Define the classes PSPACE, EXPTIME

### PSPACE

1. **Definition**: PSPACE is the class of decision problems that can be solved by a Turing machine using a polynomial amount of memory (space) with respect to the size of the input.

2. **Computational Power**: PSPACE includes all problems that can be solved using polynomial space, regardless of the time it takes.

3. **Examples**: Problems like TQBF (True Quantified Boolean Formula) and certain types of games like generalized chess or Go on an \(n \times n\) board.

4. **Relation to Other Classes**: PSPACE encompasses other complexity classes such as P (problems solvable in polynomial time) and NP (nondeterministic polynomial time), as well as co-NP.

5. **Completeness**: A problem is PSPACE-complete if it is in PSPACE and as hard as any other problem in PSPACE, meaning if you can solve this problem in polynomial space, you can solve any PSPACE problem in polynomial space.

### EXPTIME

1. **Definition**: EXPTIME is the class of decision problems that can be solved by a deterministic Turing machine in exponential time, specifically in \(O(2^{p(n)})\) for some polynomial \(p(n)\), where \(n\) is the size of the input.

2. **Computational Power**: EXPTIME includes problems that are solvable within an exponential amount of time, which is much larger than polynomial time.

3. **Examples**: Problems like the generalized version of the game Checkers, certain combinatorial game theories, and solving certain types of exponential Diophantine equations.

4. **Relation to Other Classes**: EXPTIME strictly contains P and is believed to be strictly larger than NP and PSPACE, although it has not been proven that PSPACE is not equal to EXPTIME.

5. **Completeness**: A problem is EXPTIME-complete if it is in EXPTIME and is as hard as any other problem in EXPTIME, meaning solving it efficiently would imply an efficient solution for all problems in EXPTIME.

Both PSPACE and EXPTIME are used to classify problems based on their resource requirements, providing a framework for understanding the limits of computational feasibility.