PDA for $L = \{ w c w^R \mid w \in \{a,b\}^* \}$

Idea

1. In state $q_0$, push every symbol of $w$ on the stack.
2. On reading the middle marker c, move to $q_1$ without touching the stack.
3. In $q_1$, each input symbol must match the stack top; pop it. This checks the second half is $w^R$ (the stack reverses $w$).
4. When the input is finished and only $Z_0$ remains, move to the accepting state $q_2$.

Formal definition: $M = (\{q_0,q_1,q_2\}, \{a,b,c\}, \{a,b,Z_0\}, \delta, q_0, Z_0, \{q_2\})$ (acceptance by final state).

Transition function $\delta$ (notation: input, stack top / new stack top)

#	Move
1	$\delta(q_0, a, Z_0) = (q_0, aZ_0)$
2	$\delta(q_0, b, Z_0) = (q_0, bZ_0)$
3	$\delta(q_0, a, a) = (q_0, aa)$
4	$\delta(q_0, a, b) = (q_0, ab)$
5	$\delta(q_0, b, a) = (q_0, ba)$
6	$\delta(q_0, b, b) = (q_0, bb)$
7	$\delta(q_0, c, X) = (q_1, X)$ for $X \in \{Z_0, a, b\}$
8	$\delta(q_1, a, a) = (q_1, \varepsilon)$
9	$\delta(q_1, b, b) = (q_1, \varepsilon)$
10	$\delta(q_1, \varepsilon, Z_0) = (q_2, Z_0)$

Transition diagram

Trace for abcba

State	Remaining input	Stack (top left)	Move
$q_0$	abcba	$Z_0$	push a
$q_0$	bcba	$aZ_0$	push b
$q_0$	cba	$baZ_0$	read c, go to $q_1$
$q_1$	ba	$baZ_0$	b matches, pop
$q_1$	a	$aZ_0$	a matches, pop
$q_1$	$\varepsilon$	$Z_0$	$\varepsilon$-move to $q_2$
$q_2$	$\varepsilon$	$Z_0$	accept

A string such as abcab is rejected: after c the stack top is b but the input is a, so no move exists. The PDA is deterministic because c marks the middle.