context free grammar, Slides of Theory of Automata

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Today’s Topic

Context Free grammar (CFG)

UNIT – 3

Regular and Non regular Grammar

Definition − A context-free grammar (CFG) consisting of a

finite set of grammar rules is a quadruple (N, T, P, S) where

N is a set of non-terminal symbols.

T is a set of terminals where N ∩ T = NULL.

P is a set of rules, P: N → (N ∪ T), i.e., the left-hand side of T), i.e., the left-hand side of

the production rule P does have any right context or left

context.

S is the start symbol.

Generation of Derivation Tree

A derivation tree or parse tree is an ordered rooted tree that

graphically represents the semantic information a string

derived from a context-free grammar.

Representation Technique

Root vertex − Must be labeled by the start symbol.

Vertex − Labeled by a non-terminal symbol.

Leaves − Labeled by a terminal symbol or ε.

If S → x1x2 …… xn is a production rule in a CFG, then the parse tree / derivation tree will be as follows −

Derivation Tree

There are two different approaches to draw a derivation tree − Top-down Approach −

1. Starts with the starting symbol S
2. Goes down to tree leaves using productions

Sentential Form and Partial Derivation Tree

A partial derivation tree is a sub-tree of a derivation

tree/parse tree such that either all of its children are in the

sub-tree or none of them are in the sub-tree.

Example:

If in any CFG the productions are −

S → AB, A → aaA | ε, B → Bb| ε

the partial derivation tree can be the following −

Leftmost and Rightmost Derivation of a String Leftmost derivation − A leftmost derivation is obtained by applying production to the leftmost variable in each step. Rightmost derivation − A rightmost derivation is obtained by applying production to the rightmost variable in each step. Example Let any set of production rules in a CFG be X → X+X | XX |X| a over an alphabet {a}. The leftmost derivation for the string "a+aa" may be − X → X+X → a+X → a + XX → a+aX → a+a*a The stepwise derivation of the above string is shown as-

Left and Right Recursive Grammars In a context-free grammar G, if there is a production in the form X → Xa where X is a non-terminal and ‘a’ is a string of terminals, it is called a left recursive production. The grammar having a left recursive production is called a left recursive grammar.

And if in a context-free grammar G, if there is a production is in the form X → aX where X is a non-terminal and ‘a’ is a string of terminals, it is called a right recursive production. The grammar having a right recursive production is called a right recursive grammar.

UNIT – 3

Regular and Non regular Grammar

Today’s Topic

AMBUIGUITY, CFG for regular

language and expressions

Ambiguity in Grammar A grammar is said to be ambiguous if there exists more than one leftmost derivation or more than one rightmost derivation or more than one parse tree for the given input string. If the grammar is not ambiguous, then it is called unambiguous. If the grammar has ambiguity, then it is not good for compiler construction. No method can automatically detect and remove the ambiguity, but we can remove ambiguity by re-writing the whole grammar without ambiguity. Example 1: Let us consider a grammar G with the production rule

Since there are two parse trees for a single string "3 * 2 + 5", the grammar G is ambiguous. QUES 2: Check whether the given grammar G is ambiguous or not. E → E + E E → E - E E → id Solution: From the above grammar String "id + id - id" can be derived in 2 ways: First Leftmost derivation E → E + E → id + E → id + E - E → id + id - E → id + id- id

Second Leftmost derivation E → E - E → E + E - E → id + E - E → id + id - E → id + id - id Since there are two leftmost derivation for a single string "id + id - id", the grammar G is ambiguous.

3. Check whether the given grammar G is ambiguous or not. S → aSb | SS S → ε FOR INPUT STRING “aabb”