- pattern matching over strings

Since tokens must be known at compile time, the above operations may not be applied to run-time variables (i.e. variables that stand for function arguments in linearization rules).

Hence it is not legal to write

        cat Noun ;
        fun Plural : Noun -> Noun ;
        lin Plural n = {s = n.s + "s"} ;

because `n` is a run-time variable. Also

        lin Plural n = {s = (regNoun n).s ! Pl} ;

is incorrect with `regNoun` as defined here, because the run-time variable is eventually sent to string pattern matching and gluing.

How to write tokens together without a space?

        lin Question p = {s = p + "?"} ;

is incorrect.

The way to go is to use an unlexer that creates correct spacing after linearization.

Correspondingly, a lexer that e.g. analyses `"warm?"` into to tokens is needed before parsing. This topic will be covered in here.

### Supplementary constructs for concrete syntax

#### Record extension and subtyping

The symbol `**` is used for both record types and record objects.

        lincat TV = Verb ** {c : Case} ;

        lin Follow = regVerb "folgen" ** {c = Dative} ;

`TV` becomes a subtype of `Verb`.

If T is a subtype of R, an object of T can be used whenever an object of R is required.

Covariance: a function returning a record T as value can also be used to return a value of a supertype R.

Contravariance: 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.

#### Prefix-dependent choices

English indefinite article:

        oper artIndef : Str =
          pre {"a" ; "an" / strs {"a" ; "e" ; "i" ; "o"}} ;

Thus

        artIndef ++ "cheese"  --->  "a" ++ "cheese"
        artIndef ++ "apple"   --->  "an" ++ "apple"

# Lesson 4: Using the resource grammar library

Goals:

- navigate in the GF resource grammar library and use it in applications
- get acquainted with basic linguistic categories
- write functors to achieve maximal sharing of code in multilingual grammars

## The coverage of the library

The current 12 resource languages are

- `Bul`garian
- `Cat`alan
- `Dan`ish
- `Eng`lish
- `Fin`nish
- `Fre`nch
- `Ger`man
- `Ita`lian
- `Nor`wegian
- `Rus`sian
- `Spa`nish
- `Swe`dish

The first three letters (`Eng` etc) are used in grammar module names (ISO 639 standard).

## The structure of the library

Semantic grammars (up to now in this tutorial): a grammar defines a system of meanings (abstract syntax) and tells how they are expressed(concrete syntax).

Resource grammars (as usual in linguistic tradition): a grammar specifies the grammatically correct combinations of words, whatever their meanings are.

With resource grammars, we can achieve a wider coverage than with semantic grammars.

### Lexical vs. phrasal rules

A resource grammar has two kinds of categories and two kinds of rules:

- lexical:
  - lexical categories, to classify words
  - lexical rules, to define words and their properties

- phrasal (combinatorial, syntactic):
  - phrasal categories, to classify phrases of arbitrary size
  - phrasal rules, to combine phrases into larger phrases

GE makes no formal distinction between these two kinds.

But it is a good discipline to follow.

### Lexical categories

Two kinds of lexical categories:

- closed:
  - a finite number of words
  - seldom extended in the history of language
  - structural words / function words, e.g.

              Conj ;     -- conjunction           e.g. "and"
              QuantSg ;  -- singular quantifier   e.g. "this"
              QuantPl ;  -- plural quantifier     e.g. "this"

- open:
  - new words are added all the time
  - content words, e.g.

              N ;        -- noun         e.g. "pizza"
              A ;        -- adjective    e.g. "good"
              V ;        -- verb         e.g. "sleep"

### Lexical rules

Closed classes: module `Syntax`. In the `Foods` grammar, we need

        this_QuantSg, that_QuantSg : QuantSg ;
        these_QuantPl, those_QuantPl : QuantPl ;
        very_AdA  : AdA ;

Naming convention: word followed by the category (so we can distinguish the quantifier that from the conjunction that).

Open classes have no objects in `Syntax`. Words are built as they are needed in applications: if we have

        fun Wine : Kind ;

we will define

        lin Wine = mkN "wine" ;

where we use `mkN` from `ParadigmsEng`:

### Resource lexicon

Alternative concrete syntax for

        fun Wine : Kind ;

is to provide a resource lexicon, which contains definitions such as

        oper wine_N : N = mkN "wine" ;

so that we can write

        lin Wine = wine_N ;

Advantages:

- we accumulate a reusable lexicon
- we can use a here to speed up multilingual grammar implementation

### Phrasal categories

In `Foods`, we need just four phrasal categories:

        Cl ;   -- clause             e.g. "this pizza is good"
        NP ;   -- noun phrase        e.g. "this pizza"
        CN ;   -- common noun        e.g. "warm pizza"
        AP ;   -- adjectival phrase  e.g. "very warm"

Clauses are similar to sentences (`S`), but without a fixed tense and mood; see here for how they relate.

Common nouns are made into noun phrases by adding determiners.

### Syntactic combinations

We need the following combinations:

        mkCl : NP -> AP -> Cl ;      -- e.g. "this pizza is very warm"
        mkNP : QuantSg -> CN -> NP ; -- e.g. "this pizza"
        mkNP : QuantPl -> CN -> NP ; -- e.g. "these pizzas"
        mkCN : AP -> CN -> CN ;      -- e.g. "warm pizza"
        mkAP : AdA -> AP -> AP ;     -- e.g. "very warm"

We also need lexical insertion, to form phrases from single words:

        mkCN : N -> NP ;
        mkAP : A -> AP ;

Naming convention: to construct a C, use a function `mk` C.

Heavy overloading: the current library (version 1.2) has 23 operations named `mkNP`!

