CFG Equivalent to PDA Conversion

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In automata theory and context-free grammars, the relationship between Pushdown Automata (PDA) and Context-Free Grammars (CFG) is fundamental.

Here in this chapter, we will explain how to construct a CFG that is equivalent to a given PDA. In other words, we will explain how to convert PDA to CFG. We will see the definitions and the corresponding examples to understand the process in a better way.

Relation Between PDA and CFG

A Pushdown Automaton (PDA) is a type of automaton that uses a stack to manage additional information. Using memory as stack makes it more powerful than a finite automaton.

Pushdown automata can recognize context-free languages, which are languages generated by Context-Free Grammars (CFGs).

A Context-free Grammar consists of:

• A set of variables or non-terminal symbols.
• A set of terminal symbols.
• A start symbol.
• A set of production rules that define how terminals and non-terminals can be combined.

The goal here is to construct a CFG equivalent to a PDA, which means the CFG generates the same language that the PDA recognizes.

Conversion of PDA to CFG

To convert a PDA into an equivalent CFG, we assume the PDA M = (Q, , , , q0, z0, F) accepts a language L by an empty stack.

The CFG G = (V_N, , P, S) is constructed following specific rules that ensure it generates the same language as the PDA.

Let us see the steps one by one to convert PDA to CFG.

Start Symbol Production

For every state q_i in the PDA, we add a production rule to the CFG:

$$\mathrm{S \: \rightarrow \:[q_0, \:z_0,\:q_i]}$$

If the PDA has n states, there will be n productions originating from the start symbol S.

Handling Transitions with Empty Stack Operation

For each transition function of the form (q, a, Y) → (r, ), where (q, r) ∈ Q, a ∈ , and Y ∈ , a production rule is added to the CFG as follows −

$$\mathrm{[qYr] \: \rightarrow \: a}$$

This rule handles cases where the PDA reads a symbol and pops the stack without pushing anything.

Handling Transitions with Stack Operations

For transition functions of the form (q, a, Y) → (r, Y_1, Y_2, , Y_k), a production rule is added for each combination of intermediate states q₁,q₂, ... ,q_k as:

$$\mathrm{[qYq_k ] \: \rightarrow \:a [rY_1 q_1 ] [q_1 Y_2 q_2 ] \: \dotso \: [q_{k - 1} Y_k q_k]}$$

This rule addresses scenarios where the PDA pushes multiple symbols onto the stack in response to an input symbol.

Transition Table

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

Example 1: Constructing a CFG from PDA

Lets apply these steps to an example to make the conversion process clearer.

Given PDA,

$$\mathrm{\delta(q_0,\: a,\: z_0 ) \: \rightarrow \: (q_0,\: z_1\: z_0 )}$$

$$\mathrm{\delta(q_0,\: a,\: z_1 ) \: \rightarrow \: (q_0,\: z_1\: z_1)}$$

$$\mathrm{\delta(q_0,\: b,\: z_1 ) \: \rightarrow \: (q_1,\: \lambda)}$$

$$\mathrm{\delta(q_1,\: b,\: z_1 ) \: \rightarrow \: (q_1,\: \lambda)}$$

$$\mathrm{\delta(q_1,\: b,\: z_0 ) \: \rightarrow \: (q_1,\: z_2\: z_0)}$$

$$\mathrm{\delta(q_1,\: c,\: z_2 ) \: \rightarrow \: (q_2,\: \lambda)}$$

$$\mathrm{\delta(q_2,\: \lambda,\: z_0 ) \: \rightarrow \: (q_2,\: \lambda)}$$

This PDA has three states: q₀, q₁, and q₂.

Step 1: Start Symbol Production

The following productions are added −

$$\mathrm{S \:\rightarrow\:[q_0 \:z_0\: q_0 ] \: /\: [q_0\: z_0\: q_1 ]\: / \:[q_0\: z_0\: q_2 ]}$$

