# Natural Language Syntax in Coq: Baby Steps.

In a slight diversion from my path toward NL semantics in Martin-Löf Type Theory, I'd like to delve a bit into formulating NL syntax in Coq, a dependently typed theorem prover based on the Calculus of Inductive Constructions.

We'll begin with a very simple (and underpowered) theory of syntax including the following:

1. Syntactic categories: things like Noun, Verb, and Determiner. These classify items in our lexicon.
2. Merge: an operation that combines two nodes to make a larger node. Phrases are built from the lowest head upwards, by merging new nodes onto the head.
3. X-Bar Schema: There are at most two levels of projection from a single head; a head X may merge with an internal argument, projecting a X' node, which may in turn merge with an external argument to project the maximal XP node.

The internal argument is the sister of X, and is called its complement; the external argument is the aunt of X (the sister of X'), and is called the specifier. Therefore, the two kinds of merges that we have discussed are referred to as C-Merge and S-Merge respectively.

The schema looks something like this:

              XP                             X
/ \                            / \
/   \                          /   \
/     \                        /     \
ZP     X'       (or just)      Z      X
/ \                            / \
/   \                          /   \
/     \                        /     \
YP      X                      Y       X


The latter representation is the same, except the projection-level is left to be inferred.

In later posts, I'll revisit our theory with enhancements that will make it more able to express natural language syntax.

## The Implementation

Require Import String EquivDec Bool.
Set Implicit Arguments.

Section CoreTheory.

We'll start with a few syntactic categories. This is of course not sufficient to do real syntax, but it will do for now.
Inductive category := D | N | V.

Each constituent is marked by a set of features; in our current theory, features are just a syntactic category, and specifications for internal and external arguments.

Inductive features : Set :=
{ cat : category ;
iArg : option features ;
eArg : option features
}.

Coq allows us to derive decidable equality for categories. This will prove useful below.
Theorem eq_cat_dec (a : category) (b : category) : {a = b} + {a <> b}.
Proof.
decide equality.
Defined.
Program Instance cat_eq_eqdec : EqDec category eq := eq_cat_dec.

In our current theory, there are two kinds of arguments: internal and external; an internal argument is lower in the tree than the head, whereas an external argument is higher. We provide selectors for arguments.

Inductive position := internal | external.
Definition argument_at (p : position) :=
match p with
| internal => iArg
| external => eArg
end.

This is where it starts to get interesting. We need a predicate that decides whether or not the features hfs of the head license the features fs of a constituent which wishes to be merged to it at some position p. The predicate is satisfied if fs is saturated, and there is an argument arg at p in hfs whose category is equal to the category of fs.
We can say that a node is saturated at a position if there is no specification for an argument present. This is because these specifications are erased after each merge.

Definition saturated_at (fs : features) (p : position) :=
match argument_at p fs with
| Some _ => false
| None => true
end.

Definition fully_saturated (fs : features) :=
(saturated_at fs internal) && (saturated_at fs external).

First, we determine if a merge is even plausible: for C-merge, the internal argument must not already be saturated; for S-merge, the internal argument must be saturated, and the external argument must not be.

Definition can_merge_at (hfs : features) (p : position) :=
match p with
| internal => negb (saturated_at hfs internal)
| external => (saturated_at hfs internal) && (negb (saturated_at hfs external))
end.

Now, we can check if the merge will be successful, given the syntactic category of each of the participants.

Definition selects (p : position) (hfs : features) (fs : features) :=
match argument_at p hfs with
| Some arg =>
match cat arg == cat fs with
| left _ => Is_true ((fully_saturated fs) && (can_merge_at hfs p))
| right _ => False
end
| None => False
end.

We now compute the type of a merge at position p. We require a proof that the head selects for the new node. The resulting node inherits the category of the head, and has its iArg saturated; if p is external, that means that the eArg has also been saturated.

Definition gen_merge (N : features -> Set) (p : position) :=
forall (hfs : _) (fs : _)
(h : N hfs) (n : N fs)
(sel : selects p hfs fs),
N {| cat := cat hfs ;
iArg := None ;
eArg := match p with
| internal => eArg hfs
| external => None
end
|}.

Finally, we are ready to model nodes. A node is indexed by its features, and may be either a head (minimal projection), or the result of a cmerge (merging of a complement into internal argument position), or the result of an smerge (merginf of a specifier into external argument position). The types of the latter two constructors are computed using gen_merge above.

Inductive node : features -> Set :=
| head : forall (s : string), forall (fs : features), node fs
| cmerge : gen_merge node internal
| smerge : gen_merge node external.

As a bonus, we provide a function to fold a node into a string.
Fixpoint to_string {fs : _} (n : node fs) : string :=
match n with
| head s _ => s
| cmerge _ _ h c _ => append (to_string h) (append " " (to_string c))
| smerge _ _ h s _ => append (to_string s) (append " " (to_string h))
end.

End CoreTheory.

Section Examples.

Let's make some convenient notation for merges. Note that I is the single constructor for the type True, and serves as the proof-witness that the head selects the merged node.

(right associativity, at level 100).
(left associativity, at level 101).

Let's build up a lexicon.
Definition dog :=
{| cat := N ; iArg := None ; eArg := None |}.

Definition love :=
{| cat := V ;
iArg := Some {| cat := D ; iArg := None ; eArg := None |} ;
eArg := Some {| cat := D ; iArg := None ; eArg := None |}
|}.

Definition the :=
{| cat := D ;
iArg := Some {| cat := N ; iArg := None ; eArg := None |} ;
eArg := None
|}.

Definition I :=
{| cat := D ; iArg := None ; eArg := None |}.

We can now build up some phrases. If they type check, then they are grammatical within our theory.

Definition the_dog := the |- dog.
Definition love_the_dog := love |- the |- dog.
Definition I_love_the_dog := I -| love |- the |- dog.

The last phrase that we constructed represents the following tree:
           V
/ \
/   \
/     \
I      V
/ \
/   \
/     \
love    D
/ \
/   \
/     \
the    dog

Evaluating our phrase as a string yields "I love the dog".
Eval simpl in to_string I_love_the_dog.
End Examples.

## Next steps

Basically, all that this theory provides us is the ability to base grammaticality-judgements of merges on the syntactic category of consituents and their arguments. This is obvously not enough! Not only do we need to make more interesting specifications than syntactic category, we also need to include the following notions: phrasal movement, agreement, and head movement, to name a few.

In addition, it would be prudent to escape the bonds that our current argument representation has left us in, and move toward a more minimal approach based on a single merge operation; in addition, a notion of feature strength (that is, whether a feature must be satisfied locally or not) would be an enormous improvement.

In the meanwhile, play around with the sources to this post and see what you can improve!