### Example syntactic combination

The sentence

these very warm pizzas are Italian
can be built as follows:

        mkCl
          (mkNP these_QuantPl
             (mkCN (mkAP very_AdA (mkAP warm_A)) (mkCN pizza_CN)))
          (mkAP italian_AP)

The task now: to define the concrete syntax of `Foods` so that this syntactic tree gives the value of linearizing the semantic tree

        Is (These (QKind (Very Warm) Pizza)) Italian

## The resource API

Language-specific and language-independent parts - roughly,

- the syntax API `Syntax` L has the same types and functions for all languages L
- the morphology API `Paradigms` L has partly different types and functions for different languages L

Full API documentation on-line: the resource synopsis,

`digitalgrammars.com/gf/lib/resource/doc/synopsis.html`

### A miniature resource API: categories

Category | Explanation | Example
--- | --- | ---
`Cl` | clause (sentence), with all tenses | she looks at this
`AP` | adjectival phrase | very warm
`CN` | common noun (without determiner) | red house
`NP` | noun phrase (subject or object) | the red house
`AdA` | adjective-modifying adverb, | very
`QuantSg` | singular quantifier | these
`QuantPl` | plural quantifier | this
`A` | one-place adjective | warm
`N` | common noun | house

### A miniature resource API: rules

Function | Type | Example
--- | --- | ---
`mkCl` | `NP -> AP -> Cl` | John is very old
`mkNP` | `QuantSg -> CN -> NP` | this old man
`mkNP` | `QuantPl -> CN -> NP` | these old man
`mkCN` | `N -> CN` | house
`mkCN` | `AP -> CN -> CN` | very big blue house
`mkAP` | `A -> AP` | old
`mkAP` | `AdA -> AP -> AP` | very very old

### A miniature resource API: structural words

Function | Type | In English
--- | --- | ---
`this_QuantSg` | `QuantSg` | this
`that_QuantSg` | `QuantSg` | that
`these_QuantPl` | `QuantPl` | this
`those_QuantPl` | `QuantPl` | that
`very_AdA` | `AdA` | very

### A miniature resource API: paradigms

From `ParadigmsEng`:

Function | Type
--- | ---
`mkN` | `(dog : Str) -> N`
`mkN` | `(man,men : Str) -> N`
`mkA` | `(cold : Str) -> A`

From `ParadigmsIta`:

Function | Type
--- | ---
`mkN` | `(vino : Str) -> N`
`mkA` | `(caro : Str) -> A`

### A miniature resource API: more paradigms

From `ParadigmsGer`:

Function | Type
--- | ---
`Gender` | `Type`
`masculine` | `Gender`
`feminine` | `Gender`
`neuter` | `Gender`
`mkN` | `(Stufe : Str) -> N`
`mkN` | `(Bild,Bilder : Str) -> Gender -> N`
`mkA` | `(klein : Str) -> A`
`mkA` | `(gut,besser,beste : Str) -> A`

From `ParadigmsFin`:

Function | Type
--- | ---
`mkN` | `(talo : Str) -> N`
`mkA` | `(hieno : Str) -> A`

### Exercises

1. Try out the morphological paradigms in different languages. Do as follows:

        > i -path=alltenses -retain alltenses/ParadigmsGer.gfo
        > cc -table mkN "Farbe"
        > cc -table mkA "gut" "besser" "beste"

## Example: English

We assume the abstract syntax `Foods` from Lesson 3.

We don't need to think about inflection and agreement, but just pick functions from the resource grammar library.

We need a path with

- the current directory `.`
- the directory `../foods`, in which `Foods.gf` resides.
- the library directory `present`, which is relative to the environment variable `GF_LIB_PATH`

Thus the beginning of the module is

        --# -path=.:../foods:present

        concrete FoodsEng of Foods = open SyntaxEng,ParadigmsEng in {

### English example: linearization types and combination rules

As linearization types, we use clauses for `Phrase`, noun phrases for `Item`, common nouns for `Kind`, and adjectival phrases for `Quality`.

        lincat
          Phrase = Cl ;
          Item = NP ;
          Kind = CN ;
          Quality = AP ;

Now the combination rules we need almost write themselves automatically:

        lin
          Is item quality = mkCl item quality ;
          This kind = mkNP this_QuantSg kind ;
          That kind = mkNP that_QuantSg kind ;
          These kind = mkNP these_QuantPl kind ;
          Those kind = mkNP those_QuantPl kind ;
          QKind quality kind = mkCN quality kind ;
          Very quality = mkAP very_AdA quality ;

### English example: lexical rules

We use resource paradigms and lexical insertion rules.

The two-place noun paradigm is needed only once, for fish - everythins else is regular.

          Wine = mkCN (mkN "wine") ;
          Pizza = mkCN (mkN "pizza") ;
          Cheese = mkCN (mkN "cheese") ;
          Fish = mkCN (mkN "fish" "fish") ;
          Fresh = mkAP (mkA "fresh") ;
          Warm = mkAP (mkA "warm") ;
          Italian = mkAP (mkA "Italian") ;
          Expensive = mkAP (mkA "expensive") ;
          Delicious = mkAP (mkA "delicious") ;
          Boring = mkAP (mkA "boring") ;
        }

### English example: exercises

1. Compile the grammar `FoodsEng` and generate and parse some sentences.

2. Write a concrete syntax of `Foods` for Italian or some other language included in the resource library. You can compare the results with the hand-written grammars presented earlier in this tutorial.

## Functor implementation of multilingual grammars

### New language by copy and paste

If you write a concrete syntax of `Foods` for some other language, much of the code will look exactly the same as for English. This is because

- the `Syntax` API is the same for all languages (because all languages in the resource package do implement the same syntactic structures)
- languages tend to use the syntactic structures in similar ways

But lexical rules are more language-dependent.

Thus, to port a grammar to a new language, you

1. copy the concrete syntax of a given language
2. change the words (strings and inflection paradigms)

Can we avoid this programming by copy-and-paste?

### Functors: functions on the module level

Functors familiar from the functional programming languages ML and OCaml, also known as parametrized modules.

In GF, a functor is a module that `open`s one or more interfaces.

An `interface` is a module similar to a `resource`, but it only contains the types of `oper`s, not (necessarily) their definitions.

Syntax for functors: add the keyword `incomplete`. We will use the header

        incomplete concrete FoodsI of Foods = open Syntax, LexFoods in

where

        interface Syntax    -- the resource grammar interface
        interface LexFoods  -- the domain lexicon interface

When we moreover have

        instance SyntaxEng of Syntax     -- the English resource grammar
        instance LexFoodsEng of LexFoods -- the English domain lexicon

we can write a functor instantiation,

        concrete FoodsGer of Foods = FoodsI with
          (Syntax = SyntaxGer),
          (LexFoods = LexFoodsGer) ;

### Code for the Foods functor

        --# -path=.:../foods

        incomplete concrete FoodsI of Foods = open Syntax, LexFoods in {
        lincat
          Phrase = Cl ;
          Item = NP ;
          Kind = CN ;
          Quality = AP ;
        lin
          Is item quality = mkCl item quality ;
          This kind = mkNP this_QuantSg kind ;
          That kind = mkNP that_QuantSg kind ;
          These kind = mkNP these_QuantPl kind ;
          Those kind = mkNP those_QuantPl kind ;
          QKind quality kind = mkCN quality kind ;
          Very quality = mkAP very_AdA quality ;

