Language and Grammar in Discrete mathematics

Formal Language

A formal language can be described as a language that is specified with the help of a well-defined set of syntax rules.

Formal Grammar

A formal grammar G can be described as a compact, precise definition of a Language L.

Now we will learn about some definitions so that we can clearly understand the grammar of phrase structure.

Vocabulary

A vocabulary V or alphabet can be described as a finite non-empty set of elements, which is also known as the symbols.

Word

A word over V can be described as a string containing the finite length of elements of V. Word is also known as a sentence.

Empty String

The empty string can be described as the string which is used to have no symbols. This string is also known as the empty string, which can be indicated by the symbol λ.

Set of Words and Set of Language

With the help of symbol V*, we can indicate the set of words over V. A language over V can be described as a subset of V*.

Phrase Structure Grammar

A phrase structure grammar G can be described in the following way:

Where

V is used to indicate the vocabulary

T is used to indicate the subsets T of V, which is used to contain the terminal symbols.

S is used to indicate the start symbol from V

P is used to indicate a finite set of production

With the help of N, we can indicate the set V-T. The elements of N are also known as non-terminal symbols. The left side of every production in P will have a minimum of one non-terminal.

Derivability

Suppose there is a phrase structure grammar G = (V, T, S, P). Suppose w₀ = lz₀r (concentration of l, z₀, and r) and w₁ = lz₁r be string over V. If there is a production of G: z_o → z₁, in this case, w₁ will be directly derivable from w₁ and in other words it can be written as w₀ ⇒ w₁. If there are some strings w₀, w₁, w₂, w₃, ...., w_n over V in such a way that w₀ ⇒ w₁, w₁ ⇒ w₂, w₂ ⇒ w₃, ...., w_n-1 ⇒ w_n, in this case, w_n will be derivable from w₀, and it can be written as w₀ ⇒ w_n. If there is a sequence of steps that are used to get w_n from w₀, then it will be known as derivation.

Language Generated by G, L

Suppose there is a phrase structure grammar G = (V, T, S, P). With the help of symbol L(G), we can indicate the language generated by G. It is used to contain the set of all strings of terminals, and we can drive it from the starting state S. The language generated by G can be shown in one another way, which is described as follows:

L(G) = {w ∈ T* | S ⇒ w}

Types of Grammar

There are various types of grammar and restrictions on production, which are described as follows:

Derivation Tree

With the help of context-free grammar, we can generate a derivation in language. We can represent the derivative tree graphically with the help of an ordered rooted tree, which is also known as the derivation or parse tree. The starting symbol of this tree can be represented with the help of root. The non-terminal symbols of the tree are used to indicate the internal vertices. The terminal symbols are indicated by the leaves of a tree. If there is a derivation that contains the production A → w, then the vertex that shows A will contain the children's vertices that are used to show each symbol in w, where w is used to indicate a word. The order of this process will be from left to right.

Backus-Naur form (BNF)

The BNF can be described as the syntactic rules of many computer languages, such as Java. If there is a type 2 grammar, the production will have a single non-terminal symbol on the LHS. So in place of listing all the productions one by one, we will combine all those productions with the help of same non-terminal symbol on LHS into one statement. To indicate the production, we use the symbol →, but here we will use the symbol ::=. All the non-terminal symbols will be enclosed with the help of bracket (), and all the RHS of productions will be listed in the same statement, which will be separated with the help of bars. The following example can be used to show the BNF: