Introduction to Computational Linguistics

18. Context Free Grammars 67
18 Context Free Grammars
Definition 22 Let A be as usual an alphabet. Let N be a set of symbols, N disjoint
with A, Σ ⊆ N. A context free rule over N as the set of nonterminal symbols and
A the set of terminal symbols is a pair 〈X, ⃗α〉, where X ∈ N and ⃗α is a string over
N ∪ A. We write X → ⃗α for the rule. A context free grammar is a quadruple
〈A, N, Σ, R〉, where R is a finite set of context free rules. The set Σ is called the set
of start symbols
Often it is assumed that a context free grammar has just one start symbol, but in
actual practice this is mostly not observed. Notice that context free rules allow
the use of nonterminals and terminals alike. The implicit convention we use is as
follows. A string of terminals is denoted by a Roman letter with a vector arrow,
a mixed string containing both terminals and nonterminals is denoted by a Greek
letter with a vector arrow. (The vector is generally used to denote strings.) If the
difference is not important, we shall use Roman letters.
The following are examples of context free rules. The conventions that apply
here are as follows. Symbols in print are terminal symbols; they are denoted using
typewriter font. Nonterminal symbols are produced by enclosing an arbitrary
string in brackets: < · · · >.
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< digit >→ 0
< digit >→ 1
< number >→< digit >
< number >→< number >< digit >
There are some conventions that apply to display the rules in a more compact
form. The vertical slash ‘|’ is used to merge two rules with the same left hand
symbol. So, when X → ⃗α and X → ⃗γ are rules, you can write X → ⃗α | ⃗γ. Notice
that one speaks of one rule in the latter case, but technically we have two rules
now. This allows us to write as follows.
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< digit >→ 0 | 1 | 2 | . . . | 9
And this stands for ten rules. Concatenation is not written. It is understood that
symbols that follow each other are concatenated the way they are written down.