          Wine = mkCN wine_N ;
          Pizza = mkCN pizza_N ;
          Cheese = mkCN cheese_N ;
          Fish = mkCN fish_N ;
          Fresh = mkAP fresh_A ;
          Warm = mkAP warm_A ;
          Italian = mkAP italian_A ;
          Expensive = mkAP expensive_A ;
          Delicious = mkAP delicious_A ;
          Boring = mkAP boring_A ;
        }

### Code for the LexFoods interface

        interface LexFoods = open Syntax in {
        oper
          wine_N : N ;
          pizza_N : N ;
          cheese_N : N ;
          fish_N : N ;
          fresh_A : A ;
          warm_A : A ;
          italian_A : A ;
          expensive_A : A ;
          delicious_A : A ;
          boring_A : A ;
        }

### Code for a German instance of the lexicon

        instance LexFoodsGer of LexFoods = open SyntaxGer, ParadigmsGer in {
        oper
          wine_N = mkN "Wein" ;
          pizza_N = mkN "Pizza" "Pizzen" feminine ;
          cheese_N = mkN "Käse" "Käsen" masculine ;
          fish_N = mkN "Fisch" ;
          fresh_A = mkA "frisch" ;
          warm_A = mkA "warm" "wärmer" "wärmste" ;
          italian_A = mkA "italienisch" ;
          expensive_A = mkA "teuer" ;
          delicious_A = mkA "köstlich" ;
          boring_A = mkA "langweilig" ;
        }

### Code for a German functor instantiation

        --# -path=.:../foods:present

        concrete FoodsGer of Foods = FoodsI with
          (Syntax = SyntaxGer),
          (LexFoods = LexFoodsGer) ;

### Adding languages to a functor implementation

Just two modules are needed:

- a domain lexicon instance
- a functor instantiation

The functor instantiation is completely mechanical to write.

The domain lexicon instance requires some knowledge of the words of the language:

- what words are used for which concepts
- how the words are
- features such as genders

### Example: adding Finnish

Lexicon instance

        instance LexFoodsFin of LexFoods = open SyntaxFin, ParadigmsFin in {
        oper
          wine_N = mkN "viini" ;
          pizza_N = mkN "pizza" ;
          cheese_N = mkN "juusto" ;
          fish_N = mkN "kala" ;
          fresh_A = mkA "tuore" ;
          warm_A = mkA "lämmin" ;
          italian_A = mkA "italialainen" ;
          expensive_A = mkA "kallis" ;
          delicious_A = mkA "herkullinen" ;
          boring_A = mkA "tylsä" ;
        }

Functor instantiation

        --# -path=.:../foods:present

        concrete FoodsFin of Foods = FoodsI with
          (Syntax = SyntaxFin),
          (LexFoods = LexFoodsFin) ;

### A design pattern

This can be seen as a design pattern for multilingual grammars:

                            concrete DomainL*

          instance LexDomainL                 instance SyntaxL*

                       incomplete concrete DomainI
                       /           |              \
         interface LexDomain   abstract Domain    interface Syntax*

Modules marked with `*` are either given in the library, or trivial.

Of the hand-written modules, only `LexDomainL` is language-dependent.

### Functors: exercises

1. Compile and test `FoodsGer`.

2. Refactor `FoodsEng` into a functor instantiation.

3. Instantiate the functor `FoodsI` to some language of your choice.

4. Design a small grammar that can be used for controlling an MP3 player. The grammar should be able to recognize commands such as play this song, with the following variations:

- verbs: play, remove
- objects: song, artist
- determiners: this, the previous
- verbs without arguments: stop, pause

The implementation goes in the following phases:

1. abstract syntax
2. (optional:) prototype string-based concrete syntax
3. functor over resource syntax and lexicon interface
4. lexicon instance for the first language
5. functor instantiation for the first language
6. lexicon instance for the second language
7. functor instantiation for the second language
8. ...

## Restricted inheritance

### A problem with functors

Problem: a functor only works when all languages use the resource `Syntax` in the same way.

Example (contrived): assume that English has no word for `Pizza`, but has to use the paraphrase Italian pie. This is no longer a noun `N`, but a complex phrase in the category `CN`.

Possible solution: change interface the `LexFoods` with

        oper pizza_CN : CN ;

Problem with this solution:

- we may end up changing the interface and the function with each new language
- we must every time also change the instances for the old languages to maintain type correctness

### Restricted inheritance: include or exclude

A module may inherit just a selection of names.

Example: the `FoodMarket` example "Rsecarchitecture:

        abstract Foodmarket = Food, Fruit [Peach], Mushroom - [Agaric]

Here, from `Fruit` we include `Peach` only, and from `Mushroom` we exclude `Agaric`.

A concrete syntax of `Foodmarket` must make the analogous restrictions.

### The functor problem solved

The English instantiation inherits the functor implementation except for the constant `Pizza`. This constant is defined in the body instead:

        --# -path=.:../foods:present

        concrete FoodsEng of Foods = FoodsI - [Pizza] with
          (Syntax = SyntaxEng),
          (LexFoods = LexFoodsEng) **
            open SyntaxEng, ParadigmsEng in {

          lin Pizza = mkCN (mkA "Italian") (mkN "pie") ;
        }

## Grammar reuse

Abstract syntax modules can be used as interfaces, and concrete syntaxes as their instances.

The following correspondencies are then applied:

        cat C         <--->  oper C : Type

        fun f : A     <--->  oper f : A

        lincat C = T  <--->  oper C : Type = T

        lin f = t     <--->  oper f : A = t

### Library exercises

1. Find resource grammar terms for the following English phrases (in the category `Phr`). You can first try to build the terms manually.

every man loves a woman

this grammar speaks more than ten languages

which languages aren't in the grammar

which languages did you want to speak

Then translate the phrases to other languages.

## Tenses

In `Foods` grammars, we have used the path

        --# -path=.:../foods

The library subdirectory `present` is a restricted version of the resource, with only present tense of verbs and sentences.

