Understanding Computational Methods and the Halting Problem

What did Alan Turing's halting problem demonstrate about computability?

• All problems can be solved with enough computing power.

• The only limits on algorithms are hardware constraints.

• Some problems cannot be solved by any algorithm.

• Computers can solve any problem given enough time.

What is a key characteristic of a tractable problem?

• It cannot be solved by any algorithm.

• It can be solved in polynomial time.

• It requires infinite memory to solve.

• It can be solved in exponential time.

What does the concept of intractable problems imply?

• They are always unsolvable by any algorithm.

• They can be solved in polynomial time.

• They require infinite memory to solve.

• They cannot be solved in polynomial time.

What is a common approach to dealing with intractable problems?

• Solving them using polynomial time algorithms.

• Ignoring them as they cannot be solved.

• Increasing hardware capabilities to solve them.

• Using heuristic methods to find a good enough solution.

Which method involves breaking down a problem into simpler parts to make it easier to solve?

• Enumeration

• Simulation

• Abstraction and Decomposition

• Automation

What does the method of enumeration involve?

• Designing an algorithm that performs an exhaustive search.

• Using mathematical equations to solve problems.

• Automating the problem-solving process.

• Creating a model to simulate real-world systems.

What is the main focus of simulation in computational methods?

• To automate the problem-solving process.

• To design a model of a real system to understand its behavior.

• To use mathematical equations for problem-solving.

• To perform an exhaustive search for solutions.

What is the role of automation in computational methods?

• To build a problem-solving model and put it into action.

• To perform an exhaustive search for solutions.

• To create a model for simulating real-world systems.

• To simplify problems using abstraction.

How did Turing prove that some problems cannot be solved by computers?

• By demonstrating that some algorithms contradict themselves.

• By solving the Wang tile problem.

• By proving that hardware is the only limitation.

• By showing that all problems can be solved with enough time.

What is the significance of the halting problem in computer science?

• It proves that some problems are inherently unsolvable by computers.

• It shows that all problems are solvable with enough memory.

• It indicates that all problems can be automated.

• It demonstrates that faster processors can solve any problem.