Formal Grammars

Formal Grammars

1. Context free grammars are a notation for describing sets of strings (each phrase type is really just the name of a set of strings). So, we start our formal study of grammars with some basic definitions concerning strings and sets of strings:

   Alphabet

   A finite, non-empty set of symbols.

   String

   A string over some alphabet () is a finite, possibly empty sequence of symbols from . We will use eps to denote the empty string.

   Language

   A language over an alphabet is just a set of strings over that alphabet.

   Concatenation

   If x and y are two strings over , then their concatenation, xy, is the sequence of characters obtained by placing the sequence of characters in x before the sequence y. If z = xy is a string, we say that x is a prefix of z and y is a suffix of z.

   Products

   If X and Y are languages over some alphabet, then their product, XY, is defined to be:

   { xy | x X and    y y }

   Powers

   If X is a language over we define Xⁿ to be the language containing only the empty string if n = 0 and XX^n-1 otherwise.

   Closures

   If X is a language over we define X⁺, the positive closure of X, to be the union of the sets X¹, X², X³, . . . and X^*, the closure of X, to be the union of X⁰ and X⁺.

2. Now, the definition you have all been waiting for:

   Context-free Grammar

   A context free grammar is composed of:

   1. A finite alphabet V_t called the terminal symbols.
   2. A finite alphabet V_n called the non-terminal symbols.
   3. A distinguished element of the set V_n denoted by the symbol S and referred to as the goal symbol or start symbol.
   4. A finite set P of pairs composed of one element from V_n and one element from (V_t U V_n)^* called productions. Productions are written in the form:

      A X₁ X₂ . . . X_m

3. There is an interesting, alternate approach to interpreting the productions of a grammar. Rather than viewing them as rules for producing strings that belong to the language defined, we can view them as set inequalities over a set of variables composed of the non-terminal symbols.
   • In this interpretation,

     < stmt > -> while < expr > do < stmt >

     is interpreted as

     < stmt > >= while < expr > do < stmt >

     where < stmt > and < expr > are viewed as the names of sets of strings (i.e. sub-languages) and keywords and other terminal symbols denote the set containing just that symbol ( i.e. do denotes { "do" } ).
   • The "interesting" thing about this view is that there are multiple solutions to the set of inequalities corresponding to the typical set of productions. In particular, letting all non-terminals denote Sigma^* generally does the trick. The " standard " interpretation of productions corresponds to the smallest sets that satisfy the inequalities interpretation.

     The "standard" interpretation is based on the next concept we consider, the derivation.

4. The association between a context free grammar and the language it describes is formalized through the notion of a derivation:

   Direct derivation

   Given a grammar, G = (V_t , V_n , S, P), and two strings x and y in (V_t U V_n)^* such that x = A and y = where , , (V_t U V_n)^* and ( A, ) P we say that x directly derives y. In this case we write

   x y

5. Examples of direct derivations.

   Consider G =

   < blob >	x < glob > < blob > y
   < blob >	z
   < glob >	a < glob >
   < glob >	eps

   • < blob > x < glob > < blob > y.
   • x < glob > a x a < glob > a
6. More on the notion of a derivation:

   Derivation

   Given a grammar, G = (V_t , V_n , S, P), and two strings x and y in (V_t U V_n)^* we say that x derives y if there exists a sequence of string ₀, ₁, ₂, ... ,_m all in (V_t U V_n)^* such that

   1. for all i < m, _{i i+1} ,
   2. x = ₀ , and
   3. y = _m .

   In this case we write

   x y

   The sequence ₀, ₁, ₂, ... ,_m is called a derivation of length m of y from x.

7. Using the grammar G shown above we can say that that < blob > xaxzyy since:
   • < blob > x < glob > < blob > y
   • x < glob > < blob > y x a < blob > y
   • x a < blob > y x a x < glob > < blob > y y
   • x a x < glob > < blob > y y x a x < blob > y y
   • x a x < blob > y y x a x z y y
8. Time for more definitions:

   Sentential form

   Given a grammar, G, a string is called a sentential form of G if it is derivable from the start symbol of G.

   Sentence

   A sentential form containing only symbols from the terminal vocabulary of a language is called a sentence.

   L(G)

   The language defined by a grammar G is the set of all sentences.

   L(G) = { s | S s  &  s V_t^* }

Formal Grammars