By just changing the path, we get all tenses:

        --# -path=.:../foods:alltenses

Now we can see all the tenses of phrases, by using the `-all` flag in linearization:

        > gr | l -all
        This wine is delicious
        Is this wine delicious
        This wine isn't delicious
        Isn't this wine delicious
        This wine is not delicious
        Is this wine not delicious
        This wine has been delicious
        Has this wine been delicious
        This wine hasn't been delicious
        Hasn't this wine been delicious
        This wine has not been delicious
        Has this wine not been delicious
        This wine was delicious
        Was this wine delicious
        This wine wasn't delicious
        Wasn't this wine delicious
        This wine was not delicious
        Was this wine not delicious
        This wine had been delicious
        Had this wine been delicious
        This wine hadn't been delicious
        Hadn't this wine been delicious
        This wine had not been delicious
        Had this wine not been delicious
        This wine will be delicious
        Will this wine be delicious
        This wine won't be delicious
        Won't this wine be delicious
        This wine will not be delicious
        Will this wine not be delicious
        This wine will have been delicious
        Will this wine have been delicious
        This wine won't have been delicious
        Won't this wine have been delicious
        This wine will not have been delicious
        Will this wine not have been delicious
        This wine would be delicious
        Would this wine be delicious
        This wine wouldn't be delicious
        Wouldn't this wine be delicious
        This wine would not be delicious
        Would this wine not be delicious
        This wine would have been delicious
        Would this wine have been delicious
        This wine wouldn't have been delicious
        Wouldn't this wine have been delicious
        This wine would not have been delicious
        Would this wine not have been delicious

We also see

- polarity (positive vs. negative)
- word order (direct vs. inverted)
- variation between contracted and full negation

The list is even longer in languages that have more tenses and moods, e.g. the Romance languages.

# Lesson 5: Refining semantics in abstract syntax

Goals:

- include semantic conditions in grammars, by using
  - dependent types
  - higher order abstract syntax
  - proof objects
  - semantic definitions

    These concepts are inherited from type theory (more precisely: constructive type theory, or Martin-Löf type theory).

    Type theory is the basis logical frameworks.

    GF = logical framework + concrete syntax.

## Dependent types

Problem: to express conditions of semantic well-formedness.

Example: a voice command system for a "smart house" wants to eliminate meaningless commands.

Thus we want to restrict particular actions to particular devices - we can dim a light, but we cannot dim a fan.

The following example is borrowed from the Regulus Book (Rayner & al. 2006).

A simple example is a "smart house" system, which defines voice commands for household appliances.

### A dependent type system

Ontology:

- there are commands and device kinds
- for each kind of device, there are devices and actions
- a command concerns an action of some kind on a device of the same kind

Abstract syntax formalizing this:

        cat
          Command ;
          Kind ;
          Device Kind ; -- argument type Kind
          Action Kind ;
        fun
          CAction : (k : Kind) -> Action k -> Device k -> Command ;

`Device` and `Action` are both dependent types.

### Examples of devices and actions

Assume the kinds `light` and `fan`,

        light, fan : Kind ;
        dim : Action light ;

Given a kind, k, you can form the device the k.

        DKindOne  : (k : Kind) -> Device k ;  -- the light

Now we can form the syntax tree

        CAction light dim (DKindOne light)

but we cannot form the trees

        CAction light dim (DKindOne fan)
        CAction fan   dim (DKindOne light)
        CAction fan   dim (DKindOne fan)

### Linearization and parsing with dependent types

Concrete syntax does not know if a category is a dependent type.

        lincat Action = {s : Str} ;
        lin CAction _ act dev = {s = act.s ++ dev.s} ;

Notice that the `Kind` argument is suppressed in linearization.

Parsing with dependent types is performed in two phases:

1. context-free parsing
2. filtering through type checker

By just doing the first phase, the `kind` argument is not found:

        > parse "dim the light"
        CAction ? dim (DKindOne light)

Moreover, type-incorrect commands are not rejected:

        > parse "dim the fan"
        CAction ? dim (DKindOne fan)

The term `?` is a metavariable, returned by the parser for any subtree that is suppressed by a linearization rule. These are the same kind of metavariables as were used here to mark incomplete parts of trees in the syntax editor.

### Solving metavariables

Use the command `put_tree = pt` with the option `-typecheck`:

        > parse "dim the light" | put_tree -typecheck
        CAction light dim (DKindOne light)

The `typecheck` process may fail, in which case an error message is shown and no tree is returned:

        > parse "dim the fan" | put_tree -typecheck

        Error in tree UCommand (CAction ? 0 dim (DKindOne fan)) :
          (? 0 <> fan) (? 0 <> light)

## Polymorphism

Sometimes an action can be performed on all kinds of devices.

This is represented as a function that takes a `Kind` as an argument and produce an `Action` for that `Kind`:

        fun switchOn, switchOff : (k : Kind) -> Action k ;

Functions of this kind are called polymorphic.

We can use this kind of polymorphism in concrete syntax as well, to express Haskell-type library functions:

        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 ;

### Dependent types: exercises

1. Write an abstract syntax module with above contents and an appropriate English concrete syntax. Try to parse the commands dim the light and dim the fan, with and without `solve` filtering.

2. Perform random and exhaustive generation, with and without `solve` filtering.

3. Add some device kinds and actions to the grammar.

## Proof objects

Curry-Howard isomorphism = propositions as types principle: a proposition is a type of proofs (= proof objects).

Example: define the less than proposition for natural numbers,

        cat Nat ;
        fun Zero : Nat ;
        fun Succ : Nat -> Nat ;

Define inductively what it means for a number x to be less than a number y:

- `Zero` is less than `Succ` y for any y.
- If x is less than y, then `Succ` x is less than `Succ` y.

Expressing these axioms in type theory with a dependent type `Less` x y and two functions constructing its objects:

        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) ;

Example: 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)))
         : Less (Succ (Succ Zero)) (Succ (Succ (Succ (Succ Zero))))

### Proof-carrying documents

Idea: 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.

Example: 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 ;

To extend the conditions to flight connections, we introduce a category of proofs that a change is possible:

        cat IsPossible (x,y,z : City)(Flight x y)(Flight y z) ;

A legal connection is formed by the function

        fun Connect : (x,y,z : City) ->
          (u : Flight x y) -> (v : Flight y z) ->
            IsPossible x y z u v -> Flight x z ;

## Restricted polymorphism

Above, all Actions were either of

- monomorphic: defined for one Kind
- polymorphic: defined for all Kinds

To make this scale up for new Kinds, we can refine this to restricted polymorphism: defined for Kinds of a certain class

The notion of class uses the Curry-Howard isomorphism as follows:

- a class is a predicate of Kinds --- i.e. a type depending of Kinds
- a Kind is in a class if there is a proof object of this type

### Example: classes for switching and dimming

We modify the smart house grammar:

      cat
        Switchable Kind ;
        Dimmable   Kind ;
      fun
        switchable_light : Switchable light ;
        switchable_fan   : Switchable fan ;
        dimmable_light   : Dimmable light ;

        switchOn : (k : Kind) -> Switchable k -> Action k ;
        dim      : (k : Kind) -> Dimmable k -> Action k ;

Classes for new actions can be added incrementally.

