How to convert context free grammar to push down automata?

A context-free grammar (CFG) consisting of a finite set of grammar rules is a quadruple (V, T, P, S) where,

• V is a variable (non terminals).

• T is a set of terminals.

• P is a set of rules, P: V→ (V ∪ T)*, i.e., the left-hand side of the production rules P does have any right context or left context.

• S is the start symbol.

Push down automata

A push down automata (PDA) consists of the following −

• A finite non-empty set of states denoted by Q.

• A finite non empty set of input symbols denoted by∑.

• A finite non empty set of push down symbol ┌.

• A special state is called the initial state denoted by q0.

• A special symbol called the initial symbol on the push down store denoted by Z0.

• The set of the final subset of Q denoted by F.

• The transition function ∂= QX(∑U{^})X┌ to the set of finite subsets of QX┌*.

Example

The CFG is as follows −

S->aSa

S->aSa

S->c

Design a Push down automata to accept the string

To convert CFG to PDA first write the production rules and then pop.

The rules for the conversion are as follows −

• S(q0, ɛ, ɛ)=(q0, ɛ)

• S(q0, ɛ,S)=(q0,aSa)

• S(q0, ɛ,S)=(q0,bsb)

• (q0, ɛ,S)=(q0,c)

Now start pop to convert

• S(q0,a,a)=(q1, ɛ)

• S(q1,b,b)=(q2, ɛ)

• S(q2,c,c)=(q3, ɛ)

Transition table

The transition table with regards to converting CFG to PDA is as follows −

Whenever you reach the final state by using PDA then we can say that the context free grammar converts into Push down automata.