Lecture 8: Context Free Grammars

Context free grammars:
* strictly more powerful way of expressing languages than a regular expression
* it's languages are called context free languages
* many human languages are approximately context-free (there are always exceptions), while most are not regular

Context free grammars consist of:
* An alphabet (the "words" in the language)
* A set of non-terminals
* A "start" non-terminal
* A set of rewrite rules for getting from non-terminals to either other non-terminals or terminals

Simple context free grammar:

S  -> NP VP
NP -> D N
VP -> V NP
D  -> "the"
D  -> "a"
N  -> "man"
N  -> "sandwich"
V  -> "ate"
V  -> "likes"

This is actually a finite language (what are the sentences in this language)?

Derivations are "proofs" that a sentence belongs to the language by giving a/the sequence of productions that leads to that sentence.

We'll usually draw trees as proofs, but other formats are more useful as text, for instance: (S (NP (D the) (N sandwich)) (VP (V ate) (NP (D the) (N man))))

In such grammars, we will often call the things just above the terminals (parts of speech) "pre-terminals".

We can make this grammar more fun by adding rules:

NP -> NP PP
VP -> VP PP
PP -> P NP
P  -> "with"

Now we have an infinite grammar that can generate strings like "the man with the man with the sandwich with the man with the ..."

Note also that this grammar is now ambiguous. "the man likes the man with the sandwich" could either be (S (NP the man) (VP (V likes) (NP (NP the man) (PP (P with) (NP the sandwich))))) or (S (NP the man) (VP (VP (V likes) (NP the man)) (PP (P with) (NP the sandwich)))). The first one has the "correct" meaning, while the second would mean that he uses the sandwich to like the man.

A simple context free language that is not regular is:

S -> a S b
S -> ab

Which describes the language "{ a^n b^n : n > 0}" of strings of equal numbers of as and bs. You can show this language is not regular formally, or you can recognize that a FSM would kind of have to have a N-many states to recognize the finite version of this language up to sequences of length N.

A popular approach for context free models of human language is "X-bar" theory.  Here, we have base categories (X0 = N,V,A,P), then "projections" of these categories: X' (aka X-bar), and X'' (aka X-double-bar aka XP).  The general form of a phrase is (XP (YP (X' (X0 ZP)))), where YP is the "specifier", X0 is the head, and ZP is the complement (argument).  Only full projections can fit in as adjunctions, specifiers or complements.

The idea is that the arguments fit in the "bar" slots, and the adjuncts fit in the "phrase" slots. Roughly, an argument is something that is sort-of necessary and an adjunct is sort-of optional. Usually you can have arbitrarily many adjuncts, but only a fixed number of arguments. For instance in "the man ate a sandwich of turkey with lettuce on a plate with gusto at the restaurant" the verb "ate" has one argument (sandwich) and two adjuncts ("with gusto" and "at the restaurant") while sandwich has three adjunctions "with turkey, with lettuce and on a plate).

Here are some example sentences in X-bar style analyses:

1. (PP (DP (D' (D so))) (P' (ADVP (ADV' (ADV completely))) (P' (P in) (NP (N' (DP (D' (D the))) (N wrong))))))
2. 
   
3. (IP (NP Poirot) (I' (Infl will) (VP (V' (V abandom)) (NP (N' (DP (D' (D the))) (N investigation))))))
4. Identical structure for the second half of "I did not expect Poirot to abandon the investigation."
5. Under some theories, (IP (NP Poirot) (I' (Infl +ed) (VP (V abandon) (NP the investigation))) is appropriate

In 5, we see an example of movement, a very linguisticy notion that has not been adopted much by computationalists.  Another example of movement is wh-movement ("Hal likes CL" vs "What does Hal like ____").  There can also be duplication, as in "The man who I saw _____".

(Warning: by now, lots of people no longer think NPs are really headed by nouns, but are rather headed by determiners, so that what we usually think of as an "NP" is really a "DP." That is, instead of "(NP (N' (DP (D' (D the))) (N man)))" you would have "(DP (D' (D the) (NP (N' (N man)))))". In this class, we'll always use NP and not DP, but you should know that this is slightly outdated.)