## Variable bindings

Mathematical notation and programming languages have expressions that bind variables.

Example: universal quantifier formula

        (All x)B(x)

The variable `x` has a binding `(All x)`, and occurs bound in the body `B(x)`.

Examples from informal mathematical language:

        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

        Let x be a natural number. Assume that x is even. Then x + 3 is odd.

### Higher-order abstract syntax

Abstract syntax can use functions as arguments:

        cat Ind ; Prop ;
        fun All : (Ind -> Prop) -> Prop

where `Ind` is the type of individuals and `Prop`, the type of propositions.

Let us add an equality predicate

        fun Eq : Ind -> Ind -> Prop

Now we can form the tree

        All (\x -> Eq x x)

which we want to relate to the ordinary notation

        (All x)(x = x)

In higher-order abstract syntax (HOAS), all variable bindings are expressed using higher-order syntactic constructors.

### Higher-order abstract syntax: linearization

HOAS has proved to be useful in the semantics and computer implementation of variable-binding expressions.

How do we relate HOAS to the concrete syntax?

In GF, we write

        fun All : (Ind -> Prop) -> Prop
        lin All B = {s = "(" ++ "All" ++ B.$0 ++ ")" ++ B.s}

General rule: if an argument type of a `fun` function is a function type `A -> C`, the linearization type of this argument is the linearization type of `C` together with a new field `$0 : Str`.

The argument `B` thus has the linearization type

        {s : Str ; $0 : Str},

If there are more bindings, we add `$1`, `$2`, etc.

### Eta expansion

To make sense of linearization, syntax trees must be eta-expanded: for any function of type

        A -> B

an eta-expanded syntax tree has the form

        \x -> b

where `b : B` under the assumption `x : A`.

Given the linearization rule

        lin Eq a b = {s = "(" ++ a.s ++ "=" ++ b.s ++ ")"}

the linearization of the tree

        \x -> Eq x x

is the record

        {$0 = "x", s = ["( x = x )"]}

Then we can compute the linearization of the formula,

        All (\x -> Eq x x)  --> {s = "[( All x ) ( x = x )]"}.

The linearization of the variable `x` is, "automagically", the string `"x"`.

### Parsing variable bindings

GF can treat any one-word string as a variable symbol.

        > p -cat=Prop "( All x ) ( x = x )"
        All (\x -> Eq x x)

Variables must be bound if they are used:

        > p -cat=Prop "( All x ) ( x = y )"
        no tree found

### Exercises on variable bindings

1. Write an abstract syntax of the whole predicate calculus, with the connectives "and", "or", "implies", and "not", and the quantifiers "exists" and "for all". Use higher-order functions to guarantee that unbounded variables do not occur.

2. Write a concrete syntax for your favourite notation of predicate calculus. Use Latex as target language if you want nice output. You can also try producing boolean expressions of some programming language. Use as many parenthesis as you need to guarantee non-ambiguity.

## Semantic definitions

The `fun` judgements of GF are declarations of functions, giving their types.

Can we compute `fun` functions?

Mostly we are not interested, since functions are seen as constructors, i.e. data forms - as usual with

        fun Zero : Nat ;
        fun Succ : Nat -> Nat ;

But it is also possible to give semantic definitions to functions. The key word is `def`:

        fun one : Nat ;
        def one = Succ Zero ;

        fun twice : Nat -> Nat ;
        def twice x = plus x x ;

        fun plus : Nat -> Nat -> Nat ;
        def
          plus x Zero = x ;
          plus x (Succ y) = Succ (Sum x y) ;

### Computing a tree

Computation: follow a chain of definition until no definition can be applied,

        plus one one -->
        plus (Succ Zero) (Succ Zero) -->
        Succ (plus (Succ Zero) Zero) -->
        Succ (Succ Zero)

Computation in GF is performed with the `put_term` command and the `compute` transformation, e.g.

        > parse -tr "1 + 1" | put_term -transform=compute -tr | l
        plus one one
        Succ (Succ Zero)
        s(s(0))

### Definitional equality

Two trees are definitionally equal if they compute into the same tree.

Definitional equality does not guarantee sameness of linearization:

        plus one one     ===> 1 + 1
        Succ (Succ Zero) ===> s(s(0))

The main use of this concept is in type checking: sameness of types.

Thus e.g. the following types are equal

        Less Zero one
        Less Zero (Succ Zero))

so that an object of one also is an object of the other.

### Judgement forms for constructors

The judgement form `data` tells that a category has certain functions as constructors:

        data Nat = Succ | Zero ;

The type signatures of constructors are given separately,

        fun Zero : Nat ;
        fun Succ : Nat -> Nat ;

There is also a shorthand:

        data Succ : Nat -> Nat ;    ===   fun Succ : Nat -> Nat ;
                                          data Nat = Succ ;

Notice: in `def` definitions, identifier patterns not marked as `data` will be treated as variables.

### Exercises on semantic definitions

1. Implement an interpreter of a small functional programming language with natural numbers, lists, pairs, lambdas, etc. Use higher-order abstract syntax with semantic definitions. As concrete syntax, use your favourite programming language.

2. There is no termination checking for `def` definitions. Construct an examples that makes type checking loop. Type checking can be invoked with `put_term -transform=solve`.

## Lesson 6: Grammars of formal languages

Goals:

- write grammars for formal languages (mathematical notation, programming languages)
- interface between formal and natural langauges
- implement a compiler by using GF

### Arithmetic expressions

We construct a calculator with addition, subtraction, multiplication, and division of integers.

        abstract Calculator = {

        cat Exp ;

        fun
          EPlus, EMinus, ETimes, EDiv : Exp -> Exp -> Exp ;
          EInt : Int -> Exp ;
        }

The category `Int` is a built-in category of integers. Its syntax trees integer literals, i.e. sequences of digits:

        5457455814608954681 : Int

These are the only objects of type `Int`: grammars are not allowed to declare functions with `Int` as value type.

