CS500, Theory of Computation: Midterm Solutions

(a) Prove that L is not regular. (b) Prove that L cannot be recognized in real time by a one-counter machine, i.e., i. M reads its input from left to right with no -transitions. ii. M has a finite set of states Q. At each step it may update its state and may increment or decrement its counter by any integer. iii. M ’s only dependence on the current value of its counter is whether or not it is zero; thus its transition function takes the form δ : Q×Σ×{zero, nonzero} → Q× Z where Z denotes the integers. iv. M accepts the input if the counter is zero and/or its final state is in some accepting subset Qaccept ⊆ Q. (c) Prove that L is not context-free.