Construct a Pushdown Automaton Transition Diagram for (L={wcw^R \mid w\in{a,b}^*})

We want a Pushdown Automaton (PDA) that recognizes strings of the form $wcw^R$ over input alphabet ({a,b,c}), where the middle symbol is exactly one (c), and the suffix is the reverse of the prefix (w). The standard PDA pattern for “store the first half, then match the second half in reverse” is: push each symbol of (w) onto the stack until you nondeterministically decide you have reached the (c); then pop while reading the remaining input to verify it equals (w^R). This PDA idea is described for palindromic/“(w w^R)” style languages in PDA lecture notes.

A transition diagram for a PDA is built from transitions of the form [ \delta(q,,\text{input or }\varepsilon,,\text{stack-top}) ;=; (q',,\text{stack-rewrite}) ] where the stack rewrite replaces the stack top with either a new symbol string or (\varepsilon). PDA transition-diagram conventions (arc labels show ((\text{input},\text{stack-top})/(\text{stack-rewrite}))) are given in instructional material.

Key terms for this construction:

• Pushdown Automaton
• Instantaneous Description
• Acceptance (final state/empty stack)
• Stack symbol
• Transition diagram

Step 1: Fix a PDA model (and design goals)

We will construct a PDA that accepts by empty stack after consuming the entire input (equivalently, by final state with appropriate modification). PDA configurations and the accepted language of a PDA are covered in PDA lecture notes.

Design goals:

1. Before seeing the middle (c), push every (a) and (b) of (w) onto the stack.
2. Ensure there is exactly one (c) in the middle:
   • consume one (c) while in a “switch to pop mode” state.
3. After that (c), for each next input symbol ((a) or (b)), pop the matching symbol from the stack.
4. Accept iff the stack becomes empty exactly when input is finished, meaning the suffix equals (w^R).

We use a bottom-of-stack marker (Z_0) so the PDA knows when the stack is empty. PDA examples use an initial stack symbol like (Z_0).

Step 2: Define the PDA tuple and alphabets

Let the PDA be [ M=(Q,\Sigma,\Gamma,\delta,q_0,Z_0,F) ] consistent with standard PDA definitions.

Choose:

• (Q={q_{\text{push}}, q_{\text{pop}}})
• (\Sigma={a,b,c})
• (\Gamma={a,b,Z_0})
• Start state: (q_0=q_{\text{push}})
• Initial stack symbol: (Z_0)
• Accepting: (F=\emptyset) with empty stack acceptance (we will explicitly pop down to (Z_0) then remove it via an (\varepsilon)-move)

This “push then pop matching symbols” approach follows the same principle used for languages like (ww^R) and palindromes: push symbols, then pop to match reversed order.

Step 3: Write the explicit transition rules (\delta)

Let stack-top be the third argument of (\delta(q,\text{input},\text{top})). We’ll list transitions in a way that directly becomes PDA-diagram arc labels.

(A) Push mode transitions in (q_{\text{push}})

For all stack-top symbols (X\in{a,b,Z_0}):

1. Push (a): [ \delta(q_{\text{push}}, a, X)=(q_{\text{push}}, aX) ]
2. Push (b): [ \delta(q_{\text{push}}, b, X)=(q_{\text{push}}, bX) ]

These transitions ensure that the stack contents after reading (w) equals (w) in the same left-to-right order as it was read, but the pop phase will match in reverse, due to LIFO behavior. This is the core mechanism of PDA-based reversal recognition used in “(ww^R)” PDA constructions.

(B) Middle separator transition on (c)

3. On reading (c), switch to pop mode without changing the stack: [ \delta(q_{\text{push}}, c, X)=(q_{\text{pop}}, X) ] for all (X\in{a,b,Z_0}).

This guarantees that the single symbol (c) is the pivot between push and pop.

(C) Pop mode transitions in (q_{\text{pop}})

4. Match and pop (a): [ \delta(q_{\text{pop}}, a, a)=(q_{\text{pop}}, \varepsilon) ]
5. Match and pop (b): [ \delta(q_{\text{pop}}, b, b)=(q_{\text{pop}}, \varepsilon) ]

(D) Finish/accept transition: remove bottom marker

6. When input is empty (use (\varepsilon)) and only (Z_0) remains: [ \delta(q_{\text{pop}}, \varepsilon, Z_0)=(q_{\text{pop}}, \varepsilon) ]

If we use empty-stack acceptance, popping (Z_0) with no remaining input completes acceptance.

Why this works: if the stack stores exactly (w) while reading the prefix, then popping the stack while reading the suffix forces the suffix to equal (w^R). PDA lecture notes emphasize that PDA’s stack matching (push then pop) is how it recognizes “reverse-linked” strings like (ww^R).

Step 4: Draw the transition diagram (Mermaid)

Use two states: (q_{\text{push}}) (push until (c)) and (q_{\text{pop}}) (pop to match the reverse).

This diagram uses the same “arcs labeled by (input, stack-top)/(stack-rewrite)” convention described for PDA transition diagrams.