### Concrete syntax: a simple approach

We begin with a concrete syntax that always uses parentheses around binary operator applications:

        concrete CalculatorP of Calculator = {

        lincat
          Exp = SS ;
        lin
          EPlus  = infix "+" ;
          EMinus = infix "-" ;
          ETimes = infix "*" ;
          EDiv   = infix "/" ;
          EInt i = i ;

        oper
          infix : Str -> SS -> SS -> SS = \f,x,y ->
            ss ("(" ++ x.s ++ f ++ y.s ++ ")") ;
        }

Now we have

        > linearize EPlus (EInt 2) (ETimes (EInt 3) (EInt 4))
        ( 2 + ( 3 * 4 ) )

First problems:

- to get rid of superfluous spaces and
- to recognize integer literals in the parser

## Lexing and unlexing

The input of parsing in GF is not just a string, but a list of tokens, returned by a lexer.

The default lexer in GF returns chunks separated by spaces:

        "(12 + (3 * 4))"  ===>  "(12", "+", "(3". "*". "4))"

The proper way would be

        "(", "12", "+", "(", "3", "*", "4", ")", ")"

Moreover, the tokens `"12"`, `"3"`, and `"4"` should be recognized as integer literals - they cannot be found in the grammar.

Lexers are invoked by flags to the command `put_string = ps`.

        > put_string -lexcode "(2 + (3 * 4))"
        ( 2 + ( 3 * 4 ) )

This can be piped into a parser, as usual:

        > ps -lexcode "(2 + (3 * 4))" | parse
        EPlus (EInt 2) (ETimes (EInt 3) (EInt 4))

In linearization, we use a corresponding unlexer:

        > linearize EPlus (EInt 2) (ETimes (EInt 3) (EInt 4)) | ps -unlexcode
        (2 + (3 * 4))

### Most common lexers and unlexers

lexer | unlexer | description
--- | --- | ---
`chars` | `unchars` | each character is a token
`lexcode` | `unlexcode` | program code conventions (uses Haskell's lex)
`lexmixed` | `unlexmixed` | like text, but between $ signs like code
`lextext` | `unlextext` | with conventions on punctuation and capitals
`words` | `unwords` | (default) tokens separated by space characters

## Precedence and fixity

Arithmetic expressions should be unambiguous. If we write

        2 + 3 * 4

it should be parsed as one, but not both, of

        EPlus (EInt 2) (ETimes (EInt 3) (EInt 4))
        ETimes (EPlus (EInt 2) (EInt 3)) (EInt 4)

We choose the former tree, because multiplication has higher precedence than addition.

To express the latter tree, we have to use parentheses:

        (2 + 3) * 4

The usual precedence rules:

- Integer constants and expressions in parentheses have the highest precedence.
- Multiplication and division have equal precedence, lower than the highest but higher than addition and subtraction, which are again equal.
- All the four binary operations are left-associative: `1 + 2 + 3` means the same as `(1 + 2) + 3`.

### Precedence as a parameter

Precedence can be made into an inherent feature of expressions:

        oper
          Prec : PType = Ints 2 ;
          TermPrec : Type = {s : Str ; p : Prec} ;

          mkPrec : Prec -> Str -> TermPrec = \p,s -> {s = s ; p = p} ;

        lincat
          Exp = TermPrec ;

Notice `Ints 2`: a parameter type, whose values are the integers `0,1,2`.

Using precedence levels: compare the inherent precedence of an expression with the expected precedence.

- if the inherent precedence is lower than the expected precedence, use parentheses
- otherwise, no parentheses are needed

This idea is encoded in the operation

        oper usePrec : TermPrec -> Prec -> Str = \x,p ->
          case lessPrec x.p p of {
            True  => "(" x.s ")" ;
            False => x.s
          } ;

(We use `lessPrec` from `lib/prelude/Formal`.)

### Fixities

We can define left-associative infix expressions:

        infixl : Prec -> Str -> (_,_ : TermPrec) -> TermPrec = \p,f,x,y ->
          mkPrec p (usePrec x p ++ f ++ usePrec y (nextPrec p)) ;

Constant-like expressions (the highest level):

        constant : Str -> TermPrec = mkPrec 2 ;

All these operations can be found in `lib/prelude/Formal`, which has 5 levels.

Now we can write the whole concrete syntax of `Calculator` compactly:

        concrete CalculatorC of Calculator = open Formal, Prelude in {

        flags lexer = codelit ; unlexer = code ; startcat = Exp ;

        lincat Exp = TermPrec ;

        lin
          EPlus  = infixl 0 "+" ;
          EMinus = infixl 0 "-" ;
          ETimes = infixl 1 "*" ;
          EDiv   = infixl 1 "/" ;
          EInt i = constant i.s ;
        }

### Exercises on precedence

1. Define non-associative and right-associative infix operations analogous to `infixl`.

2. Add a constructor that puts parentheses around expressions to raise their precedence, but that is eliminated by a `def` definition. Test parsing with and without a pipe to `pt -transform=compute`.

## Code generation as linearization

Translate arithmetic (infix) to JVM (postfix):

        2 + 3 * 4

          ===>

        iconst 2 : iconst 3 ; iconst 4 ; imul ; iadd

Just give linearization rules for JVM:

        lin
          EPlus  = postfix "iadd" ;
          EMinus = postfix "isub" ;
          ETimes = postfix "imul" ;
          EDiv   = postfix "idiv" ;
          EInt i = ss ("iconst" ++ i.s) ;
        oper
          postfix : Str -> SS -> SS -> SS = \op,x,y ->
            ss (x.s ++ ";" ++ y.s ++ ";" ++ op) ;

### Programs with variables

A straight code programming language, with initializations and assignments:

        int x = 2 + 3 ;
        int y = x + 1 ;
        x = x + 9 * y ;

We define programs by the following constructors:

        fun
          PEmpty : Prog ;
          PInit  : Exp -> (Var -> Prog) -> Prog ;
          PAss   : Var -> Exp  -> Prog  -> Prog ;

`PInit` uses higher-order abstract syntax for making the initialized variable available in the continuation of the program.

The abstract syntax tree for the above code is

        PInit (EPlus (EInt 2) (EInt 3)) (\x ->
          PInit (EPlus (EVar x) (EInt 1)) (\y ->
            PAss x (EPlus (EVar x) (ETimes (EInt 9) (EVar y)))
              PEmpty))

No uninitialized variables are allowed - there are no constructors for `Var`! But we do have the rule

        fun EVar : Var -> Exp ;

The rest of the grammar is just the same as for arithmetic expressions here. The best way to implement it is perhaps by writing a module that extends the expression module. The most natural start category of the extension is `Prog`.

### Exercises on code generation

1. Define a C-like concrete syntax of the straight-code language.

2. Extend the straight-code language to expressions of type `float`. To guarantee type safety, you can define a category `Typ` of types, and make `Exp` and `Var` dependent on `Typ`. Basic floating point expressions can be formed from literal of the built-in GF type `Float`. The arithmetic operations should be made polymorphic (as here).