Step 2: Transitions with Empty Stack

The transitions δ(q₀, b, z₁) → (q₁, λ), δ(q₁, b, z₁) → (q₁, λ), and δ(q₁, c, z₂) → (q₂, λ) lead to the following productions:

$$\mathrm{[q_0 \: z_0 \: q_1 ] \: \rightarrow \: b}$$

$$\mathrm{[q_1 \: z_1 \: q_1 ] \: \rightarrow \: b}$$

$$\mathrm{[q_1 \: z_2 \: q_2 ] \: \rightarrow \: c}$$

$$\mathrm{[q_2 \: z_2 \: q_2 ] \: \rightarrow \: c}$$

$$\mathrm{[q_2 \: z_0 \: q_2 ] \: \rightarrow \: \epsilon}$$

Step 3: Transitions with Stack Operations

For the transition (q₀, a, z₀) → (q₀,z₁, z₀), the following productions are added −

$$\mathrm{[q_0 \: z_0 \: q_0] \: \rightarrow \: a [q_0 \: z_1 \: q_0] \: [q_0 \: z_0 \: q_0]}$$

$$\mathrm{[q_0 \: z_0 \: q_0] \: \rightarrow \: a [q_0 \: z_1 \: q_0] \: [q_1 \: z_0 \: q_0]}$$

$$\mathrm{[q_0 \: z_0 \: q_1] \: \rightarrow \: a [q_0 \: z_1 \: q_0] \: [q_0 \: z_0 \: q_1]}$$

$$\mathrm{[q_0 \: z_0 \: q_1] \: \rightarrow \: a [q_0 \: z_1 \: q_1] \: [q_1 \: z_0 \: q_1]}$$

$$\mathrm{[q_0 \: z_0 \: q_2] \: \rightarrow \: a [q_0 \: z_1 \: q_0] \: [q_0 \: z_0 \: q_2]}$$

$$\mathrm{[q_0 \: z_0 \: q_2] \: \rightarrow \: a [q_0 \: z_1 \: q_1] \: [q_1 \: z_0 \: q_2]}$$

The CFG that generates the same language as the PDA is a collection of all these production rules.

Example 2: Another PDA to CFG Conversion

Let's consider another example for further clarity.

Given PDA:

$$\mathrm{\delta(q_0, \: a, \: z_0 ) \: \rightarrow \: (q_1, \: z_1 \: z_0)}$$

$$\mathrm{\delta(q_0, \: b, \: z_0 ) \: \rightarrow \: (q_1, \: z_2 \: z_0)}$$

$$\mathrm{\delta(q_1, \: a, \: z_1 ) \: \rightarrow \: (q_1, \: z_1 \: z_1)}$$

$$\mathrm{\delta(q_1, \: b, \: z_1 ) \: \rightarrow \: (q_1, \: \lambda)}$$

$$\mathrm{\delta(q_1, \: b, \: z_2 ) \: \rightarrow \: (q_1, \: z_2 \: z_2)}$$

$$\mathrm{\delta(q_1, \: \lambda, \: z_0 ) \: \rightarrow \: (q_1, \: \lambda)}$$

The start symbol production is:

$$\mathrm{S \: \rightarrow \: [q_0 \: z_0 \: q_0] \: / \: [q_0 \: z_0 \: q_1]}$$

Transitions with Empty Stack

The relevant transitions provide −

$$\mathrm{[q_1 \: z_1 \: q_1] \: \rightarrow \: b}$$

$$\mathrm{[q_1 \: z_2 \: q_1] \: \rightarrow \: a}$$

$$\mathrm{[q_1 \: z_0 \: q_1] \: \rightarrow \: \epsilon}$$

Transitions with Stack Operations

Transitions such as δ(q₀, a, z₀) → (q₁, z₁ z₀) result in:

$$\mathrm{[q_0 \: z_0 \: q_0] \: \rightarrow \: a[q_1 \: z_1 \: q_0] \: [q_0 \: z_0 \: q_0]}$$

$$\mathrm{[q_0 \: z_0 \: q_1] \: \rightarrow \: a[q_1 \: z_1 \: q_0] \: [q_0 \: z_0 \: q_1]}$$

$$\mathrm{[q_0 \: z_0 \: q_1] \: \rightarrow \: a[q_1 \: z_2 \: q_0] \: [q_0 \: z_0 \: q_1]}$$

In this second example too, we used the same methodology for converting PDAs into CFGs.

Conclusion

The construction of a CFG equivalent to a PDA is a systematic process that involves start symbol production, handling transitions with empty stack operations, and stack operations. In this chapter, we explained the concept through examples and understood how CFG generates the same language recognized by the PDA.