Then (q_2): ( \delta(q_2, b, X) = (q_2, \varepsilon) ) ( \delta(q_2, \varepsilon, Z_0) = (q_3, Z_0) ) (accept when stack empty and no more (b)) : (a^2 b^5 c^3) is rejected since 5 ≠ 2+3=5 actually 5=5 ✔ so accepted. Wait j=5, i=2,k=3 sum=5, so accepted. Good. (a^2 b^4 c^3) (reject: 4≠5) Run: (q_0): read aa: stack XXZ0 ε→q1: stack XXZ0 q1: read cc: stack XXXXZ0 ε→q2: stack XXXXZ0 q2: read b (1st): stack XXXZ0 b (2nd): stack XXZ0 b (3rd): stack XZ0 b (4th): stack Z0, no more b, ε→q3 accept. Wait, that accepts even though 4≠5? That's wrong — mistake!
Let's do clearly:
We popped only 4 (X)’s for 4 (b)’s, but stack had (i+k=5) symbols (X). Remaining (X) on top? No — after 4 pops, stack = (XZ_0) not (Z_0). So ε-transition to (q_3) not allowed because stack top is (X), not (Z_0). So dead. So correct. Corrected, Cleaner PDA States: (q_0) (read a), (q_1) (read c), (q_2) (read b), (q_3) (accept).
Actually better structure:
[ \beginaligned &\delta(q_0, a, Z_0) = (q_0, XZ_0) \ &\delta(q_0, a, X) = (q_0, XX) \ &\delta(q_0, \varepsilon, X) = (q_1, X) \ &\delta(q_0, \varepsilon, Z_0) = (q_1, Z_0) \ \ &\delta(q_1, c, Z_0) = (q_1, XZ_0) \ &\delta(q_1, c, X) = (q_1, XX) \ &\delta(q_1, \varepsilon, X) = (q_2, X) \ &\delta(q_1, \varepsilon, Z_0) = (q_2, Z_0) \ \ &\delta(q_2, b, X) = (q_2, \varepsilon) \ &\delta(q_2, \varepsilon, Z_0) = (q_3, Z_0) \endaligned ]
So strings are: Example: ( a^2 b^5 c^3 ) → 2 + 3 = 5 ✔ ( a^3 b^3 c^0 ) → 3 + 0 = 3 ✔ ( a^0 b^4 c^4 ) → 0 + 4 = 4 ✔ Empty string? ( i=j=k=0 ) → 0 = 0+0 ✔ but may be excluded if you require positive counts. We push for ( a )’s, then push again for ( c )’s, then pop for ( b )’s.
Start: (q_0), stack (Z_0). Accept: (q_3), stack (Z_0). This PDA accepts ( a^i b^j c^k \mid j = i + k ), with the stack keeping count of (i+k) and popping once per (b) to verify equality.