3. Extend JVM generation to the straight-code language, using two more instructions

- `iload` x, which loads the value of the variable x
- `istore` x which stores a value to the variable x

Thus the code for the example in the previous section is

        iconst 2 ; iconst 3 ; iadd ; istore x ;
        iload x ; iconst 1 ; iadd ; istore y ;
        iload x ; iconst 9 ; iload y ; imul ; iadd ; istore x ;

4. If you made the exercise of adding floating point numbers to the language, you can now cash out the main advantage of type checking for code generation: selecting type-correct JVM instructions. The floating point instructions are precisely the same as the integer one, except that the prefix is `f` instead of `i`, and that `fconst` takes floating point literals as arguments.

# Lesson 7: Embedded grammars

Goals:

- use grammars as parts of programs written in Haskell and JavaScript
- implement stand-alone question-answering systems and translators based on GF grammars
- generate language models for speech recognition from GF grammars

## Functionalities of an embedded grammar format

GF grammars can be used as parts of programs written in other programming languages, to be called host languages. This facility is based on several components:

- PGF: a portable format for multilingual GF grammars
- a PGF interpreter written in the host language
- a library in the host language that enables calling the interpreter
- a way to manipulate abstract syntax trees in the host language

## The portable grammar format

The portable format is called PGF, "Portable Grammar Format".

This format is produced by the GF batch compiler `gfc`, executable from the operative system shell:

        % gfc --make SOURCE.gf

PGF is the recommended format in which final grammar products are distributed, because they are stripped from superfluous information and can be started and applied faster than sets of separate modules.

Application programmers have never any need to read or modify PGF files.

PGF thus plays the same role as machine code in general-purpose programming (or bytecode in Java).

### Haskell: the EmbedAPI module

The Haskell API contains (among other things) the following types and functions:

        readPGF   :: FilePath -> IO PGF

        linearize :: PGF -> Language -> Tree -> String
        parse     :: PGF -> Language -> Category -> String -> [Tree]

        linearizeAll     :: PGF -> Tree -> [String]
        linearizeAllLang :: PGF -> Tree -> [(Language,String)]

        parseAll     :: PGF -> Category -> String -> [[Tree]]
        parseAllLang :: PGF -> Category -> String -> [(Language,[Tree])]

        languages    :: PGF -> [Language]
        categories   :: PGF -> [Category]
        startCat     :: PGF -> Category

This is the only module that needs to be imported in the Haskell application. It is available as a part of the GF distribution, in the file `src/PGF.hs`.

### First application: a translator

Let us first build a stand-alone translator, which can translate in any multilingual grammar between any languages in the grammar.

      module Main where

      import PGF
      import System (getArgs)

      main :: IO ()
      main = do
        file:_ <- getArgs
        gr     <- readPGF file
        interact (translate gr)

      translate :: PGF -> String -> String
      translate gr s = case parseAllLang gr (startCat gr) s of
        (lg,t:_):_ -> unlines [linearize gr l t | l <- languages gr, l /= lg]
        _ -> "NO PARSE"

To run the translator, first compile it by

        % ghc --make -o trans Translator.hs

For this, you need the Haskell compiler GHC.

### Producing GFCC for the translator

Then produce a GFCC file. For instance, the `Food` grammar set can be compiled as follows:

        % gfc --make FoodEng.gf FoodIta.gf

This produces the file `Food.pgf` (its name comes from the abstract syntax).

The Haskell library function `interact` makes the `trans` program work like a Unix filter, which reads from standard input and writes to standard output. Therefore it can be a part of a pipe and read and write files. The simplest way to translate is to `echo` input to the program:

        % echo "this wine is delicious" | ./trans Food.pgf
        questo vino è delizioso

The result is given in all languages except the input language.

### A translator loop

To avoid starting the translator over and over again: change `interact` in the main function to `loop`, defined as follows:

      loop :: (String -> String) -> IO ()
      loop trans = do
        s <- getLine
        if s == "quit" then putStrLn "bye" else do
          putStrLn $ trans s
          loop trans

The loop keeps on translating line by line until the input line is `quit`.

### A question-answer system

The next application is also a translator, but it adds a transfer component - a function that transforms syntax trees.

The transfer function we use is one that computes a question into an answer.

The program accepts simple questions about arithmetic and answers "yes" or "no" in the language in which the question was made:

        Is 123 prime?
        No.
        77 est impair ?
        Oui.

We change the pure translator by giving the `translate` function the transfer as an extra argument:

        translate :: (Tree -> Tree) -> PGF -> String -> String

Ordinary translation as a special case where transfer is the identity function (`id` in Haskell).

To reply in the same language as the question:

        translate tr gr = case parseAllLang gr (startCat gr) s of
          (lg,t:_):_ -> linearize gr lg (tr t)
          _ -> "NO PARSE"

### Abstract syntax of the query system

Input: abstract syntax judgements

      abstract Query = {

        flags startcat=Question ;

        cat
          Answer ; Question ; Object ;

        fun
          Even   : Object -> Question ;
          Odd    : Object -> Question ;
          Prime  : Object -> Question ;
          Number : Int -> Object ;

          Yes : Answer ;
          No  : Answer ;
      }

### Exporting GF datatypes to Haskell

To make it easy to define a transfer function, we export the abstract syntax to a system of Haskell datatypes:

        % gfc --output-format=haskell Query.pgf

It is also possible to produce the Haskell file together with GFCC, by

        % gfc --make --output-format=haskell QueryEng.gf

The result is a file named `Query.hs`, containing a module named `Query`.

Output: Haskell definitions

      module Query where
      import PGF

      data GAnswer =
         GYes
       | GNo

      data GObject = GNumber GInt

      data GQuestion =
         GPrime GObject
       | GOdd GObject
       | GEven GObject

      newtype GInt = GInt Integer

All type and constructor names are prefixed with a `G` to prevent clashes.

The Haskell module name is the same as the abstract syntax name.

