Leftmost Derivation in Automata

Getting a string by expanding the leftmost non terminal at each step is called as leftmost derivation in Automata.

• The representation of leftmost derivation in the form of tree is called as a leftmost derivation tree.
• In the leftmost derivation tree, we read the given input string (W) from left to right.

Example of Leftmost Derivation

Consider the following production rules for the grammar (G)

S → xB / yA

S → xS / yAA / x

B → yS / xBB / y

Consider a string W = xxxyyxyyyx

Let us derive the given string (W= xxxyyxyyyx) by using given production rules of grammar. And will also draw the derivation tree of each step as given below

Leftmost Derivation and Tree

Let us see derivation through production rules and their tree as given below

Step 01: S   → xB                    (W=xB)

Step 02: W= xxBB                   (Using B → xBB)

Step 03: W= xxxBBB                (Using B → xBB)

Step 04: W= xxxyBB                (Using B → y)

Step 05: W= xxxyyB                (Using B → y)

Step 06: W= xxxyyxBB            (Using B → xBB)

Step 07: W= xxxyyxyB            (Using B → y)

Step 08: W= xxxyyxyyS          (Using B → yS)

Step 09: W= xxxyyxyyyA        (Using S → yA)

Step 10: W= xxxyyxyyyx        (Using A → x)

Hence proved the given string (W= xxxyyxyyyx) is derived through leftmost derivation. So, given string belongs to Given Grammar (G).