### The question-answer function

Haskell's type checker guarantees that the functions are well-typed also with respect to GF.

      answer :: GQuestion -> GAnswer
      answer p = case p of
        GOdd x   -> test odd x
        GEven x  -> test even x
        GPrime x -> test prime x

      value :: GObject -> Int
      value e = case e of
        GNumber (GInt i) -> fromInteger i

      test :: (Int -> Bool) -> GObject -> GAnswer
      test f x = if f (value x) then GYes else GNo

### Converting between Haskell and GF trees

The generated Haskell module also contains

      class Gf a where
        gf :: a -> Tree
        fg :: Tree -> a

      instance Gf GQuestion where
        gf (GEven x1) = DTr [] (AC (CId "Even")) [gf x1]
        gf (GOdd x1) = DTr [] (AC (CId "Odd")) [gf x1]
        gf (GPrime x1) = DTr [] (AC (CId "Prime")) [gf x1]
        fg t =
          case t of
            DTr [] (AC (CId "Even")) [x1] -> GEven (fg x1)
            DTr [] (AC (CId "Odd")) [x1] -> GOdd (fg x1)
            DTr [] (AC (CId "Prime")) [x1] -> GPrime (fg x1)
            _ -> error ("no Question " ++ show t)

For the programmer, it is enougo to know:

- all GF names are in Haskell prefixed with `G`
- `gf` translates from Haskell objects to GF trees
- `fg` translates from GF trees to Haskell objects

### Putting it all together: the transfer definition

      module TransferDef where

      import PGF (Tree)
      import Query   -- generated from GF

      transfer :: Tree -> Tree
      transfer = gf . answer . fg

      answer :: GQuestion -> GAnswer
      answer p = case p of
        GOdd x   -> test odd x
        GEven x  -> test even x
        GPrime x -> test prime x

      value :: GObject -> Int
      value e = case e of
        GNumber (GInt i) -> fromInteger i

      test :: (Int -> Bool) -> GObject -> GAnswer
      test f x = if f (value x) then GYes else GNo

      prime :: Int -> Bool
      prime x = elem x primes where
        primes = sieve [2 .. x]
        sieve (p:xs) = p : sieve [ n | n <- xs, n `mod` p > 0 ]
        sieve [] = []

### Putting it all together: the Main module

Here is the complete code in the Haskell file `TransferLoop.hs`.

      module Main where

      import PGF
      import TransferDef (transfer)

      main :: IO ()
      main = do
        gr <- readPGF "Query.pgf"
        loop (translate transfer gr)

      loop :: (String -> String) -> IO ()
      loop trans = do
        s <- getLine
        if s == "quit" then putStrLn "bye" else do
          putStrLn $ trans s
          loop trans

      translate :: (Tree -> Tree) -> PGF -> String -> String
      translate tr gr s = case parseAllLang gr (startCat gr) s of
        (lg,t:_):_ -> linearize gr lg (tr t)
        _ -> "NO PARSE"

### Putting it all together: the Makefile

To automate the production of the system, we write a `Makefile` as follows:

      all:
              gfc --make --output-format=haskell QueryEng
              ghc --make -o ./math TransferLoop.hs
              strip math

(The empty segments starting the command lines in a Makefile must be tabs.) Now we can compile the whole system by just typing

        make

Then you can run it by typing

        ./math

Just to summarize, the source of the application consists of the following files:

        Makefile         -- a makefile
        Math.gf          -- abstract syntax
        Math???.gf       -- concrete syntaxes
        TransferDef.hs   -- definition of question-to-answer function
        TransferLoop.hs  -- Haskell Main module

## Web server applications

PGF files can be used in web servers, for which there is a Haskell library included in `src/server/`. How to build a server for tasks like translators is explained in the `README` file in that directory.

One of the servers that can be readily built with the library (without any programming required) is fridge poetry magnets. It is an application that uses an incremental parser to suggest grammatically correct next words. Here is an example of its application to the `Foods` grammars.

## JavaScript applications

JavaScript is a programming language that has interpreters built in in most web browsers. It is therefore usable for client side web programs, which can even be run without access to the internet. The following figure shows a JavaScript program compiled from GF grammars as run on an iPhone.

### Compiling to JavaScript

JavaScript is one of the output formats of the GF batch compiler. Thus the following command generates a JavaScript file from two `Food` grammars.

        % gfc --make --output-format=js FoodEng.gf FoodIta.gf

The name of the generated file is `Food.js`, derived from the top-most abstract syntax name. This file contains the multilingual grammar as a JavaScript object.

### Using the JavaScript grammar

To perform parsing and linearization, the run-time library `gflib.js` is used. It is included in `GF/lib/javascript/`, together with some other JavaScript and HTML files; these files can be used as templates for building applications.

An example of usage is `translator.html`, which is in fact initialized with a pointer to the Food grammar, so that it provides translation between the English and Italian grammars:

The grammar must have the name `grammar.js`. The abstract syntax and start category names in `translator.html` must match the ones in the grammar. With these changes, the translator works for any multilingual GF grammar.

## Language models for speech recognition

The standard way of using GF in speech recognition is by building grammar-based language models.

GF supports several formats, including GSL, the formatused in the Nuance speech recognizer.

GSL is produced from GF by running `gfc` with the flag `--output-format=gsl`.

Example: GSL generated from `FoodsEng.gf`.

        % gfc --make --output-format=gsl FoodsEng.gf
        % more FoodsEng.gsl

        ;GSL2.0
        ; Nuance speech recognition grammar for FoodsEng
        ; Generated by GF

        .MAIN Phrase_cat

        Item_1 [("that" Kind_1) ("this" Kind_1)]
        Item_2 [("these" Kind_2) ("those" Kind_2)]
        Item_cat [Item_1 Item_2]
        Kind_1 ["cheese" "fish" "pizza" (Quality_1 Kind_1)
                "wine"]
        Kind_2 ["cheeses" "fish" "pizzas"
                (Quality_1 Kind_2) "wines"]
        Kind_cat [Kind_1 Kind_2]
        Phrase_1 [(Item_1 "is" Quality_1)
                  (Item_2 "are" Quality_1)]
        Phrase_cat Phrase_1

        Quality_1 ["boring" "delicious" "expensive"
                   "fresh" "italian" ("very" Quality_1) "warm"]
        Quality_cat Quality_1

### More speech recognition grammar formats

Other formats available via the `--output-format` flag include:

Format | Description
--- | ---
`gsl` | Nuance GSL speech recognition grammar
`jsgf` | Java Speech Grammar Format (JSGF)
`jsgf_sisr_old` | JSGF with semantic tags in SISR WD 20030401 format
`srgs_abnf` | SRGS ABNF format
`srgs_xml` | SRGS XML format
`srgs_xml_prob` | SRGS XML format, with weights
`slf` | finite automaton in the HTK SLF format
`slf_sub` | finite automaton with sub-automata in HTK SLF

All currently available formats can be seen with `gfc --help`.